Bayesian Econometrics (Advances in Econometrics)

ADVANCES IN ECONOMETRICS Series Editors: Thomas B. Fomby and R. Carter Hill Recent Volumes: Volume 15: Nonstationary Pa...

Author: Siddhartha Chib | Gary Koop | Bill Griffiths | Dek Terrell

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ADVANCES IN ECONOMETRICS Series Editors: Thomas B. Fomby and R. Carter Hill Recent Volumes: Volume 15:

Nonstationary Panels, Panel Cointegration, and Dynamic Panels, Edited by Badi Baltagi

Volume 16:

Econometric Models in Marketing, Edited by P. H. Franses and A. L. Montgomery

Volume 17:

Maximum Likelihood Estimation of Misspeciﬁed Models: Twenty Years Later, Edited by Thomas B. Fomby and R. Carter Hill

Volume 18:

Spatial and Spatiotemporal Econometrics, Edited by J. P. LeSage and R. Kelley Pace

Volume 19:

Applications of Artiﬁcial Intelligence in Finance and Economics, Edited by J. M. Binner, G. Kendall and S. H. Chen

Volume 20A:

Econometric Analysis of Financial and Economic Time Series, Edited by Dek Terrell and Thomas B. Fomby

Volume 20B:

Econometric Analysis of Financial and Economic Time Series, Edited by Thomas B. Fomby and Dek Terrell

Volume 21:

Modelling and Evaluating Treatment Effects in Econometrics, Edited by Daniel L. Millimet, Jeffrey A. Smith and Edward J. Vytlacil

Volume 22:

Econometrics and Risk Management, Edited by Thomas B. Fomby, Knut Solna and Jean-Pierre Fouque

ADVANCES IN ECONOMETRICS

VOLUME 23

BAYESIAN ECONOMETRICS EDITED BY

SIDDHARTHA CHIB Olin Business School, Washington University

WILLIAM GRIFFITHS Department of Economics, University of Melbourne

GARY KOOP Department of Economics, University of Strathclyde

DEK TERRELL Department of Economics, Louisiana State University

United Kingdom – North America – Japan India – Malaysia – China

JAI Press is an imprint of Emerald Group Publishing Limited Howard House, Wagon Lane, Bingley BD16 1WA, UK First edition 2008 Copyright r 2008 Emerald Group Publishing Limited Reprints and permission service Contact: [email protected] No part of this book may be reproduced, stored in a retrieval system, transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without either the prior written permission of the publisher or a licence permitting restricted copying issued in the UK by The Copyright Licensing Agency and in the USA by The Copyright Clearance Center. No responsibility is accepted for the accuracy of information contained in the text, illustrations or advertisements. The opinions expressed in these chapters are not necessarily those of the Editor or the publisher. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-1-84855-308-8 ISSN: 0731-9053 (Series)

Awarded in recognition of Emerald’s production department’s adherence to quality systems and processes when preparing scholarly journals for print

LIST OF CONTRIBUTORS Michael K. Andersson

Sveriges Riksbank, Stockholm, Sweden

Veni Arakelian

Department of Economics, University of Crete, Rethymno, Greece

Chun-man Chan

Hong Kong Community College, Kowloon, Hong Kong, China

Cathy W. S. Chen

Department of Statistics, Feng Chia University, Taiwan

Siddhartha Chib

Olin Business School, Washington University, St. Louis, MO

S. T. Boris Choy

Discipline of Operations Management and Econometrics, University of Sydney, NSW, Australia

Michiel de Pooter

Division of International Finance, Financial Markets, Board of Governors of the Federal Reserve System, Washington, DC

Dipak K. Dey

Department of Statistics, University of Connecticut, Storrs, CT

Deborah Gefang

Department of Economics, University of Leicester, Leicester, UK

Richard Gerlach

Discipline of Operations Management and Econometrics, University of Sydney, NSW, Australia

Paolo Giordani

Research Department, Sveriges Riksbank, Stockholm, Sweden

Jennifer Graves

Department of Economics, University of California, Irvine, CA ix

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LIST OF CONTRIBUTORS

William Grifﬁths

Department of Economics, University of Melbourne, Vic., Australia

Ariun Ishdorj

Department of Economics, Iowa State University, Ames, IA

Liana Jacobi

Department of Economics, University of Melbourne, Vic., Australia

Ivan Jeliazkov

Department of Economics, University of California, Irvine, CA

Helen H. Jensen

Department of Economics, Iowa State University, Ames, IA Swedish Business School, O¨rebo University, O¨rebo, Sweden

Sune Karlsson Robert Kohn

Department of Economics, Australian School of Business, University of New South Wales, Sydney, Australia

Gary Koop

Department of Economics, University of Strathclyde, Glasgow, UK

Dimitris Korobilis

Department of Economics, University of Strathclyde, Glasgow, UK

Subal C. Kumbhakar

Department of Economics, State University of New York, Binghamton, NY

Mark Kutzbach

Department of Economics, University of California, Irvine, CA

Roberto Leon-Gonzalez

National Graduate Institute for Policy Studies (GRIPS), Tokyo, Japan De´partement de Sciences E´conomiques, Universite´ de Montre´al, CIREQ, Canada

Brahim Lgui

Arto Luoma

Department of Mathematics and Statistics, University of Tampere, Tampere, Finland

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List of Contributors

Jani Luoto

William J. McCausland

School of Business and Economics, University of Jyva¨skyla¨, Jyva¨skyla¨, Finland De´partement de Sciences E´conomiques, Universite´ de Montre´al, CIREQ and CIRANO, Montre´al, QC, Canada

Nadine McCloud

Department of Economics, The University of the West Indies, Mona, Kingston, Jamaica

Murat K. Munkin

Department of Economics, University of South Florida, Tampa, FL

Christopher J. O’Donnell

School of Economics, University of Queensland, Brisbane, Australia

Francesco Ravazzolo

Norges Bank, Oslo, Norway

Vanessa Rayner

School of Economics, University of Queensland, Brisbane, Australia

Rene Segers

Tinbergen Institute and Econometric Institute, Erasmus University Rotterdam, Rotterdam, The Netherlands

Mike K. P. So

Department of ISOM, Hong Kong University of Science and Technology, Kowloon, Hong Kong

Rodney Strachan

School of Economics, The University of Queensland, Brisbane, Australia

Sylvie Tchumtchoua

Department of Statistics, University of Connecticut, Storrs, CT

Dek Terrell

Department of Economics, Louisiana State University, Baton Rouge, LA

Justin Tobias

Department of Economics, Purdue University, West Lafayette, IN

Pravin K. Trivedi

Department of Economics, Wylie Hall, Indiana University, Bloomington, IN

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LIST OF CONTRIBUTORS

Efthymios G. Tsionas

Department of Economics, Athens University of Economics and Business, Athens, Greece

Herman K. van Dijk

Tinbergen Institute and Econometric Institute, Erasmus University Rotterdam, Rotterdam, The Netherlands

Wai-yin Wan

School of Mathematics and Statistics, University of Sydney, NSW, Australia

Arnold Zellner

Graduate School of Business, University of Chicago, Chicago, IL

BAYESIAN ECONOMETRICS: AN INTRODUCTION Siddhartha Chib, William Grifﬁths, Gary Koop and Dek Terrell ABSTRACT Bayesian Econometrics is a volume in the series Advances in Econometrics that illustrates the scope and diversity of modern Bayesian econometric applications, reviews some recent advances in Bayesian econometrics, and highlights many of the characteristics of Bayesian inference and computations. This ﬁrst paper in the volume is the Editors’ introduction in which we summarize the contributions of each of the papers.

1. INTRODUCTION In 1996 two volumes of Advances in Econometrics were devoted to Bayesian econometrics. One was on computational methods and applications and the other on time-series applications. This was a time when Markov chain Monte Carlo (MCMC) techniques, which have revolutionized applications of Bayesian econometrics, had started to take hold. The adaptability of MCMC to problems previously considered too difﬁcult was generating a revival of interest in the Bayesian paradigm. Now, 12 years later, it is time for another Advances volume on Bayesian econometrics. Use of Bayesian techniques has Bayesian Econometrics Advances in Econometrics, Volume 23, 3–9 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23021-5

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become widespread across all areas of empirical economics. Previously intractable problems are being solved and more ﬂexible models are being introduced. The purpose of this volume is to illustrate today’s scope and diversity of Bayesian econometric applications, to review some of the recent advances, and to highlight various aspects of Bayesian inference and computations. The book is divided into three parts. In addition to this introduction, Part I contains papers by Arnold Zellner, and by Paolo Giordani and Robert Kohn. In his paper ‘‘Bayesian Econometrics: Past, Present, and Future,’’ Arnold Zellner reviews problems faced by the Federal Reserve System, as described by its former chairman, Alan Greenspan, and links these problems to a summary of past and current Bayesian activity. Some key contributions to the development of Bayesian econometrics are highlighted. Future research directions are discussed with a view to improving current econometric models, methods, and applications of them. The other paper in Part I is a general one on a computational strategy for improving MCMC. Under the title ‘‘Bayesian Inference using Adaptive Sampling,’’ Paolo Giordani and Robert Kohn discuss simulation-based Bayesian inference methods that draw on information from previous samples to build the proposal distributions in a given family of distributions. The article covers approaches along these lines and the intuition behind some of the theory for proving that the procedures work. They also discuss strategies for making adaptive sampling more effective and provide illustrations for variable selection in the linear regression model and for time-series models subject to interventions.

2. MICROECONOMETRIC MODELING Part II of the book, entitled ‘‘Microeconometric Modeling’’ contains applications that use cross-section or panel data. The paper by Murat K. Munkin and Pravin K. Trivedi, ‘‘A Bayesian Analysis of the OPES Model with a Nonparametric Component: An Application to Dental Insurance and Dental Care,’’ is a good example of how Bayesian methods are increasingly being used in important empirical work. The empirical focus is on the impact of dental insurance on the use of dental services. Addressing this issue is complicated by the potential endogeneity of insurance uptake and the fact that insurance uptake may depend on explanatory variables in a nonlinear fashion. The authors develop an appropriate model which addresses both these issues and carry out an empirical analysis which ﬁnds strong evidence

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that having dental insurance encourages use of dentists, but also of adverse selection into the insured state. MCMC simulation techniques are particularly powerful in discrete-data models with latent variable representations. In their paper ‘‘Fitting and Comparison of Models for Multivariate Ordinal Outcomes,’’ Ivan Jeliazkov, Jennifer Graves, and Mark Kutzbach review several alternative modeling and identiﬁcation schemes for ordinal data models and evaluate how each aids or hampers estimation using MCMC. Model comparison via marginal likelihoods and an analysis of the effects of covariates on category probabilities is considered for each parameterization. The methods are applied to examples in educational attainment, voter opinions, and consumers’ reliance on alternative sources of medical information. In ‘‘Intra-Household Allocation and Consumption of WIC-Approved Foods: A Bayesian Approach,’’ Ariun Ishdorj, Helen H. Jensen, and Justin Tobias consider the Special Supplemental Nutrition Program for Women, Infants, and Children (WIC) that aims to provide food, nutrition education, and other services to at-risk, low-income children and pregnant, breastfeeding, and postpartum women. They assess the extent to which the WIC program improves the nutritional outcomes of WIC families as a whole, including the targeted and nontargeted individuals within the household. This question is considered under the possibility that participation in the program (which is voluntary) is endogenous. They develop an appropriate treatment–response model and conclude that WIC participation does not lead to increased levels of calcium intake from milk. A second paper that illustrates the use of Bayesian techniques for analyzing treatment–response problems is that by Siddhartha Chib and Liana Jacobi. In their paper ‘‘Causal Effects from Panel Data in Randomized Experiments with Partial Compliance,’’ the authors describe how to calculate the causal impacts from a training program when noncompliance exists in the training arm. Two primary models are considered, with one model including a random effects speciﬁcation. Prior elicitation is carefully done by simulating from a prior predictive density on outcomes, using a hold out sample. Estimation and model comparison are considered in detail. The methods are employed to assess the impact of a job training program on mental health scores. Basic equilibrium job search models often yield wage densities that do not accord well with empirical regularities. When extensions to basic models are made and analyzed using kernel-smoothed nonparametric forms, it is difﬁcult to assess these extensions via model comparisons. In ‘‘Parametric and Nonparametric Inference in Equilibrium Job Search Models,’’ Gary Koop

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develops Bayesian parametric and nonparametric methods that are comparable to those in the existing non-Bayesian literature. He then shows how Bayesian methods can be used to compare the different parametric and nonparametric equilibrium search models in a statistically rigorous sense. In the paper ‘‘Do Subsidies Drive Productivity? A Cross-Country Analysis of Nordic Dairy Farms,’’ Nadine McCloud and Subal C. Kumbhakar develop a Bayesian hierarchical model of farm production which allows for the calculation of input productivity, efﬁciency, and technical change. The key research questions relate to whether and how these are inﬂuenced by subsidies. Using a large panel of Nordic dairy farms, they ﬁnd that subsidies drive productivity through technical efﬁciency and input elasticities, although the magnitude of these effects differs across countries. The richness of available data and the scope for building ﬂexible models makes marketing a popular area for Bayesian applications. In ‘‘Semiparametric Bayesian Estimation of Random Coefﬁcients Discrete Choice Models,’’ Sylvie Tchumtchoua and Dipak K. Dey propose a semiparametric Bayesian framework for the analysis of random coefﬁcients discrete choice models that can be applied to both individual as well as aggregate data. Heterogeneity is modeled using a Dirichlet process prior which (importantly) varies with consumer characteristics through covariates. The authors employ a MCMC algorithm for ﬁtting their model, and illustrate the methodology using a household level panel dataset of peanut butter purchases, and supermarket chain level data for 31 ready-to-eat breakfast cereals brands. When diffuse priors are used to estimate simultaneous equation models, the resulting posterior density can possess inﬁnite asymptotes at points of local nonidentiﬁcation. Kleibergen and Zivot (2003) introduced a prior to overcome this problem in the context of a restricted reduced form speciﬁcation, and investigated the relationship between the resulting Bayesian estimators and their classical counterparts. Arto Luoma and Jani Luoto, in their paper ‘‘Bayesian Two-Stage Regression with Parametric Heteroscedasticity,’’ extend the analysis of Kleibergen and Zivot to a simultaneous equation model with unequal error variances. They apply their techniques to a cross-country Cobb–Douglas production function.

3. TIME-SERIES MODELING Part III of the volume is devoted to models and applications that use timeseries data. The ﬁrst paper in this part is ‘‘Bayesian Near-Boundary Analysis in Basic Macroeconomic Time-Series Models’’ by Michiel D. de Pooter,

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Francesco Ravazzolo, Rene Segers, and Herman K. van Dijk. The boundary issues considered by these authors are similar to that encountered by Arto Luoma and Jani Luoto in their paper. There are a number of models where the use of particular types of noninformative priors can lead to improper posterior densities with estimation breaking down at boundary values of parameters. The circumstances under which such problems arise, and how the problems can be solved using regularizing or truncated priors, are examined in detail by de Pooter et al. in the context of dynamic linear regression models, autoregressive and error correction models, instrumental variable models, variance component models, and state space models. Analytical, graphical, and empirical results using U.S. macroeconomic data are presented. In his paper ‘‘Forecasting in Vector Autoregressions with Many Predictors,’’ Dimitris Korobilis introduces Bayesian model selection methods in a VAR setting, focusing on the problem of drawing inferences from a dataset with a very large number of potential predictors. A stochastic search variable selection algorithm is used to implement Bayesian model selection. An empirical application using 124 potential predictors to forecast eight U.S. macroeconomic variables is included to demonstrate the methodology. Results indicate an improvement in forecasting accuracy over model selection based on the Bayesian Information Criteria. In ‘‘Bayesian Inference in a Cointegrating Panel Data Model,’’ Gary Koop, Robert Leon-Gonzalez, and Rodney Strachan focus on cointegration in the context of a cointegrating panel data model. Their approach allows both short-run dynamics and the cointegrating rank to vary across crosssectional units. In addition to an uninformative prior, they propose an informative prior with ‘‘soft homogeneity’’ restrictions. This informative prior can be used to include information from economic theory that crosssectional units are likely to share the same cointegrating rank without forcing that assumption on the data. Empirical applications using simulated data and a long-run model for bilateral exchange rates are used to demonstrate the methodology. Cointegration is also considered by Deborah Gefang who develops tests of purchasing power parity (PPP) within an exponential smooth transition (ESVECM) framework. The Bayesian approach offers a substantial methodological advantage in this application because the Gibbs sampling scheme is not affected by the multi-mode problem created by nuisance parameters. Results based on Bayesian model averaging and Bayesian model selection ﬁnd evidence that PPP holds between the United States and each of the remaining G7 countries.

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‘‘Bayesian Forecast Combination for VAR Models’’ by Michael K. Andersson and Sune Karlsson addresses the issue of how to forecast a variable (or variables) of interest (e.g., GDP) when there is uncertainty about the dimension of the VAR and uncertainty about which set of explanatory variables should be used. This uncertainty leads to a huge set of models. The authors do model averaging over the resulting high-dimensional model space using predictive likelihoods as weights. For forecast horizons greater than one, the predictive likelihoods will not have analytical forms and the authors develop a simulation method for estimating them. An empirical analysis involving U.S. GDP shows the beneﬁts of their approach. In ‘‘Bayesian Inference on Time-Varying Proportions,’’ William J. McCausland and Brahim Lgui derive a highly efﬁcient algorithm for simulating the states in state space models where the dependent variables are proportions. The authors argue in favor of a model which is parameterized such that the measurement equation has the proportions (conditional on the states) following a Dirichlet distribution, but the state equation is a standard linear Gaussian one. The authors develop a Metropolis–Hastings algorithm which draws states as a block from a multivariate Gaussian proposal distribution. Extensive empirical evidence indicates that their approach works well and, in particular, is very efﬁcient. Christopher J. O’Donnell and Vanessa Rayner use Bayesian methodology to impose inequality restrictions on ARCH and GARCH models in their paper ‘‘Imposing Stationarity Constraints on the Parameters of ARCH and GARCH Models.’’ Bayesian model averaging is used to resolve uncertainty with regard to model selection. The authors apply the methodology to data from the London Metals Exchange and ﬁnd that results are generally insensitive to the imposition of inequality restrictions. In ‘‘Bayesian Model Selection for Heteroskedastic Models,’’ Cathy W. S. Chen, Richard Gerlach, and Mike K. P. So discuss Bayesian model selection for a wide variety of ﬁnancial volatility models that exhibit asymmetries (e.g., threshold GARCH models). Model selection problems are complicated by the fact that there are many contending models and marginal likelihood calculation can be difﬁcult. They discuss this problem in an empirical application involving daily data from three Asian stock markets and calculate the empirical support for their competing models. Using a scale mixture of uniform densities representation of the Student-t density, S. T. Boris Choy, Wai-yin Wan, and Chun-man Chan provide a Bayesian analysis of a Student-t stochastic volatility model in ‘‘Bayesian Student-t Stochastic Volatility Models via Scale Mixtures.’’ They develop a Gibbs sampler for their model and show how their approach can be extended

Bayesian Econometrics: An Introduction

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to the important class of Student-t stochastic volatility models with leverage. The different models are ﬁt to returns on exchange rates of the Australian dollar against 10 currencies. In ‘‘Bayesian Analysis of the Consumption CAPM,’’ Veni Arakelian and Efthymios G. Tsionas show that Labadie’s (1989) solution to the CAPM can be applied to obtain a closed form solution and to provide a traditional econometric interpretation. They then apply Bayesian inference to both simulated data and the Mehra and Prescott (1985) dataset. Results generally conform to theory, but also reveal asymmetric marginal densities for key parameters. The asymmetry suggests that techniques such as generalized method of moments, which rely on asymptotical approximations, may be unreliable.

REFERENCES Kleibergen, F., & Zivot, E. (2003). Bayesian and classical approaches to instrumental variable regression. Journal of Econometrics, 114, 29–72. Labadie, P. (1989). Stochastic inﬂation and the equity premium. Journal of Monetary Economics, 24, 195–205. Mehra, R., & Prescott, E. C. (1985). The equity premium: A puzzle. Journal of Monetary Economics, 15, 145–162.

BAYESIAN ECONOMETRICS: PAST, PRESENT, AND FUTURE Arnold Zellner ABSTRACT After brieﬂy reviewing the past history of Bayesian econometrics and Alan Greenspan’s (2004) recent description of his use of Bayesian methods in managing policy-making risk, some of the issues and needs that he mentions are discussed and linked to past and present Bayesian econometric research. Then a review of some recent Bayesian econometric research and needs is presented. Finally, some thoughts are presented that relate to the future of Bayesian econometrics.

1. INTRODUCTION In the ﬁrst two sentences of her paper, ‘‘Bayesian Econometrics, The First Twenty Years,’’ Qin (1996) wrote, ‘‘Bayesian econometrics has been a controversial area in the development of econometric methodology. Although the Bayesian approach has been constantly dismissed by many mainstream econometricians for its subjectivism, Bayesian methods have been adopted widely in current econometric research’’ (p. 500). This was written more than 10 years ago. Now more mainstream econometricians and many others have adopted the Bayesian approach and are using it to solve a broad range of

Bayesian Econometrics Advances in Econometrics, Volume 23, 11–60 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23001-X

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econometric problems in line with my forecast in Zellner (1974), ‘‘Further, it must be recognized that the B approach is in a stage of rapid development with work going ahead on many new problems and applications. While this is recognized, it does not seem overly risky to conclude that the B approach, which already has had some impact on econometric work, will have a much more powerful inﬂuence in the next few years’’ (p. 54). See also, Zellner (1981, 1988b, 1991, 2006) for more on the past, present, and future of Bayesian econometrics in which it is emphasized that all econometricians use and misuse prior information, subjectively, objectively, or otherwise. And it has been pointed out that Bayesian econometricians learn using an explicit model, Bayes’ Theorem that allows prior information to be employed in a formal and reproducible manner whereas non-Bayesian econometricians learn in an informal, subjective manner. For empirical evidence on the rapid growth of Bayesian publications over the years in economics and other ﬁelds that will be discussed below see Poirier (1989, 1992, 2004) and Poirier (1991) for an interesting set of Bayesian empirical papers dealing with problems in economics and ﬁnance. In the early 1990s, both the International Society for Bayesian Analysis (http://www.bayesian.org) and the Section on Bayesian Statistical Science of the American Statistical Association (http://www.amstat.org) were formed and have been very active and successful in encouraging the growth of Bayesian theoretical and applied research and publications. Similarly, the NBER-NSF Seminar on Bayesian Inference in Econometrics and Statistics (SBIES) that commenced operation in 1970, has been effective for many years in sponsoring research meetings, publishing a number of Bayesian books and actively supporting the creation of ISBA and SBSS in the early 1990s. In Berry, Chaloner, and Geweke (1996), some history of the SBIES and a large number of Bayesian research papers are presented. Also, under the current leadership of Sid Chib, very productive meetings of this seminar in 2004 and 2005 have been held that were organized by him and John Geweke. In August 2006, the European–Japanese Bayesian Workshop held a meeting in Vienna organized by Wolfgang Polasek that had a very interesting program. In 2005, the Indian Bayesian Society and the Indian Bayesian Chapter of ISBA had an international Bayesian meeting at Varanasi with many of the papers presented that have appeared in a conference volume. In September 2006, a Bayesian research meeting was held at the Royal Bank of Sweden, organized by Mattias Villani that attracted leading Bayesian econometricians from all over the world to present reports on their current work on Bayesian econometric methodology. And now, this Advances in Econometrics volume features additional valuable Bayesian econometric research. And last, but not

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least, the International Society for Bayesian Analysis has commenced publication of an online Bayesian journal called Bayesian Analysis; see http://www.bayesian.org for more information about this journal with R. Kass the founding editor and listings of articles for several years that are downloadable. These and many more Bayesian activities that have taken place over the years attest to the growth and vitality of Bayesian analysis in many sciences, industries, and governments worldwide.

1.1. An Example of Bayesian Monetary Policy-Making As an example of extremely important work involving the use of Bayesian methodology and analysis, Alan Greenspan, former Chairman of the U.S. Federal Reserve System presented an invited paper, ‘‘Risk and Uncertainty in Monetary Policy’’ at the 2004 Meeting of the American Economic Association that was published in the American Economic Review in 2004 along with very knowledgeable discussion by Martin Feldstein, Harvard Professor of Economics and President of the National Bureau of Economic Research, Mervyn King of the Bank of England, and Professor Janet L. Yellen of the Haas School of Business, University of California, Berkeley. The paper is notable in that it presents a comprehensive description of the ways in which he approached and solved monetary policy problems ‘‘ . . . from the perspective of someone who has been in the policy trenches’’ (p. 33). Greenspan’s account should be of interest to Bayesians econometricians and many others since he states, ‘‘In essence, the risk management approach to policymaking is an application of Bayesian decision-making’’ (p. 37). In addition, he writes, ‘‘Our problem is not, as is sometimes alleged, the complexity of our policy-making process, but the far greater complexity of a world economy whose underlying linkages appear to be continuously evolving. Our response to that continuous evolution has been disciplined by the Bayesian type of decision-making in which we have been engaged’’ (p. 39). Feldstein (2004), after providing an excellent review of Greenspan’s successful policy-making in the past wrote, ‘‘Chairman Greenspan emphasized that dealing with uncertainty is the essence of making monetary policy (see also Feldstein, 2002). The key to what he called the risk-management approach to monetary policy is the Bayesian theory of decision-making’’ (p. 42). After providing a brief, knowledgeable description of Bayesian decision theory, Feldstein provides the following example to illustrate a case of asymmetric loss in connection with a person making a decision whether to carry an umbrella when the probability of rain is not high. ‘‘If he carries the

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umbrella and it does not rain, he is mildly inconvenienced. But if he does not carry the umbrella and it rains, he will suffer getting wet. A good Bayesian ﬁnds himself carrying an umbrella on many days when it does not rain. The policy actions of the past year were very much in this spirit. The Fed cut the interest rate to 1 percent to prevent the low-probability outcome of spiraling deﬂation because it regarded that outcome as potentially very damaging while the alternative possible outcome of a rise of the inﬂation rate from 1.5 percent to 2.5 percent was deemed less damaging and more easily reversed’’ (p. 42). Mervyn King of the Bank of England commented knowingly about model quality and policy-making, ‘‘Greenspan suggests that the risk-management approach is an application of Bayesian decision-making when there is uncertainty about the true model of the economy. Policy that is optimal in one particular model of the economy may not be ‘robust’ across a class of other models. In fact, it may lead to a very bad outcome should an alternative model turn out to be true . . . Of course, although such an approach is sensible, it is still vulnerable to policymakers giving excessive weight to misleading models of the economy. . . . But, in the end, there is no escaping the need to make judgments about which models are more plausible than others’’ (pp. 42–43). These are indeed very thoughtful remarks about problems of model uncertainty in making policy but do not recognize that just as with Feldstein’s umbrella example above, a Bayesian analysis can utilize posterior probabilities associated with alternative models that reﬂect the quality of past performance that have been shown to be useful in producing useful combined forecasts and probably will be helpful in dealing with model uncertainty in policy-making. 1.2. Greenspan’s Policy-Making Problems Below, I list and label important problems that Greenspan mentioned in connection with his successful policy-making over the years that reveal his deep understanding of both obvious and very subtle problems associated with model-building, economic analyses, forecasting, and policy-making. 1. Structural changes: For example, ‘‘ . . . increased political support for stable prices, globalization which unleashed powerful new forces of competition, and an acceleration of productivity which at least for a time held down cost pressures’’ (p. 33). ‘‘I believe that we at the Fed, to our credit, did gradually come to recognize the structural economic changes that we were living through and accordingly altered our understanding

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of the key parameters of the economic system and our policy stance . . . . But as we lived through it, there was much uncertainty about the evolving structure of the economy and about the inﬂuence of monetary policy’’ (p. 33). Forecasting: ‘‘In recognition of the lag in monetary policy’s impact on economic activity, a preemptive response to the potential for building inﬂationary pressures was made an important feature of policy. As a consequence, this approach elevated forecasting to an even more prominent place in policy deliberations’’ (p. 33). Unintended consequences: ‘‘Perhaps the greatest irony of the past decade is that the gradually unfolding success against inﬂation may well have contributed to the stock price bubble of the latter part of the 1990s . . . The sharp rise in stock prices and their subsequent fall were, thus, an especial challenge to the Federal Reserve’’ (p. 35). ‘‘The notion that a well-timed incremental tightening could have been calibrated to prevent the late 1990s bubble while preserving economic stability is almost surely an illusion. Instead of trying to contain a putative bubble by drastic actions with largely unpredictable consequences, we chose . . . to focus on policies to mitigate the fallout when it occurs and, hopefully, ease the transition to the next expansion’’ (p. 36). Uncertainty: ‘‘The Federal Reserve’s experiences over the past two decades make it clear that uncertainty is not just a pervasive feature of the monetary landscape; it is the deﬁning characteristic of that landscape. The term ‘‘uncertainty’’ is meant here to encompass both ‘Knightian uncertainty,’ in which the probability distribution of outcomes is unknown, and ‘risk,’ in which uncertainty of outcomes is delimited by a known probability distribution. In practice, one is never quite sure what type of uncertainty one is dealing with in real time, and it may be best to think of a continuum ranging from well-deﬁned risks to the truly unknown’’ (pp. 36–37). Risk management: ‘‘As a consequence, the conduct of monetary policy in the United States has come to involve, at its core, crucial elements of risk management. This conceptual framework emphasizes understanding as much as possible the many sources of risk and uncertainty that policymakers face, quantifying those risks, when possible, and assessing costs associated with each of the risks. In essence, the risk-management approach to monetary policymaking is an application of Bayesian decision-making’’ (p. 37). Objectives: ‘‘This [risk management] framework also entails devising, in light of those risks, a strategy for policy directed at maximizing the

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probabilities of achieving over time our goals of price stability and the maximum sustainable economic growth that we associate with it’’ (p. 37). Expert opinion: ‘‘In designing strategies to meet our policy objectives, we have drawn on the work of analysts, both inside and outside the Fed, who over the past half century have devoted much effort to improving our understanding of the economy and its monetary transmission mechanism’’ (p. 37). Model uncertainty: ‘‘A critical result [of efforts to improve our understanding of the economy and its monetary transmission mechanism] has been the identiﬁcation of a relatively small set of key relationships that, taken together, provide a useful approximation of our economy’s dynamics. Such an approximation underlies the statistical models that we at the Federal Reserve employ to assess the likely inﬂuence of our policy decisions. However, despite extensive efforts to capture and quantify what we perceive as the key macroeconomic relationships, our knowledge about many of the important linkages is far from complete and, in all likelihood will always remain so. Every model, no matter how detailed or how well designed, conceptually and empirically, is a vastly simpliﬁed representation of the world that we experience with all its intricacies on a day-today basis’’ (p. 37). Loss structures: ‘‘Given our inevitably incomplete knowledge about key structural aspects of an ever-changing economy and the sometimes asymmetric costs or beneﬁts of particular outcomes, a central bank needs to consider not only the most likely future path for the economy, but also the distribution of possible outcomes about that path. The decision-makers then need to reach a judgment about the probabilities, costs and beneﬁts of the various possible outcomes under alternative choices for policy’’ (p. 37). Robustness of policy: ‘‘In general, different policies will exhibit different degrees of robustness with respect to the true underlying structure of the economy’’ (p. 37). Cost–beneﬁt analysis: ‘‘As this episode illustrates, policy practitioners operating under a risk-management paradigm may, at times, be led to undertake actions intended to provide insurance against [low probability] especially adverse outcomes . . . . The product of a lowprobability event and a potentially severe outcome was judged a more serious threat to economic performance than the higher inﬂation that might ensue in the more probable scenario’’ (p. 37).

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12. Knightian uncertainty: ‘‘When confronted with uncertainty, especially Knightian uncertainty, human beings invariably attempt to disengage from medium- to long-term commitments in favor of safety and liquidity. Because economies, of necessity, are net long (that is, have net real assets) attempts to ﬂee these assets causes prices of equity assets to fall, in some cases dramatically . . . The immediate response on the part of the central bank to such ﬁnancial implosions must be to inject large quantities of liquidity . . . ’’ (p. 38). 13. Parameters (ﬁxed- and time-varying): ‘‘The economic world in which we function is best described by a structure whose parameters are continuously changing. . . . We often ﬁt simple models [with ﬁxed parameters] only because we cannot estimate a continuously changing set of parameters without vastly more observations than are currently available to us’’ (p. 38). 14. Multiple risks: ‘‘In pursuing a risk-management approach to policy, we must confront the fact that only a limited number of risks can be quantiﬁed with any conﬁdence . . . . Policy makers often have to act, or choose not to act, even though we may not fully understand the full range of possible outcomes, let alone each possible outcome’s likelihood. As a result, risk management often involves signiﬁcant judgment as we evaluate the risks of different events and the probability that our actions will alter those risks’’ (p. 38). 15. Policy rules: ‘‘For such judgment [mentioned above], policymakers have needed to reach beyond models to broader, though less mathematically precise, hypotheses about how the world works. For example, inferences about how market participants and, hence, the economy might respond to a monetary policy initiative may need to be drawn from evidence about past behavior during a period only roughly comparable to the current situation. Some critics have argued that such an approach to policy is too undisciplined – judgmental, seemingly discretionary, and difﬁcult to explain. The Federal Reserve, they conclude, should attempt to be more formal in its operations by tying its actions, solely, on the weaker paradigm, largely, to the prescriptions of a simple policy rule. Indeed, rules that relate the setting of the federal funds rate to the deviations of output and inﬂation from their respective targets, in some conﬁgurations, do seem to capture the broad contours of what we did over the past decade and a half. And the prescriptions of formal rules can, in fact, serve as helpful adjuncts to policy, as many of the proponents of these

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rules have suggested. But at crucial points, like those of our recent policy history (the stock market crash of 1987, the crises of 1997–1998, and the events that followed September, 2001), simple rules will be inadequate as either descriptions or prescriptions for policy. Moreover, such rules suffer from much of the same ﬁxed-coefﬁcient difﬁculties we have with our large-scale models’’ (pp. 38–39). Forecasting: ‘‘While all, no doubt, would prefer that it were otherwise, there is no way to dismiss what has to be obvious to every monetary policymaker. The success of monetary policy depends importantly on the quality of forecasting. The ability to gauge risks implies some judgment about how current economic imbalances will ultimately play out . . . . Thus, both econometric and qualitative models need to be continually tested’’ (p. 39). Monetary policy: ‘‘In practice, most central banks, at least those not bound by an exchange-rate peg, behave in roughly the same way. They seek price stability as their long term goal and, accounting for the lag in monetary policy, calibrate the setting of the policy rate accordingly. . . . All banks ease when economic conditions ease and tighten when economic conditions tighten, even if in differing degrees, regardless of whether they are guided by formal or informal inﬂation targets’’ (p. 39). Uncontrolled outcomes and targets: ‘‘Most prominent is the appropriate role of asset prices in policy. In addition to the narrower issue of product price stability, asset prices will remain high on the research agenda of central banks for years to come. . . . There is little dispute that the prices of stocks, bonds, homes, real estate, and exchange rates affect GDP. But most central banks have chosen, at least to date, to view asset prices not as targets of policy, but as economic variables to be considered through the prism of the policy’s ultimate objective’’ (p. 40). Performance rating: ‘‘We were fortunate . . . to have worked in a particularly favorable structural and political environment. But we trust that monetary policy has meaningfully contributed to the impressive performance of our economy in recent decades’’ (p. 40). Further evaluation of current monetary policies dealing with the 2007–2008 credit crisis is an important issue.

1.3. Greenspan’s Problems and Econometric Research It is of interest to relate Greenspan’s problem areas to current and past Bayesian econometric research. In econometric research, along with other

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scientiﬁc research, three main areas of activity have been recognized, namely, deduction, induction, and reduction, see Jeffreys (1957, 1939 [1998]) and Zellner (1985, pp. 3–10 and 1996, Chapter 1) for discussions of these topics and references to the huge literature on the deﬁnitions and other aspects of these research areas. Brieﬂy, deduction involves use of logic and mathematics to prove propositions given certain assumptions. Induction involves development and use of measurement, description, estimation, testing, prediction, and decision-making procedures, while reduction involves creating new models and methods that are helpful in explaining the past, predicting as yet unobserved outcomes at various places and/or times and in solving private and public decision problems. While much more can be and has been said about deduction, induction, and reduction, most will agree about the difﬁculty of producing good new or improved models that work well in explanation, prediction, and decisionmaking. However, as we improve our understanding of these three areas and their interrelations in past and current work and engage in more empirical predictive and other testing of alternative models and methods, testing that is much needed in evaluation of alternative macroeconomic models, as emphasized by Christ (1951, 1975), Fair (1992), and many others, more rapid progress will undoubtedly result. A categorization of Greenspan’s problems by their nature is shown in Table 1. It is seen that many of Greenspan’s problems have a deductive or theoretical aspect to them but, as recognized in the literature, deduction alone is inadequate for scientiﬁc work for a variety of reasons, perhaps best summarized by the old adage, ‘‘Logical proof does not imply complete certainty of outcomes,’’ as widely appreciated in the philosophical literature and elsewhere. Perhaps, the most striking aspect of Table 1 is the large number of entries in category III, reduction. Economic theorists, econometricians, and others have to get busy producing new models and methods that are effective in helping to solve former Chairman Greenspan’s and now Chairman Bernanke’s problems. See Hadamard (1945) for the results of a Table 1. Categories (I) Deduction (II) Induction (III) Reduction

Tabulation of Greenspan’s Problems Listed Above. Problem Numbers 3, 4, 6, 9, 10, 11, 12, 13, 14, 16, 17, 19 2, 3, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19 1, 4, 8, 12, 13, 15, 16, 18

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survey of mathematicians that provides information on how major breakthroughs in mathematics occurred and tips on how to create new theories in mathematics that may also be helpful in reductive econometric work as discussed in Zellner (1985, pp. 8–10). Also, in Zellner and Palm (2004) some methods for creating new econometric models and checking old econometric models and applications of them by a number of researchers are presented that may be helpful in the production of new econometric models that perform well in explanation, prediction, and policy-making. More will be said about this reductive problem area below.

1.4. Overview of Paper With this in the way of an introduction to some current problems facing us, in Sections 2 and 3 we shall review some early and recent work in Bayesian econometrics and relate it to some of the problems mentioned by Chairman Greenspan and consider future possible developments in Bayesian econometrics in Section 4.

2. THE PAST 2.1. Early Bayesian Econometrics As is the case with many others who commenced study of econometrics in the 1950s, in my graduate econometrics courses at the University of California at Berkeley there was no mention of Bayesian topics except in a game theory course that I took with David Blackwell (who many years later introduced an elementary Bayesian statistics course at Berkeley using Berry’s (1996) text). Also, there was no mention of Bayesian analysis in Tintner’s (1952) popular text or in most Cowles Commission publications. Although, in Klein’s Textbook of Econometrics (1953, p. 62) some discussion of Bayesian decision theory along with a reservation about prior distributions appeared that he apparently abandoned later in an invited paper, ‘‘Whither Econometrics?’’ published in JASA in which Klein (1971) wrote, ‘‘Bayesian methods attempt to treat a priori information in a systematic way. As a pure and passive forecaster of econometric methodology I can see a great deal of future research effort being channeled in that direction. Systematic ways of introducing a priori information are to be desired’’ (p. 420). Also Theil’s (1978) econometrics text included a chapter titled, ‘‘Bayesian Inference and

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Rational Random Behavior’’ in which he explained Bayes’ theorem and provided some interesting applications of it. However, he expressed strong reservations about improper prior distributions and also wrote, ‘‘The Bayesian approach is itself a matter of considerable controversy. This is not surprising, given that the approach takes a fundamentally different view of the nature of the parameters by treating them as random variables’’ (p. 254). There is no question but that Klein’s forecast regarding future econometric methodology, presented above, has been quite accurate. Much past and current Bayesian research is indeed focused on how to formulate and use prior distributions and models that incorporate ‘‘a priori information’’ in analyses of a wide range of estimation, prediction, and control problems with applications in many ﬁelds using ﬁxed and random parameter models. See the early papers by Dre`ze (1962), Rothenberg (1963), and Zellner (1965) presented at the ﬁrst World Congress of the Econometric Society in Rome, 1965 for some Bayesian results for analyzing the important simultaneous equations model. In my paper, I presented some numerical integration results, obtained using ‘‘old-fashioned’’ numerical integration methods that were of great interest to Malinvaud whose well-known 1964 (translated from French into English in 1966) econometrics text, along with many others, made no mention of Bayes and Bayesian methods, nor of the early Bayesian papers that Qin (1996) cites: ‘‘The early 1960s saw pioneering Bayesian applications in econometrics. These included published works by Fisher (1962), Hildreth (1963), Tiao and Zellner (1964, 1965) and Zellner and Tiao (1964) and unpublished works by Dre`ze (1962) and Rothenberg (1963)’’ (pp. 503–504). See also Chetty (1968) for an early Bayesian analysis of macroeconomic models introduced by Haavelmo. In spite of these and a number of other theoretical and applied Bayesian publications that appeared in the 1960s and early 1970s, in the 1974, completely revised, 2nd edition of Klein’s Textbook Of Econometrics, he wrote: Bayes’ theorem gives a logical method of making probability inferences if the a priori probabilities are known. They seldom are known and this is the objection to the use of this theorem for most problems of statistical inference. A major contribution of decision function theory [that he ably describes in this chapter of his book] is to show the relation of various inferences to Bayes’ type solutions. The beauty of the theory is that it includes hypothesis testing and estimation methods as special cases of a more general approach to inference. (p. 64)

It is clear that Klein, along with Dre`ze, Leamer, and some other econometricians, had a deep understanding of the decision theoretic approach to Bayesian statistical inference that Ramsey, Savage, Friedman, Raiffa,

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Schlaifer, and others had developed. However, he along with many others was either unfamiliar with or possibly dissatisﬁed with the early Bayesian econometric work, cited above, and with Jeffreys’ important Bayesian books, The Theory of Probability that ﬁrst appeared in 1939 with new versions in 1948, 1961, 1967, and 1998 and his Scientiﬁc Method that ﬁrst appeared in 1931 followed by later editions in 1937 and 1957. This work provided his natural science colleagues and those in other disciplines an effective method for learning from their data and experience by use of Bayesian procedures for estimation, testing, and prediction that he illustrated in many applications. In particular, he put forward an approach to inference using diffuse or noninformative priors to represent ‘‘knowing little’’ or ‘‘ignorance’’ that he used in solving many central statistical problems facing scientists and noted that his approach and results could be used to good advantage in decisionmaking. Most importantly, Jeffreys’ work involved not just theory, including among other things, a new general method for generating diffuse, invariant priors, new Bayesian statistical testing methods, and an axiom system for scientiﬁc induction, but also analyses of many important applied problems. See papers in Zellner (1980) for more information about Jeffreys and his theoretical and applied research. Thus, while some did have a good understanding of Bayesian theory, there was limited use of Bayesian techniques in the 1960s and 1970s.

2.2. The Cowles Commission Activities Modern econometrics had its beginnings in the 1940s by workers of the Cowles Commission at the University of Chicago, strongly inﬂuenced by Haavelmo’s (1944) monograph, ‘‘The Probability Approach in Econometrics’’; see Christ (1994), Epstein (1987), Morgan (1990), and Qin (1993) for more details. Marschak (1950, 1953) provided introductory chapters for the inﬂuential Cowles Commission volumes 10 and 14 in which he discussed general econometric methodology, including deductive, inductive, and reductive issues, in a most knowledgeable and thoughtful manner. What is not emphasized enough in the historical works, cited above, in my opinion, is that the situation at the Cowles Commission was not a case of peaceful, innovative research devoted to carrying out the research programs so thoughtfully described in Haavelmo’s, Koopmans’, and Marschak’s works, but was literally a battle between Cowles Commission researchers and Friedman, and others who questioned the usefulness of the elaborate simultaneous equations models and complicated, non-Bayesian statistical

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methods considered and developed by Haavelmo, Koopmans, Klein, Anderson, Rubin, and others. Christ (1951) performed forecasting tests of one of Klein’s models and found that it did not perform very well and a similar conclusion was reached vis-a`-vis one of Tinbergen’s early models, results discussed by Friedman (1951), who regarded Tinbergen’s model as a valuable source of hypotheses for future work and Klein’s model as requiring some disaggregation but thought that we lacked the knowledge about the economic dynamics of an economy needed to produce a successful disaggregated model. Further, there was much ‘‘reductive’’ controversy about the Keynesian and monetarist approaches to modeling economies. Work by Friedman and Meiselman (1963) compared simple one equation ‘‘reduced form’’ Keynesian and monetarist multiplier models in empirical analyses with the ﬁnding that the monetarist model was strongly favored by data for many historical periods except for the period including the Great Depression of the 1930s when both models worked about equally well. Later these results were conﬁrmed by Geisel (1975) using posterior odds in a pioneering application of Bayesian methods to evaluate alternative models. Also, in my doctoral dissertation and in Zellner (1957) and later in Zellner, Huang, and Chau (1965), using U.S. quarterly data, it was found empirically that a Pigou or monetarist real balance effect on consumer spending, much emphasized in the monetarist literature, exists and is important particularly with respect to the effects of changes in consumer money holdings on consumer spending on durable goods and services. That these important monetarist real balance effects were not included in Cowles Commission models, that tended to be Keynesian in general, was another point of contention. Last, and perhaps most important, the large, dynamic simultaneous equations models that the Cowles Commission researchers and others produced using much prior information, were implemented using nonBayesian estimation and testing procedures. Many contained a large number of nonlinear stochastic difference equations, were very complicated and not very well understood by the model-builders and others. See Adelman and Adelman (1959) and Zellner and Peck (1973) for some results illustrating these points produced by use of simulation experiments with one of Klein’s and Goldberger’s early models of the U.S. economy and the Federal Reserve-MIT-PENN model of the U.S. economy containing about 170 equations, some of them nonlinear, that revealed a number of unusual features of these two models. These issues relating to the complexity of Cowles Commission type models and their failure to explain the past and forecast well, as compared to simple random walk and other time series

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benchmark models, were central in very heated exchanges during the Cowles Commission days and beyond. As Greenspan notes above, having models that forecast well and are helpful in explaining the past is most important. The above reductive inference issues involved important disagreements with respect to the relative virtues of simplicity and complexity in building models, with Friedman and many of his colleagues favoring starting simply and complicating if necessary while Klein and others appear to have taken the position that the world is complicated and thus we need complicated models. Even today, this range of issues is still being debated by Sims, Hendry, Diebold, Lu¨tkepohl, Pagan, myself, and many others. What was not explicitly recognized is that if there is a need for capturing much detail, models that are large and sophisticatedly simple can be constructed rather than large and complicated models and that the former will be more understandable and will probably work better in explanation and prediction. Also, these issues of simplicity and complexity in model-building in science have been considered at length in the literature; see, for example, the papers and references in Zellner, Kuezenkampf, and McAleer (2001), presented and discussed at a conference on simplicity and complexity in the Netherlands. It seems that over the years many more have come to favor KISS, Keep it Sophisticatedly Simple, in model formulation, in part based on esthetics but more importantly on empirical results that indicate that sophisticatedly simple models often work better in forecasting and explanation than complicated models. In this connection, see the quotation from Greenspan’s (2004) paper in point number 7 above regarding his successful use of simple models and the results of a survey of economics Nobel Prize winners in Zellner et al. (2001) that showed that almost all of them favored keeping models sophisticatedly simple and starting simply and complicating if necessary.

2.3. Bayesian Decision Theoretic Approach In addition, in important articles Friedman and Savage (1948, 1952) developed a decision theoretic Bayesian statistical approach that combined economic utility theory and statistical theory, deeply appreciated by Klein (1953, 1971), and many others that is very different from the sampling theory statistical estimation, testing, prediction, and decision-making approaches utilized by researchers at the Cowles Commission. Thus, while Haavelmo and others did emphasize and produce a probabilistic approach to econometrics (without, however, thoroughly deﬁning and defending their concept of

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probability), their implementation of it was quite different from that favored by Friedman, Savage, and others, namely the decision theoretic approach mentioned above in the quotations from Klein’s text and invited JASA paper. Also, the econometric models that they developed were found not to forecast very well, were not easy to justify in terms of alternative macroeconomic theories and found to be very complicated with dynamic properties that were not well understood by many, including the model-builders. All of this was fuel for prolonged ‘‘discussions’’ and debates held during the Cowles Commission days at Chicago. The work on Bayesian decision theory by Savage, Friedman, and others led to a very impressive book by Raiffa and Schlaifer (1961), on methods and applications of Bayesian decision theory that inﬂuenced Martin Feldstein, who worked with Raiffa and Schlaifer at Harvard. Most of this work went on recognizing earlier theoretical Bayesian research by Bayes (1763), Edgeworth (see Bowley (1928) for a description of Edgeworth’s impressive Bayesian research results), de Finetti (1974), Savage (1954), and Ramsey (1931) but in large part overlooking Jeffreys’ (1939 [1998]) book, Theory of Probability, and his earlier and later work. Note that physicists do not like the word ‘‘statistics’’ and thus use ‘‘probability theory’’ instead. Also, in the inﬂuential Cowles Commission Monographs 10, Statistical Inference in Dynamic Economic Models, edited by T. C. Koopmans (1950) and 14, Studies in Econometric Method, edited by Hood and Koopmans (1953), including articles by leading statisticians, T. W. Anderson, H. Hotelling, H. Rubin, A. Wald et al., there is not a single reference to Bayes’ theorem and Jeffreys’ important work on Bayesian philosophy, estimation, testing, and other topics. Nor are there any references to the decision theoretic work of Savage, Friedman, and others.

2.4. The Jeffreys’ Approach I was introduced to Jeffreys’ work when participating in a University of Wisconsin Department of Statistics seminar devoted to a reading of Jeffreys’ book, Theory of Probability, in the early 1960s along with George Box, George Tiao, Irwin Guttman, Norman Draper, Mervyn Stone, and other statisticians. Each week at the seminar meeting one of the participants would report on a chapter of Jeffreys’ book followed by questions, answers, and discussion. I reported on one of the most difﬁcult chapters of the book, the chapter on hypothesis testing, a topic that is even quite controversial today; see Berger (2003), Ziliak and McCloskey (2004), and accompanying

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published comments on both papers that deal with issues of whether to test or not and if so, how and comparisons of various approaches to testing. It is indeed surprising that after all these years that there should be such controversy about an important, widely used inference technique, namely testing, with most current Bayesians agreeing that a posterior odds approach, along the lines suggested by Jeffreys, is favored. See Press (2003) who wrote at the end of his chapter on Bayesian hypothesis testing, ‘‘The Bayesian (Jeffreys) approach is now the preferred method of comparing scientiﬁc theories . . . Richard Feynman suggests that to compare theories (in physics) one should use the Bayesian approach’’ (p. 230). This extensive and thorough reading and discussion of Jeffreys’ book provided much material on the philosophy of science, including guidance on issues in deductive inference (namely, an axiom system for systematic learning from data and experience), reductive inference (his and Wrinch’s ‘‘simplicity postulate’’ and a quantitative measure of the simplicity of models), and inductive inference, his Bayesian procedures for analyzing and learning from data that are applicable in all areas of science. In addition to these foundational issues, he provided important critiques of the Fisherian, Neyman-Pearson, and other approaches to statistical inference, that led me to the conclusion that it would be worthwhile to undertake a program of research to evaluate these alternative approaches, theoretically and in applied econometric studies as well as to evaluate his positions on simplicity versus complexity, how to learn from data, and how to formulate good models that are successful in explaining the past, predicting the future, and in making decisions. This emphasis on all three areas, deduction, induction, and reduction is a unique feature of Jeffreys’ work and was not emphasized enough in early Bayesian and non-Bayesian econometric research. After this seminar experience, I determined to try to remedy the situation in my future research to improve learning, model-building, estimation, testing, prediction, and decision procedures in econometrics. As stated at the end of Chapter 1 of Zellner (1971), ‘‘Since, in the past, most econometricians have employed non-Bayesian techniques in their work, it is useful and interesting to compare Bayesian and non-Bayesian analyses of a range of models and problems. In the following chapters this comparative approach is pursued, since, as Anscombe (1961) remarked some years ago about the state of statistics, ‘A just appreciation of the situation can only be had by studying the orthodox and the Bayesian approaches to a variety of statistical problems, noting what each one does and how well it does it.’ ’’ (p. 11). Over the years many of my graduate students, fellow researchers, and I found Anscombe’s approach very fruitful

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and much to our liking. Running Bayesian versus non-Bayesian ‘‘horse races’’ is fun and scientiﬁc too. 2.5. The NBER-NSF Seminar on Bayesian Inference in Econometrics In my research program beginning in the early 1960s and to the present, as noted above, I pursued Anscombe’s approach by developing and comparing Bayesian and non-Bayesian solutions to a number of theoretical and applied model formulation, estimation, testing, prediction, and decision problems drawing on Bayesian research results derived with the help of colleagues and graduate students. These competitive ‘‘horse races’’ were of interest to both Bayesians and non-Bayesians since many wished to learn how to be more effective in their econometric work. And this approach that concentrated on relative performance and not on deep, drawn-out philosophical discussions, not only appealed to many but was also generally recognized as being an important part of a scientiﬁc approach to the evaluation of alternative, econometric methods, and approaches. In the late 1960s, it occurred to me that it would be useful to set up a seminar group that would meet semiannually to discuss Bayesian econometric research. Thus, I recommended to the National Bureau of Economic Research board members that a Seminar on Bayesian Inference in Econometrics be established. Fortunately, my recommendation was approved and the NBER-NSF Seminar on Bayesian Inference in Econometrics was established that held its ﬁrst meeting at the University of Chicago in 1970. Since this meeting and later ones involved both econometricians and statisticians, the name was changed to Seminar on Bayesian Inference in Econometrics and Statistics (SBIES). The seminar met two times a year from 1970 to 1996; see Berry et al. (1996) for a listing of SBIES meetings and some historical information about its activities and accomplishments. In particular, they write in their Preface, ‘‘For many years the seminar was unique in providing a sizable audience of receptive yet scientiﬁcally critical Bayesians to whom investigators could bring new research. These meetings have witnessed the conception or birth of many path breaking ideas later published in scientiﬁc journals, and the death of a few others. Both processes have contributed to the continued progress and vitality of Bayesian econometrics and statistics’’ (p. xvii). Currently, the SBIES is operating under the direction of Sid Chib and has had two very successful recent meetings, one arranged by Chib in 2004 and the other by Geweke in 2005. An overview of new, early Bayesian econometric research in the 1970s can be obtained by viewing the programs of the early meetings of the SBIES, the

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titles and authors of Savage Award winning and honorable mention Bayesian doctoral dissertations presented in Fienberg and Zellner (1975) and in the International Society for Bayesian Analysis (ISBA) web page, and the titles of books published in the SBIES sponsored series, Studies in Bayesian Econometrics and Statistics, published by North-Holland. It is the case that research reported at the early meetings of the SBIES and in the Savage Award theses and North-Holland volumes covered a wide range of topics and was not limited to just research on the ‘‘standard’’ simultaneous equations model. Indeed, there were important sessions of the meetings and a volume devoted to economic theory and Bayesian analysis. Further, as the titles of the volumes indicate, there was much interest in the comparative evaluation of not only Bayesian and non-Bayesian approaches, but also alternative Bayesian approaches, in particular those of Savage, Jeffreys, and de Finettti as well as the Jaynes’ Bayes-max-entropy approach of the Bayes–Maxent Workshop group with which our SBIES group maintained close ties. As an example of this close interaction, in Jaynes’ (1986) paper, published in the SBIES volume honoring de Finetti, he generalized the de Finetti representation theorem in a signiﬁcant way and showed how the theorem has been used in analyses of physical systems. As stated in the Preface of this volume, ‘‘Professor Jaynes discusses the extended de Finetti representation theorem for both ﬁnite and inﬁnite exchangeable sequences. The discussion and the proof are simple and concise. Jaynes also gives an example of the profound impact of this representation theorem in statistical mechanics’’ (p. 3). Thus the deductive part of de Finetti’s Bayesian approach was linked to important results in physics. Also, Jaynes, who was fond of solving paradoxes, published a 1980 article, ‘‘Marginalization and Prior Probabilities’’ in Zellner (1980) in which he showed that the Dawid– Stone–Zidek ‘‘marginalization paradox,’’ widely interpreted to reveal a defect in Bayesian methods, was based on a ‘‘subtle failure . . . to take account of all the relevant information’’ (p. 43). See also the response of Dawid, Stone, and Zidek (1980) followed by Jaynes’ rejoinder. In a similar vein, Leamer (1986) in his de Finetti volume paper provided valuable analysis linking the de Finetti representation of speculative markets to economists’ analyses of such markets. As stated in the Preface, ‘‘Professor Leamer critically examines de Finetti’s demonstration of subjective probabilities being equivalent to prices of lottery tickets which are ‘coherent.’ . . . Leamer suggests that most exchanges of lottery tickets involve bid-ask spreads and thus we need to elicit upper and lower probabilities (intervals of probabilities), since these can be elicited economically and can be more stable than sharp probabilities. He uses several models

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for intervals of probabilities to explore the accuracy and economy of various elicitation games. At least in an economic context, Leamer presents an interesting point of view which needs further investigation’’ (p. 5). These are but two examples of how deductive or axiom systems underlying Bayesian analysis have been subjected to critical analysis in the early stages of the development of Bayesian econometrics. Other axiom systems, for example, Jeffreys’ and Savage’s have undergone similar close scrutiny, see Machina (2004) for a fascinating attempt to provide an axiom system for inductive inference that takes account of both elements of Jeffreys’ axiom system and ‘‘utility-based’’ axiom systems, such as those of Savage and de Finetti. How to represent good or optimal learning behavior and decisionmaking behavior in a reasonable and effective way by producing a uniﬁed set of axioms indicating how individuals ‘‘should’’ behave so that they learn coherently and effectively and make good decisions is indeed a great challenge. Be that as it may, the important foundational work of de Finetti, Jeffreys, Savage, and other Bayesians has contributed substantially to improving our learning and decision-making capabilities and is reﬂected in many early papers in Bayesian Econometrics. The programs of the ﬁrst six meetings of the NBER-NSF Bayesian Seminar, 1970–1972, published in Fienberg and Zellner (1975) reveal the breadth of early Bayesian econometric research that included research in the areas of reduction, deduction, and induction. For example, the session titles for the ﬁrst meeting were: ‘‘Bayesian Methods for Comparing and Choosing Among Models,’’ ‘‘Bayesian Analyses of Regression and Simultaneous Equation Models,’’ and ‘‘Bayesian Adaptive Control Problems,’’ with papers analyzing a wide range of problems. There was a desire to show that the Bayesian approach produced better estimation, testing, model comparison, prediction, and decision procedures. Note for example, the papers by E. C. Prescott on ‘‘The Multi-Period Control Problems Under Uncertainty’’ and H. Woods Bowman, ‘‘Bayesian Control Theory and the St. Louis Model,’’ on very important decision problems, the sort that Greenspan and other policy-makers face. In later meetings of the SBIES, research results on a broad range of topics are also reported including one session on ‘‘Bayesian Methods in Economic Theory’’ including a paper by Richard M. Cyert, an economist and former president of Carnegie-Mellon University and the famous statistician Morris M. DeGroot, ‘‘Analysis of Cooperation and Learning in a Duoply Context.’’ And at the third meeting at Harvard University, arranged by Edward Leamer, there were papers on Bayes–Stein estimation, pre-testing, identiﬁcation in probability, and a Bayesian computer program. In addition, we had

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the pleasure of listening to Leonard J. Savage talk on ‘‘Elicitation of Personal Probabilities and Expectations.’’ Also, at the ﬁfth and sixth meetings of the Seminar a number of Bayesian papers on a variety of topics were presented and discussed including stimulating papers by E. Leamer, ‘‘Multicollearity: A Bayesian Interpretation;’’ C. Sims, ‘‘Post-Data Model Construction as Estimation with Inﬁnitely Many Parameters;’’ S. Grossman, ‘‘A Bayesian Approach to Static, Stochastic Equilibria;’’ and G. C. Chow, ‘‘Effect of Uncertainty on Optimal Control Policies.’’ In a later paper, Grossman (1975) introduced a self-fulﬁlling, rational expectations distribution of price rather than the usual Muthian assumption that an anticipated price is equal to its conditional mean. Grossman utilized the distribution of price, given the data, model and prior information, rather than just its conditional mean in representing price expectations, a very novel and ingenious idea. Last, it is noteworthy that at the 6th meeting at the University of Wisconsin in Madison, May 4–5, 1973, there was a Panel Discussion on Bayesian Inference in Econometrics with G. E. P. Box, J. H. Dre`ze, and S. Geisser as the discussants, who provided very interesting and useful comments that were much appreciated. All recognized the need for good models that perform well and suggested procedures for checking current models. These cogent remarks were of particular interest to me since Franz Palm and I had just recently become engaged in research on how to build good dynamic econometric models that resulted in our early papers, Zellner and Palm (1974, 1975) that have been reprinted along with many other papers using the Structural Econometric Time Series Analysis Approach in Zellner and Palm (2004). In the 1974 paper, the approach was applied to evaluate dynamic variants of a small Keynesian model put forward and estimated by Haavelmo and in the second paper to evaluate variants of a dynamic monetary model formulated by Friedman. In both cases, we found the two initial models to be inadequate and in the second case, we elaborated the model to produce a variant that appeared to be compatible with the information in the data by use of various diagnostic checks, etc. Over the years, the approach has been developed further, augmented by use of Bayesian predictive testing of models’ point and turning point predictive performance and is thought to be helpful in building good dynamic econometric models. Those attending the SBIES meetings appreciated the work of many different types of Bayesians and some non-Bayesians, including the famous statistician George Barnard who has worked on a broad range of problems, was familiar with many statistical approaches and contributed insightful and constructive comments at several of our meetings. When the proposal to

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produce a volume in his honor came up, some one said, ‘‘We can’t do that, he’s not a Bayesian.’’ Then another person responded, ‘‘That doesn’t matter. He’s a great guy!’’ After that we voted in favor of producing a volume honoring that ‘‘great guy’’ George Barnard in which likelihood and Bayesian methods were discussed and compared. Apparently, Barnard was hesitant to introduce formal priors in his analyses but was willing to introduce ‘‘weights’’ in his ‘‘weighted likelihood’’ approach. When I asked him whether the weights could be interpreted as probabilities, he responded, ‘‘In your Bayesian framework they may be probabilities but in mine they are weights.’’ And when he wrote to me inquiring about why a Bayesian group would publish a volume in honor of him, a ‘‘likelihood advocate,’’ I wrote to him explaining our discussion described above. He responded in a letter dated, January 31, 1997 as follows: I had wondered why I should have been judged ﬁt for a Bayesian Festschrift; though in showing back in 1946 that the proper approach to sampling inspection was Bayesian perhaps I did help to draw Bayesian theory back into the limelight. And I’m proud, too, of being the ﬁrst to tell Jimmie Savage about the likelihood principle (as an obvious consequence of his Bayesian approach) . . . He said he wished he’d thought of it himself.

In these early meetings and subsequent meetings, there were not only econometricians in attendance but also statisticians, general economists, and a few from the natural and other social sciences. This diversity tended to produce rather interesting productive results at times. For example, when Ed Jaynes, a well-known physicist, who had an interest in various systems, including economic systems, requested and read a current graduate macroeconomics text that I recommended, he mentioned that models like those in the text would not work in physics since they do not take account of turbulence. That is, he pointed out that the macroeconomic models in the text give the same results whether there is 6% unemployed with no hiring and ﬁring or 6% unemployed with a lot of hiring and ﬁring. Also, his comments on paradoxes, mentioned above, and impressive work on maximum entropy and information theoretic techniques and their applications, see Jaynes (2003), livened up some of our meetings. Last, with respect to the important inﬂuence that the seminar exerted, the Bayesian econometrician Leamer (1978) wrote in the Preface of his wellknown book, ‘‘Many of my ideas have been inﬂuenced by attendees at the semi-annual Seminars in Bayesian Inference in Econometrics. Arnold Zellner especially should be mentioned. He and Jacques Dre`ze were the ﬁrst to carry the Bayesian fasces into the econometrics arena. Another attendee at these seminars (and a Harvard colleague), John Pratt, has had a signiﬁcant

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inﬂuence on my thoughts. Though they may wish to deny it, I have discovered kindred souls in the form of James Dickey and Thomas Rothenberg’’ (pp. viii–ix). All of the above indicates that Bayesian econometricians in the early period were very concerned with and working on a wide range of topics and intent upon showing that Bayesian methods do indeed solve inductive problems better than non-Bayesian methods do. For example, optimal Bayesian estimates for parameters of many models, including the simultaneous equations model, were derived that have been shown to have better sampling properties than leading non-Bayesian estimators; see Diebold and Lamb (1997) for an example involving estimation of a key parameter in the famous Nerlove agricultural supply model for which empirical workers were having difﬁculty in obtaining good estimates using maximum likelihood (ML) and other techniques while unaware that these procedures produced estimators having sampling densities with thick tails and sometimes bimodal that produced many outlying estimates in applications, a general property of limited information maximum likelihood (LIML) estimators for simultaneous equation models’ parameters in small samples. They showed that rather simple Bayesian methods led to a complete posterior distribution for the key supply parameter and an optimal point estimate that has very good sampling properties relative to those of widely used sampling theory estimators that many empirical econometricians had been using for many years. See also Monte Carlo evidence from a number of studies summarized in Zellner (1998) that demonstrate the good sampling properties of Bayesian estimators vis-a`-vis those of popular non-Bayesian estimators, for example 2SLS, OLS, LIML, Fuller’s modiﬁed LIML, etc. Another very important result was that of Stein (1956, 1962), who used Bayesian methods to produce his famous shrinkage estimators that uniformly dominate least squares, ML, and diffuse prior Bayesian estimators relative to quadratic loss for the multimean, regression, and other models under fairly broad conditions. Given the importance of Stein’s results, Walter Vandaele and I spent most of a summer trying to understand Stein’s results and their relations to traditional ML and Bayesian results, with results reported in our 1975 Savage volume paper. Also, we reviewed Lindley’s and others’ alternative methods for deriving shrinkage estimators. And in later work on forecasting annual growth rates for 18 industrialized countries using single equation third order autoregressive relations containing lagged leading indicator variables, it was found that shrinkage, particularly when combined with joint estimation of the 18 countries’ relations led to reasonably good forecasting results in experiments conducted

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during the 1980s and 1990s and reported in papers in Zellner and Palm (2004). At a Bayesian Valencia meeting in the late 1980s, I heard Jose´ Quintana report on his Bayesian portfolio analysis work in which he and his colleagues employed dynamic equations for individual stock prices and ﬁtted them one by one by single equation Bayesian techniques. I commented that it might be useful to ﬁt the equations jointly using Bayesian SUR and Stein shrinkage techniques. He very kindly accepted my suggestion and on his return to Wall Street implemented it in his impressive computer program, along with the use of time-varying coefﬁcients and a time-varying error term covariance matrix to derive optimal Bayesian portfolios, month by month, building on the Bayesian portfolio work of H. Markowitz, R. Litterman, S. Brown, P. Jorion, V. K. Chetty, A. Zellner, and others discussed in a volume, edited by Bawa, Brown, and Klein (1979). Use of shrinkage, joint estimation, and sequential optimization techniques improved the performance of Quintana’s investment ﬁrm’s rates of return, as reported in a series of papers during the 1990s published in the annual proceedings volumes of the ASA’s Section on Bayesian Statistical Science available at www.amstat.org. See, among other striking papers, ‘‘Global Asset Allocation: Stretching Returns by Shrinking Forecasts’’ by Quintana, Chopra, and Putnam (1995), Putnam and Quintana (1995), and Quintana, Putnam, and Wilford (1997). There is no question but that Bayesian portfolio analysis on Wall Street and elsewhere has been helpful in dealing with risk and possibly uncertainty. Perhaps as some state-space engineers claim, allowing parameters to be time varying helps models to adapt quickly to unforeseen structural changes, a problem of great concern to Alan Greenspan, as noted above.

2.6. Dynamic Model Formulation and Effective Forecasting and Decision-Making As regards the very difﬁcult area of producing dynamic multivariate time series models that work well in explanation, prediction, and decision-making, there was much disagreement among Bayesians and others about how to proceed. Also, there was very little methodological guidance on this difﬁcult problem in the Cowles Commission and in the Bayesian and non-Bayesian econometric literature of the 1960s and 1970s. Some wished to push ahead with the old Cowles Commission structural equation modeling approach while others, particularly the ‘‘Minnesota crew,’’ Sims, Litterman, and others turned to ‘‘a theoretical’’ vector autoregressions (VARs), models that were

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advertised as involving little prior misinformation from economic theory and elsewhere, ﬁrst unrestricted and then restricted by the clever Litterman ‘‘Minnesota’’ prior, called Bayesian VARs or BVARs. These early VARs, some involving seven macroeconomic variables with 6 lags on each, giving each equation of the system a relation involving 42 input variables and an intercept that were criticized by many for being over-parameterized, as well as the newly developed Cowles Commission type models did not forecast as well as some naive benchmark models (e.g., random walk or simple BoxJenkins time series models); see Christ (1975), Cooper (1972), Litterman (1986a, 1986b), McNees (1986), Nelson (1972), and Nelson and Plosser (1982) for results relating to the quality of alternative models’ point forecasts. Further, in 2001, when I visited the Bank of England and talked with the research staff, they informed me that their experiments with BVARs did not produce very good forecasts. More recently, Adolfson, Laseen, Linde, and Villani (2005) reported on the relative forecasting performance of an open economy dynamic stochastic general equilibrium (DSGE) model for the Euro area, implemented using Bayesian methods, vis-a`-vis a wide range of reduced form forecasting models such as VARs, BVARs, univariate random walks, and naive forecasts based on the means of most recent data. They found that the DSGE model performed well relative to competing models, particularly at horizons four to eight quarters ahead and state that a possible reason for this is that the DSGE model has a richer theoretical structure that probably has an impact on forecasts in the long-run, where historical patterns captured in the VAR systems can lead to more erroneous forecasts, at least without a prior on the steady state. In addition, many forecasters seem to be in general agreement that all models tended to perform poorly in forecasting economic turning points, that is, downturns and upturns in economic activity. Perhaps one of the most cogent, early evaluations of the a theoretical, time series approach via VARs, etc., was provided in the comments by Klein (1975) on a paper by Sargent and Sims, ‘‘If they do not introduce some more aspects of system structure, both from economic theory and knowledge of economic institutions, for restricting the parametric speciﬁcations of their models, I am afraid that all is lost. All the problems of collinearity, shortage of degrees of freedom, and structural change will so confound their interpretation of their results that we shall not know what to make of them. In this respect, I ﬁnd their approach to be disappointingly retrogressive and contrary to the main stream of econometric analysis’’ (p. 203). Here we ﬁnd Klein, an experienced model-builder calling for the wise use of prior information in building models. Indeed, there is a great need for combining

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prior knowledge and data in a fruitful way to produce good models, as recognized by many, including Leamer (1978), Litterman (1986a, 1986b), Phillips (1991), Zellner and Palm (1974, 2004), and many others. Similarly, the somewhat a theoretical ‘‘General-to-Speciﬁc’’ (gets) modelbuilding strategy of Hendry and others does not appear as yet to have produced models that work well in explanation, prediction and policymaking. Note, however, that in a recent paper, Lu¨tkepohl (2007) writes, after pointing out that few actually employ a gets approach, ‘‘ . . . I have argued that the leading approach used for modeling multiple time series is a spec [speciﬁc to general] approach. It has theoretical and practical advantages in particular if cointegrated variables are involved. In fact, a bottom-up approach to multiple time series analysis that starts from analyzing individual variables or small groups of variables and uses the results from that analysis to construct a larger overall model has a long tradition (see, e.g. Zellner & Palm, 1974; Wallis, 1977, for early contributions to this literature and Min & Zellner, 1993 and Zellner & Palm, 2004, for more recent related work)’’ (p. 323). In recent work, described in Zellner and Chen (2001), Zellner and Israilevich (2005), and Kim (2006), it is shown how the ‘‘bottom-up approach’’ has been applied to produce disaggregated, dynamic macroeconometric models that incorporate prior information from economic theory and other sources and have provided encouraging performance in some forecasting tests and simulation experiments. This model-building activity is part of reductive inference, and as stated earlier, reductive inference procedures are not well-deﬁned and remain to a large extent an art. What apparently is needed and may be emerging is a useful model-building strategy that makes use of data, prior information, economic theory, institutional knowledge, and mathematical, computer simulation and statistical procedures in a logical sequential fashion to produce models that perform well.

2.7. The Bayesian Learning Model With respect to the Bayesian learning model, Bayes’ theorem, it was recognized by many that the theorem, as with all theorems, is based on certain assumptions, in particular ‘‘the law of insufﬁcient reason,’’ that may not be satisﬁed in all circumstances; see Jeffreys (1939 [1998], pp. 24–25), Stigler (1986), and Zellner (2004) for discussion and additional references. For example, the product rule of probability that is used by many to prove

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Bayes’ theorem is based on the assumption that the probability of drawing a particular element of a set containing n elements, is 1/n, the same for all elements, an assumption that is supposed to reﬂect ignorance and no a priori prejudice, according to Jeffreys and others. Jeffreys (1939 [1998], pp. 23–25) was so concerned about the general validity of this assumption that he introduced the product rule as an axiom rather than as a theorem in his axiom system. This concern about the usual proof of Bayes’ theorem, led me in my 1988 paper (Zellner, 1988a) to put forward an information theoretic ‘‘optimization approach’’ for deriving optimal learning models, including the Bayesian learning model, Bayes’ theorem. When standard inputs, the information in a prior density and a likelihood function, and outputs, the information in a post data density for the parameters and a marginal density for the parameters are employed along with the Gibbs–Shannon information measure, minimization of the difference between the output and input information with respect to the form of the output density for the parameters, the optimal solution turned out to be the result provided by Bayes’ theorem, namely, take the posterior pdf ¼ (prior pdf) (likelihood function)/(marginal density for the observations) and when this is done, input information ¼ output information and thus the information processing rule, Bayes’ theorem is 100% efﬁcient. See comments on this result by the discussants of my paper, Jaynes, Hill, Kullback, and Bernardo and my response. Fortunately, the commentators were rather positive about my approach and results and suggested that it can serve as a basis for much additional fruitful work, since it provides an optimization approach for deriving optimal information processing or learning rules, including Bayes’ theorem, that are very useful in practice and provide additional theoretical justiﬁcation for the Bayesian learning model, Bayes’ theorem. In later work, I have used the optimization approach along with a variety of differing inputs to produce a range of optimal information processing rules, all of them 100% efﬁcient. See Table 2 for some results that involve using just a likelihood function as an input, or inputting weighted or discounted likelihood functions and quality adjusted priors, called ‘‘power priors’’ in the literature. The optimal output post data densities for the parameters are operational and optimal variants of Bayes’ theorem that have been employed in a number of applied studies. Now it is the case that we have a set of optimal information processing rules ‘‘on the shelf,’’ for use, many of which have been shown to be effective in applied studies; for references see my invited ASA 2003 meetings paper, ‘‘Some Aspects of the History of Bayesian Information Processing,’’ in Golan and Kitamura (2007)

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Table 2.

Optimal Bayesian Information Processing Results.

Inputs

Output: Optimal Information Processing Rule

(1) Prior density, p; likelihood function, l (2) Likelihood function, l (3) Post data moments,a R mi ¼ yi g dy; i ¼ 1, . . . , m

gppl gpl

(4) Prior density, p Post data moments, R i y gðyjDÞdy ¼ mi i ¼ 1, 2, . . . , m (5) Quality adjusted inputs pw1 and l w2 , 0ow1 ; w2 1 (6) Inputs for time period t,b t ¼ 1, 2, . . . , T gt1,lt (with g0 ¼ p0 ; the initial prior density)

m P g / p exp li yi

m P g / exp li yi 1

1

g / pw1 l w2

gt / gt1 l t ;

t ¼ 1; 2; :::; T

a g denotes the post data density and li’s are Lagrange multipliers. Extensions to cases in which vectors and matrices of parameters are employed, as in multiple and multivariate regressions are available; see references at end of paper. b See Zellner (2000) for discussion of the solution to this multiperiod information processing problem, a dynamic programming problem.

along with other papers on information theory and econometrics. These developments in Bayesian information processing bring together traditional Bayesian inference techniques and elements of information theory that Jaynes (1988) described as follows: ‘‘ . . . entropy has been a recognized part of probability theory since the work of Shannon 40 years ago . . . This makes it seem scandalous that the exact relation of entropy to the other principles of probability theory is still rather obscure and confused. But now we see that there is, after all, a close relation between entropy and Bayes’s theorem. Having seen this start, other such connections may be found, leading to a more uniﬁed theory of inference in general. Thus, in my view, Zellner’s work is probably not the end of an old story but the beginning of a new one’’ (pp. 280–281). 2.8. Bayesian Computing The story of the development of Bayesian computing techniques since the 1960s is phenomenal and involved going from desk tops, to mainframes, to PCs, and to hand helds and that involved enormous gains in capacity, speed,

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broader applicability, and many successful applications. Indeed, Herman van Dijk used the phrase, ‘‘The Simulation Revolution in Bayesian Econometrics’’ to describe the enormous impact of new developments in computation on Bayesian inferences techniques. See Zellner, Bauwens, and van Dijk (1988) for one of the early applications of a direct simulation approach. In Zellner (1971), Appendix C, titled ‘‘FORTRAN Programs for Numerical Integration,’’ written by my then graduate research assistant, Martin S. Geisel, it is explained that ‘‘The purpose of this write-up is to explain FORTRAN programs for numerical integration of univariate and bivariate integrals that use Simpson’s rule. Simpson’s rule is chosen because it combines reasonable accuracy with simplicity’’ (p. 400). Alternatives considered at the time included use of series expansion techniques, along with term by term integration and the trapezoidal and Gaussian quadrature rules. In the years ahead, these and more sophisticated techniques were incorporated in a number of Bayesian computer programs for analyzing regression, time series, simultaneous equations, and other models; see Press (1980) for a listing and discussion of early programs, for example, BAYREG, SEARCH, BRP, BRAP, RATS, etc., all mainframe programs followed later by PC programs, for example PCBRAP, etc. The development of these computer programs made it possible to apply Bayesian methods and data in analyses of many standard models, for example, multiple regression, VAR, linear time series, and other models. However, for many ‘‘nonstandard’’ problems, the available numerical integration techniques were inadequate and there was reliance on asymptotic expansion techniques or asymptotic normal and other approximate results. Fortunately, developments in numerical analysis occurred that yielded a number of new, effective numerical integration techniques. As regards this success of modern numerical integration methods, van Dijk, Montfort, and Brown (1995) wrote, ‘‘In Bayesian inference, the evaluation of the integrals deﬁned in posterior and predictive moments became a topic of interest in the work of Kloek and van Dijk (1978, 1994), Dre`ze and Richard (1983), Zellner and Rossi (1984), Geweke (1989), and Gelfand and Smith (1990) for example. Simulation methods, in particular Gibbs sampling and importance sampling, allow researchers to study more ﬂexible families of probability distributions in the parameter space . . . It also allows the incorporation of non-linear restrictions on the parameters and it enables researchers to study functions of interest of parameter values. The trade-off between analytical tractability and richness of families of probability distributions has been moved more in favor of the latter. Recently applications deal with ﬁnancial models, unemployment analysis and cointegration models’’ (p. vi). An early,

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pioneering study in this area is Kloek and van Dijk (1978). Further, there is now widespread recognition of the great value of MCMC techniques in Bayesian econometrics; see discussions of these techniques in the current Bayesian econometrics texts listed in the next section and in many statistical texts and papers (for references to Bayesian computing and programs, e.g., BATS, BUGS, etc., see Berger (2000) and material at the web sites of ISBA, http://www.bayesian.org and SBSS (http://www.amstat.org). For valuable discussions of this important area of activity, see Geweke (1999) and Chib (2001) and for a good example of how to use these new techniques to solve a very difﬁcult Bayesian estimation problem, see Chib and Greenberg (1994), as well as many other examples in the Bayesian texts listed in the next section. It is accurate to state that modern computing techniques have permitted most numerical Bayesian econometric estimation, testing, prediction, and decision problems to be solved, as shown in many publications on numerical analyses of most difﬁcult problems using MCMC and other ingenious techniques. Also, many have been involved in research that has produced important convergence and other checks on the accuracy of MCMC and other computational algorithms. Some dangers that these ingenious checks detect are convergence to the wrong value, incomplete convergence, proper density output of a calculation when in fact the output density is improper, etc. For material on convergence checks with many references, see the Bayesian texts listed below. Another important, well-known procedure in computing is what is called the direct Monte Carlo or direct sampling approach; see Geweke (2005, pp. 106–109) for a discussion of it. In this approach, the joint posterior density for the parameters is analytically converted into a product of conditional and marginal densities from which draws can be made thereby yielding a draw from the joint density. Say our joint density is f(x, y, z) ¼ g(x)h(y|x)j(z|x,y). Then x can be drawn from g and substituted in h and a draw of y made from h, and then the x and y draws are substituted into j and a z is drawn from j. Repeating this procedure many times provides a large sample from the joint density from which moments, intervals, etc. can be computed. An early example of this procedure applied to calculating the predictive density of three (or more) future observations generated from a linear autoregressive model is provided in Miller and Thompson (1986). For calculating posterior densities for parameters and predictive densities in instrumental variable and simultaneous equations models by the direct MC approach with an application, see Zellner and Chen (2001) and for the SUR model, see Zellner and Ando (2008). Unfortunately, all problems cannot be

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reduced to the form needed for the direct MC approach and hence, as recognized in the literature, the need for MCMC and other techniques that have been applied successfully in many studies.

2.9. Growth of Bayesian Econometrics There is no question but that the computing ‘‘revolution’’ has indeed been a major factor in the recent rapid growth of Bayesian econometrics since the early 1960s, as Qin (1996) and many others have emphasized along with the growth in the number of successful Bayesian studies and the introduction of many more Bayesian econometrics and statistics courses. As an example, Hey (1985) wrote in the foreword of his Bayesian text, ‘‘The fundamental malaise with most statistics and econometrics courses is that they use the ‘Classical’ approach to inference. Students ﬁnd this unnatural and contorted. It is not intuitively acceptable and does not accord with the way that people assimilate information (statistical or otherwise) in their everyday life. In practice, people hold (whether consciously or not) probabilistic views about uncertain theories, hypotheses and phenomena, and they update these views in the light of new information. This is precisely the view of statistics adopted by the Bayesian approach’’ (p. xi). He goes on to explain, ‘‘I ﬁrst taught a Bayesian course two years ago . . . To my delight, I discovered that the ‘fundamental problem’ had disappeared . . . I am now more than convinced, in the light of my experience of these two years, that the Bayesian approach is the ‘correct’ one to adopt. (Though this is not to assert that the Bayesian approach is free from all problems; indeed, it is clear that it is not, but these are problems common to all approaches.)’’ (p. xi). In an attempt to measure the progress of Bayesian econometrics, Poirier (1989, 1992) reported the percentages of pages devoted to Bayesian articles in statistics and econometrics journals for the 5 year periods 1982–1988 and 1987–1991 as well as for individual years. He concludes in his 1992 paper, ‘‘The results for the 1987–1991 period support the earlier 1982–86 results. On average statistics journals have a larger Bayesian content than econometrics journals. A representative ﬁgure for statistics journals is 11.5%, and a representative ﬁgure for econometrics journals is 7%. Only Journal of Econometrics has a Bayesian content representative of the statistics journals [10.0% and 11.9% for the earlier and latter periods, respectively, an increase of about 20% in the proportion of Bayesian articles in this journal].’’ Since Dennis Aigner and I were co-editors of the Journal of Econometrics during this period and both of us appreciated the activities of the NBER-NSF

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SBIES that involved many statisticians, this may account for the fact that the Bayesian coverage in the Journal of Econometrics was somewhat higher than for other econometrics journals. Most important is the fact that the percentages of pages devoted to Bayesian articles in many statistics and econometrics journals went from very low levels in the 1960s and 1970s to much higher levels in the 1980s and early 1990s. In a later paper, Poirier (2004), provides data on ‘‘the annual proportion of journal ‘articles’ containing in their text either the words ‘Bayes’ or ‘Bayesian.’ While not all such articles are ‘Bayesian,’ their numbers provide an upper bound on the number of Bayesian articles, and they capture the impact of Bayesian thinking on authors.’’ (p. 1). From his Fig. 1, with the annual data relating to seven disciplines, including statistics, economics, philosophy, sociology, political science, general science, and ﬁnance, over the period 1970–1998, the trend is upward in all ﬁelds with statistics going from about 12% in 1970 to about 37% in 1998, a huge increase. As regards economics, the percentage of papers including the words Bayes or Bayesian in 1970 was about 2% and rose to about 10%, that is 1 paper in 10, in 1998. Some theoretical statistics journals (JASA, JRSSB, and Statistical Science) started at about 15–25% in 1970 and rose to about 60% in 1998! For ‘‘All-Purpose’’ Economics Journals (Econometrica, IER, ReStat, and ReStud), data in Fig. 5 indicate an upward trend from about 0 to 5% in 1970 to about 20 to 30% in 1998. The proportions in Fig. 6 for econometric articles in Econometrica start at a very high level in 1970, close to 50% and end up the period with several years in the late 1990s varying between 40 and 58%, very high percentages indeed, that may be due to the growing use of the words Bayes and Bayesian in economic theory articles appearing in Econometrica, as Poirier points out, and perhaps to the inﬂuence of two invited papers published in Econometrica by Dre`ze (1972) and Zellner (1985). In the data for the Journal of Econometrics in Fig. 8, the data are quite variable indicating a slight upward trend from about 20% in the 1970s to about 30% at the end of the last century. That the move to Bayesian methodology and the use of the words Bayes and Bayesian has been strongly upward since the 1960s and quite variable, reminds one of studies of the introduction of new products and services in the marketing literature by Bass and others as well as in the economic literature where individuals’ reservation prices relative to the market price play an important role in determining how quickly sales grow. Also, technical changes can impact on sales of new products and even cause ‘‘shake out’’ periods. With respect to the cost or price of becoming a Bayesian, it has probably been falling over the years, especially with the introduction of new

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Bayesian courses, papers, texts, books, meetings, computing techniques, etc.; see the list of recent Bayesian publications on the ISBA web site, www.bayesian.org. However, technical change, for example new computing techniques requiring new knowledge, can cause turbulence and slow the transition of many from non-Bayesian to Bayesian status and even cause some to revert from Bayesian to non-Bayesian status. Finally, competition from new non-Bayesian approaches may negatively impact the rate at which individuals become Bayesians. Or as widely recognized, things do not always go smoothly in a Kuhn-like paradigm shift. Given that the Bayesian paradigm has become much more widely used in recent years, we now turn to consider the present state of Bayesian econometrics. Since the present is just a point in time between the past and the future, this section will be brief.

3. THE PRESENT STATE OF BAYESIAN ECONOMETRICS At present, there are many Bayesian econometric researchers, papers, texts, courses, computer programs, meetings, chapters on Bayesian methods in general econometrics texts, etc. The following are some publications that provide recent theoretical and applied research results, information about available Bayesian computing procedures, and much information about past Bayesian research that is useful to researchers, decision-makers, those lecturing Bayesian econometrics courses, students, and others: 1. 2. 3. 4. 5. 6. 7. 8.

Bauwens, Lubrano, and Richard (1999). Dorfman (1997). Geweke (2005). Koop (2003). Lancaster (2004). Poirier (1995). Rossi, Allenby, and McCulloch (2005). Zellner (1997a).

These eight publications are but a fraction of the total number of Bayesian publications available in econometrics, statistics, and many other ﬁelds of study and application and more items are downloadable from a number of web sites. They include material on Bayesian philosophy, methods, and solutions to many frequently encountered estimation, testing, prediction, and decision problems that compare very favorably with non-Bayesian solutions

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to these problems, a Bayesian victory in the continuing, competitive ‘‘horse races’’ that I mentioned earlier. The range of applications presented in these works is indeed very broad. In Lancaster (2004), Geweke (2005), and Rossi et al. (2005) both datasets and computing algorithms are available either in the texts and/or on web sites. These works include impressive analyses of standard and some very difﬁcult econometric problems using MCMC and other modern computing techniques. Dorfman (1997) and Poirier (1995) present perceptive comparisons of Bayesian and non-Bayesian solutions to many problems that bring to the fore the relative strengths and weaknesses of alternative inference and decision-making techniques. Koop’s (2003) presentation of Bayesian solutions to many econometric problems and his last chapter that provides an overview of many topics currently being researched by Bayesian analysts are most impressive. And in Zellner (1997a), there is material on an overview of Bayesian analysis followed by theoretical and applied analyses of inference, forecasting, and decision problems, including some that relate to the interface of Bayesian analysis and information theory. While much more can be said about the books listed above and other aspects of the current state of Bayesian econometrics, see Qin (1996) and Zellner (2000, 2004) for overviews with many references to the literature, it is perhaps better to recognize that Bayesian econometricians are working hard, in the spirit of Deming, to improve current Bayesian econometric methods and to produce many more successful applications of them. Also, more work on new reductive methods to produce new models would be most worthwhile. With this in mind, in the next section on the future of Bayesian econometrics, I shall discuss and make suggestions as to how old and new Bayesian methods as well as other inductive and reductive methods may help solve some of Greenspan’s broad-ranging problems listed above and to promote more progress in Bayesian econometrics. Given that many of these problems are rather general, getting good solutions to them may not ‘‘merely’’ result in better monetary policies but may also be of value in many other areas of Bayesian theoretical and applied research.

4. THE FUTURE The future of Bayesian econometric research, broadly deﬁned to include reduction, induction, and deduction, will be considered below with the ‘‘narrow’’ purpose of attempting to make suggestions that will help improve current econometric models, methods, and applications of them. Future developments will be discussed with the qualiﬁcations that not everything has been covered and that my predictions are just that, my predictions.

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4.1. Sectoral Disaggregation and Modeling Future model formulation will involve, in my opinion, use of much more disaggregation, the virtues of which have been extolled by many over the years, including Tinbergen, Leontief, Stone, Orcutt, Friedman, Modigliani, and others. The issue has been how to disaggregate. An approach that I favor and have employed is to disaggregate by industrial sectors and to model each sector using, output demand and supply relations and a ﬁrm entry–exit relation in connection with product markets, as Marshall did many years ago, and sectors’ demands for factors, and individuals’ supplies of factors in factor markets along with a government sector that taxes, supplies money, and produces goods and services. Note that most macroeconomic models do not incorporate ﬁrm entry and exit relations, a serious omission since, as emphasized by Marshall and many others, ﬁrm entry and exit behavior is a key element in the process of dynamic industry adjustments to various proﬁt shocks, innovations, etc., as shown in many industry studies. Also, on aggregating individual ﬁrm supply and ﬁrm factor demand equations, the aggregate output supply and factor demand equations depend on the number of ﬁrms in operation, a variable that is often not included in these relations in macroeconomic models. Then too, in real business cycle models, there is the representative ﬁrm and one wonders what happens when it shuts down. Further, when entry and exit relations are omitted, usually spurious lags in supply equations occur that are sometimes misinterpreted. These considerations, which are referenced in our papers and earlier studies, led us to introduce a ﬁrm entry–exit relation in each of our industrial sector product market models, as mentioned above, and lead to nonlinearities and cointegration effects as shown in one sector, two sector and n sector versions of our Marshallian macroeconometric model in Zellner and Israilevich (2005). By disaggregating, the very different behavior of, for example, mining, construction, durable goods, agricultural, service, retail, wholesale, and government sectors can be explained and has been forecasted using sector data and models and Bayesian and other methods. In forecasting, for example, different sectors’ outputs and prices, etc., note that joint Bayesian estimation and shrinkage techniques can and have been employed to forecast not only individual sectors’ variables but also aggregate output and the general price level with results that are more precise than those obtained from aggregate and sector benchmark models, for example, an AR(3) model and an aggregate one sector model. See such results in Zellner and Chen (2001), Zellner and Israilevich (2005), and Kim (2006). In Zellner (2002), an analysis

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of effects of Bayesian shrinkage on the estimation and forecasts of totals as well as sector speciﬁc outcomes is presented that shows that it pays to shrink when forecasting, for example, not only individual sector outcomes but also totals of variables across sectors in most circumstances. As yet, no sector and aggregate forecasting results have been obtained using models with timevarying parameters and Bayesian shrinkage as in the ﬁnancial portfolio formation work of Quintana, Putnam, and others mentioned earlier. Also in the statistical inﬂation and GDP forecasting of Espasa (2005), Espasa, Senra, and Albacete (2002), Espasa and Albacete (2004), Albacete and Espasa (2005), and Minguez and Espasa (2003), for European countries and the United States, they have found that disaggregation of inﬂation into its components, for example, inﬂation in agricultural goods, durable goods, services, imported goods, etc., and combining the disaggregated forecasts to obtain forecasts of total inﬂation provides more precise forecasts than those of models based on the aggregate price data. In one of these papers, it is mentioned that it is possible to derive these successful statistical inﬂation forecasting equations from the Marshallian macroeconometric model presented in our papers, a good example of the approach to structural, causal dynamic model-building described and applied in Zellner and Palm (2004). And as shown in recent work by Kim (2006), by use of two reduced form equations for output and price per sector derived from demand, supply, and ﬁrm entry–exit models for 11 sectors of the U.S. economy, using Bayesian MCMC methods, he was able to improve on the quality of output forecasts produced when using just one equation per sector. Kim is currently working on possible additional improvements associated with use of Bayesian shrinkage along with joint estimation and prediction of two reduced form equations per sector. Also, when data become available on the number of ﬁrms in operation in each of the 11 sectors, an opportunity to use 3 reduced form equations per sector or three restricted structural equations per sector will be available that will probably result in greater explanatory and predictive accuracy. No results are as yet available on the turning point forecasting ability of these disaggregated sector models. For earlier work on Bayesian methods and applied results for forecasting turning points, see Zellner and Min (1999) and the references cited therein. And, as mentioned earlier, the joint predictive density for future outcomes can be utilized to obtain not only point forecasts of future outcomes but also ‘‘fan-like’’ predictive densities for individual future periods’ outcomes, as has been recognized in the literature. Finally, in past work we have used annual data relating to sectors. In the future, use of monthly or quarterly data that will permit more adequate

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modeling of seasonal behavior along with cyclical and trend behavior will probably lead to better models for use in explanation, prediction, and policy-making. It seems highly likely that disaggregating carefully and thoughtfully, and including ﬁrm entry and exit relations can probably help to improve the forecasting performance of the economically motivated, structural DGSE models that have been shown to forecast better than various VAR models in work by Adolfson et al. (2005), cited above and by Ireland (2004), who allowed for error term autocorrelation by use of a VAR model for the error terms of his DGSE, a ‘‘hybrid model.’’ In a review paper, Allen and Morzuch (2006) wrote, ‘‘Forecast accuracy was better for the hybrid model for almost all horizons and variables, and was frequently signiﬁcantly better, indicating that the restrictions imposed by the DSGE model contained valuable information for the forecaster’’ (p. 482). Also, they note that in work by Del Negro, Schorﬂeide, Smets, and Wouters (2004), it is mentioned that with use of a combined DGSE and vector error correction model, denoted by DSGE-VECM, ‘‘For most of the variables and for most of the horizons (up to 12 steps ahead), out-ofsample forecasts are most accurate for the DGSE-VECM . . . followed by the pure DGSE model, followed by the VECM. It appears that the prior economic theoretical information embedded in the DSGE model does appear useful, not only in an explanatory context, but also in improving forecasts’’ (p. 482). See also Allen and Morzuch (2006) who mention that DGSE models that have grown in size will probably grow larger and cite Diebold who earlier predicted that this will involve ‘‘ . . . greater use of techniques to ‘shrink’ coefﬁcients in particular directions, e.g., via Bayesian methods’’ (p. 482). With disaggregation by sectors and possibly by regions of economies, it seems probable not only that improved forecasting precision will be attained with use of sector economic models but also that much better results for explaining past movements in economies and the effects of policy changes will result. Note that with disaggregation, taking account of the emergence of new sectors, for example the information technology (IT) sector in the 1990s, is possible and may have helped to avert the serious under forecasts of U.S. real GDP growth rates during the late 1990s and the complete missing of the huge drop in the U.S. growth rate in the 1999–2000 downturn by Blue Chip Economic Forecasters reported dramatically in Krane’s (2003) Federal Reserve Bank of Chicago paper. Also, having good sector forecasts along with explanatory sector models can be invaluable to those making sector industrial policies and in integrating sector and overall economic policies that may help to avoid the surprise effects of overall policies on the ﬁnancial sector that Greenspan mentioned above.

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4.2. Temporal Disaggregation and Seasonality With respect to both aggregate and disaggregate models, the thorny problem of seasonality arises. Most workers using quarterly or monthly data work with seasonally adjusted series without fully or even partially understanding the prior assumptions regarding trend, cyclical, and seasonal components underlying various ofﬁcial seasonal adjustment procedures that may be in conﬂict with assumptions used in formulating models. Also, seasonal adjustment of time series data near the beginning of the important current period is complicated since future values of the series are not available for use of moving average and other seasonal adjustment procedures, a problem ‘‘solved’’ many years ago by use of moving averages of past values and forecasted future values, usually produced by seasonal arima models. Much more can be said with respect to the problems of the economics of seasonality, econometric models of seasonality, seasonal policy-making, seasonal adjustment methods, Bayesian seasonal models, relationships among seasonal, cyclical and trend effects, etc. For a short and extensive report on many of these issues, see ‘‘Seasonal Adjustment of the Monetary Aggregates: Report of the Committee of Experts on Seasonal Adjustment Techniques, Board of Governors of the Federal Reserve System’’ by Moore et al. (1981) that provides many references to the earlier literature, descriptions of ModelBased Approaches to Seasonal Adjustment including statistical ARIMA and regression approaches and a causal, structural approach including control variables that may be helpful in approaching the problem of introducing seasonal elements in disaggregated structural economic models. In addition, there are many structural, dynamic analyses of seasonality in the agricultural and resource economic and econometric literature that involve the demand and supply of storage, futures markets, etc. that can be incorporated in disaggregated dynamic statistical and structural econometric models. And last but not least, temporally disaggregated data are probably needed in modeling an economy’s ﬁnancial sector that has been of great concern to Greenspan in his policy-making activities along with additional work with asymmetric and multiperiod, sequentially dependent loss or social welfare functions.

4.3. Databases Current economic data are much better than those available years ago. And it is to be expected that future databases will be even better. While there are many issues that are present on the scene, it is my personal view that better

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datasets relating to industrial sectors would be very valuable. On the consumer side, for example, there might be large monthly panel datasets, as Orcutt suggested many years ago, providing information on buy, not buy decisions with respect to durable goods and other large purchases and amounts spent when buying. Current aggregate data sums purchases over buyers and nonbuyers and thus gives rise to temporal relations that are often misinterpreted. The same can be said with respect to panel data on ﬁrms’ investment outlays on large durable goods, for example electric generators, etc. and changes in ﬁrms’ dividend payments, studied by Peck (1974) and Laub (1972), respectively, who found that mixing together the zero outlays with the nonzero outlays, or the zero dividend changes with the nonzero dividend changes in modeling behavior led to most unusual long lags in behavior that are spurious. Using two part models, for example, one relation explaining the variation in a discrete ‘‘zero one’’ random variable, representing buy or not buy or do not change or change the dividend rate and the other relation explaining the size of the investment outlay or the dividend change indicated that once the individual ﬁrm decided to buy a generator, it did so without a lag and similarly with respect to dividend changes, once they were made, ﬁrms moved right up to their target dividend rate without a ‘‘partial adjustment’’ process that had been featured in many earlier studies. With large datasets, as many have emphasized over the years, it will be possible to obtain better models of consumer, ﬁrm, industry, government, and other economic agents’ and sectors’ behavior. As an impressive illustration of this point, note that in the successful portfolio formation work of Quintana, Putnam, and their colleagues, they used monthly data on individual stocks to predict returns and form portfolios using relations for large numbers of individual stocks with a variety of ﬁrm speciﬁc and other input variables, time-varying coefﬁcients, and a time-varying error term covariance matrix. Using Bayesian state space and shrinkage methods, they computed monthly predictive densities for the returns and used them to form optimal portfolios month by month with rather good returns. This is a good example of how use of a large database and appropriate ﬁnancial economic theory and Bayesian prediction and decision procedures led to very successful outcomes involving high returns for many. Hopefully, such analyses that involve creation of new models and improvement of old models, can be implemented with large future databases relating to various sectors of many economies, as Stone, Orcutt, and others suggested many years ago. The Bayesian tools are available but there is a need for better models and improved databases worldwide.

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4.4. Simulation Experiments In many cases, simulation experiments with econometric models have revealed many interesting and valuable results. For example, with our Marshallian Macroeconomic Model, that has nonlinearities giving rise to dynamic ‘‘chaotic’’ properties, simulation experiments with it revealed interesting ‘‘bubbles and bust’’ behavior of output that resembled the boom and bust of the late 1990s. Also, strenuous simulation experiments with models can reveal unusual and unsatisfactory features of models that require study and possible the need to revise the formulation of the model. Further, as in Hamilton et al. (1969) and many other studies, simulation experiments can be formulated to evaluate alternative economic and other policies, for example, to evaluate the possible effects of proposed dam construction on a river on future regional economic growth, as in Hamilton et al. (1969) or to evaluate the possible effects of changes in the money supply or changes in tax rates on an economy’s behavior, as in Zellner and Peck (1973) and Zellner and Israilevich (2005). The advice given by many in this area is to ‘‘simulate, simulate, and simulate’’ since such experiments are needed to explore the properties of multiequation, nonlinear stochastic models that are very difﬁcult to solve analytically. Even nonlinear single equation stochastic equations, called ‘‘chaotic models’’ are difﬁcult to solve analytically. There is a need for new mathematical methods to obtain analytic solutions for these so-called chaotic equations. In the meantime, more extensive simulation studies to determine the cyclical, trend, dynamic responses, policy characteristics, etc. of currently formulated models will be very valuable. Also, policy simulation experiments can be designed to provide information relative to some of the issues that Greenspan has mentioned above and many others.

4.5. Bayesian Methods There can be no doubt but that there will be a continuation of the great impact of modern computing techniques on Bayesian analyses of many problems. Whether, it will be possible to develop and implement more direct MC techniques is an issue that interests me. For the SUR model, there is the well-known MCMC approach, apparently ﬁrst put forward by Percy (1992), and incorporated in Bayesian texts, see Geweke (2005) and Koop (2003). Currently, a direct Monte Carlo approach for implementing SUR Bayesian estimation and prediction procedures has been devised, applied, and

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compared to the MCMC approach that has produced good results, see Zellner and Ando (2008). The direct Monte Carlo approach has also been applied to structural models to calculate predictive densities in Zellner and Chen (2001). Problems in which there are difﬁculties in formulating appropriate likelihood functions and/or prior densities for parameters of equations arise quite frequently. To help solve such problems, the Bayesian method of moments (BMOM), that provides post data densities for parameters and predictive densities for as yet unobserved data, has been shown to be useful. See Zellner (1994, 1996, 1997b, 1998) and Zellner and Tobias (2001) for the theory and some applications of the approach and Green and Strawderman (1996), Currie (1996), and Zellner and Chen (2001) for additional theory and applications. In the BMOM approach, given a relation for the data, assumptions are made about the properties of the realized error terms and functions of them that imply certain moment constraints on the parameters of the equation. Given these moment constraints, a least informative density for the models’ parameters can be derived by minimizing the expected log height of the density relative to uniform measure, or equivalently maximizing the Shannon entropy of the density for the parameters subject to the given moment constraints. Also, moments and distributions of future observations can be obtained given these BMOM densities for the parameters, assumptions about the properties of future error terms and maxent procedures for deriving least informative predictive densities. In Zellner and Tobias (2001), posterior odds involving BMOM and traditional Bayesian predictive densities have been derived and are useful in choosing between the models or combining them. These and other methods for dealing with problems involving unknown pdf’s for the parameters and/or observations are discussed and applied in a pioneering book by Davis (1941) and in later works by Mittelhammer, Judge, and Miller (2000), Golan (2002), Golan and Kitamura (2007), and the many references cited in these publications. This combination of information theory and Bayesian analysis has been shown to be useful in past work and will probably be very useful in future work, especially when there is great uncertainty about how to formulate prior densities for parameters and probability density functions for error terms appearing in equations of a model. In addition, much work, cited in the above Bayesian texts, is going ahead on the development of Bayesian semiparametric and nonparametric models that is very useful in coping with lack of knowledge regarding forms of likelihood functions, somewhat akin to the ‘‘Knightian uncertainty’’ mentioned by Greenspan.

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As regards optimal, rational behavior under assumed ignorance, Arrow, Hurwicz, and others have deduced that it is rational to go to extremes in making choices under ignorance, say in choosing optimal forecasts, behavior quite different from rational behavior for informed forecasters and decision-makers. And given that there is a mix of ignorant and informed economic agents in an economy, are markets efﬁcient and how does a policy-maker formulate and execute good policies in such a situation? It is clear that more deductive, reductive, and inductive research is needed to provide good or even adequate solutions to these problems.

4.6. Information Processing and Bayesian Inference As remarked above, there are now a number of different optimal information processing rules, including Bayes’ theorem, on the scene that are in use, some involving use of discounted or quality corrected likelihood functions and/or quality corrected or ‘‘power’’ priors. Other dynamic optimal information processing rules involve allowance for costs of changing beliefs and/or costs of acquiring information. Thus, a whole new set of optimal information processing rules, related to some scientiﬁc research on how learning takes place, are ‘‘on the shelf’’ for possible use that will make the Bayesian approach applicable to a broader range of problems and provide good solutions to them. And these technical information processing procedures, as with various production functions in econometrics, can be used to solve a variety of new problems, some involving expected utility maximization and others proﬁt maximization. Indeed in an information industry, with ﬁrms producing information operating in markets for input and output information with unit prices for inputs and output assumed available, one can use Marshall’s and others’ theories to formulate econometric models of such an industry that may be useful in explanation, prediction, and policy-making. With respect to some broader issues regarding the future of Bayesian statistics, in my prepared address, titled ‘‘A Bayesian Era,’’ at the International Conference on Bayesian Statistics and its Applications at the Banaras Hindu University in Varanasi, January 2005, I stated, ‘‘While a Bayesian Era has arrived, it is important that we do not become too complacent. Clearly, many more good Bayesian applied studies are needed. In addition, we have to continue research on the formulation of noninformative and informative prior densities, as J. K. Ghosh pointed out in his talk at the ISBA meeting in Istanbul, Aug. 1997; see also, Kass and Wasserman (1996) and Zellner (1997a) for reviews of past and recent work on

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this topic. In addition, further Bayesian research on model formulation is needed. I. J. Good pointed out that we usually take the likelihood function’s form to be known. Getting more Bayesians involved in using subject matter theory and other information to formulate good models would be very useful. More work is needed to explore properties of various approaches designed to deal with analyses of data when the form of the likelihood function is unknown, e.g., Bayesian bootstrap, empirical likelihood, Bayesian method of moments, maximum entropy, data mining, information theoretic, nonparametric, semi-parametric, mixture and other methods. Comparing results, e.g. predictive densities and predictions of such approaches to those provided by analyses in which an assumed form of the likelihood function is employed is a procedure that appeals to many including George Barnard. Another area that deserves more attention is the link between Bayesian analysis and information theory; see, e.g., Sooﬁ (1996, 2000), Zellner (2003) and the extensive references cited in these papers. As mentioned in these papers, work has been done to get improved, more general information processing rules that will permit Bayesian analyses of a much broader range of problems. These basic links of Bayesian analysis to the philosophy of science, information theory, learning theory, utility theory and economic theory will enrich Bayesian analysis and permit it to be applied to a wider range of problems. Finally, while progress on the computing front has been very impressive, we do need more ‘user friendly’ Bayesian computer packages or components that can be easily downloaded from the web as in the software that Sid Chib and John Geweke, among others, have developed. With such systems, users world-wide in many disciplines, industries and governments will have access to the latest Bayesian computing techniques at ‘the ﬂip of a switch,’ as Geweke and Chib have emphasized. While much Bayesian research remains to be done, I am conﬁdent that results will be forthcoming, particularly with the help of Indian researchers who have been amazingly productive over the years’’ (pp. 4–5). Most of the considerations above are relevant for the future of Bayesian econometrics in general which will, I believe with high conﬁdence, continue to grow and to win many Bayes–non-Bayes ‘‘horse races’’ in the future. The results of such Bayesian econometric research will lead to better explanations, forecasts, predictions, and economic policies that will beneﬁt many worldwide.

ACKNOWLEDGMENTS I am thankful to Herman van Dijk for his helpful comments on an earlier draft of this paper that was a keynote address to the Bank of Sweden’s

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Research Conference on Bayesian Econometric Methodology, Stockholm, September 8–9, 2006.

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Zellner, A., & Ando, T. (2008). A direct Monte Carlo approach for Bayesian analysis of the seemingly unrelated regression model (p. 59). Working Paper. H.G.B. Alexander Research Foundation, Graduate School of Business, University of Chicago. Zellner, A., Bauwens, L., & van Dijk, H. (1988). Bayesian speciﬁcation analysis and estimation of simultaneous equation models using Monte Carlo methods. Journal of Econometics, 38, 39–72. Zellner, A., & Chen, B. (2001). Bayesian modeling of economies and data requirements. Invited keynote address to International Institute of Forecasters meeting, Lisbon, 2000, and published in Macroeconomic Dynamics (2002), 5, 673–700. Zellner, A., Huang, D. S., & Chau, L. C. (1965). Further analysis of the short-run consumption function with emphasis on the role of liquid assets. Econometrica, 382–394. Zellner, A., & Israilevich, G. (2005). The Marshallian macroeconomic model: A progress report. International Journal of Forecasting, 21, 627–645 (with discussion by A. Espasa, pp. 647–650). Zellner, A., Kuezenkampf, H., & McAleer, M. (2001). Simplicity, inference and modeling: Keeping it sophisticatedly simple. Cambridge, UK: Cambridge University Press. Zellner, A., & Min, C.-K. (1999). Forecasting turning points in countries’ output growth rates: A response to Milton Friedman. Journal of Econometrics, 88, 203–206. Zellner, A., & Palm, F. C. (1974). Time series analysis and simultaneous equation econometric models. Journal of Econometrics, 2, 17–54 (reprinted in Zellner & Palm, 2004). Zellner, A., & Palm, F. C. (1975). Time series analysis of structural models of the U.S. economy. Sankya, Series C, 27, 12–56 (reprinted in Zellner and Palm (2004)). Zellner, A., & Palm, F. C. (Eds). (2004). The structural econometric modeling, time series approach. Cambridge, UK: Cambridge University Press. Zellner, A., & Peck, S. (1973). Simulation experiments with a quarterly model of the U.S. economy. In: A. Power & R. Williams (Eds), Econometric studies of macro and monetary relations (pp. 149–168). Amsterdam, The Netherlands: North-Holland. Zellner, A., & Rossi, P. (1984). Bayesian analysis of dichotomous quantal response models. Journal of Econometrics, 25, 365–394. Zellner, A., & Tiao, G. C. (1964). Bayesian analysis of the regression model with autocorrelated errors. Journal of the American Statistical Association, 59, 763–778. Zellner, A., & Tobias, J. (2001). Further results on Bayesian method of moments analysis of the multiple regression model. International Economic Review, 42(1), 121–140. Ziliak, S. J., & McCloskey, D. M. (2004). Size matters: The standard error of regression in the American Economic Review. Journal of Socio-Economics, 1, 331–358 (with discussion by W. Fuller, C. Granger, E. Leamer, A. Zellner and others on whether to test or not, and if so, how).

BAYESIAN INFERENCE USING ADAPTIVE SAMPLING Paolo Giordani and Robert Kohn ABSTRACT Our paper discusses simulation-based Bayesian inference using information from previous draws to build the proposals. The aim is to produce samplers that are easy to implement, that explore the target distribution effectively, and that are computationally efﬁcient and mix well.

1. INTRODUCTION Bayesian approaches to statistical and econometric analysis are used extensively because they provide ﬁnite sample inference and because tractable and ﬂexible methods have been developed to approximate the posterior distributions of interest in complex problems. The main computational approach is the Markov Chain Monte Carlo (MCMC) sampling methodology whose origins go back to the work of Metropolis, Rosenbluth, Rosenbluth, Teller, and Teller (1953) and Hastings (1970). There are a number of ways to implement MCMC methods, e.g., Gibbs sampling, Metropolis–Hastings (MH) sampling, and slice sampling. Explanations of these methods are given by Tierney (1994) and Neal (2003). In MCMC simulation a sequence of iterates of unknown parameters or unobserved

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(latent) variables is generated with the iterates forming a Markov chain whose invariant distribution is the posterior distribution. To make this simulation methodology tractable in complex problems the unknown parameters and latent variables are often partitioned into blocks with each block generated conditionally on the data and the other blocks. If we generate from each of the conditional distributions then the MCMC sampling scheme is called Gibbs sampling, which was popularized by Gelfand and Smith (1990). However, we may often decide to sample from an approximation to the conditional density and then ensure that the invariant distribution of the resulting Markov chain is correct by using the MH method. There are several reasons why we may wish to generate from such an approximation (or approximations) to the correct conditional distribution, including: (a) it may be computationally intractable to sample from the exact conditional; (b) it may be computationally cheaper to sample from an approximation even if it is feasible to sample from the exact distribution; (c) generating from an approximation may result in a Markov chain that mixes better than the corresponding Gibbs sampler; an example is the Metropolized Gibbs sampler discussed by Liu (2001) and applied in the regression context by Kohn, Smith, and Chan (2001). An important idea mentioned above is that in MCMC sampling we often wish to generate from an approximation to an exact distribution and we would like the approximation to be easy and cheap to compute while also giving good mixing of the iterates. There are a number of approaches for doing so in an MCMC framework. However, the MCMC framework does not allow the use of past iterates to form the proposal distributions as using them contravenes the Markov assumption. The purpose of our article is to discuss approaches for forming proposal distributions that are based on past iterates of the parameters or latent variables. More precisely, we wish to use the past history of the iterates to try and obtain a good proposal density, or more generally proposal densities, while working within a given class of proposal densities. For example, in many problems the class of proposal densities we consider is Gaussian and we wish to obtain the best Gaussian proposal for the model and data at hand. This does not imply that the target (posterior) distribution is Gaussian or even close to Gaussian. We follow the literature and call such sampling ‘‘adaptive sampling’’ to convey that we are trying to adapt the proposal density to its optimal form within a given class of proposals. For a discussion and motivation of adaptive sampling see the articles by Andrieu and Moulines (2006), Roberts and Rosenthal (2007), and Roberts and Rosenthal (2006).

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We follow Giordani and Kohn (2007) and deﬁne ‘‘strict’’ adaptation as adaptation of the proposal distribution which is subject to theoretical rules that ensure that the iterates converge to realizations from the correct target (posterior) distribution. We call all other adaptations ‘‘informal.’’ Strict adaptation may include an initial phase where sampling is initialized with a nonadaptive MCMC scheme and then sampling is switched to an adaptive phase without presuming that the scheme has converged at the end of the ﬁrst phase. In informal adaptation we may wish to intensively explore the target distribution without formally proving that the sampling scheme converges to the target if run sufﬁciently long. Although informal and strict adaptation are technically different, it is desirable in both strategies to learn about the target distribution relatively quickly even when it is nonstandard, e.g., non-normal, and to carry out this learning without extensive model-speciﬁc coding. That is, the important idea is to have a general approach for constructing proposals that works in practice for a large number of models and datasets, is sufﬁciently ﬂexible to be able to learn about nonstandard target distributions, and requires as little model-speciﬁc code as possible. Usually, the minimum model-speciﬁc code that is required is the computation of the likelihood and the priors. Our article discusses some approaches to adaptive estimation and the intuition behind some of the theory for proving that the adaptive estimation is strict. In particular, we exposit the approach proposed by Nott and Kohn (2005) for adaptive sampling of discrete variables, and the approach of Giordani and Kohn (2007) for sampling continuous variables which is based on approximating the target density by a mixture of normals. Our article also discusses how to make adaptive sampling more effective by integrating out parameters and latent variables where possible. Such an approach is illustrated for variable selection in the linear regression model and for time series models subject to interventions.

2. ADAPTIVE SAMPLING FOR VARIABLE SELECTION IN A GAUSSIAN LINEAR REGRESSION MODEL Consider the linear regression model yt ¼ x0t b þ et;

t ¼ 1; :::; n

(1)

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the errors et are N(0, s2) and independent, yt is the scalar dependent variable and xt the r 1 covariate vector which usually includes 1 as a ﬁrst element. We are interested in subset selection, variable selection, and model averaging in linear regression and in more general contexts, e.g., logistic regression. Subset selection in linear regression involves choosing a subset of variables that is optimal according to some criterion, e.g., minimizing the prediction error. We may also view subset selection as model selection, where we consider each subset of variables as a model. This important topic has a large literature (see Miller, 2001). Variable selection considers which variables ought to be in the regression, either unconditionally, or conditionally on other variables being in; e.g., t statistics address the latter question in a frequentist setting. Model averaging means estimating a functional as a weighted average of the estimates of that functional based on various models or subsets of the variables. In particular, the model averaged predictive regression density is obtained by taking a weighted average of the predictive densities of the various models under consideration. It is convenient to refer to all three ideas under the heading of variable selection and to refer to the corresponding prior on b as a variable selection prior. We introduce a vector of binary latent variables g of dimension r so that gi ¼ 1 if the ith covariate is included in the model and gi ¼ 0 if it is not. Thus g indicates which subset of the variables is in the model or, equivalently, the model under consideration. We may wish some variables to be included in all models in which case the corresponding gi are set identically to 1, e.g., the intercept in the regression. Subset selection involves selecting an optimal g based on the data. One way of doing so in the Bayesian context is to estimate the g that maximizes p(g|y). Bayesian model averaging (BMA) (see Hoeting, Madigan, Raftery, & Volinsky, 1999; Raftery & Zhang, 2003) means taking a weighted average of the regression densities for each subset or model, with the weights being the posterior model probabilities. Bayesian variable selection considers the inclusion of variables based on their posterior distributions, and in particular the marginal posterior densities Pr(gi ¼ 1|y). This section motivates adaptive sampling by applying it to Bayesian variable selection problems in the linear regression model (1). For a modern discussion of Bayesian variable and subset selection for the linear regression model (see George & McCulloch, 1993, 1997; Liang, Paulo, Molina, Clyde, & Berger, 2008). To carry out a Bayesian analysis of the regression model it is necessary to specify priors for b, s2, and g. For given g, let bg be the subvector of b corresponding to the gi that are 1. Let X be the n r design matrix having

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tth row xut and let Xg consist of those columns of X for which gi ¼ 1. Thus, given g, Xb ¼ Xgbg. For given g and s2 we take the prior for bg as normal with mean 0 and covariance matrix gs2 ðX 0g X g Þ1 (see Zellner, 1986). A number of choices of g are suggested in the literature (see Liang et al., 2008), but we take g ¼ n for simplicity. This choice of g is asymptotically equivalent to BIC; see Kass and Wasserman (1995) for a general discussion and Fernandez, Ley, and Steele (2001) for the application to Bayesian variable selection. We take the prior for s2 as proportional to 1/s2. Under these assumptions it is possible to integrate out both bg and s2 to obtain the marginal likelihood p(y|X, g) as in Smith and Kohn (1996). Let p be the prior probability that gi ¼ 1 when gi is not identically 1. We assume that those gi which are not identically 1 are independent conditional on p. We also place a prior on p whose details are given for the examples. What is important is that inference on the linear regression model is reduced to computations involving only g which is discrete. This greatly simpliﬁes both the sampling scheme and the theory underlying adaptive estimation as explained in Nott and Kohn (2005) and in Section 5. When r is small it is possible to compute the marginal likelihood for all possible allowable conﬁgurations of g and hence carry out model selection or model averaging. When r is large it is usually necessary to use simulation methods for inference, such as Gibbs sampling and Metropolized Gibbs (see Kohn et al., 2001). However, these approaches are computationally expensive as every time g changes it is necessary to update the Cholesky decomposition of X 0g X g as detailed in Smith and Kohn (1996). Nott and Kohn (2005) demonstrate the usefulness of adaptive sampling for variable selection in the linear regression model. To understand why it is useful suppose that we run a nonadaptive sampling scheme such as the Gibbs sampler to obtain iterates of g. We will often observe that some gi are mostly generated as a 1, meaning that the ith variable is nearly always in, or as a 0, meaning that the ith variable is nearly always out. Thus, for such gi we will ‘‘almost’’ know after some iterations whether gi is likely to be generated as a 1 or a 0. This is what happens in the two examples discussed in Section 2.1. However, in standard MCMC we cannot take advantage of this knowledge because using it contravenes the Markov property. One way to use such knowledge without contravening the Markov property is to run a two stage adaptive sampling scheme. The ﬁrst stage consists of running some MCMC scheme to obtain iterates of g and then use these iterates to form a proposal density which is used as a ﬁxed proposal in the second stage. We call this ‘‘two stage’’ adaptation. We follow Nott and Kohn (2005) and outline how to construct such a ﬁxed proposal after the

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last iterate of the ﬁrst stage. We note that Prðgi ¼ 1jgfjaig; yÞ ¼ Eðgi jgfjaig; yÞ and approximate the right side by the best linear predictor E~ of gi, that is, ~ i jgfjaig ; yÞ ¼ Eðgi jyÞ þ covðgi ; gfjaig jyÞvarðgfjaig jyÞ1 ðgfjaig Eðgfjaig jyÞ Eðg By using estimates of E(g|y) and var(g|y) from the ﬁrst sampling stage we can approximate Pr(gi ¼ 1|g{ j6¼i}, y) and use this approximation as a proposal density to generate the gi at the second stage. In practice, we would instead use Eq. (2) below to stabilize the estimate. Two stage adaptation does not use the approximation to the best linear ~ i jgf jaig ; yÞ during the ﬁrst stage and keeps the approximation predictor Eðg ﬁxed during the second stage. It seems more efﬁcient to keep improving the estimate of the best linear predictor as more iterates become available. This is the motivation for the adaptive sampling approach of Nott and Kohn (2005) who ﬁrst run a nonadaptive scheme for d iterations and estimate E(g|y) and var(g|y) by the sample mean g and the sample covariance matrix S based on the ﬁrst d iterates of g. After that they use estimates of Pr(gi ¼ 1|g{ j6¼i}, y) almost as outlined above as proposal densities in a MH scheme to generate the gi one at a time, and update the P estimates every d iterations. To stabilize the inversion of estimates of , Nott and Kohn (2005) use W ¼ (SþlI )1 with l a small positive number. Their proposal density for gi is W i;f jaig gf jaig g f jaig (2) qðgi ¼ 1jgf jaig Þ ¼ min 1 d; max d; g i W ii where 0odo0.5 to ensure that the proposal densities lie strictly between 0 and 1. To reduce computation they also update the sample mean g and the sample covariance S every d iterations rather than at every iteration.

2.1. Applications We motivate the use of adaptive sampling using two related applications. The dependent variable for the ﬁrst application is Swedish CPI inﬂation and the dependent variable for the second application is Swedish GDP growth. The data are quarterly and cover the period 1986Q1–2007Q1. The covariates are 3 seasonal dummies (all data are demeaned), 4 lags of inﬂation (or GDP growth), and 2 lags of 8 macroeconomic variables listed in

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Table 1.

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Variables Used in the Gaussian Linear Regression Models.

Variable Name Inﬂation Real DGP growth Unemployment Interest rate Exchange rate Money growth US interest rate US real GDP growth Euro real GDP growth

Series Name and Number Log difference of the consumer price index, not sa, SECPIQ, 25 Log difference of DGP at ﬁxed prices, sa, SEYSAQ, 1 Unemployment rate, sa, SEURQ, 11 One month Treasury bill, not sa, SETCWQ, 33 Log of the weighted exchange rate, not sa, SETCWQ, 38 Log difference of M3, not sa, SEM3Q, 44 Three month Deposit rate, USA, USR3MQ, 39 Log difference of GDP at ﬁxed prices, sa, USYSAQ, 47 Log difference of GDP at ﬁxed prices, sa, EUROYSAQ, series 51

Note: The series name and number refer to the Statistics Sweden archive. sa, seasonally adjusted.

Table 1, for a total of 23 variables. In both examples the prior for p is b(1, 4) giving an a priori expected number of included variables of 4.8. We show that carrying out adaptive sampling is sensible for these two datasets by using the following nonadaptive sampling scheme. At each iteration, the sampling scheme runs through all 23 variables such that a variable is proposed to be in with a probability of 0.5 if it is currently out and vice versa, and with probability 0.5 a switch of indicators is proposed with another randomly picked variable; this latter feature is introduced to improve mixing among highly correlated variables. Figs. 1 and 2 summarize the output from the sampling scheme for inﬂation and GDP growth. The top panels in Figs. 1 and 2 show the data, the OLS estimates of the ﬁtted values, and the means of the ﬁtted values obtained by BMA. For the inﬂation regression, the only variable with a high probability of inclusion is the fourth lag of inﬂation, whereas the ﬁrst lag of output growth is always included for output growth. The bottom two panels in Figs. 1 and 2 plot the recursive posterior means of inclusion and show that around 200 iterations are more than sufﬁcient to form good estimates of the inclusion probabilities for an adaptive scheme for these two datasets.

2.2. Discussion It is useful to summarize the approach of Nott and Kohn (2005) who consider adaptive sampling with proposal densities based on best linear predictors. These proposal densities are not the target distributions, even in large samples. To obtain the proposal densities Nott and Kohn (2005)

68

PAOLO GIORDANI AND ROBERT KOHN Inflation, average fitted value from BMA (dashed) and from OLS (dotted)

14

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Fig. 1. Inﬂation Regression: The Top Panel Plots the Data, the Fitted Values Obtained by Model Averaging (Dashed Line) and the Fitted Values Obtained by OLS Estimation (Dotted Line). The Bottom Panel Plots the Recursive Estimates of the Inclusion Probabilities for all Variables for the 2000 Iterations.

69

Bayesian Inference using Adaptive Sampling GDP growth, average fitted value from BMA (dashed) and from OLS (dotted)

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approximate the joint posterior density of g by a Gaussian and use that approximation to obtain low dimensional conditional proposal densities.

3. MODELING INTERVENTIONS IN TIME SERIES Adaptive sampling is useful in time series models that allow for interventions such as parameter shifts and outliers where we expect the number of such interventions to be small. Giordani and Kohn (2008) document considerable efﬁciency gains (in the order of 300%) from simple adaptive schemes to draw various types of interventions in time series models. Here we study one of their examples to show how quickly one can obtain information on the time points at which outliers and parameter shifts are likely to have occurred to suggest that an adaptive sampling scheme will be useful. For the actual application of adaptive sampling to this example see Giordani and Kohn (2008). We model quarterly U.S. CPI inﬂation for the period 1951Q1–2004Q4 as a ﬁrst order autoregression (AR(1)) with infrequent random shifts in intercept, autoregressive parameter, and residual log variance, and allow for innovation outliers in the observations. The model is yt ¼ ct þ bt yt1 þ st K e;t et ct ¼ ct1 þ K c;t uct bt ¼ bt1 þ K b;t ubt logðs2t Þ ¼ logðs2t1 Þ þ K v;t vt pðK t Þ pðK m;t ; K v;t Þ ¼ pðK m;t ÞpðK v;t Þ ¼ pðK t jK gat Þ

ð3aÞ (3b)

where Km,t ¼ (Ke,t, Kc,t, Kb,t). Eq. (3b) assumes that the Kt are independent a priori and that Km,t is independent of Kv,t. The latent variable Ke,t takes the value 2.5 with prior probability 0.02 and is 1 otherwise. That is, the prior probability of an innovation outlier is 2%. Kc,t can take the values (0, 0.2, 1) and Kb,t can take the values (0, 0.5). We assume that a parameter shift and an outlier cannot occur simultaneously, but shifts in ct and bt can occur both separately and jointly. The prior probability that a shift in ct or bt occurs at any time t is 0.01. Kv,t takes the value 1.39 with probability 0.005 and is 0 otherwise. This implies that st ¼ st1 with probability 0.995 but if Kv,t6¼0, then with 68% probability 0.5rst/st1r2, while with 5% probability st/st1W3.9 or st/st1o0.26.

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We now outline some of the ideas behind sampling from this model with further details in Giordani and Kohn (2008). Let ht ¼ log s2t . Conditional on the Kt and the ht, the model is a Gaussian state space model with state vector (ct, bt) so that it is possible to integrate out the states and generate the sequence of Km,t using the reduced conditional sampler in Gerlach, Carter, and Kohn (2000). Each Km,t is generated from a multinomial distribution. It is clear that after initializing the sampling using the reduced conditional sampler, it may be useful to switch to some form of adaptive sampling. There are a number of ways to do so. The simplest is to deﬁne Jm,t as 0 if there is no intervention at time t and 1 otherwise. That is, Jm,t ¼ 0 if Km,t ¼ (1,0,0) and 1 otherwise. We can now apply adaptive sampling to Jm,t. If Jm,t is currently 0 and is generated as 0 then there is nothing to do. This occurs for most t because we assume that interventions are rare. If Jm,t is generated as a 1 then Km,t is generated from a multinomial density similarly to the reduced conditional case. A reﬁnement would use past iterates to form a multinomial approximation to the posterior distribution of Km,t as in Giordani and Kohn (2008). Procedures such as those outlined above correspond to carrying out adaptive sampling for variable selection using estimates of the marginal inclusion probabilities of the variables. It is usually impractical to approximate the distribution of the Km,t or the Jm,t jointly for all t as we did in the variable selection case. However, it may be interesting to experiment with proposal densities for Jm,t or Km,t conditional on 2l of their immediate neighbors, that is based on Jm,tl , . . . , Jm,t1, Jm,tþ1, . . . , Jm,tþl for some l or Km,tl, . . . , Km,t1, Km,tþ1, . . . , Km,tþl for some l. For example, if we take l ¼ 1 then the idea is that if there is a structural shift at time t1 or tþ1 then there is less likely to be a shift at time t. Once the Km,t are generated in any given iteration we generate the states vectors {(ct, bt), t ¼ 1, . . . , n} and compute y~t ¼ yt ct bt yt1 ¼ st K e;t et Let zt ¼ log y~2t log K 2e;t so that zt ¼ ht þ log e2t . The distribution of log e2t is log w21 which is closely approximated by a mixture of normals; in our work we use the mixture of ﬁve normals in Carter and Kohn (1997). Thus we write zt ¼ ht þ f 1 ðK 1;t Þ þ f 2 ðK 1;t Þat ht ¼ ht1 þ K v;t vt

(4)

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where atBN(0,1) and K1,t is a discrete variable taking the values 1–5 which determine the component of the mixture of normals approximation to logðw21 Þ. The terms f1(K1,t) and f2(K1,t) are the mean and standard deviation of the K1,t component of the mixture of ﬁve normals. Eq. (4) is then a conditionally Gaussian state space model with state vector ht. It is possible to integrate out ht and generate the K1,t and Kv,t and generate the Kv,t adaptively similarly to above. Fig. 3 shows some results obtained from a nonadaptive sampling scheme on 80 iterations after a burn-in of 20 iterations. This brief running time is sufﬁcient to develop a reasonable picture of the location of outliers and shifts. As expected, posterior probabilities of outliers show a handful of rather clear cases, near zero probabilities for most observations, and some intermediate values. For shifts in ct and bt the posterior distributions clearly concentrate most of the mass in certain periods, while other periods are quieter. There are also clear shifts in log residual variance, but the posterior distribution of their location is more evenly spread out, suggesting a process with more frequent and smaller shifts as an interesting alternative (see Koop & Potter, 2004 for related ﬁndings). We refer to Giordani and Kohn (2008) for the actual application of adaptive sampling to this dataset.

4. ADAPTIVE SAMPLING FOR CONTINUOUS PARAMETERS Sections 2 and 3 discuss adaptive sampling for discrete latent variables in applications where it is feasible to integrate out all the continuous parameters and latent variables. In fact, we can show more generally that as long as we can generate the other parameters in a Gibbs step it is unnecessary to integrate out such parameters. This section considers adaptive sampling for continuous parameters. Standard approaches either generate all the parameters simultaneously from the posterior or in blocks (Gibbs sampling) if such generation is feasible. If it is not, then MH sampling is frequently used with the proposal densities being multivariate normal and centered at the posterior mode (or conditional posterior modes) and with the covariance matrix based on the second derivative of the posterior or the conditional density. To provide thicker tails a multivariate t density with a small number of degrees of freedom is often used instead of a multivariate normal.

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Fig. 3. Inﬂation Data: Panel (a) Plots the Inﬂation Rate and the Median Values of Each Local Mean ct/(1bt); Panel (b) Plots the Mean Values of the st; Panel (c) Plots the Means of the Ke,t; Panel (d) Plots the Means of the Kc,t; Panel (e) Plots the Means of the Kb,t; Panel (f) Plots the Means of the Kv,t.

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Although such strategies are successful for a number of models and applications, they have the following potential problems. First, if the target densities are very non-normal then the sampling can be inefﬁcient. Second, most sampling approaches require computing the ﬁrst and second derivatives of the likelihood and prior, usually at each iteration, and this can be time consuming in complex models. Third, coding up the ﬁrst and second derivatives of the target densities may also be highly time consuming and may deter users from experimenting with different models and priors. The attraction of using adaptive sampling is to try and overcome at least the ﬁrst problem and hopefully all three. The above drawbacks of current MCMC simulation methods for continuous variables may seriously slow the adoption of Bayesian methods by both statisticians and subject matter practitioners, making it important to develop improved sampling methods. Adaptive sampling offers one such possibility. We give some more details of the ﬁrst problem outlined above as the second and third problems are easier to understand. First, consider a model with likelihood p(y|y), where y is the vector of parameters for the model and let p(y) be the prior density of y. One approach to carry out posterior inference for this model is to approximate p(y|y)pp(y|y)p(y) by some distribution from which it is straightforward to sample, e.g., a multivariate normal or t distribution. We can then use this approximation as the proposal density in an independent MH scheme. However, if the dimension of y is high or the density p(y|y) is far from a multivariate normal or a t then this strategy may result in high rejection rates in the MH method and hence poor mixing of the resulting Markov chain. A reﬁnement of the above strategy would split y into two or more components with each component then sampled conditionally on the others. Suppose we partition y into two components y1 and y2 and approximate the density p(y1|y2, y) by a proposal density qðy1 ; yc1 ; yc2 Þ where yc1 and yc2 are the current values of y1 and y2. Similarly we approximate the density p(y2|y1, y) by a proposal density qðy2 ; yc2 ; y1 Þ, where y1 is the new value of y1 generated in the previous step. The approximation qðy1 ; yc1 ; yc2 Þ is often obtained as a normal or t approximation to the exact density and requires computing at least the ﬁrst derivatives of the log target density p(y1|y2, y) and at least partially iterating to its mode. Similar remarks apply to the proposal density qðy2 ; yc2 ; y1 Þ. These calculations are required not just once as in maximum likelihood estimation, but for every iteration, which may be time consuming. There are a number of models for which obtaining suitable proposal densities is fast, e.g., when it is possible to generate from the exact conditionals p(y1|y2, y) and p(y2|y1,y). There are also models for which it

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may be straightforward to calculate the ﬁrst and second derivatives and then take one or two steps toward the mode to get adequate proposals using normal or t densities (see Gamerman, 1997). In general, however, it is necessary to calculate derivatives of the log likelihood and the prior at every iteration. Two-stage sampling partially overcomes the ﬁrst problem. Here a sampling scheme is run in the ﬁrst stage at the end of which the iterates are used to construct a proposal density that is kept ﬁxed in the second stage. Such a strategy is proposed by Gelman and Rubin (1992) and Hastie (2005) who use a mixture of normals to approximate the posterior density at the end of the ﬁrst stage. Although such a two-stage strategy is useful, it may be wasteful because the ﬁrst stage may require a long running time before an adequate sample size is obtained to construct a good proposal and there is no attempt to improve the proposal during the second stage. We note that in the ﬁrst stage we can run any sampling scheme or a combination of sampling schemes to construct the ﬁxed proposal for the second stage. One way to tackle the second and third problem with MCMC computation mentioned above is to use a random walk Metropolis method to generate the parameters y. Thus, if y[ j] is the jth iterate of the Markov chain, then the proposal for the jþ1st iterate is a (5) y½ jþ1;p N y½ j ; S r where r is the dimension of y and the choice of a ¼ 2.34 is optimal if the target is normal. Tuning a on shorter runs for non-normal targets is common and can be viewed as informal adaptation. S is often set to the Laplace approximation to the posterior. Using such a random walk Metropolis proposal is simple to code and does not require expensive runtime evaluations. However, the proposal may be very inefﬁcient if a reasonable matrix S is unavailable or if the target is far from Gaussian. Haario, Saksman, and Tamminen (2001) propose adaptively estimating S in Eq. (5). Roberts and Rosenthal (2006) modify the adaptive random walk Metropolis algorithm proposed by Haario et al. (2001) and deﬁne the proposal at the jth iterate as 0:01 I ; j 2r y½ jþ1;p N y½ j ; r (6) a 0:01 y½ jþ1;p ð1 bÞN y½ j ; Sj þ bN y½ j ; I ; j42r r r

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where Sj is the sample covariance matrix of iterates y[1], . . . , y[ j–1]. Roberts and Rosenthal (2006) show that Eq. (6) converges to the target distribution so their sampling scheme is mixture of normals with both components having a mean of y[ j] but with covariances matrices ða=rÞSj and ð0:01=rÞI r and with mixing probabilities 1b and b, with 1WbW0. The inclusion of the second component makes it easier to prove convergence to the posterior distribution. We note that the sequence of proposal densities in Eq. (6) converges to the ‘‘optimal’’ proposal density a 0:01 I (7) y½ jþ1;p N y½ j ; ð1 bÞ varðyjyÞ þ b r r which is not in general the posterior density of y and may be far from it. That is, the adaptation proposed by Roberts and Rosenthal (2006) only considers a limited class of densities. If the target density is far from Gaussian then this sampler may experience high rejection rates. We note that instead of generating y in one block, it may be better to use a multiple component version of the random walk Metropolis algorithm or the adaptive random walk Metropolis algorithm. To illustrate, it is sufﬁcient to consider two components with y ¼ (y1, y2). The proposal (5) is a qðyc ! yÞ N yc ; S (8) r A two-component version would generate from the conditional versions of Eq. (8) in the obvious way. An outstanding problem is how to partition y into components. One way of doing so is to consider an estimate of the posterior variance of y and try and place highly correlated elements of y into the same component. We now discuss the adaptive sampling approach taken by Giordani and Kohn (2007) who use a sequence of multivariate normal mixtures as the class of proposal densities. The hope is that this class of densities is sufﬁciently ﬂexible to produce good proposal densities and even to provide a good approximation to the target density. The sampling is done in two stages. An informal stage and a strict sampling stage with the transition between the two determined endogenously. Let p(y)=p(y|y) be the target density. In the initial phase, let g0(y) be a mixture of the Laplace approximation of p(y) with a heavier tailed version of that approximation, by which we mean a normal distribution with the same mean but with a covariance matrix that is 25 times larger. The Laplace approximation gets 0.6 of the weight and the heavier tailed version gets 0.4. If the Laplace approximation is unavailable then they use the prior. At the

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end of the initial phase g0(y) is constructed as a mixture of the last estimated mixture of normals, which we call glast(y) together with a heavier tailed version of it in the same way that g0(y) was initially constructed from the Laplace approximation. That is, the component means of the heavier tailed version are the same as those of glast(y) while the variances are 25 times larger. The proposal density at the mth iterate is of the form qm ðy; lm Þ ¼ o1 g0 ðyÞ þ ð1 o1 Þgm ðy; l m Þ

(9)

where we take o1 ¼ 0.05 and the vector parameter l denotes the means, variances, mixture weights, and the number of components in the mixture. We now describe how gm ðy; l m Þ is constructed. Let gnm ðy; lnm Þ be a mixture of normals estimated from the iterates at the mth stage. Giordani and Kohn (2007) show how to obtain such a mixture estimate using the harmonic K-means algorithm and have checked through experimentation that their implementation is both fast and robust especially in the informal sampling phase. However, other methods can also be used to estimate the mixture, especially when there are many iterates already available in the strict adaptation phase. For example, stochastic approximation can be used. Let g~nm ðy; lnm Þ be a second mixture of normals having the same means and component weights as gnm ðy; lnm Þ but variances that are inﬂated by a factor of k; usually we take k ¼ 16. Let gnm ðy; lnm Þ ¼ o02 g~nm ðy; lnm Þ þ ð1 o02 Þgnm ðy; lnm Þ We assume that o02 1 and in practice take o02 ¼ 3=19 for the reason given below. If we take gm ðy; l m Þ ¼ gnm ðy; lnm Þ in Eq. (9), then qm ðy; lm Þ ¼ o1 g0 ðyÞ þ o2 g~n ðy; lnm Þ þ ð1 o1 o2 Þgn ðy; lnm Þ

(10)

where o2 ¼ o02 ð1 o1 Þ. Thus, when o02 ¼ 3=19; o2 ¼ 0:15, and 1o1o2 ¼ 0.8. The density qm(y; lm) is a mixture of three components. The ﬁrst component g0 does not adapt with m, has quite a small weight (in our applications that weight is 0.05), and is designed to be very heavy tailed relative to the target. Including this component corresponds to the defensive strategy used by Hesterberg (1998) for importance sampling. The third component gn ðy; lnm Þ is a mixture of normals estimate of the target distribution and in our applications has weight 0.8. The second component g~n ðy; lnm Þ is a heavy tailed version of the third component and in our implementation has weight 0.15. Each of these three components serves

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a purpose. The third component is the correct one to use if the density estimate coincides with the target. The second component is a heavy tailed version of the third to cater for the tails in the target that are not captured by the third component. This is the three component strategy for forming proposals in Giordani and Kohn (2007), with implementation details given in that article. Here we mention three implementation issues. First, Giordani and Kohn (2007) update the mixture of normals every dW1 iterations because it is too costly to update every iteration when using their approach. Second, at each update they estimate the number of components in the estimate gn ðy; lnm Þ by BIC. Third, to conform to theory they enforce the diminishing adaptation property on the sequence of proposal densities. This means roughly that as the number of iterates increases the sequence of estimators tends to adapt less and less, with a more precise deﬁnition given in Section 5. There are several ways to implement diminishing adaptation. Giordani and Kohn (2007) use the following approach. Let bm, mZ1 be a scalar sequence such that 0rbmr1 and bm-1 as m-N. Deﬁne gm ðy; l m Þ ¼ ð1 bm Þgnm ðy; lnm Þ þ bm gm1 ðy; l m1 Þ

(11)

The parameter l m is well deﬁned because the right side of Eq. (11) is a mixture of normals and so the left side is as well. The sequence gm ðy; l m Þ is initialized by g0 ðy; l 0 Þ ¼ g0 ðyÞ. From Eq. (11), gm ðy; l m Þ gm1 ðy; l m1 Þ ¼ ð1 bm Þðgnm ðy; lnm Þ gm1 ðy; l m1 ÞÞ

(12)

which suggests that jgm ðy; l m Þ gm ðy; l m1 Þj ! 0

as m ! 1

(13)

We make this statement more precise in Section 5. Some remarks and conjectures are useful here. 1. We take bm ¼ 0 in the informal adaptation phase because it is unnecessary to conform to any rules in this stage. 2. We take bm ¼ 1 in the strict adaptation stage for the iterates m when lnm is not being updated, which means for the overwhelming number of iterates, in which case gm ðy; l m Þ ¼ gm1 ðy; l m1 Þ. 3. We ﬁnd empirically in our examples that lnm tends to converge to l as m increases so that we obtain good results just taking bm ¼ 0 for all m. This raises the conjecture that if the adaptation does not converge as the

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number of iterates tends to N then perhaps the target space may not have been explored adequately. 4. Instead of Eq. (11), we can also deﬁne lm directly as lm ¼ ð1 bm Þlnm þ bm lm1

(14)

where bm ¼ 0 whenever lnm is not updated and bm-1.

5. THEORY FOR ADAPTIVE SAMPLING This section outlines the theoretical framework for strict adaptive sampling in Giordani and Kohn (2007) and relates it to actual practice in that paper and our article. Let Y be the state space of y. Giordani and Kohn (2007) assume that there exists a constant KW0 such that for all yAY pðyÞ K g0 ðyÞ

(15a)

gm ðy; lm Þ K g0 ðyÞ

(15b)

They call Eqs. (15a) and (15b) the dominance conditions because under these condition g0(y) dominates both the target p(y) ¼ p(y|y) and the adaptive part of the proposal density gm ðy; lm Þ. One of the two reasons that we make g0(y) very heavy tailed is to help satisfy these dominance conditions. The other reason is computational as g0(y) gives us a heavy tailed component in the proposal density that is not subject to the vagaries of the adaptation process. Condition (15a) implies that p(y)/qm(y; lm) is bounded because pðyÞ pðyÞ qm ðy; lm Þ o1 g0 ðyÞ Such a condition is necessary, at least in principle, to ensure that the adaptive sampling scheme does not get stuck. When Y is closed and bounded (compact) the dominance conditions hold almost automatically. This is true in the discrete cases discussed in Sections 2 and 3. It also applies in a large number of problems where it is plausible to bound the parameter space.

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Consider now the general case. Condition (15a) holds if there exists a KuW0 such that for all yAY, pðyÞ K0 g0 ðyÞ

(16a)

pðyjyÞ K 0

(16b)

Suitable choices of p(y) and g0(y) will ensure that Eq. (16a) holds and Eq. (16b) holds if the maximum likelihood estimate exists. We now argue by induction that Eq. (15b) can be satisﬁed by construction. First, Eq. (15b) holds for m ¼ 0 because g0 ðy; l 0 Þ=g0 ðyÞ ¼ 1 K. Suppose further that by construction gnm ðy; lnm Þ=g0 ðyÞ K for all yAY and all m. This condition can be met by an appropriate choice of g0(y) and lnm . Suppose now that Eq. (15b) holds for m ¼ k1. Then gk ðy; l k Þ gn ðy; ln Þ bk gk1 ðy; l k1 Þ ¼ ð1 bk Þ k þ K g0 ðyÞ g0 ðyÞ g0 ðyÞ so that Eq. (15b) holds for m ¼ k. It is harder to check that Eq. (15a) holds in general and we do so only informally as the simulation progresses. The second condition imposed by Giordani and Kohn (2007) is the diminishing adaptation condition

g ðy; l Þ g

m m m1 ðy; lm1 Þ

(17) sup

¼ am

g0 ðyÞ y2Y

where am=O(mr) for some rW0 almost surely. From Eqs. (12) and (15b)

g ðy; l Þ g

m m m1 ðy; lm1 Þ

sup

2Kð1 bm Þ

g0 ðyÞ y2Y

¼ Oð1 bm Þ so that Eq. (17) holds with am ¼ (1bm). Giordani and Kohn (2007) show that if the dominance conditions (Eqs. (15a) and (15b)) and the diminishing adaptation (Eq. (17)) condition hold then the iterates from the adaptive sampler converge to draws from the target distribution. That is, Prðy½m 2 AÞ ! pðAÞ

as m ! 1

where A is a subset of Y and y[m] is the mth iterate of y in the adaptive sampling scheme.

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Furthermore, if h(y) is a measurable function that is square integrable with respect to the density g0(y), then, almost surely, m 1X hðy½j Þ ! E p ðhðyÞÞ m j¼1

as

m!1

where y[j] are the iterates of y.

6. ADAPTATION FOR GENERAL VARIABLE SELECTION PROBLEMS In Sections 2 and 3 all other parameters can be integrated out when doing the variable selection or identiﬁcation of interventions, or at least can be easily generated once the discrete adaptation takes place. This section outlines how to combine adaptation for discrete and continuous variables in more general contexts. To illustrate, consider the logistic regression model where the dependent variable yt is binary and xt is the vector of independent variables such that Prðyt ¼ 1jxt ; bÞ ¼

expðx0t bÞ 1 þ expðx0t bÞ

(18)

Suppose that we would like to carry out subset selection or model averaging for the model (18) similarly to the linear regression model (1). To carry out variable selection it is again convenient to introduce the binary indicators gi as in Section 2. However, for the logistic regression model we cannot integrate out the parameters b and just sample the indicators. Instead, it is necessary to generate both the ‘‘active’’ b parameters (those whose indicators are 1) and the indicator variables. If the number of active variables is likely to be small there is a real computational advantage in carrying out variable selection when sampling the b coefﬁcients because only a small number of b coefﬁcients is likely to be in the model in any one iteration. We combine the ideas of Sections 2 and 4 by ﬁrst running an adaptive sampler without any variable selection to approximate the posterior of b as a mixture of normals which we call q(b|y). Once we have this approximation, we can form the conditional density q(b|y, g) for any conﬁguration g because a mixture of normals conditional on some variables being zero is still a mixture of normals. For a given g, let bg consist of all those bi for

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which the corresponding gi=1. Then, q(bg |bg=0, y, g) is also a mixture of normals, where bg consists of those gi for which gi=0. In the second phase of the sampling we learn about the posterior distribution of g by using the sampling scheme outlined in Section 2 to form an approximation q(g|y) to the true posterior p(g|y). In the third phase of the sampling scheme we continue to learn about p(g|y) by further adapting q(g|y). We illustrate these ideas with a simulated example of 100 observations and 20 i.i.d. normal covariates plus constant. The data were generated by a logistic regression model with coefﬁcients b=(2, 2, 1, 1, 0, 0, . . . , 0)u. We draw 5000 iterates (plus 1000 burn-in) of the entire vector b as in Giordani and Kohn (2007). The average acceptance rate in this phase was 26%. We then performed 5000 iterations with variable selection (with ﬁxed inclusion probability of 0.2), where in each iteration we cycle through all the variables, generating a total of 100,000 draws of the parameters. We keep each 20th draw in this phase. The average (across parameters) inefﬁciency factors in the variable selection phase is just 1.63 due to the low number of nonzero coefﬁcients (six on average) in most draws. A challenging issue is how to sample adaptively with a variable selection prior when there are a large number of coefﬁcients available for selection so that generating all the coefﬁcients simultaneously in the ﬁrst phase is infeasible.

ACKNOWLEDGMENT The research in this article was partially supported by an Australian Research Council Grant DP0667069.

REFERENCES Andrieu, C., & Moulines, D. (2006). On the ergodicity properties of some adaptive MCMC algorithms. Annals of Applied Probability, 16(3), 1462–1505. Carter, C., & Kohn, R. (1997). Semiparametric Bayesian inference for time series with mixed spectra. Journal of the Royal Statistical Society (Series B), 59, 255–268. Fernandez, C., Ley, E., & Steele, M. (2001). Benchmark priors for Bayesian model averaging. Journal of Econometrics, 100, 381–427. Gamerman, D. (1997). Sampling from the posterior distribution in generalized linear mixed models. Statistics and Computing, 7, 57–68.

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Gelfand, A., & Smith, A. (1990). Sampling based approaches to calculating marginal densities. Journal of the American Statistical Association, 85, 398–409. Gelman, A., & Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7, 473–483. George, E., & McCulloch, R. (1993). Variable selection via Gibbs sampling. Journal of the American Statistical Association, 88, 881–889. George, E., & McCulloch, R. (1997). Approaches to Bayesian variable selection. Statistica Sinica, 7, 339–373. Gerlach, R., Carter, C., & Kohn, R. (2000). Bayesian inference for dynamic mixture models. Journal of the American Statistical Association, 95, 819–828. Giordani, P., & Kohn, R. (2007). Adaptive independent Metropolis–Hastings by fast estimation of mixtures of normals. Available at http://arxiv.org/PS_cache/arxiv/pdf/0801/ 0801.1864v1.pdf Giordani, P., & Kohn, R. (2008). Efﬁcient Bayesian inference for multiple change-point and mixture innovation models. Journal of Business and Economic Statistics, 26, 66–77. Haario, H., Saksman, E., & Tamminen, J. (2001). An adaptive Metropolis algorithm. Bernoulli, 7, 223–242. Hastie, D. (2005). Towards automatic reversible jump Markov chain Monte Carlo. Unpublished Ph.D. dissertation, Department of Mathematics, University of Bristol. Hastings, W. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109. Hesterberg, T. C. (1998). Weighted average importance sampling and defensive mixture distributions. Technometrics, 37, 185–194. Hoeting, J., Madigan, D., Raftery, A., & Volinsky, C. (1999). Bayesian model averaging: A tutorial. Statistical Science, 14, 382–417. Kass, R., & Wasserman, L. (1995). A reference Bayesian test for nested hypotheses and its relationship to the Schwartz criterion. Journal of the American Statistical Association, 90, 928–934. Kohn, R., Smith, M., & Chan, D. (2001). Nonparametric regression using linear combinations of basis functions. Statistics and Computing, 11, 313–322. Koop, G., & Potter, S. (2004). Forecasting and estimating multiple change-point models with an unknown number of change-points. Unpublished manuscript, University of Leicester. Liang, F., Paulo, R., Molina, G., Clyde, M., & Berger, J. (2008). Mixtures of g-priors for Bayesian variable selection. Journal of the American Statistical Association, 103, 410–423. Liu, J. (2001). Monte Carlo strategies in scientiﬁc computing. New York: Springer. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953). Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21, 1087–1092. Miller, A. (2001). Subset selection in regression. New York: Chapman-Hall. Neal, R. (2003). Slice sampling. Annals of Statistics, 31, 705–767. Nott, D., & Kohn, R. (2005). Adaptive sampling for Bayesian variable selection. Biometrika, 92, 747–763. Raftery, A., & Zhang, Y. (2003). Discussion: Performance of Bayesian model averaging. Journal of the American Statistical Association, 98, 931–938. Roberts, G. O., & Rosenthal, J. S. (2006). Examples of adaptive MCMC. (Preprint, available at http://probability.ca/jeff/ftpdir/adaptex.pdf).

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Roberts, G. O., & Rosenthal, J. S. (2007). Coupling and ergodicity of adaptive MCMC. Journal of Applied Probability, 44(2), 458–475. Smith, M., & Kohn, R. (1996). Nonparametric regression using Bayesian variable selection. Journal of Econometrics, 75, 317–343. Tierney, L. (1994). Markov chains for exploring posterior distributions. Annals of Statistics, 22, 1701–1762. Zellner, A. (1986). On assessing prior distributions and Bayesian regression analysis with g-prior distributions. In: P. Goel & A. Zellner (Eds), Bayesian inference and decision techniques: Essays in honour of Bruno De Finetti (pp. 233–243). New York: North Holland/Elsevier.

A BAYESIAN ANALYSIS OF THE OPES MODEL WITH A NONPARAMETRIC COMPONENT: AN APPLICATION TO DENTAL INSURANCE AND DENTAL CARE Murat K. Munkin and Pravin K. Trivedi ABSTRACT This paper analyzes the effect of dental insurance on utilization of general dentist services by adult US population aged from 25 to 64 years using the ordered probit model with endogenous selection. Our econometric framework accommodates endogeneity of insurance and the ordered nature of the measure of dental utilization. The study ﬁnds strong evidence of endogeneity of dental insurance to utilization and identiﬁes interesting patterns of nonlinear dependencies between the dental insurance status and individual’s age and income. The calculated average treatment effect supports the claim of adverse selection into the treated (insured) state and indicates a strong positive incentives effect of dental insurance.

Bayesian Econometrics Advances in Econometrics, Volume 23, 87–114 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23003-3

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1. INTRODUCTION This paper analyzes the effect of dental insurance on utilization of general dentist services by adult US population, with the main focus on the role of dental insurance and income. The central research questions we confront are: What is the causal impact of dental insurance on dental use, differentiated by age and gender, and what is the relationship between the probability to have dental insurance and income? In addition, we investigate the relationship between general medical insurance and income. The Markov chain Monte Carlo (MCMC) methods are used to estimate the posterior distribution of the parameters and treatment effects. Compared with general medical insurance, the problem of endogeneity has not been properly addressed in the dental insurance literature. This is surprising given the fact that there are some speciﬁc features of dental insurance that make it econometrically easier to identify its pure treatment effects. The levels of risks against which dental insurance provides coverage are much lower than those of general medical insurance. The amount of dental coverage is usually limited and the dental coinsurance rates are much higher. As a result, the US population is almost evenly split between those with and without dental insurance coverage. In the case of general medical insurance only about 15% of the US population is uninsured, a variation with a relatively low signal-to-noise ratio, which makes it potentially difﬁcult to identify the treatment effects. One direct and commonly used impact measure is the average difference in the number of dental visits by those with and those without dental insurance. This is a valid measure for data from clinical trials with random assignment to treatment, but for observational data it is not. This is because the measured difference has two components consisting of the ‘‘pure’’ effect of insurance incentives and the added sample selection effect. Indeed, the selection problem can be ignored when the studied data come from a welldesigned social experiment in which the insurance status is randomly assigned such as in the work published by the Rand Health Insurance Experiment (RHIE) Group (Manning, Leibowitz, Goldberg, Rogers, & Newhouse, 1984). However, most empirical studies including ours are based on observational data. The sample selection effect arises because individuals are not randomly assigned to treatment, but are self-selected, in the sense that one can choose not to have insurance. Individuals make dental insurance choices based on personal characteristics and plan features. Some of these can be observed and measured but the others are not measurable. Failure to statistically control the effect of the unobservables when they also

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affect the outcome equation leads to the problem of endogeneity, generates a bias in estimates of treatment effect that is akin to the omitted variable bias, which may be positive, negative, or zero. The selection effect may be positive, negative, or zero. A positive selection effect means adverse selection and a negative one means advantageous selection. Hence ignoring it can result in biased estimates of the effect of insurance on service use. Causal models, of the type proposed in this project, attempt to separate these two components; this entails greater conceptual and computational complexity in estimation of models. By contrast, statistically simpler models of association are easier to handle but cannot identify the key parameters of interest in an economic analysis. Economic theory predicts that risk-averse individuals prefer to purchase insurance against catastrophic or simply costly evens because they value eliminating risk more than money at sufﬁciently high wealth levels. This is modeled by assuming that a risk-averse individual’s utility is a monotonically increasing function of wealth with diminishing marginal returns. This is certainly true for general medical insurance when liabilities could easily exceed any reasonable levels. However, in the context of dental insurance the potential losses have reasonable bounds with the least desirable but nevertheless cheapest prospect of simply loosing a tooth, which is not life threatening. It is interesting to see if there is a nonlinear relationship between dental insurance and wealth, and whether this relationship is monotonic. We also estimate the relationship between general medical insurance and wealth to see whether our ﬁndings are consistent with economic theory. Our estimation approach allows income to enter the insurance equation nonparametrically. However, as a baseline model we try a parametric speciﬁcation of the insurance equation with income dummies representing 20 income groups. These groups are formed by ﬁrst sorting the sample with respect to income and then evenly dividing it into 20 groups and representing each group with a dummy variable. The incentives effect of dental insurance should be positive since the economic theory predicts that when the out-of-pocket share of the cost is reduced the level of utilization increases. However, it is not clear what the sign and magnitude of the selection effect are. It is expected that less healthy with respect to dental health individuals self-select to purchase dental insurance. Therefore, the unobservable dental health status is one of the driving forces behind the selection process, which also affects utilization. On the contrary, more risk-averse people are likely to purchase dental insurance. More risk-averse people should also have more sound dental health lifetime habits which drive the level of utilization down. It is a matter

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of empirical investigation to identify the actual direction and level of selfselection. This paper ﬁnds strong evidence of diminishing marginal returns of our measure of wealth on dental insurance status and even a nonmonotonic pattern. Additionally, we identify a nonmonotonic relationship between age and dental insurance status and utilization, and ﬁnd evidence of different gender dental utilization patterns. The estimated treatment effects support the claim of adverse selection of possibly less healthy with respect to dental health individuals into the insured (treated) state. The rest of the paper is organized as follows. Section 2 reviews the literature related to the effect of dental insurance on utilization. Section 3 describes the model, selects priors, and outlines the MCMC estimation procedure. Section 4 describes the data, deals with an application, and discusses the results. Section 5 concludes the paper.

2. BACKGROUND AND LITERATURE REVIEW Following experimental literature, we interpret the purchase of dental insurance as a ‘‘treatment,’’ and nonpurchase as ‘‘control.’’ Then a useful, though crude, measure of the impact of insurance on a suitable measure of use of dental care, such as number of general dental visits, is the average difference in the number of visits between those with and those without dental insurance. Generically, we refer to such measures of impact as ‘‘average treatment effects.’’ In estimating treatment effects, controlling for the presence of systematic observed and unobserved differences between those with and those without insurance poses a challenge. A multivariate regression model, which controls for observed differences, is still potentially ﬂawed because it fails to account for mutual dependence between insurance status and use of dental services. There are several established methodologies for capturing the effects of sample selection. One is to include in the utilization equation all factors that can capture the effects of sample selection. Data limitations usually cannot guarantee such a degree of control because some determinants of selection bias are unobserved and hence cannot be controlled for. These omitted factors include personal preferences, tastes, habits, knowledge of and attitudes toward health risks. 2.1. Models of Insurance In sharp contrast to the published research on dental care use, there are relatively few studies that model the choice of private dental insurance, for

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example, Manski, Goodman, Reid, and Macek (2004). This is so despite the fact that the proportion of uninsured adult non-Medicare population is close to half, and hence much higher than that for general health insurance. On the contrary, the consequences of this high uninsured rate are generally not perceived to be as severe as those due to lack of health insurance. Employer-sponsored private dental coverage offered by some employers is the principal source of insurance. Models similar to those for health insurance choice, therefore, provide a natural starting point for a modeling exercise.

2.2. Models of Utilization A recent comprehensive survey of the economics of dental services from an international perspective is Sintonen and Linnosmaa (2000). Like much of related literature, their article emphasizes modeling individual’s demand for dental care as measured by either the number (count) of dental visits or the total dental expenditure on dental care. Both measures have been widely used; some studies that use the count measure include: Manning and Phelps (1979), who used RHIE data; and Rosenqvist, Arinen, and Sintonen (1995), Arinen, Sintonen, and Rosenqvist (1996), Melkersson and Olssen (1999), and Olssen (1999), who used Swedish cross section data. Empirical analyses based on dental care expenditure data include: Manning et al. (1984) using RHIE data; Conrad, Grembowski, and Milgrom (1985), Mueller and Monheit (1988), and Manski, Macek, and Moeller (2002) using US data; Grytten, Rongen, and Asmyhr (1996) using Norwegian data; and Sintonen and Maljanen (1995) using Finnish data. Manski et al. (2004) model the probability of a dental visit conditional on insurance status. By standard economic theory, reduction in out-of-pocket costs of care due to insurance will increase utilization. Therefore, a positive association between utilization and insurance status is expected. Several US studies cited above conﬁrm such a relationship (Manski et al., 2002). Potentially, however, these results are subject to sample selection bias of unknown magnitude. In observational studies, the recorded insurance status will generally reﬂect the optimizing behavior of a consumer. If, as seems reasonable, there exist factors (typically unmeasured in the data), such as health habits, genetic proclivities, and oral health status, that are known to individuals and that are likely to affect future dental care utilization, optimizing individuals will take them into account when insurance is purchased. Indeed, adverse selection considerations suggest that those

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purchasing dental insurance may well be less healthy on average than those who do not. As a consequence, we expect that dental insurance status and utilization will be interdependent. Sintonen and Linnosmaa (2000, p. 1273) have observed that, with some notable exceptions, ‘‘ . . . little attention has been paid to the effect of model speciﬁcation and estimation techniques in dental utilization studies, or to examining whether the distributional assumptions of the model are met.’’

2.3. Dental Insurance and Utilization in the United States Some well-established basic facts about dental insurance and utilization include the following: 1. During the past four decades, the coverage of dental insurance has increased steadily. Currently, many health plans are expanding dental beneﬁts and providing additional incentives for preventive dental services, amid evidence that it improves overall health (Wall Street Journal, September 19, 2006). In 1996, the private coverage was 51.2% overall; only 6.8% of those with Medicaid had dental insurance. According to the Report of the Surgeon General, private dental care beneﬁts are available to most full-time employees in medium-sized and large businesses, either as part of a comprehensive medical and dental plan or as a separate plan. Firms often offer employees a choice of medical plans plus a dental plan, or a dental plan that can supplement any medical plan. Employer sponsorship of dental plans is changing with the expansion of managed care and rising medical costs, see Bailit (1999). Most participants in employer-sponsored dental plans receive insurance reimbursements on a fee-for-service basis, typically by type of service performed. Coinsurance rates are typically higher than those for medical care. The coverage typically includes dental examinations, sealants, radiographs, and prophylaxes. Restorative procedures such as restorations are more commonly covered than crowns. Orthodontic care coverage is less common, and when provided may be limited to dependent children and up to set maximums. Implants and cosmetic procedures are usually not covered. 2. Certain populations, for example, African-Americans and Hispanics, register signiﬁcantly lower rates of dental insurance coverage. They also seek dental care less often. The relative importance of economic and

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noneconomic factors as determinants of utilization is a topic of continuing research (Gilbert et al., 2002). 3. Dental insurance is a strong predictor of access to care. For the entire population those with dental insurance had on average 2.65 visits and those without insurance had 2.42 visits, the difference being statistically signiﬁcant. The average measures are different in the studied dataset because additional restrictions were applied to the sample to justify the use of the instrumental variable. The average total dental expenditure for those with insurance was $417 versus $299 for those without insurance. Similar differentials between the insured and the uninsured persist in more detailed comparisons by various demographic characteristics; see Cohen, Manski, Magder, and Mullins (2002, 2003). The percentage of population with at least one annual visit to the dentist was 56.6% among the privately insured, and 28.6% among the uninsured. In the year 2000, private health insurance accounted for 42% of the total dental care expenses, and out-of-pocket payments accounted for 49.3%. 4. The positive impact of insurance on utilization persists even in a regression analysis that controls for other socioeconomic determinants of utilization; see Manski (2001a, 2001b). Recent studies do not control for sample selection, but the RHIE used randomized assignment to insurance plan. However, the RHIE studies are more than 20 years old. In the interim, considerable changes have occurred in the proportion of Americans undergoing regular dental visits (www.nidr.nih.gov/sgr/ sgrohweb/chap4.htm), losing teeth (Douglass, Shih, & Ostry, 2002), and enjoying dental insurance coverage (Bailit, 1999). Not only the composition of dental services has changed over time, but it is likely to continue to suffer transformations (Anderson, 2005) – in part as a response to evolving health care insurance and markets, as well as living standards and affordability (Macek, Cohen, Reid, & Manski, 2004), but also as a result of paradigm shifts in the understanding of common oral diseases, such as periodontitis (Papapanou, 1999).

3. ECONOMETRIC FRAMEWORK This section will outline our ordered probit model with endogenous selection (OPES) through which we will study the effect of dental insurance on the demand for general dentist services.

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3.1. The Model Assume that we observe N independent observations for individuals who choose whether to purchase dental insurance. Let di be the binary random variable (i ¼ 1, . . . , N) representing this choice such that di ¼ 1 if dental insurance is purchased and di ¼ 0 otherwise. Deﬁne this binary choice using the random utility approach that speciﬁes a latent variable representing the gain in utility received from having dental insurance relative to the alternative. To allow for income to enter it nonparametrically, we follow recent work on Bayesian semiparametric techniques by Koop and Poirier (2004) and Koop and Tobias (2006) and building on the Bayesian treatment of the OPES by Munkin and Trivedi (2008). Let the insurance equation be speciﬁed as Z i ¼ f ðsi Þ þ Wi a þ i

(1)

where Wi is a vector of regressors, a is a conformable vector of parameters, and the distribution of the error term ei is N ð0; 1Þ. Function f( ) is unknown and si is income of individual i and parameter a does not include an intercept. The recorded income variable takes almost the same number of different values as the number of observations, which potentially leads to the problem of overparameterization. It seems reasonable that the probability of purchasing dental insurance will not change much for small increments in income. We try two roundings of the income variable: up to a thousand and a hundred dollars. This gives us kg ¼ 254 and kg ¼ 1,626 distinct income values out of the total N ¼ 19,911 observations, respectively. We sort the data by values of s so that s1 is the lowest level of income (in our dataset it is zero) and sN is the largest. The main assumption that we make on function f(si) is that it is smooth such that it is differentiable and its slope changes slowly with si (Shiller, 1984). More formally, the assumption is that the ﬁrst derivative of f(s) is bounded by a constant C, or j f ðsi Þ f ðsi1 Þj Cjsi si1 j This condition includes a wide range of functions. The treatment variable is deﬁned as d i ¼ I ½0;þ1Þ ðZ i Þ where I[0,þN) is the indicator function for the set [0,þN).

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To model the ordered dependent variable, we assume that there is another latent variable Y ni that depends on the outcomes of di such that Y ni ¼ Xi b þ d i r þ ui where Xi is a vector of exogenous regressors, and b and r are conformable parameter vectors. Deﬁne Yi, an ordered variable measuring the degree of dental service utilization, as Yi ¼

M X

mI ½tm1 ;tm Þ ðY ni Þ

m¼1

where t0, t1, . . . , tM are threshold parameters and m ¼ 1, . . . , M. For identiﬁcation, we restrict t0 ¼ N, t1 ¼ 0, and tM ¼ N. The dental insurance variable is potentially endogenous to utilization and this endogeneity is modeled through correlation between ui and ei. Assume that they are jointly normally distributed with the covariance cov(ui, ei) ¼ d. Assume that Var(ui) ¼ 1 þ d2 (or stated conditionally Var(ui|ei) ¼ 1), a restriction made for identiﬁcation since Y ni is latent. Then the model can be rewritten as Y ni ¼ Xi b þ d i r þ i d þ zi where zi i

!

1 N 0; 0

i:i:d:

0

1

This representation can be interpreted as follows. Endogeneity of dental insurance is generated by latent factors not controlled for in both the dental insurance and utilization equations. Once the unexplained utility generated from dental insurance, variable ei, is included in the outcome equation, the remaining random error zi is uncorrelated with the dental insurance variable such that E(zi|di) ¼ 0.

3.2. The Priors Stacking Eq. (1) over i, we obtain Z ¼ Pc þ Wa þ e

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where 2

3 f ðs1 Þ 6 f ðs Þ 7 6 2 7 6 7 c ¼ 6 .. 7 6 . 7 4 5 f ðskg Þ and P is an N kg matrix constructed to select the appropriate element of c for each observation i. Deﬁne an kg kg matrix R such that w ¼ Rc is a vector of slope changes of function f( ), cj ¼

gj gj1 gj1 gj2 ; sj sj1 sj1 sj2

j ¼ 3; . . . ; kg

and the ﬁrst two elements are simply c1 ¼ f(s1) and c2 ¼ f(s2). Then Z ¼ PR1 w þ Wa þ e and we place a ﬂat but proper prior on (c1, c2) as N (02, I2) and an informative prior for the rest of the parameter vector cj N ð0; ZÞ;

j ¼ 3; . . . ; kg

where Z Gða; bÞ with a ¼ 3 and b ¼ 106. The priors employed embed the idea that changes in f(si) f(si1) ¼ gi gi1 should be small. We use an informative prior on the degree of smoothness, that is, differences of gj gj1 are assumed to be small. Hyperparameters a and b determine the tightness of the prior. The smoothness restriction depends on the choice of R and Z. How to select these hyperparameters is an issue. Koop and Poirier (2004) relate this choice to the classical issue of optimal bandwidth selection. If the prior of Z is selected to be too tight, it can result in the regression function to be simply linear. After experimenting with different values of the hyperparameters, we select a ¼ 3 and b ¼ 106 and ﬁnd them to produce smooth posteriors.

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We select proper prior distributions for parameters. a, b, and r a N ð0; 10Ik Þ; b N ð0; 10Ip Þ; r N ð0; 10Þ For the covariances between the error ei and the error ui, we select prior distributions 1 d N 0; 2 The priors for the threshold parameters s ¼ (t2, . . . , tM1) must respect the order restrictions placed on them. It is easier to choose priors by reparameterizing these parameters ﬁrst which we do in the next section. 3.3. The MCMC Algorithm Let Di ¼ (Xi, Wi, w, a, b, r, d, s), and denote Pi the ith row of matrix P. For each observation i, the likelihood is Pr½Y i ; Y ni ; d i ; Z i jDi ¼ ð2pÞ1=2 exp½0:5ðZ i Pi R21 w Wi aÞ2 " # M X n I fY i ¼mg I ½tm1; tm Þ ðY i Þ ½d i I ½0;þ1Þ ðZ i Þ þ ð1 d i ÞI ð1;0Þ ðZi Þ m¼1

exp½0:5ðY ni ðXi b þ d i r þ ðZi Pi R21 w Wi aÞdÞÞ2 The joint distribution for all observations is the product of such N independent observations over i ¼ 1, . . . , N. The posterior density is proportional to the product of the prior density of the parameters and the joint distribution of observables and included latent variables. We block the parameter set as Zi, ½Y ni ; s, [w,a], and [b,r,d] and adopt a hybrid Metropolis–Hastings/Gibbs algorithm, in which the threshold parameters of the block ½Y ni ; s are drawn using a Metropolis–Hastings step (Cowles, 1996). The steps of the MCMC algorithm are the following: 1. The latent vectors Zi (i ¼ 1, . . . , N) are conditionally independent with 1 bivariate normal distribution Z i i:i:d: N ½Zi ; H i where 1 H i ¼ 1 þ d2 ; Z i ¼ Pi R1 w þ Wi a þ H i ½dðY ni Xi b d i rÞ

and subject to Z ji 0

if

d ji ¼ 1

Z ji o0

if

d ji ¼ 0

and

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2. The full joint conditional density of block ½Y ni ; s is Pr½Y ni ; sjZ i ; b; r; d; w; a " # N X M Y n I fY i ¼mg I ½tm1 ;tm Þ ðY i Þ ¼ i¼1 m¼1

exp½0:5ðY ni ðXi b þ d i r þ ðZ i Pi R1 w Wi aÞdÞÞ2 which we write as Pr½Y ni ; sjZ i ; b; r; d; w; a ¼ Pr½Y ni js; Z i ; b; r; d; w; aPr½sjZ i ; b; r; d; w; a Latent variable Y ni is N ½Xi b þ d i r þ ðZi Pi R1 w Wi aÞd; 1 and truncated on the left and right. Conditional on Yi ¼ m, it is truncated on left by tm1 and on the right by tm. The full conditional density of vector s ¼ (t2, . . . ,tM1) is " # N X M Y n I fY i ¼mg Prðtm1 oY i otm jDi ; d i ; Z i Þ i¼1 m¼1

where Prðtm1 oY ni otm jDi ; d i ; Z i Þ ¼ Fðtm ðXi b þ d i r þ ðZ i Pi R1 w Wi aÞdÞÞ Fðtm1 ðXi b þ d i r þ ðZ i Pi R1 w Wi aÞdÞÞ

ð2Þ

Since the elements of vector s are ordered, the prior assigned to the threshold parameters must be restricted. Instead, we follow Chib and Hamilton (2000) and reparameterize them as g2 ¼ logðt2 Þ;

gj ¼ logðtj tj1 Þ;

3 joM 1

and assign a normal prior N ðc0 ; C0 Þ without any restrictions since elements of vector c ¼ (g2, . . . , gM1) do not have to be ordered. The full conditional for vector c is the product of the prior and the full conditional (2) after substituting tj ¼

j X

expðgk Þ

k¼2

This density is intractable and we utilize the Metropolis–Hastings algorithm to sample from it, using t-distribution centered at the modal value of the full conditional density for the proposal density. Let c^ ¼ arg max log pðcjDi ; d i ; Z i Þ

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and V g^ ¼ ðH g^ Þ1 be the negative inverse of the Hessian of log p(c|Di, di, Zi) evaluated at the mode c^ . Choose the proposal distribution qðcÞ ¼ f T ðcj^c; jV g^ ; uÞ, a t-distribution with u degrees of freedom and tuning parameter j, an adjustable constant selected to obtain reasonable acceptance rates. When a proposal value c is drawn, the chain moves to the proposal value with probability n pðc jDi ; d i ; Z i ÞqðcÞ n ;1 Prðc; c Þ ¼ min pðcjDi ; d i ; Z i Þqðcn Þ if the proposal value is rejected, then the next state of the chain is at the current value c. 1 3. Let the prior distributions of w be N ½w; H1 c and a be N ½ a; Ha . 1 Denote Gi ¼ (PiR , Wi), hu ¼ (wu, a) with the prior distribution N ½ h; H1 y . H 1 where Then the full conditional distribution of h is N ½h; y y ¼ Hy þ H h ¼

1 H y

"

N X

G0i ð1 þ d2 ÞGi

i¼1 N X Hy h þ G0i ð1 þ d2 ÞZ i G0i dðY ni Xi b d i rÞ

#

i¼1

4. Let Ci ¼ (Xi, di, (Zi PiR1wWia)), vu ¼ (bu, r, d) and denote the prior distribution v N ½v; H1 w . The full conditional distribution of v is 1 where H N ½v; w w ¼ Hw þ H v ¼

1 H w

"

N X

C 0i C i

i¼1

Hw v þ

N X

# C 0i Y ni

i¼1

5. Finally, 0 Z IG@

kg 1X

kg 2 þ a; b1 þ 2 2

This concludes the MCMC algorithm.

!1 1 A c2 j

i¼3

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MURAT K. MUNKIN AND PRAVIN K. TRIVEDI

4. APPLICATION Following Grossman’s (1972) seminal work, dental care, personal characteristics, and oral health practices are viewed by health economists as an input into a production process whose output is oral health. The inputs in the health production are determined partly by economic constraints. For example, the decision to seek dental care will partly depend upon having dental insurance. Having dental insurance is an economic decision that in turn can be modeled using demographic factors, personal health characteristics, cost of insurance, and family income. In reality oral health, dental insurance and dental utilization are intertemporally linked. For example, private investments made through preventive care or sound health habits during childhood will yield returns later in life; a past history of poor oral health may cause purchase of insurance and subsequent greater use of dental care. One needs good quality longitudinal data to model such dynamic interdependencies, especially that between dental care use and oral health. But as in much existing empirical work, we only have available individual level cross-sectional data, resulting in a limited empirical analysis. Another limitation of the database that we use is that it does not have any measures of oral health or a measure of individual wealth. Instead of wealth we use the annual total income, which includes both wage and nonwage sources.

4.1. Identiﬁcation Strategy An important element of our identiﬁcation strategy is the utilized instrumental variable or an exclusion restriction, a variable that affects the dental insurance choice but not the utilization. We propose to use the size of the ﬁrm where the individuals are employed. For that matter, we study only employed population of the United States. However, there is a great degree of heterogeneity in availability of dental beneﬁts among the employed. Since self-employed individuals are likely to have different dental insurance choices, we delete them from the simple. Further, we restrict our sample to only privately employed individuals. Governmental jobs are known to be more generous in providing insurance beneﬁts. Additionally, individuals might choose governmental jobs for the beneﬁt reasons, which makes the ﬁrm size potentially endogenous since governmental ﬁrms tend to be larger in size. Even though this arguments is usually used in regards to general medical insurance, it might still be valid for dental insurance. Firm size

Bayesian Analysis of the OPES Model with a Nonparametric Component

101

should not affect utilization but it should affect availability of dental beneﬁts with larger ﬁrms more likely to offer such beneﬁts. The restricted sample reduces a possibility for our instrument to fail as being strictly exogenous; however, it does not eliminate such a possibility entirely. We decide to rely on this instrument because of its robustness as a predictor of dental insurance, and because there is no measurable impact on general dentist visits. The instrument has also been used in health literature before (Bhattacharya, Goldman, & Sood, 2003; Johnson & Crystal, 2000; Olson, 2002; Deb, Munkin, & Trivedi, 2006).

4.2. Data The application section investigates the effect of dental insurance on a measure of the demand for dental services by the US population between the ages of 25 and 64 years using data from the Medical Expenditure Panel Survey (MEPS). Speciﬁcally, we look at the number of general dentist visits. The sample does not include children and young adults since their utilization patterns are likely to be completely different from those of the adult nonelderly population. MEPS is a nationally representative survey of health care use including dental, expenditure, sources of payment, and insurance coverage for the US civilian noninstitutionalized population, and it is publicly available at the Agency for Healthcare Research and Quality (AHRQ). We use data from the 1996, 1997, 1998, 1999, and 2000 surveys and restrict our sample to only those who are employed by private ﬁrms but not self-employed. The sampling scheme of the MEPS data is a 2-year overlapping panel; that is, in each calendar year after the ﬁrst survey year, one sample of persons is in its second year of responses while another sample of persons is in its ﬁrst year of responses. To avoid panel and clustering issues, we only use observations on ‘‘new’’ survey respondents in each year. The ﬁnal sample size is 19,911. Table 1 gives a summary statistics of all variables used in our analysis. Table 2 describes the distribution of the general dentist visit variable up to cell 6. Since it is not feasible to estimate threshold parameters for very slim cells, we combine all observations of at least six visits into the last cell, making sure that there is at least 1% of the whole sample (200 observations) in all cells. It can be seen that the dependent variable has a substantial (W62%) share of zero utilization. The maximum number of visits in the original variable is 28. The distribution has a short tail. This utilization pattern justiﬁes our use of the ordered probit framework since it is hard to

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MURAT K. MUNKIN AND PRAVIN K. TRIVEDI

Table 1.

Summary Statistics.

Utilization GDVIS Number of general dentist visits Insurance DENTAL ¼ 1, if dental insurance PRIVATE ¼ 1, if private medical insurance Demographic characteristics FAMSIZE Family size AGE Age/10 EDUC Years of schooling INCOME $ income/1,000 FEMALE ¼ 1, if female BLACK ¼ 1, if black HISPANIC ¼ 1, if Hispanic MARRIED ¼ 1, if married NOREAST ¼ 1, if northeast MIDWEST ¼ 1, if midwest SOUTH ¼ 1, if south MSA ¼ 1, if metropolitan statistical area AGE 2 ¼ AGE AGE AGE FEM ¼ AGE FEMALE Age dummies AGE35 ¼ 1, if 30oAGEr35 AGE40 ¼ 1, if 35oAGEr40 AGE45 ¼ 1, if 40oAGEr45 AGE50 ¼ 1, if 45oAGEr50 AGE55 ¼ 1, if 50oAGEr55 AGE60 ¼ 1, if 55oAGEr60 AGE65 ¼ 1, if 60oAGEo65 Employment characteristic (exclusion restriction) FIRMSIZE Firm size Health characteristics VEGOOD ¼ 1, if very good health GOOD ¼ 1, if good health FAIRPOOR ¼ 1, if fair or poor health PHYSLIM ¼ 1, if physical limitation TOTCHR Number of chronic conditions Year dummies YEAR97 ¼ 1, if year 1997 YEAR98 ¼ 1, if year 1998 YEAR99 ¼ 1, if year 1999 YEAR00 ¼ 1, if year 2000

1.763

1.272

0.580 0.774

0.494 0.418

3.185 4.100 12.713 41.041 0.474 0.132 0.210 0.660 0.177 0.220 0.363 0.810 17.768 1.950

1.571 0.978 2.916 34.151 0.499 0.338 0.407 0.474 0.382 0.414 0.481 0.393 8.432 2.160

0.173 0.178 0.155 0.131 0.101 0.062 0.031

0.378 0.383 0.362 0.337 0.301 0.241 0.174

14.248

18.181

0.338 0.267 0.088 0.058 0.500

0.473 0.443 0.284 0.234 0.777

0.167 0.184 0.197 0.153

0.373 0.388 0.398 0.360

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Bayesian Analysis of the OPES Model with a Nonparametric Component

Table 2. Frequencies Cells

0 1 2 3 4 5 Z6

GDVIS

GDVIS by Insurance

Actual

Predicted

DENTAL ¼ 1

DENTAL ¼ 0

62.38 17.49 10.54 4.71 2.29 1.10 1.49

62.49 17.52 10.40 4.62 2.27 1.12 1.58

53.40 21.05 13.39 5.85 3.05 1.39 1.87

74.78 12.57 6.60 3.13 1.24 0.71 0.97

Table 3.

Health Status by Insurance.

Insurance Plan

DENTAL ¼ 1 DENTAL ¼ 0

Utilization Patterns.

Health Status EXCELLENT

VEGOOD

GOOD

FAIRPOOR

32.98 27.53

35.25 31.72

24.61 29.66

7.17 11.09

expect that even mean preserving transformations of the Poisson model allowing for overdispersion would ﬁt such a pattern well. The model has ﬁve threshold parameters to estimate. Table 2 also presents the distribution of the dependent variable by dental insurance. The no insurance group has a larger portion of zeroes, but it also has a shorter tail. If the assignment of dental insurance were random, then one would expect that the group means are not signiﬁcantly different from each other. The Pearson test statistic for categorical independence produces w2(6) ¼ 959 (po0.0001). This gives a strong signal that the assignment of dental insurance is nonrandom and, therefore, a full investigation of the problem permitting endogeneity of insurance status is needed. Table 3 gives the distribution of self-reported health status with respect to dental insurance. The adverse selection arguments support the claim that less healthy individuals are more likely to self-select themselves into being insured. However, the frequencies presented in Table 3 tell the opposite story. The dental insured group appears to look more healthy with respect to the general health status (not dental health) than the uninsured one. This seems to favor the possibility that the main driving force behind the

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MURAT K. MUNKIN AND PRAVIN K. TRIVEDI

selection process is risk preferences and not the underlined health status with more risk-averse people choosing to purchase dental insurance. Being risk-averse has made them healthier than their less risk-averse counterparts. On the contrary, it is not clear how general health status is correlated with dental health. We perform the formal Pearson test for categorical independence, which produces w2(3) ¼ 196 (po0.0001) rejecting the null hypothesis of categorical independence. The covariate vector X consists of self-perceived health status variables VEGOOD, GOOD, FAIRPOOR; measures of chronic diseases and physical limitation TOTCHR and PHYSLIM, respectively; geographical variables NOREAST, MIDWEST, SOUTH, and MSA; demographic variables BLACK, HISPANIC, FAMSIZE, FEMALE, MARRIED, EDUC, AGE; and additional variables AGE 2, AGE FEM, which are deﬁned as the square of AGE and product of AGE and FEMALE, respectively; year dummies YEAR97, YEAR98, YEAR99, YEAR00; economic variable INCOME; and insurance variable DENTAL. The insurance equation includes all variables included in X, except for the dental insurance dummy, income, and the age-related variables, plus an additional variable, FIRMSIZE, our exclusion restriction. The exclusion restriction serves to identify the correlation between the insurance equation and the dependent variable. As a baseline model, we estimate a spline regression in which vector W contains INCOME dummies constructed to capture the effects of these variables for different income categories in the following way. We divide all individuals in the sample into 20 income groups according to percentiles based on 5% increments and include 19 income dummies in the insurance equation omitting the dummy related to the ﬁrst income group (r5%). This makes it roughly 995 observations per category. The semiparametric model corresponding to kg ¼ 254 is estimated next in which we allow INCOME to enter the insurance equation nonparametrically. Additionally, we divide our sample into 8 age categories for the ages from 25 to 64 years with each group including 5 years and include 7 age dummies in the insurance equation. This should allow one to see how the probability of purchasing dental insurance changes with the age dummies.

4.3. Results We estimate the two speciﬁcations of the OPES for the number of general dentist visits corresponding to our different treatments of INCOME

Bayesian Analysis of the OPES Model with a Nonparametric Component

105

variable in the insurance equation. It happens that the posterior distributions of all parameters not related to INCOME and including the treatment effects and predicted frequencies are practically identical for both speciﬁcations and, therefore, we report posterior means and standard deviations of parameters b, a, r, d, and s in Table 4 only for the speciﬁcation corresponding to the spline regression. The results are based on Markov chains run for 10,000 replications after discarding ﬁrst 1,000 draws of the burn-in phase. We collect every 10th iteration discarding the rest. The Markov chains have good mixing properties with autocorrelation functions of the chain dying-off after one to two lags for all parameters. The predicted cell probabilities are given in Table 2. The model does a very accurate job in predicting these probabilities. This level of precision is difﬁcult to achieve with Poisson-based models. It is interesting to notice that health status indicators GOOD and FAIRPOOR have strong negative impacts on dental insurance with worsening health conditions decreasing the probability of having dental insurance. Alternatively, this can be interpreted as individuals with better health status are more likely to have dental coverage. At the same time, the number of chronic conditions increases the probability of purchasing dental insurance and the indicator of physical limitations has no impact. However, the health indicators, except for TOTCHR, have no impact on utilization, which supports the claim that general health status is not strongly correlated with dental health. Education, being female, and married increase both the probability of being insured and the level of utilization. It is interesting that blacks are more likely to have dental coverage and at the same time are less likely to see a general dentist. Living in a metropolitan statistical area increases the likelihood of dental coverage perhaps because of more choices available to the nonrural population. FAMSIZE and HISPANIC are the only variables the impacts of which are strong and negative both on the utilization and insurance variables. 4.3.1. Role of Age The impact of the AGE dummies shows a nonlinear relationship between age and dental insurance coverage as expected. For relatively younger individuals, the probability of being insured increases with age (AGE35, AGE40, and AGE50) but then for older patients the effect of age on dental insurance becomes negative (AGE60 and AGE65) and much stronger in magnitude. Dental insurance seems to be a less desirable good for older near-elderly adults. As a comparison exercise, we estimate the model

106

MURAT K. MUNKIN AND PRAVIN K. TRIVEDI

Table 4.

CONST FAMSIZE EDUCYR FEMALE BLACK HISPANIC MARRIED NOREAST MIDWEST SOUTH MSA PHYSLIM TOTCHR VEGOOD GOOD FAIRPOOR YEAR97 YEAR98 YEAR99 YEAR00

Posterior Means and Standard Deviations of Parameters. Insurance

Visits

1.521 0.086 0.071 0.008 0.058 0.004 0.166 0.021 0.079 0.029 0.204 0.028 0.443 0.023 0.114 0.032 0.001 0.030 0.126 0.027 0.193 0.025 0.029 0.045 0.047 0.014 0.017 0.024 0.097 0.026 0.146 0.039 0.085 0.028 0.136 0.028 0.036 0.028 0.067 0.031

2.526 0.179 0.036 0.007 0.047 0.004 0.487 0.079 0.402 0.030 0.193 0.026 0.070 0.025 0.095 0.028 0.082 0.027 0.066 0.025 0.023 0.025 0.008 0.040 0.071 0.011 0.029 0.022 0.011 0.025 0.023 0.038 0.010 0.027 0.045 0.028 0.038 0.026 0.099 0.028

Insurance AGE35 AGE40 AGE45 AGE50 AGE55 AGE60 AGE65

0.100 0.032 0.098 0.031 0.054 0.033 0.102 0.035 0.074 0.040 0.134 0.045 0.148 0.058

AGE

0.392 0.081 0.033 0.009 0.049 0.018 0.0015 0.0003

AGE 2 AGE FEM INCOME FIRMSIZE

0.016 0.001

DENTAL

0.783 0.069 0.272 0.042 0.587 0.010 1.104 0.014 1.483 0.017 1.791 0.020 2.035 0.024

Covariance (d) t1 t2 t3 t4 t5 Z

Visits

1.04 l06 1.29 l07

Bayesian Analysis of the OPES Model with a Nonparametric Component

107

including AGE variable itself assuming a linear form. As a result, we ﬁnd a negative impact of age on dental insurance. Even though the true relationship is nonmonotonic, the overall effect of age is dominated by the negative impact of that for the near elderly. This shows the importance of allowing for nonlinear dependence. The interaction term AGE FEM has a strong negative impact which indicates that, indeed, there are different gender patterns of aging and its impact on dental utilization. Females with age are less likely to use dental services than their male counterparts, possibly indicating having on average better lifetime preventive habits. The effects of AGE and AGE 2 on utilization show that there is an increasing but at a diminishing rate effect of age on the number of general dentist visits. 4.3.2. Role of Income The level of utilization increases with income as expected. The effect of INCOME on the probability of having dental insurance is presented in Figs. 1 and 2 for the spline regression and nonparametric model, respectively. The nonparametric speciﬁcation does not have an intercept in deﬁnition of parameter a. In order to make the results comparable, we 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2 -0.4 0

Fig. 1.

1

2

3

4

5

6

7

8

9 10 11 12 13 14 15 16 17 18 19 20

The Effects of Income Dummies on Dental Coverage. Spline Regression. (The Dotted Error Bounds Correspond to Two Standard Deviations.)

108

MURAT K. MUNKIN AND PRAVIN K. TRIVEDI 2.5

2.0

1.5

1.0

0.5

0.0

-0.5 0

20

40

60

80 100 120 140 160 180 200 220 240 260 280

Fig. 2. The Effects of Income on Dental Coverage. Nonparametric Estimates (k ¼ 254). (The Dotted Error Bounds Correspond to Two Standard Deviations.)

subtract the posterior mean of the intercept in the spline regression (1.521) from the estimated posterior means of vector c. The solid lines in the ﬁgures correspond to the estimated parameters and the dotted lines correspond to the two standard deviation bounds. The results present different patterns in how income affects the probability of purchasing dental insurance at different income levels. At low income levels as estimated by both speciﬁcations, there is no signiﬁcant impact of income on the probability of having dental insurance that is consistent with the fact that risk-averse individuals are willing to purchase dental insurance only at high enough levels of income when diminishing marginal returns of wealth start to apply. If the income level is very low and the basic life necessities have not been satisﬁed, then even being risk-averse would not lead to valuing eliminating risk more than money. As income goes up the probability of dental coverage starts to rise but the increases occur at diminishing rates. Fig. 1 shows that given the two standard deviation error band, one cannot conclude that the probability of dental insurance drops below the previous level at any income category. This means that the spline regression supports the claim that dental insurance is always a desirable good. It is interesting to

Bayesian Analysis of the OPES Model with a Nonparametric Component

109

notice that the standard deviations for all 19 income dummies are at about the same level of 0.06. The nonparametric model provides a different pattern in which the probability of having dental insurance starts to drop at income level of $120,000 until the $200,000 level is reached. One should be cautious interpreting this result since the standard deviations increase substantially with income. In addition, only about 3.5% of individuals in the sample has income exceeding $120,000 and only 0.4% has income greater than $200,000. As a robustness check, we estimate the model for a different speciﬁcation of the income variable corresponding to rounding up to a hundred dollars, which produces kg ¼ 1,626. Fig. 3 presents the estimated relationship between dental insurance and income. The ﬁgure suggests that the probability of dental insurance coverage drops at income level of $120,000. Our interpretation for this result is that at higher income level above $120,000 dental insurance becomes a less desirable good because the monetary value of risks eliminated by the coverage decreases relative to the income. This should not be the case for general medical insurance since there is no effective upper bound in the level of potential losses due to general (not just dental) health status. We investigate the relationship between the medical insurance coverage, variable PRIVATE in Table 1, and income using the semiparametric probit model deﬁned for the insurance equation only. Figs. 4 and 5 provide estimates corresponding to kg ¼ 254 and kg ¼ 1,626, respectively. The main deﬁnite conclusion that we could reach for both speciﬁcations is that strong evidence is found in favor of diminishing marginal returns and a monotonic relationship, which are consistent with economic theory.

4.3.3. Role of Insurance The exclusion restriction FIRMSIZE is strongly correlated with the dental insurance choice variable. Larger ﬁrms are more likely to offer greater insurance beneﬁts including dental insurance which is consistent with the positive impact of this variable on the insurance status. The correlation parameter, 0.272, is separated from zero by more than six standard deviations (0.042) which is strong evidence in favor of endogeneity of dental insurance. We perform a formal test of the null hypothesis H0:d ¼ 0 against HA:d 6¼ 0. Based on the calculated Bayes factor the hypothesis of no endogeneity is overwhelmingly rejected.

110

MURAT K. MUNKIN AND PRAVIN K. TRIVEDI 2.5

2.0

1.5

1.0

0.5

0.0

-0.5 0

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800

Fig. 3. The Effects of Income on Dental Coverage. Nonparametric Estimates (k ¼ 1,626). (The Dotted Error Bounds Correspond to Two Standard Deviations.)

The effect of dental insurance on utilization is very strong and positive, 0.783 (0.069). However, based on the signs and magnitudes of this parameter and those of the covariance it is impossible to assess the direction of the incentives and selection effects in the ordered probit model. In order to do that we calculate the ATE. 4.3.4. The Average Treatment Effect Deﬁnition of dependent variable Yi establishes the link between the observed and counterfactual outcomes as Y i ¼ d i Y 1i þ ð1 d i ÞY 0i

Bayesian Analysis of the OPES Model with a Nonparametric Component

111

3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0

20

40

60

80 100 120 140 160 180 200 220 240 260 280

Fig. 4. The Effects of Income on General Medical Insurance Coverage. Nonparametric Estimates (k ¼ 254). (The Dotted Error Bounds Correspond to Two Standard Deviations.)

3.0 2.5 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 0

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800

Fig. 5. The Effects of Income on General Medical Insurance Coverage. Nonparametric Estimates (k ¼ 1,626). (The Dotted Error Bounds Correspond to Two Standard Deviations.)

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MURAT K. MUNKIN AND PRAVIN K. TRIVEDI

The average treatment effect is deﬁned as the expected outcome gain from receipt of treatment, E[Y1Y0|X], for a randomly chosen individual. The ATE is calculated as E½Y 1 Y 0 jX ¼

N 1X EðY 1i Y 0i jXi Þ N i¼1

(3)

where the average is taken with respect to the sample. Denote gi ¼ (Zi, b, r, d, w, a, s) and deﬁne the expected utilization gain evaluated at gi for a randomly selected individual i between the treated state (with dental insurance) and the baseline choice (no dental insurance) as EðY 1i Y 0i jXi ; gi Þ ¼

M X

m½PrðY i ¼ mjd i ¼ 1; gi Þ PrðY i ¼ mjd i ¼ 0; gi Þ

m¼1

(4) In order to calculate EðY 1i Y 0i jXi Þ, we integrate Zi out numerically from Eq. (4) with respect to the posterior distribution of the parameters and substitute it into Eq. (3). The estimated ATE value is 0.373 (0.003) which indicates the average number of general dentist visit by which the level of utilization is increased with availability of dental insurance for a randomly chosen individual. The observed difference in utilization between the insured individuals in the sample (1.958) and the uninsured (1.495) is 0.463 visits. This difference is composed of the pure incentives effect and the selection effect. The selection effect happens to be positive (0.090) in our application which accounts for 19% of the observed difference in the utilization rates. Positive selection effect is consistent with adverse selection. All those unobservable factors based on which individuals make their dental insurance choices contribute positively to utilization. One potential explanation for this is that, indeed, those individuals who self-select to purchase dental insurance have on average worse dental health, which contributes to their higher levels of utilization.

5. CONCLUSION This paper analyzes the effect of dental insurance on utilization of general dentist services for adult US population using the MEPS data. The ordered probit approach extended to account for potential endogeneity of dental insurance status is used. Bayesian estimation of the model allows to avoid

Bayesian Analysis of the OPES Model with a Nonparametric Component

113

the intractability of the distributional forms by using the MCMC methods to approximate the posterior distribution of the parameters in the model and treatment effects. We ﬁnd strong evidence that dental insurance is endogenous to utilization. The calculated average treatment effect shows a positive incentives effect of dental insurance. It also provides evidence of adverse selection into dental insurance. The selection effect accounts for 19% of the observed difference in utilization between insured and uninsured individuals. A semiparametric approach is used to model the effect of income on the probabilities of having dental and general medical insurance coverage. We ﬁnd a nonmonotonic relationship between income and dental insurance. The results also support the claim of diminishing marginal returns of income on the probability of general insurance coverage.

REFERENCES Anderson, M. H. (2005). Future trends in dental beneﬁts. Journal of Dental Education, 69(5), 586–594. Arinen, S. S., Sintonen, H., & Rosenqvist, G. (1996). Dental utilization by young adults before and after subsidization reform in Finland. DP # 149. Center for Health Economics, York University. Bailit, H. (1999). Dental insurance, managed care, and traditional practice. Journal of the American Dental Association, 130(12), 1721–1727. Bhattacharya, J., Goldman, D., & Sood, N. (2003). The link between public and private insurance and HIV-related mortality. Journal of Health Economics, 22, 1105–1122. Chib, S., & Hamilton, B. H. (2000). Bayesian analysis of cross-section and clustered data treatment models. Journal of Econometrics, 97, 25–50. Cohen, L. A., Manski, R. J., Magder, L. S., & Mullins, D. (2002). Adult Medicaid patients’ dental visits in hospital emergency departments. Journal of the American Dental Association, 133(6), 715–724. Cohen, L. A., Manski, R. J., Magder, L. S., & Mullins, D. (2003). A Medicaid population’s use of physicians’ ofﬁces for dental problems. American Journal of Public Health, 93(8), 1297–1301. Conrad, D. A., Grembowski, D., & Milgrom, P. (1985). Adverse selection within dental insurance markets. Advances in Health Economics and Health Services Research, Biased Selection in Health Care Markets, 6, 171–190. Cowles, M. K. (1996). Accelerating Monte Carlo Markov chain convergence for cumulativelink generalized linear models. Statistics and Computing, 6, 101–111. Deb, P., Munkin, M. K., & Trivedi, P. K. (2006). Private insurance, selection, and the health care use: A Bayesian analysis of a Roy-type model. Journal of Business and Economic Statistics, 24, 403–415. Douglass, C. W., Shih, A., & Ostry, L. (2002). Will there be a need for complete dentures in the United States in 2020? The Journal of Prosthetic Dentistry, 87(1), 5–8. Gilbert, G. H., Shah, G. R., Shelton, B. J., Heft, M. W., Bradford, E. H., & Chavers, L. S. (2002). Racial differences in predictors of dental care use. Health Services Research, 37(6), 1487–1507.

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Grossman, M. (1972). On the concept of health capital and the demand for health. Journal of Political Economy, 80(2), 223–255. Grytten, J., Rongen, G., & Asmyhr, O. (1996). Subsidized dental care for young men: Its impact on utilization and dental health. Health Economics, 5(2), 119–128. Johnson, R. W., & Crystal, S. (2000). Uninsured status and out-of-pocket costs at midlife. Health Services Research, 35, 911–932. Koop, G., & Poirier, D. J. (2004). Bayesian variants of some classical semiparametric regression techniques. Journal of Econometrics, 123, 259–282. Koop, G., & Tobias, J. L. (2006). Semiparametric Bayesian inference in smooth co-efﬁcient models. Journal of Econometrics, 134, 283–315. Macek, M. D., Cohen, L. A., Reid, B. C., & Manski, R. J. (2004). Dental visits among older US adults, 1999. Journal of the American Dental Association, 135(8), 1154–1162. Manning, W. G., Leibowitz, A., Goldberg, G. A., Rogers, W. H., & Newhouse, J. (1984). A controlled trial of the effect of a prepaid group practice on the use of services. New England Journal of Medicine, 310(23), 1505–1510. Manning, W. G., & Phelps, C. E. (1979). The demand for dental care. Bell Journal of Economics, 10(2), 503–525. Manski, R. J. (2001a). Access to dental care: An opportunity waiting. Journal of the American College of Dentists, 68(2), 12–15. Manski, R. J. (2001b). Dental insurance: Design, need and public policy. Journal of the American College of Dentists, 68(1), 29–32. Manski, R. J., Goodman, H. S., Reid, B. C., & Macek, M. D. (2004). Dental insurance visits and expenditures among older adults. American Journal of Public Health, 94(5), 759–764. Manski, R. J., Macek, M. D., & Moeller, J. F. (2002). Private dental coverage: Who has it and how does it inﬂuence dental visits and expenditures? Journal of the American Dental Association, 133(11), 1551–1559. Melkersson, M., & Olssen, C. (1999). Is visiting the dentist a good habit? Analyzing count data with excess zeros and excess ones. Umea˚ Economic Studies No. 492. Umea˚ University, Umea˚, Sweden. Mueller, C. D., & Monheit, A. C. (1988). Insurance coverage and the demand for dental care: Results for non-aged White adults. Journal of Health Economics, 7(1), 59–72. Munkin, M. K., & Trivedi, P. K. (2008). Bayesian analysis of the ordered probit model with endogenous selection. Journal of Econometrics, 143, 334–348. Olson, C. (2002). Do workers accept lower wages in exchange for health beneﬁts? Journal of Labor Economics, 20, S91–S114. Olssen, C. (1999). Visiting a dentist: Is dental insurance important? Umea˚ Economic Studies No. 490. Umea˚ University, Umea˚, Sweden. Papapanou, P. N. (1999). Epidemiology of periodontal diseases: An update. Journal of International Academy of Periodontology, 1(4), 110–116. Rosenqvist, G., Arinen, S., & Sintonen, H. (1995). Modiﬁed count data models with an application to dental care. Working Paper No. 293. Swedish School of Economics and Business Administration, Helsinki. Sintonen, H., & Linnosmaa, I. (2000). Economics of dental services. In: J. Newhouse & A. Culyer (Eds), Handbook of health economics (Vol. 1B, pp. 1251–1296). Amsterdam: Elsevier. Sintonen, H., & Maljanen, T. (1995). Explaining utilization of dental care: Experiences from Finnish dental care market. Health economics, 4, 453–456. Shiller, R. J. (1984). Smoothness priors and nonlinear regression. Journal of the American Statistical Association, 79, 609–615.

FITTING AND COMPARISON OF MODELS FOR MULTIVARIATE ORDINAL OUTCOMES Ivan Jeliazkov, Jennifer Graves and Mark Kutzbach ABSTRACT In this paper, we consider the analysis of models for univariate and multivariate ordinal outcomes in the context of the latent variable inferential framework of Albert and Chib (1993). We review several alternative modeling and identiﬁcation schemes and evaluate how each aids or hampers estimation by Markov chain Monte Carlo simulation methods. For each identiﬁcation scheme we also discuss the question of model comparison by marginal likelihoods and Bayes factors. In addition, we develop a simulation-based framework for analyzing covariate effects that can provide interpretability of the results despite the nonlinearities in the model and the different identiﬁcation restrictions that can be implemented. The methods are employed to analyze problems in labor economics (educational attainment), political economy (voter opinions), and health economics (consumers’ reliance on alternative sources of medical information).

Bayesian Econometrics Advances in Econometrics, Volume 23, 115–156 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23004-5

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1. INTRODUCTION This article considers three main inferential problems, namely those of identiﬁcation, estimation, and model comparison, in the context of models for ordinal outcomes. We exploit the inferential framework of Albert and Chib (1993), which capitalizes on the latent variable representation of binary and categorical response models to simplify the analysis of such problems. In our setting, this framework lends itself to efﬁcient ﬁtting by Markov chain Monte Carlo (MCMC) methods – in some instances it allows for direct sampling from known full-conditional distributions, and in others it facilitates the application of versatile simulation techniques such as the Metropolis–Hastings (MH) and accept–reject Metropolis–Hastings (ARMH) algorithms (Gelfand & Smith, 1990; Metropolis, Rosenbluth, Rosenbluth, Teller, & Teller, 1953; Hastings, 1970; Tierney, 1994; Chib & Greenberg, 1995). In the ordinal context, we review alternative sets of identiﬁcation constraints and evaluate how each aids or hampers estimation by MCMC methods. We then consider the issue of model comparison by showing how marginal likelihoods and Bayes factors can be computed in the ordinal data setting using the method presented in Chib (1995) and its extensions developed in Chib and Jeliazkov (2001, 2005). This, for instance, allows for the formal comparison of models with different correlation structures, covariates, or link functions. In addition, we describe a simulation-based approach for calculating the effect of covariates on the outcome, which provides interpretability of the estimates despite the nonlinearity of the model and the different identiﬁcation schemes that can be used to identify the parameters. We apply our methods to three problems in economics involving educational attainment, reliance on health care information sources, and exit poll data on voter opinions about the economy, the war in Iraq, and President George W. Bush’s performance in ofﬁce. To illustrate the setting, consider a simple univariate case where yi is a scalar response variable that takes one of the J ordered values, j ¼ 1, . . . , J, and the index i (i ¼ 1, . . . , n) refers to units in the sample (e.g., individuals, families, ﬁrms, etc.). The deﬁning feature of ordinal data is that the outcomes are arranged and measured on a monotone scale – for example, in quantifying survey responses, 1 could be assigned to ‘‘very unhappy’’, 2 to ‘‘not too happy’’, 3 to ‘‘happy’’, and 4 to ‘‘very happy’’; however, the scale is not assumed to be cardinal, so that differences between categories are not directly comparable. In other words, while the scale tells us that 4 implies more happiness than 2, this does not mean that 4 implies twice as much

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happiness as 2, or that the difference in happiness between 1 and 3 is the same as that between 2 and 4. Models for ordinal data address these features of the data by postulating a data-generating process in which the outcomes can be thought of as arising from an underlying latent variable thresholdcrossing framework. In particular, the problem can be motivated by assuming that a continuous latent random variable zi depends on a k-vector of covariates xi through the model zi ¼ x0i b þ ei ;

i ¼ 1; . . . ; n

(1)

and that the outcome yi arises according to yi ¼ j

if

gj1 ozi gj

(2)

where E(eijxi) ¼ 0 and N ¼ g0og1o . . . ogJ1ogJ ¼ N are cutpoint parameters that determine the discretization of the data into the J ordered categories. Given this representation and a cumulative distribution function (cdf) for ei, F(ei), the probability of observing yi ¼ j, conditional on b and c ¼ (g1, . . . , gJ1)u, is given by Prðyi ¼ jjb; cÞ ¼ Prðfgj1 ozi g \ fzi gj gÞ ¼ Prðfgj1 ox0i b þ i g \ fx0i b þ i gj gÞ. Letting A ¼ fgj1 ox0i b þ i g and B ¼ fx0i b þ i gj g, from set theory we know that Pr(A\B) ¼ Pr(A) þ Pr(B) Pr(A[B). Therefore, since PrðAÞ ¼ 1 Fðgj1 x0i bÞ; PrðBÞ ¼ Fðgj x0i bÞ and Pr(A[B) ¼ 1, we obtain that (3) Prðyi ¼ j b; cÞ ¼ Fðgj x0i bÞ Fðgj1 x0i bÞ Given y ¼ (y1, . . . , yn)u, the likelihood function for the model can be written as f ð yjb; cÞ ¼

n Y J Y 1fy ¼jg Fðgj x0i bÞ Fðgj1 x0i bÞ i

(4)

i¼1 j¼1

where 1{yi ¼ j} is the indicator function of the event yi ¼ j, which takes the value 1 if the event is true and 0 otherwise. Various choices for the cdf F( ) are possible (speciﬁc ones will be mentioned below), but practical applications most commonly rely on the Gaussian cdf. For the purposes of illustration, Fig. 1 depicts the probabilities of yi falling in category j as determined by Eq. (3) for a four-category setting. However, both location and scale restrictions are necessary to uniquely identify the parameters of the model. To see this, consider the probabilities in Eq. (3) and let gnj ¼ gj þ c and x0i bn ¼ x0i b þ c for some constant c (note that the latter is always possible since xi is assumed to contain a constant

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f(z) Pr(yi =3) Pr(yi = 2) Pr(yi =4) Pr(yi =1)

γ1

Fig. 1.

γ2

xi′ β

γ3

z

Outcome Probabilities for the Four-Category Case Given the Mean x0i b and the Cutpoints g1, g2, and g3.

term). Then, because gnj x0i bn ¼ gj þ c x0i b c ¼ gj x0i b, it is straightforward to verify that Pr(yi ¼ jjb, c) ¼ Pr(yi ¼ jjb, c). This identiﬁcation problem is usually easily corrected by ﬁxing a cutpoint (in addition to g0 ¼ N and gJ ¼ N) – in particular, letting g1 ¼ 0 removes the possibility for shifting the distribution without changing the probability of observing yi. While in principle it is possible to achieve identiﬁcation by dropping the intercept instead of ﬁxing a cutpoint, the choice of letting g1 ¼ 0 has the practical beneﬁt of facilitating posterior sampling (since simulating b is generally easier than simulating c) and also makes the ordinal probit model theoretically consistent with the binary data probit model when there are only two outcome categories (in the binary probit model yi ¼ 1 if ziW0 and yi ¼ 0 otherwise). For a survey on Bayesian models for ordered categorical data see Liu and Agresti (2005). The upper panel of Fig. 2 shows the above considerations regarding location restrictions; the lower panel of that ﬁgure, however, shows that even if one sets g1 ¼ 0, a second restriction is necessary in order to ﬁx the scale of the latent data that is implied by F( ). Speciﬁcally, the second panel of Fig. 2 shows that in the absence of additional constraints, one can change the scale of F( ) and simultaneously rescale the mean and the remaining free cutpoints without affecting the probabilities for yi, implying lack of likelihood identiﬁcation. This is due to the fact that so far we have only required F( ) to be the cdf of a mean zero distribution, but F n ðgj x0i bÞ Fððgj x0i bÞ=cÞ is another cdf in the same class that, given mean and cutpoint parameters that are appropriately rescaled by a positive constant c, can produce the same probabilities as F( ). The usual approach to achieving identiﬁcation in this case is to ﬁx the variance of e. For example, when e is assumed to be Gaussian, the restriction var(e) ¼ 1 is usually imposed, leading to an ordered probit model whose link function F( ) is assumed to

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f(z)

γ1

γ2 x ′ β i

γ3

γ2 xi′ β γ2*

γ3 xi′ β *

γ1*

γ2*

xi′ β *

γ3*

z

f(z)

γ 1= 0

γ3*

z

Fig. 2. Location and Scale Constraints are Required to Identify the Outcome Probabilities. Note: In the upper panel the same probabilities result when the mean and cutpoints are shifted by the same constant. In the lower panel, even though g1 ¼ 0, the remaining cutpoints and the mean can be rescaled by a multiplicative constant to produce the same probabilities.

be the standard normal cdf F( ). Other choices for the link function F(e) include the logistic cdf (1+ee)1 where the variance of e is given by p2/3, the extreme value cdf ee , that implies var(e) ¼ p2/6, or the t-link model where F( ) is taken to be the standard Student-t cdf with n degrees of freedom implying var(e) ¼ n/(n2). However, in addition to ﬁxing the variance of e, there are other possible ways to identify the scale of the model. The presence of these alternatives makes it possible to explore various approaches to estimation. We examine these methods and report on the implementation and performance of the different MCMC estimation algorithms that they determine. The remainder of this paper is organized as follows. Section 2 discusses a number of identiﬁcation restrictions together with their corresponding estimation methods in the univariate setting. Section 3 presents the MCMC ﬁtting method for multivariate settings. Section 4 is concerned with model comparison. In Section 5, we show how to compute the effects of covariates

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on the outcome probability and discuss several extensions of the methods presented in this paper. Section 6 is devoted to the analysis of data on educational attainment, voter opinions, and health information, while brief concluding remarks are presented in Section 7.

2. ORDINAL DATA MODELS FOR UNIVARIATE OUTCOMES 2.1. Estimation under Traditional Identiﬁcation Assumptions Albert and Chib (1993) showed how the latent variable representation of ordinal data models can be exploited for estimating models for ordered data such as the ordinal probit and Student-t models. Their idea was to focus not on the posterior distribution of the parameters conditioned on the data but rather on the posterior distribution of the latent data z ¼ (z1, . . . , zn)u and the parameters conditioned on the data, which is simpler and more tractable to deal with in the context of MCMC methods. In particular, under the probit model assumption that eiBN(0, 1), and given the priors bBN (b 0, B0) and p(c)p1, the posterior distribution for the latent data and the parameters is given by pðb; c; zyÞ / f ðyjb; c; zÞpðb; c; zÞ ¼ f ðyjb; c; zÞpðzjbÞpðbÞpðcÞ ( ) n Y ¼ f ðyi jzi ; cÞ pðzjbÞpðbÞpðcÞ

(5)

i¼1

where the second line used the decomposition p(b, z, c) ¼ p (zjb) p(b)p(c) that is afforded by prior independence, and the third line used the fact that given the latent zi and c, the observed yi is independent of b because Eq. (2) determines yi given zi and c with probability 1 and that relationship does not involve b. Speciﬁcally, the probability of yi ¼ j given zi and c equals 1 when gj1ozirgj and 0 otherwise, so that f(yijzi, c) ¼ 1 {gj1ozirgj}. Also note that Q p(zjb) can be obtained from Eq. (1) and is given by pðzjbÞ ¼ ni¼1 Nðzi jx0i b; 1Þ. With these considerations, the ‘‘complete data posterior’’ in Eq. (5) involving the latent data and the parameters is given by ( ) n Y 0 1 gj1 ozi gj N zi jxi b; 1 Nðbjb0 ; B0 Þ pðb; c; zjyÞ / i¼1

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Then, upon letting X ¼ (x1, . . . , xn)u, and under the ‘‘traditional’’ identiﬁcation constraints g0 ¼ N, g1 ¼ 0, gJ ¼ N, and var(e) ¼ 1, the posterior distribution of the latent data and the parameters can be sampled by MCMC methods as follows: Algorithm 1 (Albert & Chib, 1993) Sampling in the univariate ordered probit model. 1. For j ¼ 2, . . . , J1, sample gjjzBU (max{zi: yi ¼ j}, min{zi: yi ¼ j+1}), that is, sample gj from a uniform distribution bounded between the maximum zi in category j and the minimum zi in category j+1; 2. For i ¼ 1, . . . , n, sample zi jy; b; c TNðgj1 ;gj Þ ðx0i b; 1Þ, where the support for this truncated normal distribution is determined by the cutpoints gj1 and gj associated with yi ¼ j; ^ 1 b0 þX 0 zÞ1 , ^ BÞ; ^ where B^ ¼ ðB 0 þX 0 XÞ1 and b^ ¼ BðB 3. Sample bjz Nðb; 0 where given z, the full-conditional distribution of b does not depend on y or c. This algorithm forms a basis for the ﬁtting of a wide variety of univariate ordinal models because once the probit case can be worked out, a number of other link functions F( ) can be represented either as mixtures or scale mixtures of normals, including the Student-t and the logistic link functions (Albert & Chib, 1993; Wood & Kohn, 1998; Chen & Dey, 2000). While sampling from the distributions in Algorithm 1 is convenient because they are of known form (i.e., uniform, truncated normal, and normal), one remaining question concerns the sampling of the cutpoints in Step 1 conditioned on the latent data. Cowles (1996) noted the possibility that in some cases, the sampling of the cutpoints conditioned on the latent data can lead to small changes in the cutpoints between successive iterations, especially as more data become available. The resulting high autocorrelation in the MCMC sample for the cutpoints could then also affect the convergence of b. To deal with this problem, Cowles (1996) suggested that it helps to sample the latent data z and the cutpoints c jointly by sampling cBp (cjy, b), marginalized over the latent data and subsequently sampling zBp (zjy, b, c), given c and the remaining parameters and data. Although the resulting distribution of the cutpoints is not of standard form, Cowles employed a sequence of MH steps to sample each gj conditioned on (y, b, gj1, gjþ1). Nandram and Chen (1996) improved upon Cowles (1996) by noting that the cutpoints should be sampled jointly, not one at a time, and that the particular MH proposal density suggested in Cowles (1996) may be difﬁcult to tune. They suggested a reparameterization of the model

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and presented a sampler that allows for joint sampling of the reparameterized cutpoints in a single block and also marginally of the latent data using a Dirichlet proposal density that depends on the previous cutpoints, but does not depend on the other parameters or the latent data. However, Chen and Dey (2000) point out that the Dirichlet density will generally work well when the cell counts are balanced, but may fail to serve as a good proposal density when the category counts are unbalanced. Subsequent work (e.g., Chen & Dey, 2000; Albert & Chib, 2001) built upon these ideas and showed that the cutpoints c can easily be sampled jointly in a single block by well-tailored independence chains, marginally of z, to improve the efﬁciency of the MCMC algorithm. Maintaining the identiﬁcation restriction that var(e) ¼ 1, Albert and Chib (2001) simpliﬁed the sampling of the cutpoints c by transforming them so as to remove the ordering constraint by the one-to-one map dj ¼ lnðgj gj1 Þ; 2 j J 1

(6)

Other transformations have been considered (e.g., in Chen & Dey, 2000), but details of those transformations will be delayed until Section 2.2 below as they will be related to an alternative identiﬁcation of the scale of model. In either case, however, the advantage of working with the transformed quantities d ¼ (d2, . . . , dJ1)u, instead of c ¼ (g2, . . . , gJ1)u, is that the parameters of the tailored proposal density in the MH step for d can be obtained by unconstrained optimization and the prior p(d) can be unrestricted, for example, multivariate normal N(d0, D0). The algorithm is deﬁned as follows. Algorithm 2 (Albert & Chib, 2001) Sampling in the univariate ordered probit model (identiﬁcation through variance restriction). 1. Sample d, zjy, b in one block as follows: (a) Sample djy, b marginally of z by drawing duBq(djy, b) from a ^ D; ^ nÞ; where d^ ¼ Student-t proposal density qðdjy; bÞ ¼ fT ðdjd; ^ arg max f ðyjb; dÞpðdÞ; D is the inverse of the negative Hessian of ^ and n is a degrees of freedom ln{ f(yjb, d)p (d)} evaluated at d, parameter. Given the current value of d and the proposed draw du, return du with probability aMH ðd; d0 Þ ¼ minf1; ½ f ðyjb; d0 Þpðb; d0 Þ= ^ D; ^ D; ^ nÞ=f ðd0 jd; ^ nÞg; otherwise repeat the ð f ðyjb; dÞ pðb; dÞÞ½fT ðdjd; T old value d. (b) Sample zjy, b, c by drawing zi jy; b; c TN ðgj1 ;gj Þ ðx0i b; 1Þ for i ¼ 1, . . . , n, where c is obtained by the one-to-one mapping used to relate c and d.

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^ BÞ, ^ 2. Sample bjz Nðb; ^ 1 b0 þ X 0 zÞ1 . BðB 0

where

B^ ¼ ðB 0 þ X 0 XÞ1

and

b^ ¼

In Step 1 of the above algorithm, the degrees of freedom parameter n is taken to be a low number such as 5 or 10 to ensure that the proposal density has sufﬁciently heavy tails. By grouping the sampling of d and z into a single step, the above two-block algorithm produces a well-mixing Markov chain, whose performance will be illustrated in Section 6. We next consider the issue of using alternative identiﬁcation restrictions in order to ﬁx the scale of the model. Doing so results in a different blocking of the parameters that, except for cases with J ¼ 3 categories, will produce a three-block algorithm, as opposed to the two-block sampler given in Algorithm 2. For this reason, in the univariate setting one can easily estimate ordinal probit models without having to consider these alternatives (except when J ¼ 3), but the restrictions we discuss below are quite useful in multivariate settings. We emphasize that the choice of blocking is not unique and is something that should be determined by the researcher given the speciﬁc context of the model and data under consideration. In practice, it is useful to (i) group parameters that are correlated into one block and sample them jointly, and (ii) group parameters in a way that allows for easy construction of suitable MCMC samplers.

2.2. Estimation under Alternative Identiﬁcation Restrictions As discussed in the Introduction, there are a number of ways to identify the scale of the model. To gain further insight into the identiﬁcation problem, consider the ordinal probit likelihood

n Y J Y gj x0i b gj1 x0i b 1fyi ¼jg F F (7) f ðyjb; gÞ ¼ s s i¼1 j¼1 The standard identiﬁcation restriction commonly used in discrete data models, that is, var(ei) ¼ s2 ¼ 1 (so that eiBN(0, 1)), was used in Section 2.1 to preclude the possibility for arbitrary rescaling of the parameters by some constant c such that Fðcðgj x0i bÞ=ðcsÞÞ ¼ Fððgj x0i bÞ=sÞ: However, it is possible to identify the scale of the model differently. In particular, we can leave var(ei) ¼ s2 as an unrestricted parameter to be estimated, but instead ﬁx another cutpoint in addition to having g0 ¼ N, g1 ¼ 0, and gJ ¼ N in order to determine the scale of the latent data. For instance, one possibility is to let g2 ¼ 1. This restriction precludes the simultaneous rescaling of the numerator

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and denominator in Fððgj x0i bÞ=sÞ because it would violate g2 ¼ 1. In this case, using Eq. (6), one can work with d ¼ (d3, . . . , dJ1)u. Of course, any other cutpoint can be ﬁxed instead of g2, and the ﬁxing can be at any positive constant, not just 1 (e.g., Webb & Forster, 2008) even though 1 is perhaps the most natural metric to use. For example, the reparameterization considered by Nandram and Chen (1996) and Chen and Dey (2000) corresponds to a particular identiﬁcation scheme where gJ1 ¼ 1, thus ﬁxing the last free cutpoint instead of the second one. Under this identiﬁcation restriction, the ordering constraints on the interior cutpoints can be removed by any one of a number of transformations. Chen and Dey (2000) consider the mapping gj ¼ ðgj1 þ edj Þ=ð1 þ edj Þ, which implies that ( ) ðgj gj1 Þ dj ¼ ln ; 2j J2 (8) ð1 gj Þ so that now d ¼ (d2, . . . , dJ2)u. Other transformations of c to an unrestricted and real-valued d are conceivable and could include log-ratios of category bin-widths or trigonometric functions such as arctan and arcsin, but generally any monotone transformation from a compact set to the real line would work. Because these alternative identiﬁcation schemes are isomorphic, the parameters under each identiﬁcation scheme can easily be related to those under another in a one-to-one mapping. For instance, Nandram and Chen (1996) and Chen and Dey (2000) discuss how the parameters under their identiﬁcation scheme relate to those under traditional identiﬁcation using unit variance restrictions. Of course, when there are only three categories (J ¼ 3) and therefore only two cutpoints that separate them, these two cutpoints need not be sampled (since g1 ¼ 0 and g2 ¼ 1) and the different parameterizations that use ﬁxing of a second cutpoint become identical. When JW3, however, the choice of which cutpoint to ﬁx and what mapping to apply can inﬂuence the performance of the MCMC sampler. We provide some evidence on this in Section 6. One should also note that when J ¼ 2, identiﬁcation cannot be achieved by ﬁxing an additional cutpoint and the model does not automatically become identical to the binary data probit model. This is because with only two outcome categories there is no g2 that can be ﬁxed in order to determine the scale of the model, and without restrictions on s2 the model becomes unidentiﬁed. In those cases, one will have to resort to the traditional identiﬁcation restrictions discussed in Section 2 and Algorithm 2. Then, for cases where JZ3 and under a semi-conjugate inverse gamma prior on s2, that is, s2BIG(v0/2, d0/2), the resulting complete data

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posterior is given by ( pðb; d; z; s2 jyÞ /

n Y

1fgj1 ozi gj gN zi jx0i b; s

2

)

i¼1

Nðbjb0 ; B0 ÞNðdjd0 ; D0 ÞIGðs2 jv0 =2; d 0 =2Þ which results in the following three-block sampling algorithm. Algorithm 3 (Chen & Dey, 2000) Sampling in the univariate ordered probit model (identiﬁcation through cutpoint restrictions). 1. Sample d, zjy, b in one block as follows: (a) Sample djy, b marginally of z by drawing duBq(djy, b), with ^ D; ^ nÞ where d^ ¼ arg max f ðyjb; d; s2 ÞpðdÞ and D ^ qðdjy; bÞ ¼ fT ðdjd; 2 is the inverse of the negative Hessian of ln{f(yjb, d, s )p(d)} ^ Given the current value of d and the proposed evaluated at d. draw du, return du with probability aMH ðd; d0 Þ ¼ minf1; ½ f ðyjb; d0 Þ ^ D; ^ D; ^ nÞ=f ðd0 jd; ^ nÞg; otherwise pðb; d0 Þ=ð f ðyjb; dÞpðb; dÞÞ½fT ðdjd; T repeat the old value d. (b) Sample zjy, b by drawing zi jy; b; d TN ðgj1 ;gj Þ ðx0i b; s2 Þ for i ¼ 1, . . . , n, where c is obtained by the one-to-one mapping used to relate c and d. ^ 1 b0 þ ^ BÞ; ^ where B^ ¼ ðB 0 þ X 0 X=s2 Þ1 and b^ ¼ BðB 2. Sample bjz Nðb; 0 0 2 1 X z=s Þ . 3. Sample s2 IG ðv0 þ nÞ=2; ðd 0 þ ðz XbÞ0 ðz XbÞÞ=2 . The above algorithm has been applied in Chen and Dey (2000), and its performance is further illustrated in Section 6. We note, however, that this is a three-block algorithm using the grouping ({d, z}, b, s2), so that generally the ﬁtting of univariate models can be done more efﬁciently using the two-block sampler in Algorithm 2 that was developed under standard identiﬁcation restrictions and used the blocking ({d, z}, b). (We note that even when J ¼ 3 and d need not be sampled, Algorithm 3 is still a three-block algorithm involving (z, b, s2), but it does not involve an MH step for d.) Algorithm 2 is also more useful when it comes to model comparison as it allows for an easier computation of the marginal likelihood, which will be discussed in Section 4. Nonetheless, the ideas behind Algorithm 3 can be quite useful when applied to the multivariate setting as discussed in Section 3 below. In closing, we mention that another approach can be used to identify the scale of the model. We mention this approach only for the sake of completeness and not because we endorse it, as it imposes additional

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restrictions not required by the other identiﬁcation approaches discussed so far. In particular, in Eq. (7) we ﬁrst identiﬁed the model by ﬁxing s2 ¼ 1, leading to the standard version of the ordered probit model. We subsequently noted that arbitrary rescaling can be prevented by ﬁxing an additional cutpoint and suggested that ﬁxing g2 ¼ 1 or gJ1 ¼ 1, in addition to the usual g1 ¼ 0, could be implemented. But by considering Eq. (7), one can easily see that the potential for arbitrary rescaling can be removed by ﬁxing one of the elements of b, say bh ¼ 1. While this formally identiﬁes the likelihood without the need for restricting g2, . . . , gJ1 or s2, this identiﬁcation restriction imposes both a scale restriction and a sign restriction because in reality even if 6 0, we might mistakenly assign a positive effect on bh by ﬁxing it at 1, bh ¼ when its effect could be negative (so bh ¼ 1 would have been appropriate). Moreover, this restriction complicates the analysis when one is interested in performing model comparison tests whereby xh may be removed from the set of covariates (so that bh is no longer part of the model), requiring that normalization be based on a different covariate effect.

3. MULTIVARIATE ORDINAL OUTCOMES We now extend the preceding discussion to the case of multivariate ordinal outcomes. The framework for this analysis follows closely that of Chib and Greenberg (1998) and Chen and Dey (2000), who dealt with multivariate binary and ordinal outcomes, respectively. To introduce the setting, we write the multivariate version of the ordinal probit model using a thresholdcrossing representation for the latent variables zi ¼ X i b þ i where the q-dimensional vector of latent variables zi ¼ (zi1, . . . , ziq)u implies a vector of observed responses yi according to the discretization imposed by the variable-speciﬁc cutpoints, namely yik ¼ j

if

gk; j1 ozik gkj ;

for

i ¼ 1; . . . ; n;

k ¼ 1; . . . ; q

(9)

In the multivariate version of the ordered probit model, the matrix of covariates 0 1 ðxi11 xi1k1 Þ B C B C ð xi21 xi2k2 Þ B C C Xi ¼ B .. B C . B C @ A ð xiq1 xiqkq Þ

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is in the form used in seemingly unrelated regression models, b is a k-dimensional vector of corresponding regression parameters ðwith k ¼ Pq k Þ, and e i is a q-dimensional vector of disturbances where eiBN (0, X) j¼1 j with X being a symmetric positive deﬁnite covariance matrix. It should be noted that in the model, each element of yi can have a different number of categories J ¼ (J1, . . . , Jq)u and its own set of cutpoints ck ¼ ðg1 ; . . . ; gJ k Þ0 for k ¼ 1, . . . , q, that can be collected in c ¼ ðc01 ; . . . ; c0q Þ0 . Also, because X is not required to be diagonal, the model can account for the presence of unobserved factors that can correlate the latent variables in the vector zi. Despite these correlations, however, sampling of the multivariate model can be done as a straightforward extension of the univariate algorithms by using the fact that the conditional distributions of the multivariate Gaussian distribution are also Gaussian with well-known moments. Moreover, the probit model allows for further modeling extensions – for example, Chen and Dey (2000) extend the basic setup to other link functions, such as the class of scale mixtures of multivariate normal densities that is of interest in itself or can be used to approximate other link functions such as the multivariate t-link and logit models (see also O’Brien & Dunson, 2004, who develop a multivariate logit model for ordinal outcomes). For this reason, the multivariate probit model appears to be a desirable model that strikes a balance between ﬂexibility and conceptual simplicity. To identify the model, we use parallel restrictions to those discussed in Section 2, imposed on each of the marginal distributions for the individual zik, k ¼ 1, . . . , q. In particular, following the layout of Section 2, we ﬁrst present an algorithm in which the scale of the individual (marginal) latent processes is ﬁxed by imposing a unit error variance, leading to a matrix X that is in correlation form as in the multivariate probit model for binary data in Chib and Greenberg (1998). Subsequently, we discuss an alternative algorithm, where each of the marginal distributions is scaled by ﬁxing two cutpoints (e.g., either gk1 ¼ 0 and gk2 ¼ 1, or gk1 ¼ 0 and gk,J1 ¼ 1, k ¼ 1, . . . , q), so that the covariance matrix X is free and can be sampled from a known full-conditional distribution. When X is in correlation form, the sampling of the correlations is nonstandard and we approach this task by relying on the versatility of the ARMH algorithm. To introduce that algorithm, let h be a parameter vector whose density, p(h), is the target density of interest, but is possibly known only up to a normalizing constant and is not easy to simulate. Let h(h) denote a source (or proposal) density for the ARMH algorithm and let the constant c deﬁne the region of domination D ¼ fh : pðhÞ chðhÞg

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which is a subset of the support Y of the target density (because the domination condition need not be satisﬁed for all hAY, the source density h(h) is often called a pseudo-dominating density). Commonly, the choice of a pseudo-dominating density is determined by tailoring. For instance, h(h) can be given by a multivariate-t density h(h) ¼ fT (hjl, tV, n) with n degrees of freedom, mean l given by the maximum of the target density p(h), and scale matrix tV, where V is the inverse of the negative Hessian of ln p(h) evaluated at l, and t is a parameter allowing for the possibility of additional tuning. Let Dc be the complement of D, and suppose that the current state of the chain is h. Then the ARMH algorithm proceeds as follows. Algorithm 4 The ARMH algorithm. 1. AR step: Generate a draw huBh(h); accept hu with probability aAR ðh0 Þ ¼ minf1; pðh0 Þ=ðchðh0 jyÞÞg. Continue the process until a draw hu has been accepted. 2. MH step: Given the current value h and the proposed value hu: (a) If h 2 D, set aMH(h, hu) ¼ 1; (b) If h 2 Dc and h0 2 D, set aMH(h, hu) ¼ chðhÞ=pðhÞ; (c) If h 2 Dc and h0 2 Dc , set aMH(h, hu) ¼ minf1; ðpðh0 ÞhðhÞÞ=ðpðhÞhðh0 ÞÞg. Return hu with probability aMH(h, hu); otherwise return h. The ARMH algorithm is an MCMC sampling procedure which nests both the accept–reject and MH algorithms when Dc and D become empty sets, respectively. But in the intermediate case when both D and Dc are nonempty, ARMH has several attractive features that make it a useful choice for our setting. First, the algorithm is well suited to problems that do not require conjugacy and result in nonstandard full-conditional densities, which is the case for the elements of X when it is in correlation form. Second, ARMH can be less demanding and works quite well even if the proposal density h(h) is only a rough approximation of the target density (e.g., Chib & Jeliazkov, 2005). This is particularly useful in our setting because previous research suggests that standard asymptotic approximating densities can be only rough approximations when sample sizes are small (e.g., Zellner & Rossi, 1984). Third, ARMH can produce draws that are closer to i.i.d. than those from a similarly constructed MH simulator, but without requiring global domination that is needed for the simple accept– reject algorithm. Fourth, in sampling covariance or correlation matrices, only draws that satisfy positive deﬁniteness pass through the AR step of the algorithm, thus improving the performance of the MH step. Finally, the building blocks of the ARMH algorithm provide a straightforward way to

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estimate the marginal likelihood (e.g., Chib & Jeliazkov, 2005), which will be discussed in Section 4. We are now ready to proceed with estimation of the multivariate ordered probit under traditional identiﬁcation restrictions, where the scale is ﬁxed by requiring X to be in correlation form. We begin by considering the complete data posterior p(b, d, q, zjy), where q is the vector of unique correlations in X. Assuming the prior qBN(q0, R0) 1{qAS}, where S is the set of correlations that produce a positive deﬁnite matrix X with ones on the main diagonal, we have ( " # ) q n Y Y 1fgk; j1 ozik gkj g Nðzi jX i b; XÞ pðb; d; q; zjyÞ / i¼1 k¼1

Nðbjb0 ; B0 ÞNðdjd0 ; D0 ÞNðqjr0 ; R0 Þ1fq 2 Sg where the index j in the indicator functions above is determined by the value of yik according to Eq. (9). The above posterior distribution gives rise to the following MCMC estimation algorithm, where as a matter of notation, we will use ‘‘\k’’ to represent all elements in a set except the kth one (e.g., if hk is the kth block in the vector h ¼ ðh0 1 ; . . . ; h0 p Þ; then hnk ¼ ðh0 1 ; . . . ; h0 k1 ; h0 kþ1 ; . . . ; hp Þ0 ). Algorithm 5 Sampling in the multivariate ordered probit model (X is in correlation form). 1. For each equation k ¼ 1, . . . , q, sample the latent vector zk and vector of transformed cutpoints dk from dk, zkjy, b, q, z\k in one block as follows: (a) Sample dkjy, b, q, z\k marginally of zk by drawing a value ^ k ; nÞ with d0 k qðdk jy; b; q; znk Þ, where qðdk jy; b; q; znk Þ ¼ fT ðdk jd^ k ; D ^ k is the inverse of the d^ k ¼ arg max f ðyk jb; q; d; znk Þpðdk jdnk Þ and D negative Hessian of ln{f(ykjb, q, d, z\k)p(dkjd\k)} evaluated at d^ k . Given the current value of d and the proposed draw du, return du with probability ( ) ^ nÞ f ðyk jb; q; d0 k ; znk Þpðb; q; d0 k jdnk Þ f T ðdk jd^ k ; D; 0 aMH ðdk ; d k Þ ¼ min 1; ^ nÞ f ðyk jb; q; d; znk Þpðb; q; dk jdnk Þ f T ðd0 k jd^ k ; D; and otherwise repeat the old value d. (b) Sample zkjy, b, q, dk, z\k by drawing zik jy; b; q; c; znk TN ðgj1 ;gj Þ ðmkjnk ; s2kjnk Þ for i ¼ 1, . . . , n, where mk|\k and s2kjnk are the usual conditional mean and variance for a Gaussian distribution.

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^ BÞ, ^ ^ 2. Sample bjzP Nðb; b^ and given by b^ ¼ where 1 B Pare 1 1 n n 1 1 0 0 ^ b þ X X z and B ¼ B þ X . B^ B 1 i i iX X i 0 0 0 i¼1 i¼1 3. Sample qjz, b by ARMH (Algorithm 4) with proposal density ^ nÞ, where q^ and R ^ are approximations to the ^ tR; hðqjz; bÞ ¼ fT ðqjq; maximizer and inverse of the negative Hessian of lnff ðzjb; rÞpðrÞg ¼ Qn Nðz jX b; XÞNðqjq ; R Þ1fq 2 Sg ; respectively, and t is a ln i i 0 0 i¼1 tuning constant: ^ ¼ ðR1 þ C 1 Þ1 ^ 1 q0 þ C 1 cÞ and R (a) as a ﬁrst step, try q^ ¼ RðR 0 0 where c is the vector of unique elements of corr(zi Xib) and C is the BHHH outer product of gradients estimate of the modal dispersion matrix evaluated at c; (b) if the gradient at q^ above is far from zero (so that q^ is not a good approximation), ﬁne-tune q^ by further optimization and proceed with the ARMH algorithm. Computing the quantities in Step 3(a) of Algorithm 5 is particularly easy and does not require optimization; moreover, it is very fast because the sampling is conditional on z so that the computations are comparable to those in a continuous data model. Importantly, when ﬁne-tuning is required, it can generally be accomplished in a few steps by quasi-Newton methods. Alternatively, the proposal density in the ARMH step can be initialized with ^ being the BHHH q^ being the correlations from the last draw of X and R ^ estimate of modal dispersion matrix at q. Because of the standard way of identifying the parameters, the multivariate ordinal probit model simpliﬁes to the multivariate probit model for binary data that was analyzed in Chib and Greenberg (1998) when there are only two choice categories. Moreover, with this algorithm it is possible to ﬁt models with restricted covariance matrices that may involve off-diagonal zeros or various Toeplitz structures in X. The performance of this algorithm is demonstrated in Section 6. We now turn attention to the second identiﬁcation scheme that relies on ﬁxing two cutpoints for each outcome such as gk1 ¼ 0 and either gk2 ¼ 1 or gk,J1 ¼ 1 for k ¼ 1, . . . , q. As discussed in the univariate case, doing so frees up the variances of the latent variables, thus removing the requirement that the diagonal elements of X must be ones. The beneﬁt of doing so is that when X is unrestricted, we can use the usual semi-conjugate inverse Wishart prior on X, that is XBIW(v0, W0), which produces the a complete data posterior that is given by ( " # ) q n Y Y 1fgk; j1 ozik gkj g Nðzi jX i b; XÞ pðb; c; X; zjyÞ / i¼1 k¼1

Nðbjb0 ; B0 ÞNðdjd0 ; D0 ÞIWðXjv0 ; W 0 Þ

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This posterior distribution gives rise to the following MCMC estimation algorithm (note that because a second cutpoint is ﬁxed for each response variable, the vectors dk for k ¼ 1, . . . , q, have one less element than the corresponding vectors in Algorithm 5). Algorithm 6 (Chen & Dey, 2000) Sampling in the multivariate ordered probit model (identiﬁcation through cutpoint restrictions). 1. For k ¼ 1, . . . , q, sample dk, zkjy, b, z\k in one block as follows: (a) Sample dkjy, b, X, z\k marginally of z\k by drawing d0 k qðdk jy; b; ^ k ; nÞ with d^ k ¼ X; znk Þ, where qðdk jy; b; X; znk Þ ¼ fT ðdk jd^ k ; D ^ arg max f ðyk jb; X; d; znk Þpðdk jdnk Þ and Dk is the inverse of the negative Hessian of ln{f(ykjb, X, d, z\k)(dkjd\k)} evaluated at d^ k . Given the current value d and the proposed draw du, return du with probability (

^ nÞ f ðy jb; X; d0 k ;znk Þpðb; X; d0 k jdnk Þ f T ðdk jd^ k ; D; aMH ðdk ; d k Þ ¼ min 1; k 0 ^ nÞ f ðyk jb; X; d;znk Þpðb; X; dk jdnk Þ f T ðd k jd^ k ; D;

)

0

and otherwise repeat the old value d. (b) Sample zkjy, b, X, dk, z\k by drawing zik jy; b; X; c; znk TN ðgj1 ;gj Þ ðmkjnk ; s2kjnk Þ for i ¼ 1, . . . , n, where mkj\k and s2kjnk are the usual conditional mean and variance for a Gaussian distribution. ^ BÞ, ^ where b^ and B^ are given by b^ ¼ BðB ^ 1 b0 þ 2. P Sample bjz Nðb; 0 Pn n 1 1 1 1 0 1 0 ^ and B ¼ ðB0 þ i¼1 X i X X i Þ . i¼1 X i X zi Þ 3. Sample XBIW(v0+n, W0+(zXb)u(zXb)). The above algorithm makes the conditional sampling of X particularly attractive since the draws are obtained from a well-known full-conditional distribution and sampling is fast and easy to implement. Another beneﬁt of this algorithm is that no cutpoints need to be sampled for any response variable that has three categories. Unfortunately, the identiﬁcation restriction cannot be applied (a) when one or more elements in yi are binary and there is only one cutpoint between the two categories so that unit variance is required for identiﬁcation; (b) when X must be restricted in any other way (e.g., when a block of off-diagonal elements is restricted to zero); (c) when different subsets of elements of X are updated by different subsamples of data (e.g., in the case of missing or incidentally truncated outcomes); or (d) when nonconjugate priors are used to model the elements of the matrix X. In these cases, estimation can proceed under traditional identiﬁcation assumptions with X being a correlation matrix.

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4. MODEL COMPARISON A central issue in the analysis of statistical data is model formulation, since the appropriate speciﬁcation is rarely known and is subject to uncertainty. Among other considerations, the uncertainty may be due to the problem of variable selection (i.e., the speciﬁc covariates to be included in the model) or perhaps due to the functional speciﬁcation through which the covariates affect the probability of the outcome. In general, given the data y, interest centers upon a collection of models fM1 ; . . . ; ML g representing competing hypotheses about y. Each model Ml is characterized by a model-speciﬁc parameter vector hl and sampling density f ðyjMl ; hl Þ. Bayesian model selection proceeds by comparison of the models in {Ml } through their posterior odds ratio, which for any two models Mi and Mj is written as PrðMi jyÞ PrðMi Þ mðyjMi Þ ¼ PrðMj jyÞ PrðMj Þ mðyjMj Þ

(10)

R where mðyjMl Þ ¼ f ðyjMl ; hl Þpl ðhl jMl Þdhl is the marginal likelihood of Ml . The ﬁrst fraction on the right-hand side of Eq. (10) is known as the prior odds and the second as the Bayes factor. We show that the question of calculating the marginal likelihood for ordinal data models under each of the identiﬁcation schemes discussed above can be managed through straightforward application of methods that are built upon the structure of the sampling algorithms. Chib (1995) provides a method based on the recognition that the marginal likelihood can be re-expressed as mðyjMl Þ ¼

f ðyjMl ; hl Þpðhl jMl Þ pðhl jy; Ml Þ

(11)

which holds for any point hl, so that calculation of the marginal likelihood is reduced to ﬁnding an estimate of the posterior ordinate pðhnl jy; Ml Þ at a single point hnl , given that usually f ðyjMl ; hnl Þ and pðhnl jMl Þ for the numerator of Eq. (11) are either available directly or by an alternative calculation. In practice, the point hnl is often taken as the posterior mean or mode, which tends to minimize the estimation variability. In the following, we suppress the model index for notational convenience. In the context of ordinal probit models, the parameter vector h will consist of the regression coefﬁcients b, the cutpoint transformations d, and possibly s2 or X (depending on the identiﬁcation scheme and the dimensionality of yi). To keep the discussion general, let h be split into B

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components, or blocks, that emerge in constructing the MCMC sampler under the chosen identiﬁcation scheme, so that h ¼ (hl, . . . , hB). Let wni ¼ ðhn1 ; . . . ; hni Þ denote the blocks up to i, ﬁxed at their values in h, and let ci+1 ¼ (hi+1, . . . , hB) denote the blocks beyond i. Then, by the law of total probability, we have pðhn1 ; . . . ; hnB jyÞ ¼

B Y i¼1

pðhni jy; hn1 ; . . . ; hni1 Þ ¼

B Y

pðhni jy; wni1 Þ

i¼1

When the full-conditional densities are known, each ordinate pðhni jy; wni1 Þ can P be estimated by Rao-Blackwellization as pðhni jy; wni1 Þ J 1 Jj¼1 pðhni jy; wni1 ; wiþ1;ðjÞ Þ, where ci;ðjÞ pðwi jy; wni1 Þ; j ¼ 1; . . . ; J, come from a reduced run, where sampling is only over wi, with the blocks wni1 being held ﬁxed. The ordinate pðhn1 jyÞ for the ﬁrst block of parameters h1 is estimated with draws hBp(hjy) from the main MCMC run, while the ordinate pðhnB jy; wnB1 Þ is available directly. When one or more of the full-conditional densities are not of a standard form and sampling requires the MH algorithm, Chib and Jeliazkov (2001) use the local reversibility of the MH chain to show that pðhni jy; wni1 Þ ¼

E 1 faMH ðhi ; hni jy; wni1 ; wiþ1 Þqðhi ; hni jy; wni1 ; wiþ1 Þg E 2 faMH ðhni ; hi jy; wni1 ; wiþ1 Þg

(12)

where E1 is the expectation with respect to conditional posterior pðwi jy; wni1 Þ and E2 that with respect to pðwiþ1 jy; wni Þqðhni ; hi jy; wni1 ; wiþ1 Þ. In the preceding, qðh; h0 jyÞ denotes the candidate generating density of the MH chain for moving from the current value h to a proposed value hu, and aMH ðhi ; h0 i jy; wni1 ; wiþ1 Þ denotes the MH probability of move from h to hu. For blocks that are sampled using the multi-block version of the ARMH algorithm, Chib and Jeliazkov (2005) show that pðhni jy; wni1 Þ ¼

E 1 faMH ðhi ; hni jy; wni1 ; wiþ1 ÞaAR ðhni jy; wni1 ; wiþ1 Þhðhni jy; wni1 ; wiþ1 Þg E 2 faMH ðhni ; hi jy; wni1 ; wiþ1 ÞaAR ðhi jy; wni1 ; wiþ1 Þg (13)

where, similarly to above, E1 is the expectation with respect to pðwi jy; wni1 Þ and E2 that with respect to pðwiþ1 jy; wni Þhðhi jy; wni1 ; wiþ1 Þ with h( ) being the proposal density in the AR step of the ARMH algorithm. Each of these expectations can be computed from the output of appropriately chosen reduced runs, where wni1 is kept ﬁxed. Methods for evaluating the

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variability of the ordinate estimates in each of the above cases are presented in Chib (1995) and Chib and Jeliazkov (2001, 2005). Estimation of the marginal likelihoods for univariate ordinal data models is quite straightforward, regardless of which identiﬁcation scheme is used. In particular, when the model is identiﬁed by assuming var(e) ¼ 1, then the marginal likelihood can be estimated by using the posterior decomposition pðbn ; dn jyÞ ¼ pðbn jyÞpðdn jy; bn Þ P (g) n ðgÞ where pðbn jyÞ G1 G g¼1 pðb jy; z Þ, with p(b jy, z ) being the fullconditional density for sampling b in Step 2 of Algorithm 2 and z(g) are draws from the main MCMC run, so estimation of that ordinate uses random draws that are already available. The ordinate p(djy, b) is estimated by P ðgÞ n n n n G1 G g¼1 aMH ðd ; d jy; b Þqðd jy; b Þ n n pðd jy; b Þ P n ðhÞ n G1 G h¼1 aMH ðd ; d jy; b Þ where the numerator draws d(g), g ¼ 1, . . . , G, come from p(djy, b) and the denominator draws d(h), h ¼ 1, . . . , G, are draws from q(djy, b). We note that the latter ordinate is particularly easy to obtain, because b is already ﬁxed at b and z is not involved in the sampling step for d, so that maximization to determine the moments of q(djy, b) need only be performed once. Moreover, because the sampling of d uses an independence proposal density q(djy, b) that does not depend on the current value of d, the draws d(h) in the denominator quantity determine the numerator draws d(g) once they go through the MH step of the ARMH algorithm and are either accepted or rejected. Therefore, estimation of p(djy, b) is done with draws that are obtained concurrently in the same MCMC reduced run. When univariate models are identiﬁed by ﬁxing a second cutpoint but allowing var(e) ¼ s2 to be a free parameter to be estimated, the marginal likelihood can be estimated by using the decomposition pðbn ; s2n ; dn jyÞ ¼ pðbn jyÞpðs2n jy; bn Þpðdn jy; s2n ; bn Þ where the ﬁrst and last ordinates are obtained similarly to the preceding discussion, and thePsecond ordinate p(s2jy, b) is estimated as an average (g) n ðgÞ 2n pðs2n jy; bn Þ G1 G taken from sampling g¼1 pðs jy; b ; z Þ with draws z the distribution p(z, djy, b ) in a reduced run given b. For this reason, estimation of the marginal likelihood under the alternative identiﬁcation restrictions will be more cumbersome except, of course, in cases when there

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are only J ¼ 3 categories so that no cutpoints need to be sampled and this identiﬁcation scheme may be easy to implement. Turning attention to multivariate models, we note that estimation of the marginal likelihood follows by straightforward extension of the above methods. Speciﬁcally, estimation of the posterior ordinate can be done using the decomposition pðbn ; Xn ; dn jyÞ ¼ pðbn jyÞpðXn jy; bn Þpðdn jy; Xn ; bn Þ Similarly to the previously mentioned cases, estimation of p(bjy) is done by averaging the full-conditional density with draws {z(g), X(g)}Bp(z, Xjy) from the main MCMC run pðbn jyÞ G1

G X p bn jy; zðgÞ ; XðgÞ g¼1

The next ordinate, p(Xjy, b), can be estimated either by pðXn jy; bn Þ G1

G X p Xn jy; bn ; zðgÞ g¼1

if identiﬁcation is achieved by ﬁxing two cutpoints for each latent series {zik}, k ¼ 1, . . . , q, so that the full-conditional density of X is inverse Wishart. However, if one instead chooses to pursue estimation under the traditional identiﬁcation assumption that X is in correlation form with correlations given by q, then estimation of that ordinate can be done by adapting Eq. (13), so that E 1 faMH ðq; qn jy; bn ; zÞaAR ðqn jy; q; zÞhðqn jy; bn ; zÞg E 2 faMH ðqn ; qjy; bn ; zÞaAR ðqjy; bn ; zÞg Q The last ordinate p(djy, X, b) can be decomposed as qk¼1 pðdnk jy; Xn ; bn ; fdni : iokgÞ, where each term is estimated in a reduced run by Eq. (12) holding all preceding blocks ﬁxed. While in the univariate case the quantities in the numerator of Eq. (11) are available directly, implementation of these methods in the multivariate case requires the likelihood ordinate f(yjX, b, d) in Eq. (11), which we obtain by the Geweke, Hajivassiliou, and Keane (GHK) method (Geweke, 1991; Bo¨rsch-Supan & Hajivassiliou, 1993; Keane, 1994; Train, 2003). In addition, when the normalizing constant of any of the priors is needed for determining the prior ordinate in Eq. (11), as may be the case with p(q)pN(qjr0, R0) 1{qAS} when q is multivariate, that normalizing constant can be evaluated pðXn jy; bn Þ ¼ pðqn jy; bn Þ ¼

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by simulation, for example, by drawing qBN(qjr0, R0) and evaluating the frequency with which qAS is satisﬁed (e.g., Chib & Greenberg, 1998).

5. ADDITIONAL CONSIDERATIONS 5.1. Covariate Effects In the preceding sections, we have presented alternative algorithms for estimating univariate and multivariate ordinal data models. However, interpretation of the resulting parameter estimates is complicated by the nonlinear and nonmonotonic dependence of the response probability on the covariates and the model parameters. To see the possibility for nonmonotonicity, consider Fig. 1 once again. In that ﬁgure, one can see that given the cutpoints, if one increases the mean x0i b, the probability of the ﬁrst category, Pr(yi ¼ 1), will decrease and that of the last category, Pr(yi ¼ J), will increase. However, those are the only two categories for which the effect of a change in x0i b on the probability of observing that response is monotonic. It is easy to see that for the case given in Fig. 1, decreasing x0i b will actually initially increase Pr(yi ¼ 2). That probability will reach a maximum when x0i b is decreased to the midpoint between g1 and g2, while any further decrease in x0i b will cause Pr(yi ¼ 2) to fall. Due to the nonlinearity, nonmonotonicity, and the alternative identiﬁcation schemes that can be applied in this setting, we now turn to the question of evaluating the effect of a given covariate xj on the probability of observing yi. This is important for understanding the model, for determining the impact of a change in one or more of the covariates, and also for evaluating the plausibility of particular covariate values in setting priors for the model parameters. Because of the nonlinearity of the model, the effect of a change in a covariate depends on all other covariates and model parameters, as well as on the identiﬁcation restrictions used in estimation. The impact can be quite complex and can be calculated either marginalized over the remaining covariates and the parameters or conditional on some of them, for example, if we are interested in inference conditional on a particular covariate such as gender, race, or geographical location. Given the speciﬁc context, one may consider various scenarios of interest – examples of economic policy interventions may include increasing or decreasing income or taxes by some percentage, requiring an additional year of education, etc. These interventions will affect the probability of response for any one of the ordered categories, but as argued above that effect can be

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not only of unknown magnitude but also of unknown sign, for the intermediate categories. To illustrate the main ideas, consider the univariate ordinal model zi ¼ x0i b þ i ;

and

yi ¼ j

if

gj1 ozi gj

where we are interested in the effect of a particular x, say x1, on the Pr(yi ¼ j) for some 1rjrJ. Splitting x0it and b accordingly, we can re-write the above model as zi ¼ x1i b1 þ x02i b2 þ i The covariate effect can then be analyzed from a predictive perspective, similarly to thinking about inferences for a hypothetical new individual i. For speciﬁcity, suppose that one is interested in the average difference in the implied probabilities between the case when x1i is set to the value xy1i and the case when x1i is set to the value xz1i . Given the values of the other covariates and those of the model parameters h (which, in addition to b and c, can also include s2 depending on the identiﬁcation restrictions), one can obtain the probabilities Prðyi ¼ jjxy1i ; x2i ; hÞ and Prðyi ¼ jjxz1i ; x2i ; hÞ, which are available analytically. If one is interested in the distribution of the difference fPrðyi ¼ jjxy1i Þ Prðyi ¼ jjxz1i Þg marginalized over {x2i} and h given the data y ¼ (y1, . . . , yn)u, a practical procedure is to marginalize out the covariates using their empirical distribution, while the parameters are integrated out with respect to their posterior distribution. Formally, the goal is to obtain a sample of draws from the distribution n

o Z n y z Pr yi ¼ jjxy1i ;x2i ;h Pr yi ¼ jjx1i Pr yi ¼ jjx1i ¼

o Pr yi ¼ jjxz1i ; x2i ; h pðx2i ÞpðhjyÞdx2i dh A sample from the above predictive distribution can be obtained by the method of composition applied in the following way. Randomly draw an individual and extract the covariate values. Sample a value for h from the posterior and evaluate fPrðyi ¼ jjxy1i ; x2i ; hÞ Prðyi ¼ jjxz1i ; x2i ; hÞg. Repeat this for other individuals and other draws from the posterior distribution to obtain draws from the predictive distribution. The mean of that distribution gives the expected difference in the computed pointwise probabilities as xy1i is changed to xz1i , but other quantities such as quantiles and dispersion measures are also easy to obtain given the draws from the predictive distribution.

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The above approach can similarly be applied conditionally upon, instead of marginally of, certain variables (such as gender or race) that might determine a particular subsample of interest, in which case the above procedures are applied only to observations in the subsample. Importantly, the procedures can be applied to multivariate data as well because one only has to consider the marginal distribution for the response of interest. Finally, we note that these techniques can be useful in setting priors on the parameters, where one can calibrate the parameters in the prior distributions by performing the above simulations with draws from the prior (instead of the posterior) in order to see whether speciﬁc settings for the hyperparameters produce plausible outcome probabilities and covariate effects.

5.2. Extensions We now brieﬂy turn attention to some straightforward extensions of the techniques discussed in this paper to other settings. One such extension concerns the analysis of count data because in many cases count data that take a limited number of values can also be viewed and analyzed similarly to ordinal data. For example, in studying a conﬂict one may be interested in the number of days in a week that are characterized by social unrest or violence, while in transportation economics one may have an interest in the number of vehicles a household owns and uses (Goldberg, 1998; West, 2002). Fang (2008) uses data from the National Household Transportation Survey to estimate a Bayesian multivariate response system with both discrete (number of vehicles) and censored (miles driven) responses. In labor economics one may be interested in the number of children in a family – for example, Borg (1989) uses data from the Korean Institute for Family Planning to study the income–fertility relationship using an ordered probit model. In these cases, in order to adapt the ordinal structure to the analysis of count data, there has to be a category that exhausts the set of possible outcomes and is the complement of the categories listed. For this reason, there is usually a response category such as ‘‘J or more outcomes’’ – for example, a family can own 1, 2, or ‘‘3 or more’’ cars. When the inclusion of such a remainder category is sensible, ordinal data models can be quite useful in the analysis of count data. This is because they can produce more ﬂexible outcome probabilities over the chosen range of outcomes than those given by Poisson models since ordinal models are not restricted by the equidispersion property of the Poisson distribution. In multivariate settings,

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ordinal models can also produce both positive and negative correlation between the responses, whereas the multivariate Poisson model can only accommodate positive correlations; in addition, ordinal models can accommodate both over- and under-dispersion, whereas mixed models such as the Poisson-lognormal model, while capable of capturing negative correlations, can only accommodate overdispersion. Several other extensions are possible. For instance, the techniques discussed here can be applied in the analysis of longitudinal (panel) data by merging the algorithms provided above with those in Chib and Carlin (1999) for sampling the individual-speciﬁc effects. A similar sampler in the context of binary panel outcomes is presented in Chib and Jeliazkov (2006), who consider a semi-parametric model with an unknown covariate function. For panel data settings, Algorithm 5 provides an approach for dealing with correlated errors which can also accommodate, in a fairly straightforward fashion, commonly used intertemporal correlation structures, such as exponentially correlated errors, where X[t, s] ¼ exp {–ajtsjr} for scalars a and r (e.g., Diggle & Hutchinson, 1989) or various other Toeplitz-type correlation matrices. Yet other extensions can be pursued in the latent variable step; for example, as mentioned earlier, the methods can be adapted to models with mixture of normals or scale mixture of normals link functions (Albert & Chib, 1993; Wood & Kohn, 1998; Chen & Dey, 2000). A nonparametric Bayesian model based on Dirichlet process priors is presented in Kottas, Mu¨ller, and Quintana (2005).

6. APPLICATIONS Ordinal data outcomes arise frequently in survey data, where respondents may be asked to evaluate a particular issue on an ordinal scale (such as whether they disagree, agree, or strongly agree), as is common in the subjective well-being literature (Duch, Palmer, & Anderson, 2000; McBride, 2001; Di Tella, MacCulloch, & Oswald, 2003; Luechinger, Stutzer, & Winkelmann, 2006). Ordinal outcomes also result when the dependent variable is naturally classiﬁed into meaningful ordered categories (e.g., by thresholds in cost or income), such as working part-time, full-time, or overtime (Kahn & Lang, 1996; Olson, 1998). Bayesian models involving an endogenous ordinal variable have recently been considered in Li and Tobias (2006) and Herriges, Kling, Liu, and Tobias (2007). In this section, we consider several problems that are of interest in economics and the broader social sciences, and show how the techniques

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discussed in the preceding sections can be applied in practice. For the univariate case, we consider the widely studied topic of educational attainment using data from the National Longitudinal Survey of Youth (NLSY79). Subsequently, we consider two multivariate applications involving survey data on voter opinions and health information. In each application, we estimate the models under the two identiﬁcation approaches discussed previously – either by ﬁxing two cutpoints or by imposing unit variances – and illustrate and compare the performance of the alternative estimation algorithms by the inefﬁciency factors for the sampled parameters. The inefﬁciency factors are calculated as

L X Ll rk ðlÞ 1þ2 L l¼1 where rk(l ) is the sample autocorrelation for the kth parameter at lag l, and L is chosen at values where the autocorrelations taper off. The inefﬁciency factors approximate the ratio of the numerical variance of the posterior mean from the MCMC chain relative to that from hypothetical i.i.d. draws. The data sets used in our applications can be downloaded from http://www.economics.uci.edu/Bivan/.

6.1. Educational Attainment Educational attainment has been the subject of a large literature that is relevant to researchers and policy makers alike. It is also well suited for empirical study involving ordinal models because the dependent variable is naturally categorized by measurable thresholds of educational attainment such as the completion of high school or college. In our ﬁrst application the dependent variable, level of education, is measured in four categories: (i) less than a high school education; (ii) high school degree; (iii) some college or associate’s degree; and (iv) college or graduate degree. In order to facilitate comparability of our results with other research, we estimate a model of educational attainment using the NLSY79. Previous research in the area has been abundant. For example, Ermisch and Francesconi (2001) use the British Household Panel Study to estimate the inﬂuence of parental education on child educational attainment, while Dearden, Ferri, and Meghir (2002) use the British National Child Development Survey to measure the inﬂuence of school quality on educational attainment. Cameron and Heckman (2001) use a dynamic

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discrete data model and data from the NLSY to measure the inﬂuence of parental income on children’s educational attainment. A large literature attempts to estimate the returns to education using a number of methods ranging from explicit proxies for ability, twins studies, ﬁxed effects, and arguably exogenous inﬂuences on educational attainment. Bayesian studies of the return to schooling include Li and Tobias (2006, 2007), where the ordered probit analysis of the factors that inﬂuence educational attainment is embedded into a larger model that studies the subsequent returns to such educational attainment. A number of studies have analyzed the binary outcome of whether a student graduates from high school (Astone & McLanahan, 1991; Haveman, Wolfe, & Spaulding, 1991; Ribar, 1994; Wilson, 2001). However, when one is interested in more than just those individuals on the verge of graduating from high school, ordinal data models offer a natural generalization to a choice that is truly among a number of distinct levels of education. As previously noted in Section 5, covariate effects can be rather general in ordinal data models allowing for the theoretically likely possibility that factors such as a student’s parental education or income can have varying impacts on educational attainment across categories. In 1979, the NLSY began annual interviews with over 12,000 youths on a battery of demographic questions. Using these data, we estimate the effect of family background, individual, and school variables on educational attainment. The NLSY speciﬁcally asks whether the respondent has obtained various education degrees, but this information can also be inferred through the years of schooling variable. For this application, we restrict our sample to those cohorts that were of ages 14–17 in 1979 for whom a family income variable can be constructed. To create the family income variable, we average family income over age 16 and 17, when available. The income measure is given in thousands of 1980 dollars. We also restrict our sample by availability of other relevant variables in the data set. Additionally, we exclude disabled individuals and those who report more than 11 years of education at age 15. The resulting sample consists of 3,923 individuals. The data set includes variables on an individual’s family at the age of 14, including the highest grade completed by their father and mother, whether the mother worked, family income (stabilized by the square root transformation), and whether the youth lived in an urban area or in the South at the age of 14. We also include the individual’s gender and race. To control for age cohort affects, we include dummy variables to indicate an individual’s age in 1979.

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The results of our analysis are presented in Table 1. The signs of the coefﬁcients presented in the table are consistent with what is often found in the literature. Parental education and income have positive effects on educational attainment and so does the labor force participation of the mother. A mother’s work force participation could be seen as detrimental due to lack of parental supervision or could be viewed as providing a positive role model for her children to follow; the sign of the coefﬁcient indicates the latter case is a viable possibility. Conditionally on the remaining covariates, we also see that blacks and individuals from the South have higher educational attainment. To gauge the magnitudes of some of the more interesting covariate effects in this example, one can use the framework presented in Section 5.1. To illustrate these calculations, we computed the effect of an increase in family income of $1,000 on educational outcomes. For the overall sample, the effect of such increase in family income is to lower the probability of dropping out of high school by approximately 0.0050, lower the probability of only obtaining a high school degree by 0.0006, but increase the probability of having some college or associate’s degree by 0.0020 and increase the probability of getting a college or graduate degree by 0.0036. For the subsample of females, the effects of an income increase on the four outcome probabilities were comparable at Table 1. Parameter b

d

Posterior Means and Standard Deviations for the Parameters in the Educational Attainment Application. Covariate

Mean

SD

Intercept Family income (square root) Mother’s education Father’s education Mother worked Female Black Urban South Age cohort 2 Age cohort 3 Age cohort 4

1.34 0.14 0.05 0.07 0.03 0.16 0.15 0.05 0.05 0.03 0.00 0.23 0.08 –0.28

0.09 0.01 0.01 0.01 0.04 0.04 0.04 0.04 0.04 0.05 0.06 0.06 0.02 0.03

Identiﬁcation is achieved through variance restriction and estimation is performed by Algorithm 2 using a sample of 10,000 MCMC iterations following a burn-in of 1,000 iterations.

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approximately 0.0048, 0.0009, 0.0019, and 0.0038, respectively. For the subsample of blacks, the effects of income change were somewhat stronger – in that subsample, an increase of $1,000 in family income changed the four educational outcome probabilities by 0.0060, 0.0009, 0.0026, and 0.0043, respectively. While Table 1 presents results estimated under a variance restriction (var(e) ¼ 1) by Algorithm 2, we also estimated the model by Algorithm 3 using two types of cutpoint restrictions: g1 ¼ 0 and g2 ¼ 1, and g1 ¼ 0 and g3 ¼ 1, respectively. In the latter two cases, var(e) ¼ s2 is a free parameter. The point estimates for the parameters from that algorithm, when transformed by 1/s produced estimates that were virtually identical to those in Table 1 and were therefore suppressed. However, the inefﬁciency factors from the three MCMC runs differed, and are presented in Fig. 3. The ﬁrst and second panel in that ﬁgure suggest that Algorithm 2 and

Inefficiency Factors

20

σ2=1

γ1=0, γ3=1

γ1=0, γ2=1

20

20

18

18

18

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14

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12

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6

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4

4

4

2

2

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β

δ

β

δ

σ2

β

δ

σ2

Fig. 3. Boxplots of Inefﬁciency Factors in the Educational Attainment Application. Note: In the ﬁrst panel, sampling is implemented under variance restrictions (s2 ¼ 1) and b is a 12 1 vector and d is 2 1; in the second and third panels, sampling is implemented by algorithm 3 under cutpoint restrictions, using (g1 ¼ 0, g3 ¼ 1) and (g1 ¼ 0, g2 ¼ 1), respectively; in those panels, b is a 12 1, while d and s2 are scalars.

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Algorithm 3 (using transformation (8) due to Chen & Dey, 2000) perform well in this case. A comparison of the second and third panels in Fig. 3 shows that identifying the model by ﬁxing the ﬁrst and last cutpoints is preferable to ﬁxing the ﬁrst and second cutpoints, indicating that restricting a larger fraction of the latent data z to a ﬁxed range tends to identify the scale better. Regardless of which cutpoints are ﬁxed, however, it is important to keep in mind that Algorithm 3 is a three-block sampler and requires an additional reduced run when it comes to marginal likelihood estimation relative to Algorithm 2. We note that in all three cases, the MH acceptance rates for d were very good and were in the range (0.90, 0.97).

6.2. Voter Opinions Surrounding election day, political analysts often present descriptive statistics of voter opinions and demographics from exit poll surveys. Such surveys are of particular interest to political economists and politicians since they cast light on voter sentiment and help to predict future election outcomes or guide policy. Recently, an ordered probit model using economic perceptions as the dependent variable was used in Duch et al. (2000) to study the degree to which voter perceptions are objective. When a given survey respondent is asked to comment on multiple issues, his or her responses will generally be correlated; however, analysts and the media often present the results for various questions independently or as tabulations by demographic groups. In contrast, we estimate the effect of voter demographics on survey responses using a multivariate model that accounts for correlations in a voter’s set of responses. We use the National Election Pool General Election Exit Polls, 2006 (Interuniversity Consortium for Political and Social Research, 2007). Our sample consists of 6,050 voters casting a ballot in the 2006 United States general election. Election day and absentee voters were surveyed from October 27, 2006, to November 7, 2006, after leaving polling places or through phone interviews. The survey attempted to elicit voters’ opinions on current issues such as how President George W. Bush is performing in ofﬁce, approval of the war in Iraq, and status of the national economy. Responses to these questions are coded as categorical variables with four ordered categories (higher values imply more favorable opinions). The demographic variables we include are age, sex, race, ethnicity, urban location, region, household income, marital status, and whether children lived in the household. Other explanatory variables include religious afﬁliation, frequency of religious

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Fitting and Comparison of Models for Multivariate Ordinal Outcomes

attendance, and political philosophy. The survey is a stratiﬁed random sample that over-samples minorities. The results of ﬁtting this trivariate ordered probit model are presented in Table 2. The parameter estimates in Table 2 show plausible signs and magnitudes that accord well with intuition. Not surprisingly, relative to moderates, liberals have negative opinions on all three topics, in contrast to conservatives. Respondents in higher income categories, who most likely also have higher education (unavailable in this data set), have a more positive opinion of the national economy, but share a lower opinion of the Iraq war and George W. Bush’s performance as president. Those who subjectively consider themselves in a good ﬁnancial situation have a positive opinion on all three topics. Those in the youngest age category have opinions of smaller magnitude across the board than those in the older age category. Females have a negative opinion of the economy and the Iraq war relative to males, but relatively weak opinion of the president. Coefﬁcients

Table 2. Parameter

b

d

Posterior Means and Standard Deviations for the Parameters in the Voter Opinions Application. Covariate

Intercept Liberal Conservative Income 30 k–50 k Income 50 k–75 k Income 75 k up Financial situation Age o30 Age W64 Married Children Female Black Latino Attends church Born again

Economy

Iraq War

President

Mean

SD

Mean

SD

Mean

SD

0.08 0.34 0.49 0.06 0.08 0.14 0.71 0.01 0.02 0.00 0.02 0.21 0.44 0.12 0.04 0.01 0.39 0.52

0.06 0.04 0.04 0.05 0.05 0.05 0.02 0.05 0.04 0.03 0.03 0.03 0.05 0.06 0.01 0.04 0.02 0.02

0.72 0.60 0.68 0.08 0.13 0.22 0.48 0.05 0.19 0.05 0.04 0.09 0.53 0.16 0.10 0.18 0.54 0.16

0.06 0.04 0.04 0.05 0.05 0.05 0.02 0.05 0.04 0.04 0.03 0.03 0.05 0.06 0.01 0.04 0.03 0.02

1.25 0.70 0.75 0.00 0.04 0.18 0.61 0.02 0.14 0.05 0.04 0.03 0.67 0.08 0.13 0.24 0.53 0.07

0.07 0.04 0.04 0.05 0.05 0.05 0.02 0.05 0.04 0.04 0.03 0.03 0.05 0.06 0.01 0.04 0.03 0.02

Identiﬁcation is achieved through variance restrictions and estimates are obtained by Algorithm 5 using a sample of 10,000 MCMC iterations after a burn-in of 1,000 draws.

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for minorities are negative for all three topics, with larger magnitudes for blacks. Church attenders (regardless of religion) as well as born again Christians have a positive view of the Iraq war and President George W. Bush, with born again Christians having stronger positive opinions. Because both liberals and conservatives appear strikingly different than moderates (the omitted voter category), we quantify the effect of these two covariates by calculating the implied changes in the response probabilities using the techniques of Section 5.1. To see in practical terms what the effects of the two ideological opposites imply relative to being moderate, we take the subsample of moderates and calculate the response probabilities with and without each dummy. The results are presented in Table 3. We note that a similar exercise can be performed on the subsamples of liberals or conservatives or any other subsample of interest, but one has to be aware that the results need not be identical because the subsamples of respondents will differ in their covariates. Overall, all entries indicate large effects of the liberal and conservative dummies; interstingly, the magnitudes of these effects are somewhat more balanced for opinions on the national economy than those on the Iraq war and the performance of the president. Because of the joint modeling and estimation for a voter’s set of responses, the model takes into account the correlation between voter opinions. The correlation matrix, estimated by Algorithm 5 under unit

Table 3. Examples of Estimated Covariate Effects for the Parameters of the Liberal and Conservative Dummies in the Voter Opinion Application. DPr(yi ¼ 1)

DPr(yi ¼ 2)

DPr(yi ¼ 3)

DPr(yi ¼ 4)

Economy Moderate-Liberal Moderate-Conservative

0.0734 0.0699

0.0387 0.0963

0.0857 0.0999

0.0264 0.0663

Iraq War Moderate-Liberal Moderate-Conservative

0.2148 0.2178

0.1364 0.0450

0.0756 0.1573

0.0028 0.0155

President Moderate-Liberal Moderate-Conservative

0.2351 0.2323

0.1521 0.0522

0.0796 0.1610

0.0034 0.0191

The entries indicate the average change in the probability of each outcome for a given covariate change.

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variance restrictions, is given by 0

1 B 0:40 X¼@

0:40 1

0:50

0:79

1 0:50 C 0:79 A 1

These estimates suggest that the three outcomes in this application are highly positively correlated, which suggests the presence of common unobserved factors inﬂuencing all three responses. In closing, we mention a few additional considerations. As can be expected, the estimates of the parameters from Algorithms 5 and 6 agreed closely after accounting for the different scale identiﬁcation in the model. However, the inefﬁciency factors differed very widely as shown in Fig. 4, where we see that traditional identiﬁcation through variance restrictions produces improved mixing of the Markov chain. Moreover, the inefﬁciency

Inefficiency Factors

60

Ω in correlation form

γ j1=0, γ j2=1

γ j1=0, γ j3=1 60

60

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β

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Ω

β

δ

Ω

Fig. 4. Boxplots of Inefﬁciency Factors in the Voter Opinion Application. Note: In the ﬁrst panel, sampling is implemented under variance restrictions (X is in correlation form with correlations given by q) by algorithms 5 and b is 48 1, d is 6 1, and q is 3 1; in the second and third panels, sampling is implemented by algorithm 6 under cutpoint restrictions, using (gj1 ¼ 0, gj3 ¼ 1) and (gj1 ¼ 0, gj2 ¼ 1), respectively; in those panels, b is a 48 1, d is 3 1, and the unique elements of X form a 6 1 vector.

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factors presented here are higher than those in the univariate case from the educational attainment application. One reason is that while the vectors of cutpoint differences dk are sampled marginally of the corresponding zk for k ¼ 1, . . . , q, that is, dkBdkjy, b, q, z\k, the sampling still depends on the latent data z\k for the other responses, so that when X exhibits high correlations as is the case in this example, the Markov chain will mix more slowly. This intuition is conﬁrmed in our next example, where the estimated covariance matrix exhibits lower correlations that lead to improved mixing for the parameters. Finally, in both algorithms, the MH steps performed quite well with MH acceptance for d in the range (0.90, 0.95) suggesting a close match between the proposal and target densities. Also, by setting the AR constant c in the ARMH algorithm such that ch(qjz, b)/p (qjz, b) ¼ 2 at the mode of h(h) and setting the ARMH tuning parameter t ¼ 1.5, in our context we were able to obtain an AR acceptance rate of 0.4 and a corresponding MH acceptance rate of 1 indicating that these settings of the tuning parameters produce an AR sampler for X.

6.3. Sources of Medical Information An essential goal in marketing research is the identiﬁcation of the source, or sources, of information that consumers use in making decisions to buy. The question is of particular interest when it comes to identifying the sources of medical information, because of the signiﬁcant investments in drug advertising and the public health consequences of medical choices. Kravitz et al. (1996) identiﬁes doctors, family and friends, and media advertisements, as the main sources of medical information for most people. Ippolito and Mathios (1990) speciﬁcally study the inﬂuence of advertisements on health information, while Hellerstein (1998) studies the inﬂuence of a physician on patients’ use of generic versus brand-name prescription drugs. These methods of obtaining medical information may be correlated. For example, an individual’s desire to research a wide array of sources could result in positive correlation between the sources. Alternatively, someone who has a high frequency of obtaining information from a doctor may not rely as often on other sources of information, such as friends and family, or advertising. As a result, it is possible that medical advertising could either substitute for advice from doctors or encourage patients to see a doctor about a concern they have. In our study we use ordered categorical survey

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responses on the frequency of obtaining information from various sources to address this question. We use the Public Health Impact of Direct-to-Consumer Advertising of Prescription Drugs, July 2001–January 2002 (Interuniversity Consortium for Political and Social Research, 2003). The sample consists of entries from 2,879 respondents on their self-reported sources of medical information. The sample was created to be nationally representative using random digit dialing, subsequently stratiﬁed to over-sample minorities. For our dependent variables we use responses to questions on how often respondents obtain medical information from various sources, namely information obtained from friends and family, through advertisements, and from a doctor. Responses to these questions are coded as categorical variables with four ordered categories indicating frequency. We include additional covariates such as whether the individual has a health condition, medical insurance, or a regular doctor. The demographic variables we include are age, sex, race, ethnicity, education, employment status, student status, urban location (suburban is the omitted category), region, household income, marital status, and family size. Parameter estimates obtained by Algorithm 5 are presented in Table 4. The table reveals that relative to those in the lowest income bracket, individuals in higher income brackets are less likely to turn to advertisements as a source of health information, and more likely to turn to friends and family. Higher income brackets are incrementally more likely to obtain health information from a doctor. Surprisingly, those with insurance, a medical condition, or a regular doctor are all less likely to turn to a doctor for medical information. These effects are shown in Table 4, and the covariate effects in Table 5 show the average effect of insurance on the probabilities of the outcome categories for each source of medical information. One factor that may contribute to some of the surprising results presented here is that the variable for medical condition is comprised of a list of serious and well-known medical conditions but does not include minor aches and pains or less serious illnesses. For this reason, many aspects of health which may drive demand for medical information may not be captured in the available data. Additionally, obtaining medical information from a doctor likely requires a higher cost and, unlike with friends and family or advertisements, is more likely to be associated with actually having some physical concern that may not entirely be captured in the documented list of medical conditions. The absence of detailed health information is a limitation of the data.

150

Table 4. Parameter

b

IVAN JELIAZKOV ET AL.

Posterior Means and Standard Deviations for the Parameters in the Health Information Application. Covariate

Advertisement

Intercept Income 25 k–50 k Income 50 k–75 k Income 75 k up Insurance Medical condition Regular doctor Employed Student Less than HS Some college College Age Married Family size Female Minority Urban Rural South

d

Friends and Family

Doctor

Mean

SD

Mean

SD

Mean

SD

0.82 0.10 0.09 0.08 0.06 0.02 0.04 0.05 0.07 0.22 0.09 0.10 0.00 0.04 0.02 0.15 0.17 0.06 0.06 0.05 0.21 0.40

0.13 0.05 0.07 0.07 0.07 0.05 0.06 0.05 0.12 0.08 0.06 0.05 0.00 0.05 0.03 0.04 0.05 0.05 0.05 0.04 0.03 0.04

0.71 0.05 0.09 0.04 0.03 0.03 0.07 0.03 0.13 0.21 0.14 0.20 0.01 0.08 0.01 0.18 0.02 0.04 0.01 0.02 0.04 0.28

0.13 0.05 0.07 0.07 0.07 0.05 0.06 0.05 0.12 0.08 0.06 0.05 0.00 0.05 0.03 0.04 0.05 0.05 0.05 0.04 0.03 0.03

1.44 0.06 0.07 0.11 0.15 0.37 0.57 0.15 0.38 0.05 0.02 0.30 0.00 0.14 0.01 0.18 0.06 0.05 0.05 0.04 0.06 0.16

0.13 0.06 0.07 0.07 0.07 0.05 0.06 0.05 0.12 0.08 0.06 0.05 0.00 0.05 0.03 0.04 0.05 0.05 0.06 0.04 0.03 0.04

Identiﬁcation is achieved through variance restrictions and estimates are obtained by Algorithm 5 using a sample of 10,000 MCMC iterations after a burn-in of 1,000 draws.

Table 5.

Estimated Covariate Effect of Insurance. DPr(yi ¼ 1)

DPr(yi ¼ 2)

DPr(yi ¼ 3)

DPr(yi ¼ 4)

Advertisements Insurance ¼ 0-Insurance ¼ 1

0.0202

0.0035

0.0069

0.0168

Friends & Family Insurance ¼ 0-Insurance ¼ 1

0.0068

0.0031

0.0014

0.0085

Doctor Insurance ¼ 0-Insurance ¼ 1

0.0373

0.0153

0.0061

0.0465

The entries indicate the changes in the probability of each outcome based on the sample of uninsured.

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Because of the joint modeling of the responses, the model accounts for the correlation between health information sources. The correlation matrix, estimated by Algorithm 5 under unit variance restrictions, is given by 0

1

B X ¼ @ 0:21 0:03

0:21 1 0:24

0:03

1

C 0:24 A 1

These estimates suggest that certain outcomes in this application are correlated, while others are not. For example, the frequency of using friends and family to obtain medical information is correlated with both information from advertisements and from doctors (0.21 and 0.24, respectively), while the correlation between information from advertisements and doctors is very low (0.03). It may be the case that individuals use friends and family to ﬁlter information from the other two sources; this may indicate that information from friends and family serves as a complement to other sources of medical information. Upon comparing this correlation matrix to the one the voter opinion application, we see that overall the correlations here are much lower. In Fig. 5, we present the inefﬁciency factors that result from Algorithms 5 and 6 in this application. These overall inefﬁciency factors appear to be lower when identiﬁcation is achieved through variance restrictions and estimation is done by Algorithm 5. Again, in both algorithms, the MH steps performed quite well with MH acceptance for d again in the range (0.90, 0.95) suggesting a close match between the proposal and the target. As in our voter opinion application, setting the AR constant c in the ARMH algorithm such that ch(qjz, b)/p(qjz, b) ¼ 2 at the mode of h(h) and the ARMH tuning parameter t ¼ 1.5, we were able to obtain an AR acceptance rate of 0.41 and a corresponding MH acceptance rate of 1 indicating that the ARMH algorithm essentially involved AR sampling for X. In closing, we return to an interesting difference between the inefﬁciency factors in this and the previous application. Since sampling of the latent data zk and cutpoints ck is conditional on the latent data for the other responses z\k for k ¼ 1, . . . , q, the magnitude of the correlations in X plays an important role in determining the mixing of the chain. When those correlations are high (as in the voter opinion example), the chain mixes more slowly than when correlations in X are low as in the current example.

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Inefficiency Factors

γ j1=0, γ j2=1

γ j1=0, γ j3=1

Ω in correlation form 10

10

10

9

9

9

8

8

8

7

7

7

6

6

6

5

5

5

4

4

4

3

3

3

2

2

2

1

1

1

β

δ

ρ

β

δ

Ω

β

δ

Ω

Fig. 5. Boxplots of Inefﬁciency Factors in the Health Information Application. Note: In the ﬁrst panel, sampling is implemented under variance restrictions (X is in correlation form with correlations given by q) by algorithms 5 and b is 60 1, d is 6 1, and q is 3 1; in the second and third panels, sampling is implemented by algorithm 6 under cutpoint restrictions, using (gj1 ¼ 0, gj3 ¼ 1) and (gj1 ¼ 0, gj2 ¼ 1), respectively; in those panels, b is a 60 1, d is 3 1, and the unique elements of X form a 6 1 vector.

7. CONCLUSION There are alternative ways in which ordinal models can be identiﬁed. In this paper, we have discussed some possibilities and shown how they can be implemented in practice using well-tailored MH or ARMH algorithms. Our main points can be summarized as follows. First, in the univariate setting, identiﬁcation through variance restrictions appears to be preferable when the number of categories J is greater than 3. This is because Algorithm 2 allows for more efﬁcient blocking for sampling and marginal likelihood estimation. However, when J ¼ 3, a sampler built upon identiﬁcation by ﬁxing the two cutpoints can be useful as it will not involve any MH steps. Second, in multivariate settings when ordinal models are identiﬁed through

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variance restrictions (leading to a correlation matrix X), efﬁcient sampling can be made possible through the ARMH algorithm. In our examples, identiﬁcation through variance restrictions and estimation through Algorithm 5 was shown to lead to overall improved mixing of the Markov chain relative to identiﬁcation by cutpoint constraints and ﬁtting by Algorithm 6. Algorithm 5 also allows greater ﬂexibility when X involves restrictions, for example, when it is structured or involves off-diagonal zeros; however, the easier application of Algorithm 6 can also be appealing when X is not restricted. When identiﬁcation is achieved by ﬁxing cutpoints, our examples have revealed that ﬁxing the ﬁrst and last cutpoints (e.g., as in Chen and Dey, 2000) appears to result in lower inefﬁciency factors than ﬁxing the ﬁrst and second cutpoints. For these reasons, we recommend Algorithm 5 to more advanced statistical programmers dealing with complex problems, whereas Algorithm 6 can be useful for tackling standard problems by more customary Markov chains. Third, we have shown that the estimation algorithms discussed here allow for the straightforward calculation of marginal likelihoods and Bayes factors for comparing alternative ordinal models. Finally, the paper has discussed a simulation-based framework for covariate effect evaluation that can be quite useful in eliciting the impact of covariates on the probabilities of ordinal outcomes. The above issues have been illustrated in three important problems in labor economics, political science, and health economics. These studies have demonstrated the applicability and usefulness of the inferential techniques in the context of ordinal data models.

ACKNOWLEDGMENT We are much indebted to Siddhartha Chib, William Grifﬁths, and an anonymous referee for their help with earlier drafts of this paper.

REFERENCES Albert, J., & Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88, 669–679. Albert, J., & Chib, S. (2001). Sequential ordinal modeling with applications to survival data. Biometrics, 57, 829–836. Astone, M., & McLanahan, S. S. (1991). Family structure, parental practices and high school completion. American Sociological Review, 56, 309–320.

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INTRA-HOUSEHOLD ALLOCATION AND CONSUMPTION OF WIC-APPROVED FOODS: A BAYESIAN APPROACH Ariun Ishdorj, Helen H. Jensen and Justin Tobias ABSTRACT WIC, the Special Supplemental Nutrition Program for Women, Infants, and Children, is a widely studied public food assistance program that aims to provide foods, nutrition education, and other services to at-risk, lowincome children and pregnant, breastfeeding, and postpartum women. From a policy perspective, it is of interest to assess the efﬁcacy of the WIC program – how much, if at all, does the program improve the nutritional outcomes of WIC families? In this paper, we address two important issues related to the WIC program that have not been extensively addressed in the past. First, although the WIC program is primarily devised with the intent of improving the nutrition of ‘‘targeted’’ children and mothers, it is possible that WIC may also change the consumption of foods by nontargeted individuals within the household. Second, although WIC eligibility status is predetermined, participation in the program is voluntary and therefore potentially endogenous. We make use of a treatment–response model in which the dependent variable is the requirement-adjusted calcium intake from milk consumption and the Bayesian Econometrics Advances in Econometrics, Volume 23, 157–182 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23005-7

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endogenous variable is WIC participation, and estimate it using Bayesian methods. Using data from the CSFII 1994–1996, we ﬁnd that the correlation between the errors of our two equations is strong and positive, suggesting that families participating in WIC have an unobserved propensity for high calcium intake. The direct ‘‘structural’’ WIC parameters, however, do not support the idea that WIC participation leads to increased levels of calcium intake from milk.

1. INTRODUCTION In ﬁscal year 2006, the United States Department of Agriculture (USDA) spent nearly $53 billion on food assistance programs (Oliveira, 2007). The third largest of these programs, the Special Supplemental Nutrition Program for Women, Infants, and Children (commonly and henceforth denoted as WIC), has been widely studied in the health and nutrition literatures and aims to serve the public by providing supplemental foods, nutrition education and other services to foster the growth, development, and long-term health of participating individuals. For families that qualify for WIC participation, the program provides access to nutritious foods to supplement the diets of infants, children up to age ﬁve, and pregnant, breastfeeding, and postpartum women. The program beneﬁts, usually in the form of checks or vouchers, allow participants to obtain speciﬁc ‘‘packages’’ of foods. These foods include infant formula, milk, cheese, eggs, juice, cereals, peanut butter/dried beans, and, for fully breastfeeding mothers, these also include carrots and tuna. From a policy perspective, it is of primary interest to assess the efﬁcacy of the WIC program – how much, if at all, does the program improve the nutritional outcomes of WIC families? In this paper, we employ a Bayesian methodology to address this question and estimate the impact of WIC participation on a speciﬁc nutritional outcome – calcium intake via milk consumption. Our study is certainly not the ﬁrst in this regard, as other efforts using different models and maintained assumptions have been conducted in the past. For example, Oliveira and Chandran (2005) ﬁnd that participation in the WIC program increases consumption for some types of WIC-approved foods for WIC children compared to eligible nonparticipating children and children living in households with income too high to be eligible for WIC (income greater than 185% of the poverty threshold). Other efforts in this regard include the studies of Rose, Habicht, and Devaney (1998), Burstein et al. (2000), Oliveira and Gundersen (2000), Ponza,

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Devaney, Ziegler, Reidy, and Squatrito (2004), and Siega-Riz et al. (2004), who generally ﬁnd positive impacts associated with the WIC program. There are, however, two important issues related to the WIC program that have not been extensively addressed in past work, and we seek to address these issues in the current paper. First, although the WIC program is primarily devised with the intent of improving the nutrition of ‘‘targeted’’ children and mothers, it is possible that WIC may also change the consumption of foods by nontargeted individuals within the household. This has been referred to as ‘‘spillover’’ (Oliveira & Chandran, 2005) or ‘‘leakage’’ (Barrett, 2002) of WIC beneﬁts. As Oliveira and Chandran note, this might occur if (1) receipt of WIC beneﬁts frees up food dollars for use to beneﬁt other, nonparticipating children; (2) nutrition education changes food selection for all members; or (3) WIC foods are shared with non-WIC household members. Little is known about the degree to which this occurs. In the current paper, we formally address this issue by comparing the impact of WIC participation on both targeted household members as well as nontargeted members of WIC families. Second, the previous literature on this topic has certainly been aware of the potential endogeneity of WIC participation and, in some cases, has interpreted the obtained results with caution in light of this concern. To our knowledge, however, no study in the literature has dealt with this problem extensively. To address this endogeneity issue, we make use of a treatment– response model in which the dependent variables are the requirementadjusted calcium intake from milk consumption and the decision to participate in WIC. We estimate this two-equation system jointly and handle the endogeneity issue by introducing covariates that affect WIC participation directly but (presumably) are conditionally uncorrelated with levels of calcium intake. These instruments include indicators of household assets as well as variables exploiting regional variation in requirements for WIC participation. Ostensibly, WIC participation will lead to increased calcium intake from milk, though in the presence of endogenous participation, this need not be the case. For example, families who choose to participate in WIC may simultaneously (and unobservably) be quite concerned regarding the nutritional intake of each family member, and thus members of households participating in WIC may have high calcium intake even in the absence of WIC. Moreover, freed resources enable families to consume calcium through other sources, so that WIC could actually lead to a reduction in calcium intake through milk. In terms of our posterior predictive distributions of calcium intake from milk, we ﬁnd results consistent with our prior expectations and the majority

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of past work on this topic. That is, WIC-targeted individuals have higher levels of calcium intake than their non-WIC counterparts. However, the posterior predictives combine two sources of information: what we might term the ‘‘structural’’ effect of WIC participation as well as an unobserved correlation between the errors of the participation and outcome equations. As one might suspect, we ﬁnd that the correlation between errors in the WIC participation and calcium consumption equations is strong and positive, suggesting that families participating in WIC have an unobserved propensity for high calcium intake. What drives the intuitive ordering among the posterior predictives is primarily the selection effect – those families in WIC would have had large levels of calcium intake in the absence of the program. The direct ‘‘structural’’ WIC parameters do not directly support the idea that WIC participation leads to increased levels of calcium intake, a ﬁnding that is, to our knowledge, new to this literature. Indeed, these families may be substituting away from milk and toward other preferred alternatives, a ﬁnding that has signiﬁcant implications for the selection of foods within the WIC program. The paper proceeds as follows. The next section describes the model speciﬁcation and the associated Bayesian posterior simulator. The data used in the analysis are described in Section 3, followed by a description of empirical results in Section 4. The paper concludes with a summary of the ﬁndings in Section 5.

2. THE MODEL, POSTERIOR SIMULATOR AND POSTERIOR PREDICTIVES We ﬁrst let wh be a binary variable equal to one if household h participates in WIC and equal to zero otherwise. Within a given household, some members, including children under ﬁve and pregnant/breastfeeding mothers, will be targeted individuals, that is, those family members the WIC program is primarily designed to serve. To this end, we will let Tih be an exogenous binary variable denoting if individual i in household h is a WIC-targeted individual. The construction of these two variables leads to the categorization of all individuals in our sample into four mutually exclusive groups: G1,ih=whTih (targeted individual in a WIC-participating household), G2,ih=wh(1Tih) (nontargeted individual in a WIC-participating household),

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G3,ih=(1wh)Tih (targeted individual in a WIC eligible but nonparticipating household), G4,ih=(1wh)(1Tih) (nontargeted individual in a WIC eligible but nonparticipating household). Our outcome variable of interest is requirement-adjusted calcium intake through milk consumption. We represent this variable as cih. Importantly, there is a censoring problem associated with calcium intake in our data, since approximately 16% of our sample has identically zero consumption values. To this end, we follow Chib (1992) and Albert and Chib (1993) and work with latent consumption cih , which is assumed to be generated by:1 cih ¼ xih a þ ih

(1)

cih ¼ maxf0; cih g

(2)

and

The group identiﬁers G2-G42 above together with other relevant demographic variables such as age, income, gender indicators, etc., are included in the vector xih. By comparing the a coefﬁcients across these groups, we can determine if WIC participation has an important effect on calcium intake, and, moreover, we can test for the presence of the hypothesized ‘‘spillover’’ effects within a WIC household. That is, we can determine whether nontargeted members in WIC households have higher levels of calcium intake through milk consumption than nontargeted members of non-WIC households. As stressed in the introduction of this paper, WIC participation is voluntary, and thus the binary indicator wh (and associated group identiﬁers G2-G4) is not necessarily exogenous in Eq. (1). That is, household heads choosing to participate in WIC could, for example, be very concerned about the nutritional intakes of its constituents, thus leading to higher levels of calcium intake for these families on average. To this end, we ﬁrst consider the household-level decision to participate in WIC: i:i:d:

wh ¼ zh b þ nh ; where

( wh ¼

nh Nð0; 1Þ

1

if

wh 40

0

if

wh 0

and zh is a vector of instruments and household-speciﬁc characteristics.

(3)

(4)

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To account for the potential endogeneity of WIC participation, we allow the errors of Eqs. (1) and (3) to be correlated. That is, household-level unobservables that make a family more likely to participate in WIC may also make that family more likely to have high levels of calcium intake. We choose to accommodate this type of correlation by including a householdspeciﬁc error term in Eq. (1) and allowing this error to be correlated with nh in Eq. (3). The intuition behind this modeling assumption is that a household head who chooses to participate in WIC will also tend to guide meal preparation in the household and monitor the nutritional habits of the household members. Thus, unobservable factors affecting WIC participation will likely spill over and correlate with the nutritional intakes of all the family members and should probably correlate in a similar way across each member. To this end, we consider the following model: cih ¼ xih a þ csih þ uh þ ih

(5)

wh ¼ zh b þ nh

(6)

where "

uh nh

#

i:i:d:

jx; z; s N

" 0 0

;

s2u suv

suv 1

!# (7)

and i:i:d:

ih jx; z; s Nð0; s2 Þ

(8)

Eqs. (5) and (6) now represent a standard two-equation treatment–response model using only observed rather than potential outcomes.3 However, we note that Eq. (5), unlike its counterpart in Eq. (1), has included a latent variable sih . This latent variable is included, like the model of Chen, Dey, and Shao (1999), to capture possible skew in the distribution of calcium intake.4 These latent variables are speciﬁed to be generated from a known distribution with one-sided support, thereby introducing the possibility of accommodating skew in the outcome distribution beyond what is implied by normality (given that cih 40). A rather standard choice in this regard, as employed in Chen et al. (1999), is to assume that sih is generated from a half-normal distribution, i:i:d:

sih jx; z TN ð0;1Þ ð0; 1Þ with TN(a, b) (m,s2) denoting a normal distribution with mean m and variance s2 truncated to the interval (a, b). When integrating the conditional density

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for cih ðgiven sih Þ over this half-normal for sih , it can be shown that, marginally, cih will have a skew-normal distribution (e.g., Azzalini & Dalla Valle, 1996; Chen et al., 1999; Branco & Dey, 2002). The sign of the parameter c governs the direction of the skew (i.e., positive values produce a distribution with a right-skew, conversely for negative values of c, and c=0 reduces to joint normality). Since the potential for such skew exists in both the conditional and unconditional distributions of calcium intake (Fig. 1), we adopt the above procedure for handling this issue. As shown in our empirical results section, the data strongly support the hypothesis of c 6¼ 0 so that the default assumption of joint normality is not appropriate for this data. This is suggested by the intake histogram of Fig. 1. With the formulation in Eq. (5), the composite error term csih þ uh þ ih is not mean zero, since sih is not mean zero. Though this shift will be ‘‘absorbed’’ by the intercept parameter, this creates a muddled interpretation of the parameter c and may lead to slower mixing of the posterior simulations.5 We handle this issue by simply shifting the distribution of sih 300

250

200

150

100

50

0 0

500

1000

Calcium Intake (mg) | Intake>0

Fig. 1.

Distribution of Positive Calcium Intake.

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back by its mean,

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pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð2=pÞ. Thus, in our analysis, we specify6 rﬃﬃﬃ ! 2 i:i:d: ﬃ sih jx; z TN ðpﬃﬃﬃﬃﬃ ;1 2=p;1Þ p

(9)

and the model is given by Eqs. (5)–(8) together with the revised distributional assumption on sih given in Eq. (9). 2.1. The Joint Posterior For the implementation of the posterior simulator, it will be instructive to work with the population expectation of uh given nh. Given the joint normality assumption above, we can write uh ¼ suv nh þ Zh where i:i:d:

Zh Nð0; s2Z Þ;

and

s2Z s2u s2uv

Thus, we can rewrite our initial equation system in the following way: cih ¼ xih a þ csih þ suv nh þ Zh þ ih wh ¼ zh b þ nh where i:i:d:

ih Nð0; s2 Þ i:i:d:

nh Nð0; 1Þ i:i:d:

Zh Nð0; s2Z Þ Thus, conditioned on the common nh, the consumption and WIC participation equations are independent. Let d ¼ ½a0 b0 c suv s2 s2Z 0 denote the parameters of our model other than the random effects g. In addition, let nh denote the number of individuals in household P h, H denote the total number of households in the sample, NH H h¼1 nh ,

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k denote the number of explanatory variables and, ﬁnally, deﬁne 2 3 2 3 2 3 c1h s1h x1h 6 c 7 6 7 6 7 6 2h 7 6 x2h 7 6 s2h 7 6 7 6 7 6 7 ch ¼ 6 .. 7; Xh ¼ 6 .. 7; sh ¼ 6 .. 7 6. 7 6. 7 6. 7 4 4 4 5 5 5 cnh h snh h xnh h 2

3 c1 6 c 7 6 2 7 6 7 c ¼ 6 . 7; 6 .. 7 4 5 cH

2

3 s1 6 s 7 6 2 7 6 7 s ¼ 6 . 7; 6 .. 7 4 5 sH

2

3 w1 6 w 7 6 2 7 6 7 w ¼ 6 . 7; 6 .. 7 4 5 wH

2 and

Z1

6 6 Z2 6 g ¼ 6. 6 .. 4

ZH

3 7 7 7 7 7 5

where xih is a 1 k covariate vector for agent i, Xh is the NH k matrix of stacked covariate data, and c, s, w, and g are NH 1 vectors. As in Albert and Chib (1993), we will include the latent c, w, s, and vector of random effects g into our posterior and thus will work with an augmented posterior of the form pðc ; w ; s ; d; gjc; wÞ / pðc; wjc ; w ; s ; d; gÞpðc ; w ; s jd; gÞpðgjdÞpðdÞ " H Y pðwh jwh Þpðch ; wh jsh ; d; Zh ÞpðZh jdÞ ¼ pðdÞ i¼1

Y

!# pðcih jcih Þpðsih Þ

i2h

In the ﬁrst line above, we write the posterior as proportional to the full joint distribution (of parameters, latent and observed data), and decompose this joint distribution into a sequence of conditionals times marginals. The densities p(g|d) and p(d) denote prior distributions for these parameters, and, in the second line of the above, we incorporate the assumed (conditional) independence across households. Finally, in regard to the density p(c, w|c, w, s, d, g), we note that the distribution of wh depends only on wh (and is degenerate given its value), and, likewise, the distribution of cih depends only on cih (and is degenerate given its value). That is, pðwh jwh Þ ¼ Iðwh ¼ 1ÞIðwh 40Þ þ Iðwh ¼ 0ÞIðwh 0Þ

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and pðcih jcih Þ ¼ Iðcih ¼ cih ÞIðcih 40Þ þ Iðcih ¼ 0ÞIðcih 0Þ As for the joint distribution of household h’s calcium intake, ch , and WIC participation, wh , note that ! " # " !# ch xh a þ csh þ Zh inh s2 Inh þ s2uv inh i0 nh suv inh ind ; js ; d; Zh N wh h zh b suv i0 nh 1 where inh is an nh 1 vector of ones, and, likewise, Inh is the identity matrix of dimension nh. To complete our Bayesian analysis we must also introduce our priors. To this end, we let 2 3 a 6c7 c4 5 b and specify priors of the form c Nðlg ; Vg Þ

(10)

suv Nðmuv ; V uv Þ

(11)

s2 IGða ; b Þ

(12)

s2Z IGðaZ ; bZ Þ

(13)

2.2. The Posterior Simulator We ﬁt this model using recent advances in Markov Chain Monte Carlo (MCMC) techniques, namely, the Gibbs sampler (e.g., Gelfand, Hills, Racine-Poon, & Smith, 1990; Casella & George, 1992; Albert & Chib, 1993). Implementation of the Gibbs sampler involves deriving and then iteratively simulating from the conditional posterior distributions of the model’s parameters. The sequence of simulations produced from this sampling procedure forms a Markov chain that, under certain regularity conditions, converges to the targeted distribution (i.e., the joint posterior). To mitigate the effect of initial conditions on this chain, an initial set of pre-convergence

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or ‘‘burn-in’’ simulations is discarded, and the remaining set of simulations is then used to calculate posterior features of interest. Our complete Gibbs algorithm consists of eight steps, and the ﬁrst two of these form a blocking step (e.g., Chib & Carlin, 1999), where the parameters c=[au c bu]u and random effects g are sampled in a single block. We do this via the method of composition. That is, we ﬁrst sample c from its conditional posterior, where the random effects g have been integrated out. We then sample g by drawing each Zh independently from its complete conditional posterior. For simplicity in notation below, we let C=[du cu wu su]u and let Cx denote all parameters other than x. Step 1: c|Gc, c, w. First, deﬁne

" Xh

and

" Sh

Xh 0

# 0 ; zh

sh 0

" ch

ch

#

wh

½s2 I nh þ ðs2Z þ s2uv Þinh i0nh suv i0nh

suv inh 1

#

It follows that cjCg ; c; w NðDg dg ; Dg Þ

(14)

where Dg ¼

hX

0

1 ðXh S1 h Xh Þ þ Vg h

i1

and

dg ¼

X

0

1 ðXh S1 h ch Þ þ Vg lg

h

Step 2: Zh jGZh ; c; w ind

Zh jCZh ; c; w NðDZh d Zh ; DZh Þ;

h ¼ 1; 2; :::; H

where DZh ¼ d Zh ¼

s2Z s2 nh s2Z þ s2

1X ðc xih a sih c suv ½wh zh bÞ s2 i2h ih

(15)

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Step 3: suv jGsuv ; c; w First, deﬁne the NH 1 vectors V and g as follows: 2 3 2 3 in1 ½w1 z1 b in1 ½Z1 6 i ½w z b 7 6 i ½Z 7 6 n2 2 7 6 n 7 2 6 7 6 2 2 7 ; g 6. V 6. 7 7 6 .. 7 6 .. 7 4 5 4 5 inH ½wH zH b inH ½ZH It follows that suv jGsuv ; c; w NðDsuv d suv ; Dsuv Þ

(16)

where Dsuv ¼

V0 V þ V 1 uv s2

Step 4: s2 jGs2 ; c; w s2 jGs2 ; c; w IG "

1 ;

dsuv ¼

V0 ðc Xa s c g Þ þ V 1 uv muv s2

NH þ a ; 2

NH 1X ðc xih a sih c Zh suv ½wh zh bÞ2 b1 þ 2 i¼1 ih

Step 5: s2Z jGs2Z ; c; w

0

s2Z jGs2Z ; c; w IG@

"

H 1X

H þ aZ ; b1 Z þ 2 2 h¼1

#1 1 A

#1 1 A ðZ2 Þ h

ð17Þ

(18)

Step 6: w jGw ; c; w Each of the latent variables in the WIC participation equation are sampled independently as follows: 8 if wh ¼ 1 < TN ð0;1Þ ðmwh ; s2w Þ h (19) wh jGw ; c; w : TN ð1;0 ðmw ; s2w Þ if wh ¼ 0 h

h

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where mw ¼ zh b þ suv i0nh ½s2 Inh þ s2uv inh i0nh 1 ðch Xh a sh c Zh inh Þ h

and s2w ¼ 1 s2uv i0nh ½s2 Inh þ s2uv inh i0nh 1 inh h

Step 7: c jGc ; c; w Note that, conditioned on Zh and the remaining parameters of the model, each latent cih can be sampled independently from its conditional posterior: cih jGcih ; c; w TN ð1;0Þ ðmc ; s2 Þ if cih ¼ 0 ih

(20)

where mc ¼ xih a þ sih c þ Zh þ suv ðwh zh bÞ ih

When cihW0, the conditional posterior for cih is degenerate around the observed cih and does not need to be simulated. Step 8: sih jGs ; c; w ih The assumptions of our model imply that each sih can be sampled independently from its complete conditional posterior. Completing the square in sih yields a posterior conditional of the form: ind

ﬃ ðm ; s2 Þ; sih jGsih ; c; w TN ðpﬃﬃﬃﬃﬃ s 2=p;1Þ s ih

i ¼ 1; 2; . . . ; NH

where ms ¼

cðcih xih a Zh suv ½wh zh bÞ

pﬃﬃﬃﬃﬃﬃﬃﬃ 2 2=ps

s2 þ c2

ih

and s2s ¼

s2 s2 þ c2

A Gibbs sampler proceeds by iteratively sampling Eqs. (14)–(21).

(21)

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2.3. Posterior Predictive Intake Distribution In our empirical application we are primarily concerned with the calculation and comparison of intake distributions for individuals in each of the four groups of interest. To this end, we focus on posterior prediction and ﬁx the exogenous covariates’ values for simplicity. Given our model, the posterior predictive intake distribution for such a representative agent with ﬁxed covariates and wh=1, conditioned on the model parameters G, is given as pðcnþ1;h jwh ¼ 1; CÞ ¼ pðcnþ1;h jwh 40; CÞ Z 1 ¼ ½Prðwh 40jCÞ

0

1

pðcnþ1;h ; wh jCÞdwh

where the n+1 subscript is used to denote an out-of-sample ‘‘representative’’ agent. After some manageable algebra, we perform the required integration and obtain: pðcnþ1;h jwh

" # zh b þ ½suv =ðs2uv þ s2 Þðcnþ1;h xnþ1;h a csnþ1;h Zh Þ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ 1; CÞ ¼ F s2 =½s2 þ s2uv

fðcnþ1;h ; xnþ1;h a þ csnþ1;h þ Zh ; s2 þ s2uv Þ Fðzh bÞ

ð22Þ

The density in Eq. (22) is not of an immediately recognizable form, though the steps leading to its derivation suggest a method of obtaining draws directly from this density. Speciﬁcally, draws from Eq. (22) can be obtained from the following procedure: First, sample wnþ1;h TN ðzh b;1Þ ð0; 1Þ Then, set cnþ1;h ¼ p0;nþ1 þ p1 wnþ1;h þ p2 where Nð0; 1Þ p0;nþ1 ¼ xnþ1;h a þ csnþ1;h þ Zh p1 ¼ suv p2 ¼ s

(23)

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It can be shown that cnþ1;h has the density given in Eq. (22). The proof fact is reasonably straightforward, noting that pðcnþ1;h Þ ¼ Rof this pðcnþ1;h jwnþ1;h Þpðwnþ1;h Þdwnþ1;h and substituting in the formulas above to perform the necessary integration. Since this procedure obtains a draw from the posterior predictive for a given vector of parameters C, the inﬂuence of these parameters can be marginalized out of the predictive by noting: Z (24) pðcnþ1;h jwh ¼ 1; c; wÞ ¼ pðcnþ1;h jwh ¼ 1; CÞpðCjc; wÞdC Thus, for every post-convergence C draw produced from the simulator, we apply the above procedure to obtain a draw from the posterior predictive. Though the details are suppressed here, similar steps can be used to obtain the posterior predictive associated with the event that wh=0. Finally, calcium intake is linked to the latent cnþ1;h by noting: cnþ1;h ¼ maxf0; cnþ1;h g, which is calculated for each iteration of the sampler.

3. THE DATA Our empirical analysis makes use of data from the USDA 1994–1996 Continuing Survey of Food Intake by Individuals (CSFII). The CSFII is a nationally representative, cross-sectional survey of individuals in households in the United States. The survey uses a randomization strategy to select certain members of the household to participate in a complete food intake survey; thus, not all members of a WIC household are present in our sample. For each of the sampled individuals, questions involving a 24 hour recall of food intake were conducted on two nonconsecutive days. Importantly for our purposes, respondents report milk consumption and the consumption of milk-containing foods during this period. Household and individual characteristics can be used to identify WICeligible households, and we focus only on those individuals and households that are WIC eligible in our analysis. To be eligible for WIC, at least one individual in the household must be in a WIC-qualifying population group (women who are pregnant; nonbreastfeeding women up to six months postpartum; breastfeeding women up to one year postpartum; infants under one year of age; or children from one year old up to the child’s ﬁfth birthday). The household’s income must also be at or below 185% of the federal poverty guidelines, or the household must participate in other

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qualifying programs (Medicaid, Food Stamps, and Temporary Assistance for Needy Families [TANF]). Finally, individual applicants must be at nutritional risk, as determined by a health professional. Although it is not possible to determine individuals who are at nutritional risk from the CSFII data, nearly all U.S. women and children meet this criterion (IOM, 2002) so that, in practice, this additional constraint can be assumed to apply to all eligible individuals. Finally, we follow Oliveira and Chandran (2005) and deﬁne eligible households as those with incomes within 200% of the federal poverty guidelines. Our ﬁnal sample consists of 2,372 individuals from 1,036 households. As discussed in the previous section, these individuals were assigned into one of the four mutually exclusive groups (Table 1). For our analysis, we deﬁne WIC-targeted individuals as children of ages one to four and pregnant, breastfeeding, and postpartum women, and nontargeted individuals as children or adults in the household age ﬁve and older.7 All households in our ﬁnal sample are identiﬁed as WIC eligible by meeting the income criterion and having at least one targeted individual living in the household. Each of the four population groups described in Table 1 may have both adults and children. In order to compare the food intakes of individuals of varying age and gender, we convert the dependent variable, amount of milk consumed (grams), to a calcium-equivalent measure and then normalize the consumption in terms of the individuals’ dietary requirement for calcium. This is accomplished in several steps. First, the CSFII 1994–1996 data set contains information on grams of milk consumed as a single food or an ingredient in a food containing dairy products. However, milk is commonly included as an ingredient in other nondairy foods, and it is important to capture this aspect of calcium intake in the construction of our dependent variable. To this end, we consult the Pyramid Servings Database for USDA Survey Food Codes, Version 2.0, which provides information on the amount of milk per 100 grams contained within a variety of different foods.8 The amounts of milk from all foods consumed by an individual are then Table 1.

Number of Individuals in Each Group by WIC Status.

Number of Individuals 526 488 712 646 2,372

Group

WIC Status

G1 G2 G3 G4

Targeted individuals in WIC household Nontargeted individuals in WIC household Targeted individuals in non-WIC household Nontargeted individuals in non-WIC household Total individuals

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added together to produce the total amount of calcium intake from milk and milk product consumption by the individual. The Dietary Reference Intake (DRI) value expresses the average intake of calcium required by a given population subgroup (i.e., children age 1–3 years old) (IOM, 1997). The calcium requirement for children of ages one through three (500 mg of calcium/day) was used as the base or reference amount to normalize consumption by other population groups. That is, the calcium intake of the surveyed individuals was converted into an age- and gender-equivalent measure. Thus, the dependent variable is measured as a requirement-adjusted amount of calcium (mg) received from the foods consumed. For example, if a young child reported an intake of 600 mg per day of calcium, their reported intake of 600 mg would be measured relative to their DRI (500 mg) and converted to a 500 mg reference value of 600 mg: (=[600 mg/500 mg] 500 mg). For an adult with a DRI of 1,000 mg, an actual intake of 600 mg is converted to a requirement-adjusted intake of 300 mg (=[600 mg/1,000 mg] 500 mg). Table 2 lists a summary of the data for the total sample and for each of the four groups of interest observed at the individual and at the household levels. The individual-level controls that are used in the analysis include household income, household size, an indicator if an individual is currently receiving food stamps, an indicator if an individual is currently lactating or postpartum, and a set of dummies for age, main food preparer’s education level, urban residence, gender, and race. The household-level controls include household income, household size, an indicator for the presence of lactating or postpartum women in the household, an indicator for the presence of an infant, an indicator denoting if the household receives food stamps, and a set of dummies for the main food preparer’s education and race. In order to deal with the potential endogeneity of WIC program participation in our model, it is useful to have an instrument. This instrument must affect the household’s WIC participation decision but not be correlated with unobservables in the consumption equation. Our choice of instrument in this regard is to exploit state-level institutional characteristics of the WIC program in which the individual resides. Speciﬁcally, we make use of information regarding whether or not the state WIC program allows participants to self-declare their income in order to prove eligibility. Less strict states (i.e., those that allow individuals to self-declare) should generally be associated with higher participation rates in WIC. However, allowing households to self-declare income in order to establish WIC eligibility should play no structural role in calcium intake, conditioned on

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Table 2. Variable

Individual Number of individuals Milk (100 g) Income ($1,000) Household size Food stamp individual Pregnant/Lactating/ Postpartum individual Age College Urban White Male Household Number of households Income ($1,000) Household size Food stamp present College Children ages 1–5 Urban White Infant present Pregnant/lactating/ postpartum present Self-report income Savings o$5,000

Variables and Sample Mean Values. Sample

WIC

Non-WIC

Targeted

Nontargeted

Targeted

Nontargeted

2372 3.17 17.83 4.96 0.38 0.03

526 4.68 15.37 4.84 0.56 0.07

488 1.49 16.63 5.45 0.52 0.00

712 4.42 18.68 4.62 0.28 0.05

646 1.82 19.81 5.07 0.24 0.00

12.61 0.30 0.76 0.47 0.49

3.31 0.27 0.76 0.40 0.48

23.63 0.26 0.75 0.35 0.53

3.42 0.32 0.77 0.56 0.48

22.01 0.32 0.76 0.49 0.49

1036 17.19 4.56 0.44 0.30 0.94 0.76 0.60 0.20 0.15

15.07 4.64 0.60 0.27 0.98 0.75 0.54 0.25 0.20

15.21 4.85 0.54 0.29 0.76 0.73 0.51 0.42 0.16

18.69 4.48 0.33 0.33 0.99 0.78 0.64 0.08 0.12

18.72 4.71 0.34 0.34 0.98 0.78 0.61 0.08 0.11

0.16 0.94

0.16 0.98

0.17 0.95

0.16 0.91

0.15 0.91

WIC participation.9 We also make use of a second instrument, which is an indicator denoting if household savings are less than $5,000. Our argument in this regard is that families with little savings may be more likely to participate in WIC, while levels of asset accumulation should have little to do with calcium intake, conditioned on current income, WIC participation, education, and other demographic controls.

4. EMPIRICAL RESULTS Using the algorithm of Section 2, we ﬁt our model, running the Gibbs sampler for 100,000 iterations and discarding the ﬁrst 10,000 as the burn-in period.

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The prior hyperparameters used in the calculations are mg ¼ 0kg , Vg ¼ 100Ikg , muv=0, Vuv=100, ae=3, be=1/(2 0.3), aZ=3, and bZ=1/(2 0.3). Generated data experiments were also performed with large sample sizes to suggest that our code performs well and that our algorithm can adequately recover parameters of the data-generating process in these cases. Parameter posterior means, standard deviations, and probabilities of being positive associated with the model in Eqs. (5)–(9) are reported in Table 3. Table 3.

Posterior Means, Standard Deviations, and Probabilities of Being Positive.

Variable

E( |y)

Standard Deviation( |y)

Pr( W0|y)

Consumption equation Intercept G2 G3 G4 Household size Income ($1,000) Food stamp individual Pregnant/lactating/postpartum individual Age White Male College Urban

3.40 1.37 1.21 0.50 0.07 0.02 0.09 0.51 0.07 0.21 0.03 0.10 0.50

0.42 0.24 0.37 0.39 0.06 0.01 0.20 0.34 0.01 0.18 0.11 0.20 0.22

1.00 0.00 1.00 0.90 0.94 0.06 0.31 0.07 0.00 0.88 0.61 0.29 0.99

Participation equation Intercept Household size Income ($1,000) Infant present Food stamp present Pregnant/lactating/postpartum present College Children ages 1–5 White Urban Savings o$5,000 Self-report income

0.31 0.06 0.02 0.89 0.45 0.12 0.03 0.95 0.13 0.15 0.35 0.60

0.35 0.03 0.01 0.11 0.10 0.12 0.09 0.22 0.09 0.10 0.18 0.31

0.81 0.99 0.00 1.00 1.00 0.84 0.37 0.00 0.06 0.07 0.97 0.98

Covariance matrix and skew parameters r s2 s2u c

0.53 0.15 3.45 4.55

0.10 0.07 0.43 0.10

1.00 1.00 1.00 1.00

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With respect to WIC participation, the results shown in Table 3 are generally consistent with our prior expectations. Larger households with smaller incomes and infants present in the house are clearly more likely to participate in WIC. Similarly, our instruments appear to play strong roles in the WIC participation decision and operate in the direction that we expect a priori. That is, families living in regions where self-reports of income are more likely to provide sufﬁcient proof of WIC eligibility are associated with higher probabilities of WIC participation. Similarly, families with relatively small amounts of savings are also associated with higher probabilities of WIC participation. We also conduct a variant of the standard ‘‘overidentiﬁcation’’ test to investigate an aspect of the instrument’s validity. That is, conditioned on the assumption that self-report of income is a valid instrument, our savings indicator is superﬂuous in the sense that it is not needed for identiﬁcation. To this end, we re-estimate the model and include this variable in the latent calcium consumption equation. Doing this, we ﬁnd a posterior mean (and posterior standard deviation) associated with the Saving o$5,000 coefﬁcient equal to 0.15(0.32), and an associated posterior probability of being positive equal to 0.32. Thus, we do not see a strong role for our asset accumulation variable in the calcium consumption equation. Moreover, we calculate the relevant Bayes factor (via the Savage–Dickey density ratio) which, under equal prior odds and under the employed priors, gives the posterior odds in favor of the model imposing that (basset=0). The Bayes factor in this case turns out to be (approximately) 22.7, again providing the evidence that the asset accumulation variable can be omitted from the calcium consumption equation. With respect to calcium intake, few variables emerge as strong predictors. Individuals in larger households tend to consume more calcium through milk, while individuals in households with higher incomes tend to consume less. Of course, the most important of the coefﬁcients in Table 3 are the coefﬁcients associated with the group identiﬁers G2-G4.10 These ﬁndings ﬁrst suggest, quite sensibly, that nontargeted members living in WIC households (G2) have a lower (adjusted) calcium intake through milk than targeted members of WIC households (G1). Surprisingly, however, the results also suggest that non-WIC members, both targeted and nontargeted, receive more calcium intake through milk than their WIC counterparts. Although these results might seem startling, and potentially suggest that the WIC program is ineffective, this is not necessarily the correct interpretation. Individuals participating in WIC may, in fact, use the beneﬁts provided by the WIC program to purchase other products and

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receive an adequate level of calcium intake through the consumption of these alternative products. What the results do tell us, however, is that the WIC program does not appear to be effective at increasing calcium intake through milk. In short, the coefﬁcients associated with the group identiﬁers do not necessarily call into question the effectiveness of the WIC program, but at the same time, and unlike past studies in the literature, they cannot be used to speak to its virtues. At a minimum, we ﬁnd that the presence of the WIC program leads to repackaging of consumption bundles and a substitution away from milk consumption toward other possible foods providing calcium. To our knowledge, these results represent a new contribution to the existing literature on this topic. Table 3 also shows signiﬁcant evidence of skew through positive values associated with the skewness parameter c. The table also shows, quite interestingly, a large, positive value associated with the correlation parameter r. This suggests, consistent with our prior views, that unobservable factors making a family more likely to participate in WIC also lead that family to consume higher levels of calcium through milk. Table 4 presents posterior predictive means and standard deviations associated with calcium intake levels, as described in Section 2.3, while Fig. 2 plots the entire posterior predictive calcium distributions for each of the four groups. When performing these calculations, we set the continuous covariates at sample mean values speciﬁc to the ‘‘targeted’’ or ‘‘nontargeted’’ populations. (Setting age, for example, to the overall mean of 12.6 would seem inconsistent with both the targeted and nontargeted populations, leading us to set the covariates to group-speciﬁc means for this exercise). For the binary indicators, we round the targeted-/nontargetedspeciﬁc sample means to either zero or one. Since these posterior predictive densities account for both the ‘‘structural’’ impacts of WIC participation as well as the role of unobserved confounding, we would expect these predictives to match, to a reasonable degree, the

Table 4. Posterior Predictive Statistics Associated with Calcium Intake for Four Groups. Group G1 G2 G3 G4

E( |y)

Standard Deviation ( |y)

470 192 387 183

338 338 325 325

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2

x 10−3 G1 G2 G3 G4

1.8 1.6 1.4

Density

1.2 1 0.8 0.6 0.4 0.2 0

0

Fig. 2.

500 1000 Calcium Intake (mg)

1500

Predictive Posterior Intake Distributions for Four Groups.

means found in the raw data. A comparison of the entries of Tables 2 and 4 shows that (approximately) the case-targeted members of WIC households and targeted members of non-WIC households have the highest levels of calcium intake with posterior means equal to 470 and 387 mg, respectively. Similarly, nontargeted WIC and non-WIC members have lower levels of adjusted calcium intake with posterior means equal to 192 and 183, respectively, which is also broadly consistent with the mean intake levels found in the raw data. Fig. 2 and Table 4 also offer little evidence in favor of the potential ‘‘leakage’’ or ‘‘spillover’’ beneﬁts associated with the WIC program; the posterior predictives for the nontargeted WIC (G2) and nontargeted non-WIC (G4) individuals are very similar and nearly indistinguishable in Fig. 2. Finally, the posterior standard deviations of Table 4 and plots in Fig. 2 also reveal considerable heterogeneity associated with the calcium intakes for each of these four groups, with targeted individuals associated with the highest levels of uncertainty. While inspection of just the ‘‘structural’’ WIC coefﬁcients in Table 3 would appear to suggest that targeted non-WIC individuals will have more calcium intake through milk than targeted WIC individuals, the posterior

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predictives tell a different story. Like the raw data, these posterior predictives reveal that targeted WIC individuals will have the highest levels of calcium intake through milk. What is responsible for this ﬁnding is the role of unobserved correlation – those families that select into WIC possess unobserved factors that also strongly correlate with calcium intake. This ﬁnding is broadly consistent with the idea that the families participating in WIC, holding all else constant, also take great care in the nutritional intakes of their children and thus would likely consume relatively high levels of calcium even in the absence of WIC. What we have offered in this paper, which to our knowledge is new to this literature, is a model that seeks to separate the inﬂuences of unobservables and direct ‘‘structural’’ impacts. When combining these inﬂuences, we generate predictions that are consistent with the raw data and the ﬁndings of past work on this topic. When separating them, we produce no direct evidence that WIC itself is responsible for increases in calcium intake and improved overall nutrition. Again, we must interpret this ﬁnding with care, as it is certainly possible that the WIC program leads individuals to substitute away from traditional consumption bundles and meet necessary nutritional requirements through other foods. If true, this result does not seem to have been documented in the literature and has important implications for designing efﬁcient mechanisms for achieving desired nutrient intake levels.

5. CONCLUSION In this paper we have described a Bayesian posterior simulator for ﬁtting a two-equation treatment–response model and employed this method to investigate the effectiveness of a widely used food assistance program. This program, commonly denoted as WIC, seeks to improve the nutrition of at-risk, low-income children and pregnant/breastfeeding mothers. We evaluate this program by focusing on calcium intake through milk consumption and comparing such intake levels across WIC and non-WIC households and individuals. Though this metric is, admittedly, rather narrow, we also recognize that adequate calcium intake is one of the primary focuses of the WIC program, and milk is a primary vehicle through which calcium is consumed. Overall, we ﬁnd little direct evidence that speaks to the efﬁcacy of WIC. Instead, most of the beneﬁts that might potentially be attributed to the program seem to arise from differences in unobservables across WIC and non-WIC families. Furthermore, we ﬁnd little evidence associated with

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possible ‘‘spillover’’ or ‘‘leakage’’ beneﬁts that have been suggested in the literature, as nontargeted members of WIC households have consumption patterns that are quite consistent with nontargeted members of non-WIC households. We must interpret our results with caution, however, as it remains possible that WIC beneﬁts lead individuals to substitute away from milk and toward other goods that also provide adequate nutrition. To our knowledge, no studies in the area have attempted to separate the effects of unobservables and direct impacts, yet doing so has clearly been quite important in the context of our application.

NOTES 1. We follow standard conventions of using capital letters to denote matrix quantities and bold script to denote vectors or matrices. 2. Here G1 (targeted individuals participating in WIC) is the excluded category. 3. For more on related posterior simulators for such models, see Koop and Poirier (1997), Chib and Hamilton (2000, 2002), Poirier and Tobias (2003), and Chib (2007). 4. Note that, unlike adopting the log speciﬁcation, the model in Eq. (5) introduces skew without having to address potential issues such as taking the log of negative values (and simultaneously considering the mass point at zero consumption), or introducing an additional ‘‘hurdle’’ or ‘‘threshold’’ to the analysis. This representation is, of course, not as ﬂexible as other alternatives such as Gaussian mixtures or Dirichlet processes, but is a simpler alternative that may be adequately ﬂexible to capture the salient features of a given problem. 5. This issue has been pointed out by Pewsey (2000) and others. 6. In generated data experiments, this simple transformation seemed to improve the mixing of the posterior simulations. 7. Infants of age less than one year old are not included in the analysis because of their unique dietary requirements and intakes. 8. For reference, one cup of liquid milk is set equal to 244 g. 9. Owing to conﬁdentiality concerns, our data do not provide state identiﬁers but do provide region identiﬁers. To this end, we obtain an average of state policies within each region, using the fraction of WIC participants in state s within region r to weight the policy associated with state s. This instrument is not ideal but should still provide some overall degree of conditional correlation with WIC participation to aid identiﬁcation. Empirically, we ﬁnd that this is the case. 10. Given that G1 (targeted individuals participating in WIC) represents the excluded category, the coefﬁcients on G2-G4 should be interpreted relative to this base group.

ACKNOWLEDGMENT We acknowledge partial ﬁnancial support from the USDA/Economic Research Service/FANRP Small Grants Program, through the University of

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California, Davis. We thank an anonymous referee for comments on an earlier draft of the paper. All errors remain our own. The Matlab code for this paper is available upon request.

REFERENCES Albert, J., & Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88, 669–679. Azzalini, A., & Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika, 83, 715–726. Barrett, C.B. (2002). Food security and food assistance programs. In: B. Gardner, & G. C. Rausser (Eds). Handbook of Economics 18, Handbook of Agricultural Economics (2B, pp. 2103–2190). Amsterdam: Elsevier. Branco, M. D., & Dey, D. K. (2002). Regression model under skew elliptical error distribution. Journal of Mathematical Sciences, 1, 151–169. Burstein, N. R., Fox, M. K., Hiller, J. B., Kornfeld, R., Lam, K., Price, C., & Rodda, D. T. (2000). Proﬁle of WIC children. Cambridge, MA: Abt Associates, Inc. Prepared for USDA/Food and Nutrition Service. Casella, G., & George, E. (1992). Explaining the Gibbs sampler. The American Statistician, 46, 167–174. Chen, M.-H., Dey, D., & Shao, Q.-M. (1999). A new skewed link model for dichotomous quantal response data. Journal of the American Statistical Association, 94(448), 1172–1186. Chib, S. (1992). Bayes regression for the tobit censored regression model. Journal of Econometrics, 51, 79–99. Chib, S. (2007). Analysis of treatment response data without the joint distribution of potential outcomes. Journal of Econometrics, 140, 401–412. Chib, S., & Carlin, B. (1999). On MCMC sampling in hierarchical longitudinal models. Statistics and Computing, 9, 17–26. Chib, S., & Hamilton, B. (2000). Bayesian analysis of cross section and clustered data treatment models. Journal of Econometrics, 97, 25–50. Chib, S., & Hamilton, B. (2002). Semiparametric Bayes analysis of longitudinal data treatment models. Journal of Econometrics, 110, 67–89. Gelfand, A. E., Hills, S., Racine-Poon, A., & Smith, F. M. (1990). Illustration of Bayesian inference in normal data models using Gibbs sampling. Journal of the American Statistical Association, 85, 972–985. IOM. (1997). Dietary reference intakes for calcium, phosphorus, magnesium, vitamin D, and ﬂuoride. Washington, DC: National Academy Press. IOM. (2002). Dietary risk assessment in the WIC program. Washington, DC: National Academy Press. Koop, G., & Poirier, D. J. (1997). Learning about the cross-regime correlation in switching regression models. Journal of Econometrics, 78, 217–227. Oliveira, V. (2007). The food assistance landscape: FY 2006 annual report. Economic Information Bulletin 6–2, USDA, ERS.

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CAUSAL EFFECTS FROM PANEL DATA IN RANDOMIZED EXPERIMENTS WITH PARTIAL COMPLIANCE Siddhartha Chib and Liana Jacobi ABSTRACT We present Bayesian models for ﬁnding the longitudinal causal effects of a randomized two-arm training program when compliance with the randomized assignment is less than perfect in the training arm (but perfect in the non-training arm) for reasons that are potentially correlated with the outcomes. We deal with the latter confounding problem under the principal stratiﬁcation framework of Sommer and Zeger (1991) and Frangakis and Rubin (1999), and others. Building on the Bayesian contributions of Imbens and Rubin (1997), Hirano et al. (2000), Yau and Little (2001) and in particular Chib (2007) and Chib and Jacobi (2007, 2008), we construct rich models of the potential outcome sequences (with and without random effects), show how informative priors can be reasonably formulated, and present tuned computational approaches for summarizing the posterior distribution. We also discuss the computation of the marginal likelihood for comparing various versions of our models. We ﬁnd the causal effects of the observed intake from the predictive distribution of each potential outcome for Bayesian Econometrics Advances in Econometrics, Volume 23, 183–215 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23006-9

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compliers. These are calculated from the output of our estimation procedures. We illustrate the techniques and ideas with data from the 1994 JOBS II trial that was set up to test the efﬁcacy of a job training program on subsequent mental health outcomes.

1. INTRODUCTION We present Bayesian models for ﬁnding the longitudinal causal effects of a randomized two-arm training program when compliance with the randomized assignment is less than perfect in the training arm (but perfect in the non-training arm) for reasons that are potentially correlated with the outcomes. We deal with the latter confounding problem under the principal stratiﬁcation framework of Sommer and Zeger (1991) and Frangakis and Rubin (1999), further discussed and applied in Imbens and Rubin (1997), Hirano, Imbens, Rubin, and Zhou (2000), Jo (2002), Yau and Little (2001), Ten Have, Joffe, and Cary (2003), Frangakis et al. (2004), Levy, O’Malley, and Normand (2004), and Mealli, Imbens, Ferro, and Biggeri (2004). In this framework, as explained in detail in Chib and Jacobi (2008), the confounder is assumed to be a (partially observable) latent variable that represents subject type, where subject type can take one of the four values – complier, never-taker, always-taker, and deﬁer – deﬁned in terms of the potential intake for each level of the assignment. Under certain assumptions, most importantly, the absence of always-takers (because these cannot be identiﬁed in our partial compliance setup where subjects in the control arm have no possibility of getting the training), the absence of deﬁers (the monotonicity assumption), and the exclusion restriction (that the assignment variable is a proper instrumental variable that has no direct affect on the outcomes), it becomes possible to ﬁnd the effect of the actual intake on the outcome for the subclass (or strata) of compliers. In this paper we discuss how this framework can be modiﬁed to the case of panel outcomes. We take a Bayesian approach because there is much that the Bayesian perspective can offer in this context, following the developments reported in Chib (2007) and Chib and Jacobi (2007, 2008). In particular, the Bayesian perspective offers the means to develop rich (parameter-heavy) models of the potential outcomes conditioned on subject type. In this modeling it is also possible to include random effects that vary by subject type. One reason that it is possible to specify rich models of the potential outcomes is because one can include prior information about the parameters in the analysis. For instance, we discuss how information from

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another sample of subjects can be used to formulate beliefs about the timevarying (intake and subject-type speciﬁc) regression coefﬁcients and intake and subject-type speciﬁc covariance matrices. Another reason that the Bayesian perspective is helpful is because it provides a well-established way of dealing with the mixture model that emerges for subjects in the control arm (mixed over the two possible types of subjects in that case, compliers and never-takers). Mixture models are particularly well handled from the Bayesian perspective by simply including the latent subject type of each subject as an additional parameter in the prior-posterior analysis. The labelswitching problem that arises in mixture models does not occur in this problem because subject type (under our assumptions) is observed for subjects in the treatment arm who forgo the treatment (these being the never-takers) and for those in the treatment arm who take the treatment (these being the compliers). In contrast, frequentist ﬁtting of the same model is more difﬁcult because mixture models (even with latent type partially observed for some subjects) are not as easy to deal with, especially when there are many parameters (as in our problem) and random effects. Yet another appeal of the Bayesian approach is that it provides the means to calculate the causal effect from a predictive perspective. This perspective is particularly helpful because it leads to various summaries of the causal effects, for instance quantile casual effects, that are not as easily obtained by either a non-predictive formulation or non-Bayesian methods. Finally, the Bayesian approach provides a coherent procedure for comparing various versions of our models through the computation of marginal likelihoods and Bayes factors. We use this method to compare two versions of our panel data causal models, one with random effects and one without. Comparisons of this type are more difﬁcult from the frequentist tradition. The only previous discussion of the principal stratiﬁcation framework in the panel context is by Yau and Little (2001). This paper is also from the Bayesian perspective and is motivated by the same data that we analyze in this paper. But apart from those connections, the treatment in this paper is different on the following dimensions: 1. Modeling: Our modeling of the potential outcome allows for subject and time speciﬁc shocks, whereas the modeling in Yau and Little does not. In their case, therefore, there is no issue about modeling the joint distribution of the potential outcomes since the potential outcomes are generated from the same shocks. In our case, this issue is relevant. However, the recent work of Chib (2007) has shown that the joint distribution of the potential outcomes does not have to be modeled in

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3.

4.

5.

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causal models. This leads to a considerable simpliﬁcation in the modeling especially in the context of panel data and type speciﬁc distributions where the joint distribution of the potential outcomes can be very high dimensional. This complication can be bypassed as we show here and simpliﬁes both the modeling and the subsequent estimation of the model. This same point of Chib (2007) is utilized to advantage in the panel data model of Chib and Jacobi (2007). Prior: Whereas Yau and Little (2001) use diffuse, improper priors, we adopt informative priors that are constructed in a reasoned way from another sample of subjects that were exposed to the same experiment. Random effects: We propose and estimate models with random effects. Such models were not analyzed by Yau and Little (2001) but are natural in the context of panel data for dealing with individual speciﬁc inﬂuences. Inference: Although we also proceed by Bayesian means, and summarize the posterior distribution by MCMC methods, the actual ﬁtting approaches we develop are quite different from those used in Yau and Little (2001). Model comparisons: Unlike Yau and Little (2001), we go beyond the problem of estimation and consider the question of model comparisons by marginal likelihoods and Bayes factors that we estimate by the method of Chib (1995). Causal effects: Finally, our calculation of the causal effects is different and is based on a predictive perspective that provides a more complete summary of these effects than the complier-average causal effects that are reported by Yau and Little (2001).

The rest of the paper is organized as follows. In Section 2 we brieﬂy discuss the data set that we analyze in this paper. This helps to ﬁx the context for the developments we then provide in the remainder of the paper. In Section 3 we present the Bayesian formulation of the principal stratiﬁcation approach for the panel context and describe two models that we think are useful. For each model, we also discuss our prior distribution. In Section 4 we discuss how the posterior distribution from each of our models can be summarized by MCMC methods, and how the marginal likelihood of the models can be computed. We then present results for various versions of our models that are deﬁned by different assumptions about the error distributions. Section 5 deals with our predictive approach for calculating the causal effects, while Section 6 has our conclusions. Details of the ﬁtting methods are given in the appendix.

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2. DATA As motivation for the model and problem we are going to consider, we discuss a data set that we will analyze below. The data comes from the 1994 JOBS II trial that was set up to test the efﬁcacy of a job training program (see Vinokur, Price, & Schul, 1995 for a detailed description) on subsequent mental health outcomes. In the experiment, recently unemployed subjects were randomized to participate in a job training program with speciﬁc components to promote self-esteem and sense of control, job search skills and inoculation against setbacks. Those randomized into the control arm of the experiment received a booklet on job search skills that was also distributed among the treatment arm subjects after the training program. One question to be addressed by the trial was whether a training program can alleviate the negative mental health effects that are commonly associated with job loss (Clark & Oswald, 1994). The mental health of all subjects was evaluated through questionnaires at the start of the experiment and then again 2 months, 6 months, and 2 years after the start of the experiment. Subjects rated various stress symptoms from an 18-item index, each on a scale from 1 to 5. This information was used to construct a continuous outcome variable for the change in the mental health over time, measured in terms of the change in the depression score at each follow-up period compared to the baseline score. Table 1 gives the sample means and standard deviations for the changes in depressions scores for the three periods. The table also provides a sample summary in terms of other health related variables and personal characteristics Table 1. Sample Means and Standard Deviations of Our Study Data from the JOBS II Intervention Project. Variable

Explanation

Mean

Standard Deviation

y1 y2 y3 Depress0 Risk0 Age Motivate Edu Assert Marr Econ Nonw

Change in depression score (t ¼ 1) Change in depression score (t ¼ 2) Change in depression score (t ¼ 3) Baseline depression score Baseline risk score Age in years Motivation to attend School grade completed Assertiveness Marriage indicator Economic hardship Indicator for non-white

0.36 0.47 0.49 2.44 1.67 37.16 5.30 13.43 2.98 0.60 3.54 0.17

0.71 0.76 0.78 0.30 0.21 10.27 0.80 2.05 0.91 0.87

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that are used in the modeling of the outcome data. The information in the table refers to a sample of 387 subjects that were classiﬁed as being at high risk of depression at the start of the experiment and were observed in all follow-up periods. We have excluded subjects with a low risk of depression since no training effects were found for this group in previous studies.

3. BAYESIAN MODELING We need the following notation. For each subject i (irn) in the sample, let zi ¼ l (l ¼ 0, 1) denote the random assignment indicator, with l ¼ 0 indicating assignment into the no-training or control arm and l ¼ 1 assignment into the training or treatment arm. xli ¼ j ( j ¼ 0, 1) denote the potential intake when zi ¼ l, with j ¼ 0 indicating the no-training intake and j ¼ 1 indicating receipt of training. xi ¼ j ( j ¼ 0, 1) denote the actual intake given by xi ¼ x0i ð1 zi Þ þ x1i zi In the partial compliance setup we are dealing with, xi ¼ 0 if zi ¼ 0, whereas xi can be 0 or 1 when zi ¼ 1. yji ¼ (yji1, yji2, yji3) denote the vector of potential outcomes when the treatment intake at baseline is j. yi ¼ (yi1, yi2, yi3) denote the actual response given by yi ¼ y0i ð1 xi Þ þ y1i xi Now let si be an unobserved binary confounder that takes the values kA{0, 1}, where k ¼ 0 represents a never-taker and k ¼ 1 a complier. Formally, a subject is a never-taker if x0i ¼ x1i ¼ 0, and a complier if x0i ¼ 0 and x1i ¼ 1. Under the assumption that no other subject types exist, a person with (zi ¼ 0, xi ¼ 0) is either a never-taker or a complier, a person with (zi ¼ 1, xi ¼ 0) is a never-taker, and a person with (zi ¼ 1, xi ¼ 1) is a complier. Table 2 gives the distribution of these types in our sample by assignment and intake. Only 159 of the 260 subjects randomized into the treatment actually participated in the training program. These numbers reﬂect the general compliance problem that is common in such trials. Following Chib and Jacobi (2008), the modeling of this problem requires a speciﬁcation of the joint distribution pðyi ; xi ¼ jjWi ; zi ¼ l; si ¼ kÞ pðyji ; xi ¼ jjWi ; zi ¼ l; si ¼ kÞ

(1)

¼ pj ðyi jWi ; si ¼ kÞPrðxi ¼ jjyij ; Wi ; zi ¼ l; si ¼ kÞ

(2)

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Table 2.

Distribution of the Sample Subjects and their Types by Treatment Assignment and Intake.

Control arm z ¼ 0 Treatment arm z ¼ 1

No Training x ¼ 0

Training x ¼ 1

n00 ¼ 127 (compliers and never-takers) n10 ¼ 101 (never-takers)

– n11 ¼ 159 (compliers)

where pj(yi|Wi, si ¼ k) is the density of yji conditional on the latent subject type and the second term is the conditional mass function of xi ¼ j. The former density does not involve zi ¼ l on account of the so-called exclusion restriction. Notice, too, that the second term is either 0 or 1, for any value of yi or Wi. For example, if zi ¼ 0 and si ¼ 1, then xi ¼ 0, so that Pr(xi ¼ 0|yi, Wi, zi ¼ 0, si ¼ 1) ¼ 1. In addition, if zi ¼ 1 and si ¼ 0, then xi ¼ 0, implying that Pr(xi ¼ 0|yi, Wi, zi ¼ 1, si ¼ 0) ¼ 1. Thus, given zi ¼ l and si ¼ k, the intake is fully determined. It is important to keep in mind that there is no need to model the joint density pðy0i ; y1i ; xi ¼ jjWi ; zi ¼ l; si ¼ kÞ which is actually unidentiﬁed because the potential outcomes (y0i, y1i) are not observed simultaneously. That the modeling and subsequent estimation of the model can proceed without this joint distribution is due to Chib (2007). To model the joint density of the outcome and the intake, let Ilj ¼ {i: zi ¼ l and xi ¼ j} denote the sample indices of the subjects in each of the three non-empty cells of Table 2. Also, let Pr(si ¼ 1|vi) ¼ qci denote the probability that a subject is of type c, which we assume is a function of the q 1 vector of pre-treatment variables vi that is a subset of Wi. This probability is independent of zi because of the random assignment of subjects to the treatment arms. However, since we do not observe the subject type in the control arm, the joint density of yi and xi ¼ j conditional on zi ¼ l is given by appropriately averaging over possible types: 8 ð1 qci Þp0 ðyi jWi ; si ¼ 0Þ þ qci p0 ðyi jWi ; si ¼ 1Þ if i 2 I 00 > < ð1 qci Þp0 ðyi jWi ; si ¼ 0Þ if i 2 I 10 pðyi ; xi ¼ jjWi ; zi ¼ lÞ ¼ > : q p ðy jW ; s ¼ 1Þ if i 2 I 11 i i ci 1 i

(3) This expression does not involve the mass function of the intake due to the discussion surrounding Eq. (1). Note also that zi neither appears in the conditioning set of the exogenous type probability due the randomization

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argument nor in that of the outcome distribution due to the exclusion restriction. From expression (3) we see that the modeling of (yi, xi ¼ j) requires three type and treatment state speciﬁc multivariate distributions for the health outcomes, p0(yi|Wi, si ¼ 0) and pj(yi|Wi, si ¼ 1), for j ¼ 0, 1, and a model for the type probabilities qci. In the next section we introduce two model speciﬁcations that are based on different formulations for the intake and type speciﬁc distributions of the health outcomes. In each case we assume that the probability of being a complier, qci, is generated by a probit model. Previous papers such as Hirano et al. (2000), Jo (2002), Frangakis et al. (2004), and Chib and Jacobi (2008) have found that it is important to model the compliance probability in terms of baseline predictors. Here we follow Yau and Little (2001) and let qci ¼ Fðw0i0 aÞ where wi0 ¼ (1, Age, Edu, Marr, Nonw, Assert, Motivate, Econ). Our modeling is completed with a prior distribution on the parameters of the preceding distributions. We use standard distributional forms to compose the prior distribution. For example, we choose the normal distribution for the regression parameters and the Wishart distribution for the covariance matrices. A challenging component of the prior speciﬁcation is the choice of hyperparameters. We deal with this problem by constructing a prior distribution that is reasonable for the sample of low-risk subjects that is excluded from our analysis. Our strategy is to set the hyperparameters, sample the prior, and then simulate the outcome distributions. We do this many times and see whether the simulated distribution of the outcomes is similar to the empirical distribution of the outcomes in the lowrisk sample. If the two distributions are quite different, we revise our hyperparameters somewhat and repeat the process.

3.1. Model 1 One choice (which we call Model 1) is to let p0 ðyi jWi ; si ¼ 0Þ ¼ tn ðyi jWi b0n ; X0n Þ pj ðyi jWi ; si ¼ 1Þ ¼ tn ðyi jWi bjc ; Xjc Þ;

j ¼ 0; 1

ð4Þ

where b0n and bjc are intake and type speciﬁc regression parameters, O0 and Ojc are the corresponding full (3 3) dispersion matrices and tn( |m, O) is the

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multivariate Student’s t density function with n degrees of freedom, mean m, and variance matrix nO/(n2), n W2. Equivalently, under the common representation of the Student’s t distribution as a scale mixture of normal distributions, the latter model can be expressed as p0 ðyi jWi ; si ¼ 0; li Þ ¼ N ðyi jWi b0n ; l1 i X0n Þ pj ðyi jWi ; si ¼ 1; li Þ ¼ N ðyi jWi bjc ; l1 i Xjc Þ;

j ¼ 0; 1

ð5Þ

where li is distributed as gamma

n n li G ; 2 2

In our application, we parameterize the matrix Wi in a way to allow for time-varying effects for each of the covariates: 1 0 1 depressi0 riski0 0 0 0 0 0 0 B 0 0 C 0 0 1 depressi0 riski0 0 Wi ¼ @ 0 A 0 0 0 0 0 0 1 depressi0 riski0 and denote the covariate vector as bjk ¼ (bjk,1, bjk,2, bjk,3): 9 1 so that bjk,1 is the effect of the three predictors in the ﬁrst time period, bjk,2 in the second time period, and bjk,3 in the third time period. We specify the prior distribution for the vector of model parameters h as pðhÞ ¼ N p ðaja0 ; A0 Þ

1 Y Y

N 3k ðbjk jbjk;0 ; Bjk;0 ÞWðX1 jk jrjk;0 ; Rjk;0 Þ

(6)

j¼0 k2K j

and ﬁx the prior means for bjk and a: b0c;0 ¼ ð0:8; 1:1; 0:8; 1:5; 1:5; 1:0; 1:4; 1:2; 0:6Þ b0n;0 ¼ ð0:7; 1:1; 0:8; 1:5; 1:5; 1:0; 1:2; 1:2; 0:6Þ b1c;0 ¼ ð0:8; 1:1; 0:8; 0:6; 1:5; 1:0; 1:6; 1:2; 0:6Þ a0 ¼ ð5; :03; :5; :1; 0; 0; 0; 0Þ As mentioned above we want a prior distribution that generates outcomes that are reasonable in relation to those seen in the low-risk sample. For this, we set Bjk,0 ¼ 9I and A0 ¼ 9I and set the hyperparameters of the Wishart prior for X1 jk to imply a full covariance matrix with 0.5 on the diagonal and 0.25 for all off-diagonal elements. This seems a reasonable choice given our outcome variable that is measured in terms of the change in depression scores, each restricted to values between 0 and 5. To show what these

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Table 3. Model 1 – Means and Quantiles of the Empirical Distribution of the Change in Depression Score Implied by the Assumed Prior.

t¼1 t¼2 t¼3

ys0:05

ys0:25

ys0:50

ys0:75

ys0:95

6.74 6.94 6.63

3.89 4.05 3.75

0.59 0.90 0.47

2.73 2.44 2.77

5.48 5.12 5.47

assumptions imply for the outcomes, we provide in Table 3 the lower and upper quantiles of the simulated outcome distributions under this prior. We conclude that our prior assumptions are reasonably ﬂexible. 3.2. Model 2 Another option, which we call Model 2, is to derive p0(yi|Wi, si ¼ 0) and pj( yi|Wi, si ¼ 1) from a random effects formulation. Let Vi be a 3 k (ko3) matrix of covariates whose effect on the outcome is individual speciﬁc. In this particular application, where all the covariates in the outcome model are measured at baseline, Vi is a vector of constants. To allow for ﬂexibility in the covariance structure, as in Model 1, we assume that the random effects are intake and type speciﬁc. We denote these by bi0c, bilc, and bin, one for compliers under no intake, another for compliers under treatment intake, and ﬁnally one for never-takers. Conditioned on the random effects, we now let p0 ðyi jWi ; si ¼ 0; bin Þ ¼ tn ðyi jWi b0n þ Vi bin ; diagðs0n ÞÞ pj ðyi jWi ; si ¼ 1; bijc Þ ¼ tn ðyi jWi bjc þ Vi bijc ; diagðsjc ÞÞ;

j ¼ 0; 1

ð7Þ

where the dispersion matrices are in diagonal form for identiﬁcation reasons. Once again with the introduction of positive latent scale variables li Gðn=2; n=2Þ, we can express this model as p0 ðyi jWi ; si ¼ 0; bin ; li Þ ¼ N 3 ðyi jWi b0n þ Vi bin ; l1 i diagðs0n ÞÞ pj ðyi jWi ; si ¼ 1; bijc ; li Þ ¼ N 3 ðyi jWi bjc þ Vi bijc ; l1 i diagðsjc ÞÞ;

j ¼ 0; 1

If we now assume that the random effects are distributed as bijc jDc N k ð0; Djc Þ; bin jDn N k ð0; Dn Þ

j ¼ 0; 1

ð8Þ

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where the matrices Djc and Dn are unknown, it follows that marginalized over the random effects (but conditioned on li) the distributions of the outcome by intake and type are given by pj ðyi jWi ; si ¼ 1; li Þ ¼ N 3 ðyi jWi bjc ;

X

p0 ðyi jWi ; si ¼ 0; li Þ ¼ N 3 ðyi jWi b0n ;

jc

X

0 ¼ fl1 i diagðsjc Þ þ Vi Djc Vi g

0n

0 ¼ fl1 i diagðs0n Þ þ Vi Dn Vi gÞ

whereas marginalized over li these are Z

n n N 3 yi jWi bjc ; Rjc G li ; dli 2 2 Z0 1 n n p0 ðyi jWi ; si ¼ 0Þ ¼ N 3 yi jWi b0n ; R0n G li ; dli 2 2 0 1

pj ðyi jWi ; si ¼ 1Þ ¼

ð9Þ

which differ from the ones in Eq. (4). As in Model 1 we specify the prior distribution of the model parameters p(h) as N p ðaja0 ; A0 Þ

1 Y Y j¼0 k2K j

! njk;0 djk;0 ; WðD1 N 3k ðbjk jbjk;0 ; B0 Þ I G sjk;t jk jrjk;0 ; Rjk;0 Þ 2 2 t¼1 3 Y

where K0 ¼ {c, n} and K1 ¼ {c}. The prior means and variances of a and bjk are ﬁxed at the same values as in Model 1, so that b0c;0 ¼ ð0:0; 1:0; 0:6; 1:0; 1:5; 1:0; 1:8; 1:2; 0:6Þ b0n;0 ¼ ð0:0; 1:0; 0:6; 0:8; 1:5; 1:0; 0:5; 1:2; 0:6Þ b1c;0 ¼ ð0:0; 1:0; 0:6; 0:7; 1:5; 1:0; 0:5; 1:2; 0:6Þ a0 ¼ ð5; :03; :5; :1; 0; 0; 0; 0Þ and Bjk,0 ¼ 9I and A0 ¼ 9I. The parameters of the inverse gamma prior on the scalar variances are set to imply means and standard errors of 0.5 and 3, respectively (njk,0 ¼ 4.04, djk,0 ¼ 1.03). Finally, for the Wishart prior on the inverse of the variances of the random effects we let rjk,0 ¼ 5 and Rjk,0 ¼ 0.66I, which implies a prior mean of 0.5I for Djk. As we had done in the case of Model 1, we simulate the outcomes under this prior. The resulting lower and upper quantiles of these outcome distributions are given in Table 4 and again appear to be reasonable and sufﬁciently ﬂexible.

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Table 4. Model 2 – Means and Quantiles of the Empirical Distribution of the Change in Depression Score Implied by the Assumed Prior.

t¼1 t¼2 t¼3

ys0:05

ys0:50

ys0:25

ys0:75

ys0:95

7.49 7.17 7.50

4.26 3.90 4.26

0.81 0.47 0.81

2.62 2.98 2.60

5.68 6.04 5.64

4. PRIOR-POSTERIOR ANALYSIS 4.1. Model 1 We now turn to the prior-posterior analysis of the ﬁrst model. Our modeling assumptions imply that p(y,x|h,{li}), the joint density of the observed health outcomes y ¼ {y1,y2, . . . ,yn} and training intake data x ¼ {x1,x2, . . . ,xn} given the vector of model parameters and the scale parameters, is of the form N n n Y Y G li ; ½ð1 Fðw0i0 aÞÞN T ðyi jWi b0n ; X0n Þ 2 2 i2I 00 i¼1 Y þ Fðw0i0 aÞN T ðyi jWi b0c ; X0c Þ ð1 Fðw0i0 aÞÞN T ðyi jWi b0n ; X0n Þ

Y

i2I 10

Fðw0i0 aÞN T ðyi jWi b1c ; X1c Þ

ð10Þ

i2I 11

This joint density has three distinct components that correspond to the three non-empty cells in Table 2. The ﬁrst term gives the likelihood contributions for the 127 subjects in the control arm, while the second and the third product terms provide the likelihood contributions for the 101 never-takers and the 159 compliers in the treatment arm, respectively. As the type is not observed for the ﬁrst group, the likelihood contributions take the form of mixture distributions over compliers and never-takers. It may be noted that the mixture component is only present in the control arm since subject type is otherwise observed. We handle the mixture terms in the control arm by including the latent subject type of each subject as an additional parameter in the prior-posterior analysis. The label-switching problem that arises in mixture models does not occur in this problem because subject type is observed for subjects in the

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treatment arm who forgo the treatment and for those in the treatment arm who take the treatment. Let s00 denote the type indicators for control arm subjects. Then, our target posterior density of interest is p(h, s00, {li}|y, x, W, z), which is proportional to the prior density speciﬁed in Eq. (6) times the function N n n Y Y G li ; fI½si ¼ 0ð1 Fðw0i0 aÞÞN T ðyi jWi b0n ; l1 i X0n Þ 2 2 i2I i¼1 00

þ I½si ¼ 1Fðw0i0 aÞN T ðyi jWi b0c ; l1 i X0c Þg Y 0 I½si ¼ 0ð1 Fðwi0 aÞÞN T ðyi jWi b0n ; l1 i X0n Þ i2I 10

Y

I½si ¼ 1Fðw0i0 aÞN T ðyi jWi b1c ; l1 i X1c Þ

i2I 11

We summarize this density by tuned MCMC methods (see Chib & Greenberg, 1995 for details on the Metropolis–Hastings algorithm and Chib, 2001 for an extended summary of MCMC methods). The sampling scheme involves three blocks and is summarized next. Full details are supplied in the appendix. 1. Sample (s00, a, {li}|y, x, b, O) by sampling (a) si for iAI00 with Pr(si ¼ 1|yi, xi, b0c, b0n, a, O0c, O0n) (b) a|y, x, b, O, s00 with a Metropolis–Hastings step (c) li|yi, xi, b, O, s00 for iAN from a gamma density 2. Sample {bjk} by drawing bjk|y, x, s00, k, O, a from a normal density 1 3. Sample fX1 jk g by drawing Xjk jy, x, s00, k, b, a, D from a Wishart density In the ﬁrst block, to produce a well mixing chain, the type indicators s00, a and the scale parameters li are sampled jointly by the method of composition. It is also possible to proceed by sampling the si’s under the framework of Albert and Chib (1993). In the second block we update the coefﬁcients b0c, b0n, and b0n. Under our model setup, the bjk’s depend on distinct subsets of the population, (yjk: {xi ¼ j, si ¼ k}, xjk: {xi ¼ j, si ¼ k}). We can therefore sample bjk|yjk, xjk, s00, a, Ojk separately from their respective normal posterior distribution. We proceed in a similar fashion to update Ojk in the last block of the chain and sample X1 jk jyjk , xjk, s00, a, bjk from Wishart distributions.

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4.2. Model 2 For the posterior analysis of Model 2, we augment the parameter space with the random effects {bi0c}, {bi1c}, and {bin}. To improve the tractability of the posterior distribution further we follow the same strategy as in Model 1 and include the type indicators s00 ¼ {si: iAI00}, and the latent scale parameters k ¼ {li}. The posterior density of interest is then p(h, s00, {bijk}, {li}|y, x, W, z), which is proportional to the prior density times Y

I½si ¼ 0ð1 Fðw0i0 aÞÞN T ðyi jWi b0n þ Vi bin ; l1 i diagðs0n ÞÞN k ðbin j0; Dn Þ

i2I 00

þI½si ¼ 1Fðw0i0 aÞN T ðyi jWi b0c þ Vi bic ; l1 i diagðs0c ÞÞN k ðbi0c j0; D0c Þ Y I½si ¼ 0ð1 Fðw0i0 aÞÞN T ðyi jWi b0n þ Vi bin ; l1 i diagðs0n ÞÞN k ðbin j0; Dn Þ i2I 10

Y

I½si ¼ 1Fðw0i0 aÞN T ðyi jWi b1c þ Vi bic ; l1 i diagðs1c ÞÞN k ðbi1c j0; D1c Þ

i2I 11

N n n Y G li ; 2 2 i¼1

In the appendix we provide a detailed description of the MCMC algorithm we have developed to generate draws from the posterior distribution. One important point is that in Step 1a we sample the compliance indicators marginalized over the random effects, which avoids having to sample the complier and never-taker random effects for each subject in the control arm. This reduces the computational burden considerably and improves the mixing of the MCMC chain. A short version of the algorithm is given here: 1. Sample (s00, a, {bjk}, {bijk}|y, x, {sjk}, {li}, {Djk}) by sampling (a) si for iAI00 with Pr(si ¼ 1|yi, xi, b0c, b0n, a, s0c, s0n, {li}, {Djk}) (b) a|yjk, xjk, s00 by a MH step (c) bjk|y, x, s00, k, sjk, k, {Djk} from a normal density (d) bjki|yi, xi, bjk, s00, li, sjk, sjk, Djk for iAIjk from a normal density 2. Sample li|y, x, {bjk), {sjk}, a, s00, {bjk} for iAN from a gamma density 3. Sample sjk|y, x, s00, k, bjk, a, {bijk}, {Djk} from an inverse gamma density 4. Sample D1 jk jy, x, s00, k, bjk, a, {bijk}, sjk from a Wishart distribution

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4.3. Model Comparison In practice one would be interested in comparing Models 1 and 2 and variations of these models to see which model is best supported by the data. We do this comparison from the marginal likelihood/Bayes factor perspective. Following Chib (1995), the log marginal likelihood of a given model can be expressed in terms of the logs of the likelihood and the prior and posterior distribution evaluated at h as ln mðy; xÞ ¼ ln f ðy; xjz; W; hn Þ þ ln pðhn Þ ln pðhn jy; x; W; MÞ where h is a vector of the model parameters given by (say) the posterior mean. The prior ordinate at h for models 1 and 2 can, of course, be computed directly from the respective prior densities. The likelihood ordinate for Model 1 can also be computed directly from the expression in Eq. (4). However, the likelihood of Model 2, marginalized over {bi} and {li}, is not available in closed form. Since the likelihood contribution conditional on {li} is in closed form, we employ an importance sampling approach to get p j(yi|Wi, si ¼ k). We now turn to the estimation of the posterior ordinates. For Model 1, with the parameter vector h ¼ (b0c, b0n, b1c, O0c, O0n, O1c, a), we employ the decomposition pðhn jy; x; WÞ ¼ pðX1 n jy; x; WÞpðan jy; x; W; Xn Þpðbn jy; x; W; Xn ; an Þ where the ﬁrst expression can be obtained via Rao–Blackwell methods as 0 1 M X Y Y ðgÞ ðgÞ ðgÞ ðgÞ A @ ^ 1 jy; x; WÞ ¼ M 1 pðX p X1 jk jyjk ; xjk ; sjk ; Wjk ; a ; bjk ; kjk g¼1

j¼0;1 k2K j

For the second, ordinate we use the result from Chib and Jeliazkov (2001) that pðan jy; x; Xn Þ ¼

E1 ½aðan jy; x; b; Xn ; zÞqðan jy; x; Xn ; bÞ E2 ½aðan ; ajy; x; b; Xn ; zÞ

(11)

where the expectation E1 in the numerator is with respect to p(b, a|y, x, O) and the expectation E2 in the denominator is with respect to p(b|y, x, a, O) q(a|y, x, b, O). Each expectation can be estimated from the output of suitable reduced runs (Chib, 1995). To estimate the numerator, we ﬁx O at O and continue the MCMC iterations with the quantities hO and z ¼ (k, s00), and then average a(a, a|y, x, b, O, z)q(a|y, x, O, b) over the

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resulting draws. To estimate the denominator, we ﬁx (O, a) at (O, a) and continue the MCMC iterations; in each cycle of this run, we also draw a from q(a|y, x, O, b). We then average a(a, a|y, x, b, O, z) over the draws on (b, a) from this run. Simultaneously, from the output of the latter run we estimate p(b|y, x, O, a) as 0 1 M X Y Y ðgÞ ðgÞ @ ^ jy; x; WÞ ¼ M 1 p bnjk jyjk ; xjk ; sjk ; Wjk ; an ; Xnjk ; kjk A pðb g¼1

j¼0;1 k2K j

To estimate the posterior ordinate for Model 2, where h ¼ ({bjk}, {sjk},a, {Djk}), we proceed in a similar way using the decomposition n 1n 1n n n n n pðh jy; x; WÞ ¼ pðD1 0c ; D1c ; Dn jy; x; WÞpða jy; x; W; D0c ; D1c ; Dn Þ

pðbn jy; x; W; Dn0c ; Dn1c ; Dnn ; an Þ pðrn jy; x; W; Dn0c ; Dn1c ; Dnn ; an ; bn Þ 1 1 ^ 1 where pðD 0c ; D1c ; Dn jy; x; WÞ is estimated via Rao–Blackwell methods from

M 1

Y ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ p D1 jy ; x ; s ; W ; a ; b ; k ; b jk jk jk 0c jk jk jk jk

M X

i2N 0c

g¼1

Y ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ p D1 1c jyjk ; xjk ; sjk ; Wjk ; a ; bjk ; kjk ; bjk i2N1

Y ðgÞ ðgÞ ðgÞ ðgÞ ðgÞ p D1 n jyjk ; xjk ; sjk ; Wjk ; a ; bjk ; kjk ; bjk

!

i2N n

The reduced ordinates for a and b are updated in the same manner as in Model 1 in two reduced runs. Here the ﬁrst reduced run is done conditional on ðDn0c ; Dn1c ; Dnn Þ and the second reduced run, which also yields the posterior estimate of b, is done conditional on ðDn0c ; Dn1c ; Dnn ; bn Þ. A ﬁnal third reduced run with ðD0c ; D1c ; Dn ; a; bÞ ﬁxed at ðDn0c ; Dn1c ; Dnn ; an ; bn Þ is required ^ n jy; x; W; Dn0c ; Dn1c ; Dnn ; an ; bn Þ from to estimate the posterior ordinate pðr 0 1 M X Y Y ðgÞ A @ p r jk jyjk ; xjk ; sðgÞ M 1 jk ; Wjk ; a ; bjk ; kjk g¼1

j¼0;1 k2K j

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4.4. Results In this section we present the key results from ﬁtting Models 1 and 2 to the data on high-risk respondents. For each model we consider three different values of the degrees of freedom parameter (n ¼ 5, 10, 20) and the model with the highest marginal likelihood in each model class is then studied more intensively. All our results are based on 20,000 MCMC iterations following a burn-in of 1,000 iterations. Table 5 contains the estimated log marginal likelihoods for our six contending models. As can be seen, the models with n ¼ 5 provide the best ﬁt to the data. We now discuss the ﬁtting results in more detail. Table 6 summarizes the prior-posterior analysis for the covariance matrices in Model 1. One point to note is that our MCMC algorithm is well behaved as indicated by the low inefﬁciency factors that are reported P for all the parameters. The inefﬁciency factors are computed as 1 þ 2 Ll¼1 rk ðl Þ, where rk(l ) is the autocorrelation of the kth parameter at lag l and L is chosen as the value at which the autocorrelation function tapers off. The inefﬁciency factors approximate the ratio of the numerical variance of the posterior mean from the MCMC chain relative to that from hypothetical iid draws. As is evident from Tables 6–9, the inefﬁciency factors for the covariance and slope parameters of Model 1 (and 2) are small and in some case quite close to the ideal value of 1. An interesting point is that even though the prior on the covariances matrices Ojk is the same, the posterior mean of these matrices is quite different. In this connection it may be observed that the largest variances and covariances occur for compliers in the no-training state. To illustrate the differences we show image plots of the covariance matrices (see Fig. 1). To plot the posterior means we have used a gray scale that is set at black for 0 and white for 0.55. The results suggest that our extension of Yau and Little’s

Table 5.

Estimates of the Log Marginal Likelihoods for Models 1 and 2 for Different Degrees of Freedom.

Model

Degrees of Freedom n¼5

Log marginal likelihoods M1 1339.07 M2 1332.96

n ¼ 10

n ¼ 20

1345.11 1341.32

1362.13 1351.26

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SIDDHARTHA CHIB AND LIANA JACOBI

Table 6. Model 1 – Prior-Posterior Analysis for the Covariance Matrices: Prior Means, Posterior Means and Standard Deviations (in Parentheses). O0c

O0n

O1c

Prior

Post.

Ineff.

Prior

Post.

Ineff.

Prior

Post.

Ineff.

0.50

0.35 (0.08) 0.26 (0.08) 0.40 (0.10) 0.23 (0.08) 0.17 (0.08) 0.46 (0.11)

2.21

0.50

1.86

0.50

0.25

2.12

0.25

3.02

0.50

2.88

0.50

2.27

0.25

1.95

0.25

2.34

0.25

2.23

0.25

2.52

0.50

2.05

0.50

0.28 (0.04) 0.15 (0.03) 0.30 (0.04) 0.14 (0.03) 0.17 (0.03) 0.36 (0.05)

1.32

3.34

0.35 (0.05) 0.16 (0.04) 0.28 (0.04) 0.12 (0.03) 0.13 (0.03) 0.27 (0.04)

0.25 0.50 0.25 0.25 0.50

1.30 1.83 1.24 1.27 1.86

basic model to allow for type and treatment speciﬁc random shocks in the distributions of the health outcomes is useful in the context of these data. Table 7 summarizes the prior-posterior analysis for the elements in the diagonal covariance matrices sjk and the random effects variances Djk from the ﬁtting of Model 2. As in Model 1, we observe that the estimates vary by intake and type. For a better comparison of the results with Model 1 we consider the covariance matrix of the Student’s t outcome distribution marginalized over the random effects (see Eq. (9)). We obtain estimates of the posterior means (and standard deviations) by computing P 1 0 ¼ fl i diagðsjk Þ þ Vi Djk Vi g at each iteration of the MCMC algorithm jk;i for Model 2. The plots of the posterior means in Fig. 2 use a gray scale that is set to black at 0 and white at 0.9. As in the case of Model 1, the largest variances/covariances are observed for compliers under no training. In general, the random effects speciﬁcation yields higher variances than Model 1. We now turn to the inferences about the remaining parameters in Models 1 and 2. We focus on the coefﬁcients a, b0c, and b1c, which play a key role in the determination of the causal training effects on the mental health outcomes discussed in Section 5. Table 8 summarizes the prior-posterior analysis for a, b0c, and b1c from the ﬁtting of Model 1. For each parameter we report the prior and posterior means and standard deviations. We also provide the inefﬁciency factors as a measure of the autocorrelation of the

0.35(.05)

0.26(.08)

0.41(.10)

0.23(.08)

0.17(.08) (a) Ω0c

Fig. 1.

0.47(.11)

0.28(.04)

0.16(.04)

0.28(.04)

0.12(.03)

0.13(.03) (b) Ω0n

0.27(.04)

0.15(.03)

0.30(.04)

0.14(.03)

0.17(.03)

0.36(.05)

(c) Ω1c

Image Plots of the Posterior Means of the Covariance Matrices in Model 1. The Gray Scale is Set at Black for 0 and White for 0.5. The Posterior Standard Deviations are Given in Parentheses.

Causal Effects of Randomized Experiments with Partial Compliance

0.35(.08)

201

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SIDDHARTHA CHIB AND LIANA JACOBI

Table 7. Model 2 – Prior-Posterior Analysis for the Variance Parameters: Prior Means, Posterior Means and Standard Deviations (in Parentheses). Prior

sjk

0.50 0.50 0.50

Djk

0.50

j ¼ 0, k ¼ c

j ¼ 0, k ¼ n

j ¼ 1, k ¼ c

Post.

Ineff.

Post.

Ineff.

Post.

Ineff.

0.12 (0.04) 0.22 (0.07) 0.32 (0.09) 0.38 (0.09)

4.14

0.20 (0.03) 0.12 (0.02) 0.15 (0.03) 0.17 (0.03)

2.39

0.14 (0.02) 0.15 (0.03) 0.21 (0.04) 0.22 (0.04)

2.59

2.95 2.52 2.07

4.04 2.81 4.18

2.54 2.44 2.52

draws. The second column of the table gives the posterior inference on a. The reported posterior means imply a higher compliance probability for subjects that are older, more motivated to attend the program and better educated. Subjects that have a higher level of assertiveness, are married, and non-white and those who experience economic hardship are less likely to be a complier. Column 3 shows that all coefﬁcients are measured with low inefﬁciency factors. Columns 4 through 9 in the same table provide results for bjc. These parameters capture the interaction of the training intake with the coefﬁcients on the constant and the baseline depression and risk scores on the change in the depression scores in the subsequent three time periods. A comparison of the posterior means between b0c and b1c in columns 5 and 8 reveals that the actual training intake affects the health outcomes after controlling for unobserved confounders through subject type. The differences between the posterior means of compliers in both training intake states are especially pronounced in the last two time periods, shown in the last six rows of the table. The higher posterior standard deviations and inefﬁciency factors of b0c, as compared to b1c, reﬂect our earlier point that the parameters for compliers under no treatment are the most difﬁcult to estimate as they are identiﬁed from a mixture distribution. We report a similar set of results for Model 2 in Table 9. The posterior means and standard deviations for a reported in column 2 are almost identical with those reported for Model 1 in Table 8. One would expect this result as both models use the same probit speciﬁcation for the compliance probability. This is not the case for the coefﬁcients in the outcome

0.38(.09)

0.71(.33)

0.38(.09)

0.38(.09) (a) ∑0c

Fig. 2.

0.44(.25)

0.51(.44)

0.88(.48)

0.17(.03)

0.36(.26)

0.17(.03)

0.17(.03) (b) ∑0n

0.42(.34)

0.22(.04)

0.47(.27)

0.22(.04)

0.22(.04)

0.57(.38)

(c) ∑1c

Image Plots of the Posterior Means of the Covariance Matrices from Model 2. The Posterior Mean and Standard Errors (in Parentheses) are Given for Each Element.

Causal Effects of Randomized Experiments with Partial Compliance

0.58(.21)

203

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SIDDHARTHA CHIB AND LIANA JACOBI

Table 8. Model 1 – Prior-Posterior Analysis for the Coefﬁcient Vectors from the Compliance Probability and Outcome Models for Compliers: Prior Means, Posterior Means, Standard Deviations (in Parentheses) and Inefﬁciency Factors. a

b0c

b1c

Prior

Post.

Ineff.

Prior

Post.

Ineff.

Prior

Post.

Ineff.

0.5 (5.0) 0.03 (5.0) 0.5 (5.0) 0.1 (5.0) 0.0 (5.0) 0.0 (5.0) 0.0 (5.0) 0.0 (5.0)

3.39 (0.87) 0.04 (0.01) 0.42 (0.10) 0.13 (0.04) 0.26 (0.10) 0.22 (0.18) 0.03 (0.11) 0.25 (0.21)

2.13

0.8 (5.0) 1.1 (5.0) 0.8 (5.0) 1.5 (5.0) 1.5 (5.0) 1.0 (5.0) 1.4 (5.0) 1.2 (5.0) 0.6 (5.0)

0.91 (0.91) 1.00 (0.62) 0.71 (0.96) 0.77 (0.93) 1.42 (0.58) 1.39 (0.82) 1.87 (0.89) 1.25 (0.67) 0.52 (0.93)

3.51

0.8 (5.0) 1.1 (5.0) 0.8 (5.0) 0.6 (5.0) 1.5 (5.0) 1.0 (5.0) 1.6 (5.0) 1.2 (5.0) 0.6 (5.0)

0.77 (0.38) 1.00 (0.26) 0.66 (0.38) 0.58 (0.41) 1.14 (0.27) 0.92 (0.39) 1.00 (0.44) 0.95 (0.30) 0.31 (0.43)

1.00

2.15 2.05 2.03 1.99 2.89 2.25 1.96

4.72 5.18 3.89 3.49 2.65 1.77 4.47 3.42

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

distributions. The posterior means of b0c and b1c reported in columns 4 and 6 in the table differ from those reported for Model 1 in Table 8. As in the case of Model 1 we observe that the posterior means of the elements in b0c and b1c differ in all time periods. For example, in all periods the coefﬁcient on the intercept is lower for compliers under training. The posterior means of the coefﬁcient on the baseline depression score is negative for all compliers, but more negative for compliers in the no-training state. On the other hand, the posterior means of the coefﬁcients on the baseline risk score are positive and smaller for compliers in the training state. All estimates vary by time. Also note that the coefﬁcients in Model 2 come with lower inefﬁciency factors than those from Model 1. While we can conclude from these results that training intake affects the mental health outcomes, it is less easy to calculate the size and direction of the training effects from these estimates. In the next section we discuss a predictive approach that allows us to calculate the training effects.

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Causal Effects of Randomized Experiments with Partial Compliance

Table 9. Model 2 – Prior-Posterior Analysis for the Coefﬁcient Vectors from the Compliance Probability and Outcome Models for Compliers: Prior Means, Posterior Means, Standard Deviations (in Parentheses) and Inefﬁciency Factors. a

b0c

b1c

Prior

Post.

Ineff.

Prior

Post.

Ineff.

Prior

Post.

Ineff.

0.5 (5.0) 0.03 (5.0) 0.5 (5.0) 0.1 (5.0) 0.0 (5.0) 0.0 (5.0) 0.0 (5.0) 0.0 (5.0)

3.41 (0.86) 0.04 (0.01) 0.43 (0.10) 0.13 (0.04) 0.23 (0.10) 0.22 (0.18) 0.04 (0.10) 0.21 (0.21)

2.07

0.0 (5.0) 1.0 (5.0) 0.6 (5.0) 1.0 (5.0) 1.5 (5.0) 0.8 (5.0) 1.4 (5.0) 1.2 (5.0) 0.6 (5.0)

1.10 (0.88) 1.36 (0.56) 1.20 (0.88) 1.26 (0.95) 2.12 (0.61) 2.25 (0.92) 2.29 (0.99) 1.44 (0.63) 0.62 (0.94)

1.56

0.0 (5.0) 1.0 (5.0) 0.6 (5.0) 0.7 (5.0) 1.5 (5.0) 1.0 (5.0) 0.5 (5.0) 1.2 (5.0) 0.6 (5.0)

0.83 (0.42) 1.11 (0.30) 0.84 (0.41) 0.67 (0.43) 1.28 (0.30) 1.12 (0.42) 1.10 (0.46) 1.08 (0.32) 0.56 (0.45)

1.00

2.13 1.40 1.97 1.97 2.12 2.20 1.94

1.73 1.85 1.20 1.66 1.57 1.00 1.20 1.17

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

5. ANALYSIS OF TREATMENT EFFECTS The treatment effects analysis investigates whether the actual intake of the training program has a positive causal effect on the vector of mental health outcomes for compliers. A natural way to answer this question within our Bayesian modeling framework is to take a predictive perspective. Chib (2007) has shown that the Bayesian predictive approach is useful in drawing inferences about causal treatment effects. In this section we extend the methods discussed in Chib and Jacobi (2008) to the panel case. We also show how these predictive distributions can be used to compute various treatment effects, such as quantile treatment effects, and a predictive version of the complier-average causal effect that was computed in Yau and Little (2001). We begin our predictive analysis by considering a subject that is randomly drawn from the subpopulation of compliers. We let yjc,nþ1,

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SIDDHARTHA CHIB AND LIANA JACOBI

j ¼ 0, 1, denote the subject’s potential vector of depression scores under no training and under training, and p(yjc,nþ1|y, x, W, z) denote the corresponding marginal predictive distribution of interest. These marginal distributions are deﬁned as Z pðyjc;nþ1 jWnþ1 ; hÞIðsnþ1 ¼ 1Þpðsnþ1 jwi0 ; aÞpðhjy; x; W; zÞpðWnþ1 jy; x; W; zÞdsnþ1 dhdWnþ1

where pðsnþ1 jvnþ1 ; aÞ ¼ Fðw0i0 aÞsnþ1 f1 Fðw0i0 aÞg1snþ1 but are not in closed form. In this expression we take p(Wnþ1|y, x, W, z) as the empirical distribution of the covariates in our sample. From expressions (4) and (9) we know that p(yjc,nþ1|Wnþ1,h) ¼ tn(yi|Wnþ1bjc,Ojc) in Model 1 and p(yjc,nþ1|Wnþ1,h) ¼ tn(yi|Wnþ1bjcþVibijc, diag(sjc)) in Model 2. The fact that these conditional distributions are easily sampled means that the predictive distributions can be calculated by the method of composition. At each iteration g ¼ 1; 2; . . . ; M of the MCMC chain, we randomly sample ðgÞ WðgÞ nþ1 and wnþ1;0 from the full set of covariates. Next, we sample ðgÞ 0ðgÞ ðgÞ snþ1 ¼ I½wnþ1;0 aðgÞ þ uðgÞ nþ1 40, where unþ1 N ð0; 1Þ. We then check comðgÞ pliance. If sðgÞ nþ1 ¼ 1, we draw the potential outcomes yjc;nþ1 under each intake state from the Student’s t outcome density, conditional on the current sampled draw of the parameters. Otherwise we skip and move to the next ðgÞ ðJÞ step in the chain. The resulting draws ½yð1Þ jc;nþ1 ; . . . ; yjc;nþ1 ; . . . ; yjc;nþ1 ; J M are from the marginal predictive distributions of the potential outcomes. We summarize these generated draws in various ways. One is in terms of (kernel smoothed) predictive density plots. Another is in terms of the differences in means and quantiles of the sampled draws. For example, the predictive average treatment is calculated as E(ylc,nþ1|y, x, W, z) E(y0c,nþ1|y, x,W,z), where the means are computed directly as sample averages Eðyjc;nþ1 jy; x; W; zÞ ¼

J 1X yðgÞ J g¼1 jc;nþ1

5.1. Results Before providing the predictive treatment effects, we pause to examine which subjects in the control arm are a-posteriori classiﬁed as compliers and whether the subjects so classiﬁed are similar to the compliers in the treatment arm with respect to their observable characteristics. In Fig. 3 we

1.0 Compliance Probibilties

Compliance Probibilties

compliers

0.7 0.6 0.1 0.3 0.5

Fig. 3.

0.7 0.6 0.1 0.3 0.5

never takers

compliers

never takers

Control Arm Subjects

Control Arm Subjects

(a) Model 1

(b) Model 2

Posterior Mean of Probability of Compliance for Control Arm Subjects from Models 1 and 2.

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1.0

207

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present the posterior mean of the compliance probability from each bestﬁtting model for each of the 127 control arm subjects. Probabilities between 0 and 0.5 (which can be taken to indicate a never-taker) are plotted below the horizontal axis and those above 0.5 (which indicate a complier) are plotted above the horizontal axis. We see that when the compliance probability is less than 0.5, it tends to be generally less than 0.4, suggesting strongly that each of those subjects is a never-taker. Similarly, when the compliance probability is greater than 0.5, it tends to be generally greater than 0.6, suggesting again that for those subjects inference about the type is more or less decisive. There are, however, some subjects whose compliance probabilities are close to 0.5 and therefore for these subjects a precise determination of type is not possible. An interesting point is that these compliance probabilities are almost the same across the two models. That the covariates are balanced for compliers can be seen from Table 10, which reports the sample means of the covariates by intake and type. The ﬁrst 7 rows refer to the covariates in the model for the compliance probability. Comparing the sample means for compliers in the control arm with those in the treatment arm (columns 2 and 6 for Model 1, columns 3 and 7 for

Table 10.

Sample Means of the Baseline Covariates from the Probit Model for the Compliance Probabilities by Type.

Variable

Sample Means by Intake and Type j¼0 Compliers

Age (demeaned) Motivate Edu Assert Marr Econ Nonw Depress Risk

j¼1 Never-Takers

Compliers

M1

M2

M1

M2

M1

M2

20.70 5.62 13.69 2.87 0.47 3.41 0.15 2.49 1.70

21.20 5.61 13.85 2.91 0.48 3.39 0.12 2.46 1.67

13.57 5.01 13.03 3.13 0.39 3.62 0.21 2.44 1.67

13.50 5.02 12.98 3.11 0.39 3.62 0.22 2.46 1.68

19.26 5.46 13.72 2.87 0.37 3.53 0.14 2.41 1.67

19.26 5.46 13.72 2.87 0.37 3.53 0.14 2.41 1.67

Note: Compliers and never-takers in the control arm are classiﬁed based on the estimated posterior mean of compliance.

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Model 2) we see that the two groups look almost identical. The only exception is the marriage indicator covariate which in any case was estimated with a very low precision and mean of 0 (see Tables 8 and 9). A look at columns 4 and 5 shows that never-takers seem different from compliers. Never-takers are younger, less motivated, and less educated. Finally, compliers and never-takers have almost identical sample means of the covariates (baseline depression and risk scores) that are in the outcome model (but not present in the model of compliance). The three graphs in Fig. 4 show the kernel plots of the marginal predictive densities for compliers, for each of the periods t ¼ 1,2,3. The solid lines refer to the potential outcomes under no training and the dashed lines under training participation (y1c,t). All plots show an improvement in mental health from participation in the training participation compared to no training. The marginal densities under training participation have more mass for negative values. In comparison, the densities in the no-training case have more mass over positive values. The greatest difference between the two predictive densities occur in the second and third time periods. To get a better view of the magnitude of the mental health improvements caused by the participation in the training program, we compare the means and quantiles of the predictive densities in each time period. In Table 11 we report the average and quantile treatment effects for the 0.05, 0.25, 0.50, 0.75, and 0.95 quantiles for Models 1 and 2. The entries in the ﬁrst row of results show that on average the program leads to a decrease in the depression scores. In the case of Model 1 the training program decreases average depression scores between 0.20 points in the ﬁrst period, 0.32 points in the second period, and 0.36 points in the third period. The analysis for Model 1 suggests higher average treatment effects that range between 0.28 in the ﬁrst period to 0.41 in the second and third periods. As indicated by the kernel plots in Fig. 4 all quantile treatment effects are negative. For Model 1 the effects vary between 0.02 and 0.16 points at the 5% quantiles and between 0.37 and 0.60 points at the 95% quantiles. For Model 2 the estimated 5% quantile treatment effects lie between 0.04 and 0.07 points. The 95% quantiles range between 0.52 and 0.80 points. Our results for the average complier effect differ from those found by Yau and Little (2001) in that we do not ﬁnd a decrease in the treatment effect after period 2. Our estimated average complier effect at t=2 under Model 1 is similar in magnitude to that found in the study by Skrondahl and Rabe-Hesketh (2004) that focused on health outcomes in period 2.

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y0c,1 y1c,1

0.6

y0c,1 y1c,1

0.6

0.4

0.4

0.2

0.2

0

0 -2

0

2

-2

(a) Model 1: t = 1

y0c,2 y1c,2

0.6

2

y0c,2 y1c,2

0.6

0.4

0.4

0.2

0.2

0

0 -2

0

2

-2

(c) Model 1: t = 2

0

2

(d) Model 2: t = 2

y0c,3 y1c,3

0.6

y0c,3 y1c,3

0.6

0.4

0.4

0.2

0.2

0

0 -2

0 (e) Model 1: t = 3

Fig. 4.

0 (b) Model 2: t = 1

2

-2

0

2

(f ) Model 2: t = 3

Predictive Marginal Distributions of the Potential Outcomes for Compliers in Models 1 and 2.

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Table 11.

Predicted Average and Quantile Treatment Effects from Models 1 and 2 for all Time Periods.

Treatment Effect

Treatment Effects Estimates Model 1

Model 2

t¼1

t¼2

t¼3

t¼1

t¼2

t¼3

Average

0.20

0.32

0.36

0.28

0.41

0.41

Quantile 5% 25% 50% 75% 95%

0.02 0.14 0.21 0.25 0.37

0.13 0.25 0.32 0.40 0.55

0.16 0.28 0.36 0.45 0.60

0.07 0.20 0.28 0.39 0.52

0.04 0.24 0.40 0.57 0.78

0.04 0.24 0.40 0.57 0.80

6. CONCLUSION We have discussed Bayesian models for ﬁnding the longitudinal causal effects of a randomized two-arm training program when compliance with the randomized assignment is less than perfect in the training arm for reasons that are potentially correlated with the outcomes. We show how the type approach can be used to calculate interesting causal effects. An important point is that the Bayesian approach is particularly useful in this context because it provides an automatic way of dealing with the mixture outcome distribution in the control arm. The possibility of incorporating real prior information is also another advantage of the Bayesian approach. We discuss how different versions of our models can be compared by marginal likelihoods and Bayes factors and how useful summaries of the causal effects can be determined from a predictive perspective. All of our computations proceed without the joint distribution of the potential outcomes. In addition, the ﬁtting algorithms are efﬁcient and provide detailed information about the compliance status of subjects in the control arm. Because of these strengths of the techniques discussed here, we believe that the methods of this paper will prove useful in practical work.

REFERENCES Albert, J., & Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88, 669–779.

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Chib, S. (1995). Marginal likelihood from the Gibbs output. Journal of the American Statistical Association, 90, 1313–1321. Chib, S. (2001). Markov Chain Monte Carlo methods: Computation and inference. In: J. J. Heckman & E. Leamer (Eds), Handbook of econometrics (Vol. 5, pp. 3569–3649). Amsterdam: North Holland. Chib, S. (2007). Analysis of treatment response data without the joint distribution of potential outcomes. Journal of Econometrics, 140, 401–412. Chib, S., & Greenberg, E. (1995). Understanding the Metropolis–Hastings algorithm. American Statistician, 49, 327–335. Chib, S., & Jacobi, L. (2007). Modeling and calculating the effect of treatment at baseline from panel outcomes. Journal of Econometrics, 140, 781–801. Chib, S., & Jacobi, L. (2008). Analysis of treatment response data from eligibility designs. Journal of Econometrics, 144, 465–478. Chib, S., & Jeliazkov, I. (2001). Marginal likelihood from the Metropolis–Hastings output. Journal of the American Statistical Association, 96, 270–281. Clark, A. E., & Oswald, E. J. (1994). Unhappiness and unemployment. The Economic Journal, 104, 648–659. Frangakis, C. E., Brookmeyer, R. S., Varadhan, R., Safaeian, M., Vlahov, D., & Strathdee, S. A. (2004). Methodology for evaluating a partially controlled longitudinal treatment using principal stratiﬁcation, with application to a needle exchange program. Journal of the American Statistical Association, 99, 239–249. Frangakis, C. F., & Rubin, D. B. (1999). Addressing complications of intention-to-treat analysis in the combined presence of all-or-none treatment-noncompliance and subsequent missing outcomes. Biometrika, 86, 365–379. Hirano, K., Imbens, G. W., Rubin, D. B., & Zhou, X. (2000). Assessing the effect of an inﬂuenza vaccine in an encouragement design. Biostatistics, 1, 69–88. Imbens, G. W., & Rubin, D. B. (1997). Bayesian inference for causal effects in randomized trials with noncompliance. The Annals of Statistics, 25, 305–327. Jo, B. (2002). Statistical power in randomized trials with noncompliance. Psychological Methods, 7, 178–193. Levy, D. E., O’Malley, J. A., & Normand, S. T. (2004). Covariate adjustment in clinical trials with non-ignorable missing data and non-compliance. Statistics in Medicine, 23, 2319–2339. Mealli, F., Imbens, G. W., Ferro, S., & Biggeri, A. (2004). Analyzing a randomized trial on breast self-examination with non-compliance and missing outcomes. Biostatistics, 5, 207–222. Sommer, A., & Zeger, S. (1991). On estimating efﬁcacy in clinical trials. Statistics in Medicine, 10, 45–52. Skrondahl, A., & Rabe-Hesketh, S. (2004). Generalized latent variable modeling: Multilevel, longitudinal and structural equation models. New York: Chapman & Hall/CRC. Ten Have, T. R., Joffe, M., & Cary, M. (2003). Causal logistic models for non-compliance under randomized treatment with univariate binary response. Statistics in Medicine, 22, 1255–1283. Vinokur, A. D., Price, R. H., & Schul, Y. (1995). Impact of JOBS intervention on unemployed workers varying in risk for depression. American Journal of Community Psychology, 19, 543–562. Yau, L., & Little, R. (2001). Inference for the complier-average causal effect from longitudinal data subject to noncompliance and missing data, with application to a job training assessment for the unemployed. Journal of the American Statistical Association, 96, 1232–1244.

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APPENDIX. MCMC ALGORITHMS IN DETAIL Model 1 1. Sample (s00, a, {ki}|y, x, b, O) by sampling (a) si ¼ 1|y, x, b, O for iAI00 with probability Pr(si ¼ 1|yi, xi, b0c, b0n, a, O0c, O0n) given by qci tT;n ðyi jWi b0c ; X0c Þ qci tT;n ðyi jWi b0c ; X0c Þ þ ð1 qci ÞtT;n ðyi jWi b0n ; X0n Þ (b) a|y, x, b, O, s00 for iAN by a MH step by proposing aw from t20 (a |m,V ) and accepting the proposal value aw with probability 9 8 Q Q y Fðsi jw0i0 ay Þ f1 Fðsi jw0i0 aÞgt20 ðajm;VÞ> > = < pða Þ i2N 0c [N 1c i2N 0n Q Q min 1; > Fðsi jw0i0 aÞ 1 Fðsi jw0i0 aÞt20 ðay jm;VÞ > ; : pðaÞ i2N 0c [N 1c

i2N 0n

where m is the approximate mode of " # Y Y 0 0 ln Fðsi jwi0 aÞ f1 Fðsi jwi0 aÞg i2N 0c [N 1c

i2N 0n

and V is the inverse Hessian of the latter expression evaluated at m. (c) ki|y, x, b, O, s00 for iAN from ! n þ T n þ ðyi Wi bjk ÞX1 ðyi Wi bjk jk ; G ki 2 2 2. Sample b|y, x, s00, k, O, a by sampling bjk|y, x, s00, k, Ojk, a from 9 8 0 = < X N p @bjk jBjk B1 W0i ki X1 jk yi ; jk;0 bjk;0 þ ; : i2N jk 91 1 8 = < X C 1 0 Bjk ¼ B1 þ W k X W A i i i jk ; : jk;0 i2N jk

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3. Sample O|y, x, s00, k, b, a, D by sampling Ojk|y, x, s00, bjk, k, a from 0 2 31 1 X 4 1 ki ðyi Wi bjk Þðyi Wi bjk Þ0 5 A W 3 @X1 jk jrjk;0 þ njk ; Rjk;0 þ i2N jk

Model 2 1. Sample s00, a, b, bin, bic|y, x, k, s, D0c, Dlc, Dn by sampling (a) si for iAI00 with probability Pr(si ¼ 1|yi, xi, b0c, b0n, a, k, s0c, s0n) given by P qci N T ðyi jWi b0c ; 0c;i Þ P P qci N T ðyi jWi b0c ; 0c;i Þ þ ð1 qci ÞN T ðyi jWi b0n ; 0n;i Þ P 0 where jci fk1 i diagðsjc Þ þ Vi Djc Vi g: (b) a|y, x, s00 by a MH step by proposing aw from t20(a|m,V) and accepting the proposal value with probability of move given in the algorithm for Model 1. (c) bjk|y, x, s00, k, sjk, k, D0c, Dlc, Dn from 0

8 <

X

9 =

8 <

X

B N p @bjk jBjk B1 b þ W0i S1 B1 þ jk yi ; Bjk ¼ : jk;0 jk;0 i2N ; : jk;0 i2N jk

jk

91 1 = C W0i S1 A jk Wi ;

(d) bjci|yi, xi, bjc, s00, ki, s0c, s1c, Djc for iAIjc, j ¼ 0, 1, from 1 1 0 1 N q ðbjci jBjci fki V0i X1 jc ðyi Wi bjc Þg; Bjci ¼ fDjc þ ki Vi ðdiagðrjc ÞÞ Vi g Þ

(e) bin|yi, xi, b0n, s00, ki, s0n, Dn for iAI0n from 1 1 0 1 N q ðbni jBni fki V0i X1 0n ðyi Wi b0n Þg; Bni ¼ fDn þ ki Vi ðdiagðr0n ÞÞ Vi g Þ

2. ki|y, x, b, s0c, s0n, s1c, s0c, a, s00, bin, bic for iAN from ! n þ T n þ ðyi Wi bjk Vi bik Þðdiagðrjk ÞÞ1 ðyi Wi bjk Vi bik Þ ; G ki 2 2

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3. Sample s|y, x, s00, k, b, a, {bin}, {bic}, D0c, Dlc, Dn by sampling sjk,t|yjk,t, x, s00, bjk, k, a, {bin}, {bic}, D0c, Dlc, Dn from P 0 1 d jk;0 þ ki ðyi;t w0it bjk vit bi Þ2 Zjk;t0 þ njk i2N jk B C I G@rjk;t ; A 2 2

4. Sample D0c, D1c, Dn|y, x, s00, k, b, a, {bi0c}, {bi1c}, {bin}, s from 0 2 31 1 X 4 1 W k @D1 bji b0ji 5 A; j ¼ 0; 1 jc jrjc;0 þ njc ; Rjc;0 þ i2N jc

0

"

1 W k @D1 n jrn;0 þ nn ; Rn;0 þ

X

#1 1 A b i b0 i

i2N 0n

PARAMETRIC AND NONPARAMETRIC INFERENCE IN EQUILIBRIUM JOB SEARCH MODELS Gary Koop ABSTRACT Equilibrium job search models allow for labor markets with homogeneous workers and ﬁrms to yield nondegenerate wage densities. However, the resulting wage densities do not accord well with empirical regularities. Accordingly, many extensions to the basic equilibrium search model have been considered (e.g., heterogeneity in productivity, heterogeneity in the value of leisure, etc.). It is increasingly common to use nonparametric forms for these extensions and, hence, researchers can obtain a perfect ﬁt (in a kernel smoothed sense) between theoretical and empirical wage densities. This makes it difﬁcult to carry out model comparison of different model extensions. In this paper, we ﬁrst develop Bayesian parametric and nonparametric methods which are comparable to the existing non-Bayesian literature. We then show how Bayesian methods can be used to compare various nonparametric equilibrium search models in a statistically rigorous sense.

Bayesian Econometrics Advances in Econometrics, Volume 23, 217–244 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23007-0

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1. INTRODUCTION Most potential policy initiatives in labor markets (e.g., minimum wage legislation) have effects on both workers1 and employers. Hence, it is strongly desirable that policy analysis be carried out in the context of an equilibrium model which takes into account the interaction between worker and ﬁrm behavior. This consideration has motivated the development of a large literature on equilibrium job search models (see also, van den Berg & Ridder, 1998; Ridder & van den Berg, 1997; Bontemps, Robin, & van den Berg, 1999; Mortensen, 1990, 1998; Burdett, 1990; Kiefer & Neumann, 1993). Details of the equilibrium search model will be provided in the next section, sufﬁce it to note here that it assumes a ﬁxed number of homogeneous workers and ﬁrms with full information about all the parameters in the model. Workers and ﬁrms interact in a market where jobs arrive and are destroyed at random exogenous rates. Assuming workers maximize the expected value of their future income stream and ﬁrms maximize expected steady-state proﬁt ﬂow, it turns out that a nondegenerate wage offer distribution exists given with density by: 1 f ðwi Þ / pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p wi for wiA[r, h] where wi is the wage offer to individual i for i ¼ 1, . . . , N, p the productivity of workers (i.e., their marginal value product), r their reservation wage, and h the upper bound on the wage distribution. Further details will be provided in the next section, at this point the key thing to note is that the wage offer density is increasing in w. However, empirical wage offer densities are invariably roughly bell-shaped, but skewed to the right. Accordingly, a conﬂict exists. One the one hand, the equilibrium search model is structural and, hence, well suited for policy analysis. On the other hand, it does a very poor job of ﬁtting the data. In response to this conﬂict, many of the recent papers in the literature add a new component to the basic equilibrium search model (see also, Bontemps et al., 1999; Bontemps, Robin, & van den Berg, 2000; Mortensen, 1998; Nielsen & Rosholm, 1999; Koning, Ridder, & van den Berg, 1995). This component might be unobserved heterogeneity in productivity or in workers’ value of leisure, measurement error in wages, or an economic concept like an investment in training function. From a statistical point of view, the wage offer density can be written as f ðwi jy1 ; hðy2 ÞÞ, where y1 and y2 are parameter vectors and h( ) is the appropriate new component. This approach can be

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criticized on the ground that economic theory rarely is a reliable guide to the choice of functional form for h( ). For instance, Koning et al. (1995) assume h( ) is log-normal in a model which allows for heterogeneity in productivity. The choice of log-normality does not have any basis in economic theory. One of the chief advantages of equilibrium search models over earlier partial search models (Lancaster, 1997) was that the former used a wage offer distribution which was derived using an assumption of rational ﬁrm behavior, whereas the latter merely made assumptions of convenience (e.g., the wage density is assumed to be log-normal or exponential). Merely adding a new ad hoc component such as h( ) into the equilibrium search model could be interpreted as a regressive step away from structural and back toward reduced form modeling. Nevertheless, the use of a parametric form for h( ) implies a parametric likelihood function and, thus, estimation, testing, and model comparison can be done using either Bayesian or nonBayesian likelihood-based paradigms. Recent work (Bontemps et al., 1999, 2000; Nielsen & Rosholm, 1999), perhaps in realization that economic theory is rarely a guide to choice of h( ), does not specify a functional form for it. Instead nonparametric kernel smoothing algorithms are used. In essence, kernel methods are used to ﬁt f( ) (or the earnings density which is a simple transformation of f( )) and then the implied form of h( ) required to provide this perfect ﬁt can be obtained. Since f( ) is usually the key element of the likelihood function, this method can be loosely interpreted as saying ‘‘Let us ﬁt the likelihood function perfectly,2 and then see what it implies h( ) must look like.’’ Such an approach has the advantage of not making an ad hoc functional form assumption for h( ), but also has drawbacks. For instance, it attributes all of the departures from the basic equilibrium search model to a particular source. One paper might say all such departures are due to heterogeneity in productivity, another might attribute them to measurement error, another to effects relating to investment in training, etc. It is difﬁcult to choose between different models since all ﬁt the likelihood function perfectly. In the previous paragraph, the parametric approach was criticized since it involved an ad hoc functional form assumption. The nonparametric approach does not suffer from such a criticism, but does still depend on strong assumptions that are difﬁcult to test (e.g., that all departures from the basic equilibrium search model are due to unobserved heterogeneity in productivity). These considerations motivate the present paper. In previous work (Koop, 2001), it was argued that Bayesian methods provided a useful tool for analyzing the equilibrium search model and computational methods for doing so were developed. One contribution of the present paper is similar to

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this. That is, methods of Bayesian inference for the extensions of the equilibrium search model described in the previous paragraphs are developed. A particular focus is the development of a Bayesian nonparametric approach which is comparable to the non-Bayesian nonparametric approach of Bontemps et al. (2000). There is a sizeable statistical literature on Bayesian nonparametric methods, but this is only slowly making it over to the ﬁeld of econometrics (Campolieti, 1997; Jochmann & Leon-Gonzalez, 2004; Ruggiero, 1994). In the present paper, Bayesian nonparametric methods are developed for various extensions of the basic equilibrium search model. A second contribution of the paper derives from the fact that all of the models in the paper, including the nonparametric ones, involve a well-deﬁned probability density for the data. This implies that standard statistical tools for model comparison3 can be calculated. Using the Bayesian approach, two nonparametric models (i.e., with h( )s arising from different sources of heterogeneity) can both ﬁt the data perfectly yet be compared using formal statistical methods. The paper describes how to do this and how to interpret the model comparison measures which are calculated. Loosely speaking, the Bayesian approach forces the researcher to specify what a reasonable form for h( ) might be, but they do not impose this form. If the data force large departures from this reasonable form, Bayesian tools of model comparison will indicate less support for the model. We argue that this formalizes what researchers do informally. For instance, suppose the researcher believes that heterogeneity in productivity may exist, but is relatively small and unimodal. However, the nonparametric distribution exhibits large dispersion and is multimodal. In this case, the researcher might reject the model with heterogeneity in productivity. The Bayesian methods outlined below are a statistically coherent formalization of the sensible but informal actions of the researcher.4 The paper is organized as follows. The following section outlines the basic equilibrium search model. The third section derives equilibrium search models with heterogeneity introduced in various parametric ways. The fourth section is similar to the third, but heterogeneity is introduced in various nonparametric ways. The ﬁfth section contains an empirical application using the dataset of Bowlus, Kiefer, and Neumann (1995). Computational algorithms are provided in the appendix.

2. THE BASIC EQUILIBRIUM SEARCH MODEL The basic equilibrium search model assumes a ﬁxed, homogeneous population of workers and ﬁrms. The ratio of the measure (number) of workers to

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ﬁrms is m. Workers are initially unemployed and job offers arrive as events in a Poisson process at a rate of l0. Once workers are employed they can continue searching and new job offers arrive at a rate of l1. Firms lay off workers at a rate of l2. The distribution of wage offers is given by F(w), the corresponding density is denoted by f(w). The value of leisure to individuals is equal to b. When working, their productivity (i.e., their marginal value product) is p. We assume workers and ﬁrms know the values of all parameters and distributions. To derive the likelihood function worker and ﬁrm behavior must be speciﬁed. Here we provide only the ﬁnal results and a few details of the derivation. Complete details, along with extensive intuition, are given in many of the papers listed at the beginning of the Introduction. Ridder and van den Berg (1997) include a great deal of survey material which is particularly useful. If workers maximize the expected value of future income, then unemployed workers will set a reservation wage, r, and accept the ﬁrst wage offer above it. The optimal reservation wage is given by: r ¼ xb þ ð1 xÞp

(1)

where x¼

ð1 þ ðl1 =l2 ÞÞ2 ð1 þ ðl1 =l2 ÞÞ þ ððl0 =l2 Þ ðl1 =l2 ÞÞðl1 =l2 Þ 2

Employed workers accept any wage offer above their current wage. Firm behavior is described by F(w), which is determined endogenously assuming that the job market is in a steady-state equilibrium. Given that ﬁrms know the acceptance strategies of workers, the supply of labor to a ﬁrm who offers wage w every time a vacancy arises can be derived and denoted by l(w) (Ridder & van den Berg, 1997, Equation (8)). Assuming a linear production function, the steady-state proﬁt ﬂow of a ﬁrm offering wage w is given by: pðwÞ ¼ ðp wÞlðwÞ No ﬁrm will ever offer a wage below r, since no worker would accept it, hence we know the support of f(w) begins at r. Furthermore, in equilibrium all wage offers must yield the same proﬁt ﬂow (i.e., ﬁrms must have no incentive to change their offers). Using these facts, it can be shown that: f ðwÞ ¼

l2 þ l1 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2l1 pr pw

(2)

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for wA[r, h]. The highest wage offered is given by: h ¼ b2 r þ ð1 b2 Þp

(3)

where b¼

l2 l2 þ l 1

The contributions of this paper all relate to the wage offer distribution, as opposed to the duration of unemployment or employment spells. Accordingly, we simplify the analysis by assuming we only observe wi for i ¼ 1, . . . , N where wi is the wage offer accepted by initially unemployed worker i. Identiﬁcation issues relating to the structural parameters, l0, l1, l2, b, p are often of great interest in models of search (Koop & Poirier, 2001; Koop, 2001). However, to focus on the paper, we ignore these issues and work only with the identiﬁed parameters which directly affect the shape of the wage density, p and r. We also deﬁne t ¼ l2/l1, g ¼ ln(r), and p ¼ ln(p).5 Accordingly, the likelihood function for W ¼ (w1, . . . , wN)u is given by:

LðW; p; rÞ ¼

N Y

f ðwi Þ

(4)

i¼1

where f ðwi Þ ¼

1þt 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ IðBÞ 2t ðp rÞ p wi

(5)

and I(B) is an indicator function for the bounds of the wage distribution. That is, B is deﬁned by rowioh and h ¼ b2r+(1b2)p with b ¼ 1/(1+t). Note that all the parameters enter B. Bayesian inference can be carried out by specifying a prior and using a Markov Chain Monte Carlo (MCMC) algorithm for posterior simulation. Details are given in the appendix.

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3. PARAMETRIC EXTENSIONS OF THE EQUILIBRIUM SEARCH MODEL 3.1. Heterogeneity in Productivity The basic equilibrium search model assumes homogeneity of workers and ﬁrms. Heterogeneity can be introduced in many ways. Perhaps the most common is to allow for heterogeneity in productivity. Here, following much of the literature (e.g., Koning et al., 1995 or Bontemps et al., 1999), we assume heterogeneity in productivity arises since the labor market is segmented into many smaller homogeneous markets. Each of these homogeneous markets has a different productivity.6 Following Koning et al. (1995), it is assumed that productivity is log-normally distributed. That is, the log-productivities of the workers, pi for i ¼ 1, . . . , N, are i.i.d. draws from the normal distribution: pðpi Þ ¼ fN ðmp ; s2p Þ

(6)

where mp, s2p are unknown parameters to be estimated and fN(c, C) denotes the normal density with mean c and variance C. This parametric distribution for productivity can, as in Koning et al. (1995), be treated as part of the likelihood function. Alternatively (and equivalently), it can be interpreted as a hierarchical prior for pi. We ﬁnd it more natural to use the latter interpretation and, hence, refer to Eq. (6) as a prior. The likelihood function is as in the basic model, except for adding a subscript. That is, Eq. (5) becomes f ðwi Þ ¼

1þt 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ IðBÞ 2t ðpi rÞ pi wi

(7)

where pi ¼ exp(pi) and B is deﬁned by rowiohi and hi ¼ b2r+(1b2)pi. Note that the assumption of a reservation wage common to all workers, r, is retained. This can be justiﬁed if unemployed workers do not know, a priori, which market they will be working in and, hence, all use mp to select their reservations wages. Computational methods for Bayesian inference of this model are given in the appendix. Heuristically, conditional on knowing pi for i ¼ 1, . . . , N, the posterior for the remaining parameters is essentially the same as for the basic search model. Furthermore, the posterior for pi is relatively easy to work with (see appendix). Hence, an MCMC algorithm with a data augmentation

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step can be used to efﬁciently carry out Bayesian inference. In contrast, classical analysis (Koning et al., 1995) involves integrating over the productivity distribution at each iteration, which greatly adds to the computational burden.

3.2. Measurement Error in Wages The wage data used in empirical analyses of job search models is invariably taken from surveys. The measurement error which undoubtedly arises from individuals misrepresenting their wages could explain why the basic equilibrium search model ﬁts empirical wage distributions so poorly, although intuitively it seems unlikely to do so. Remember, the basic job search model implies an increasing wage density when empirical wage densities are roughly bell-shaped. For measurement error to explain this divergence it must be of a form that is somewhat unusual. That is, measurement error must work in such a way to convert an observed bell-shaped wage density into one where the true wages imply an increasing wage density. Nevertheless, measurement error is often mentioned as being important in labor market datasets. For this reason, and in order to illustrate our econometric methodology, we present an equilibrium search model which includes measurement error. Let wi be the true, unobserved, wage of individual i and assume: wi ¼ wi þ i

(8)

where ei is measurement error, assumed to be independent of wi, and i.i.d. with density pði Þ ¼ fN ðm ; s2 Þ

(9)

Note that me is allowed to be nonzero, so a systematic bias in reporting of wages is allowed for. Assuming the basic equilibrium search model holds for the true wages, a likelihood function can be derived by replacing wi by wi in Eq. (5) and using Eq. (8) to write: f ðwi Þ ¼

1þt 1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ IðBÞ 2t ðp rÞ p wi i

(10)

where B is deﬁned by rowi+eioh. Analogous to the model with heterogeneity in productivity, we can treat Eq. (9) either as a part of the

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likelihood functional or as part of a hierarchical prior. If we do the latter, ei for i ¼ 1, . . . , N, can be interpreted as parameters with prior given by Eq. (9). Hence, Eqs. (4) and (10) give the likelihood function for the observed data, wi for i ¼ 1, . . . , N. Computational methods for Bayesian inference of this model are given in the appendix. The informal motivation of these are similar to the model with heterogeneity in productivity and an MCMC algorithm with data augmentation can be derived. 3.3. Other Models In order to focus on the econometric methodology, we do not consider any further extensions to the basic equilibrium search model. However, it is worthwhile noting in passing that the issues discussed in this paper hold for a wide range of models which introduce observation-speciﬁc heterogeneity in some way (e.g., a model where individuals are heterogeneous with respect to their value of leisure and, hence, their reservation wage). Furthermore, the Bayesian computational tools developed here will, with some modiﬁcation, be suitable with many sorts of extensions to the basic job search model. It is also possible to work with models using several extensions of the basic model at one (e.g., a model with both heterogeneity in productivity and measurement error) by extending the tools of the present paper in a simple, obvious way.

4. NONPARAMETRIC EXTENSIONS OF THE EQUILIBRIUM SEARCH MODEL 4.1. Heterogeneity in Productivity The most common Bayesian nonparametric approach uses mixtures of Dirichlet processes to model a distribution with unknown form (see also, Antoniak, 1974; Ferguson, 1973; West, Muller, & Escobar, 1994; Escobar & West, 1995). In this section, we derive and motivate this approach for the case of the equilibrium search model with unobserved heterogeneity in productivity. Computational methods for Bayesian inference are given in the appendix. For notational convenience, we will suppress the dependence of the wage distribution below on all parameters other than p. The equilibrium search model with unobserved heterogeneity in productivity implies a wage distribution for a ﬁrm with log-productivity pi, which

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we denote by F(wi|pi), where Fðwi jpi Þ ¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þt pi w i 1 IðBÞ t pi r

(11)

where pi ¼ exp(pi) (i.e., this is the integral of the wage density given in Eq. (7)). In the preceding section, we assumed pi came from a known distribution (i.e., its density was given by pðpi Þ ¼ fN ðmp ; s2p Þ). In the present section, we assume pi comes from a distribution, G(pi), with unknown form and use Bayesian nonparametric methods. From a Bayesian viewpoint, G( ) is best interpreted as a prior. However, it can equally be interpreted as part of the likelihood and the non-Bayesian may prefer to interpret it that way. Formally, we assume G( ) is a Dirichlet process. A precise deﬁnition of the Dirichlet process is given in any of the citations mentioned at the beginning of this section. Here we give an intuitive motivation of how Dirichlet processes can be used to carry out nonparametric Bayesian inference. In standard Bayesian inference, one begins with a prior distribution for the unknown parameters in the model and then updates using data information to obtain a posterior. In nonparametric Bayesian inference, one begins with a prior for the unknown distribution, G( ), and then updates with data information to obtain a posterior. Let G0 be this base prior distribution which is the prior expectation for G( ).7 Here, we let G0 be normal with mean mp and variance s2p . In other words, our prior for the nonparametric model is that it is the same as the parametric model of Section 3.1. The next element in the Bayesian nonparametric analysis involves the researcher specifying the prior parameter, a, which is a scalar precision parameter which represents the weight of our belief in the prior G0. The exact role of a will be clariﬁed below. Formally, we have, for i ¼ 1, . . . , N: wi jpi Fðwi jpi Þ, pi GðÞ Gja DðaG0 Þ where ‘‘B’’ denotes ‘‘is distributed as,’’ and the last equation is notation for ‘‘G is a random distribution generated by a Dirichlet process with base measure aG0.’’ Antoniak (1974) (or the other citations in this section) give more details about precisely what the previous sentence means. However, its implications are best understood by considering what the Dirichlet process prior implies

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for certain conditional prior distributions. It can be shown that: N X

pi jpðiÞ aaN1 G0 þ aN1

dðpj Þ

(12)

j¼1; jai

where p(i) ¼ (p1, . . . , pi1, pi+1, . . . , pN)u, d(pj) ¼ 1 if pi ¼ pj ( ¼ 0 otherwise), and aN1 ¼ 1=ða þ N 1Þ. In other words, a priori, pi, is either drawn from G0 (in this paper, the normal distribution) or randomly chosen from the other realized values, p(i). Eq. (12) also makes clear the role of a. As a approaches zero, the ﬁrst term on the right-hand side of Eq. (12) vanishes and the conditional prior becomes purely nonparametric. That is, pi|p(i) approaches a discrete distribution with points of support given by the other realized values. Equivalently, Eq. (12) becomes a histogram of the other realized values, p(i). In contrast, as a approaches inﬁnity the prior becomes more and more parametric and, in the limit, pi|p(i)BG0(pi). In the present case, we approach the model of Section 3.1 with productivity being lognormally distributed across individuals. The conditional posteriors analogous to Eq. (12) provide additional intuition. It can be shown that these are: pi jpðiÞ ; W qi0 Gi0 þ

N X

qij dðpj Þ

(13)

j¼1; jai

where gi0 ¼ f ðwi jpi Þg0 ðpi Þ

(14)

Z qi0 / a

f ðwi jpi Þg0 ðpi Þdpi

qij / f ðwi jpj Þ PN

(15) (16)

l0 qil ¼ 1 and gi0( ), g0( ), and f ( ) are densities corresponding to the distributions Gi0( ), G0( ), and F( ), respectively. Note ﬁrst that Eq. (13), like Eq. (12) is a mixture of a parametric and nonparametric part. The parametric part is a posterior obtained by updating the base prior via the likelihood function for one observation, wi (i.e., it is the posterior we would have obtained using the parametric model of Section 3.1 with one data point). The nonparametric part involves merely the other realizations, p(i). Secondly, the weight placed on the parametric part is given

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by Eq. (15). Note that this is proportional to the marginal density of the realized data point wi (i.e., the marginal likelihood which would have obtained in Section 3 if we only had one data point). Loosely speaking, this measures how likely the observed data point could have come from the parametric model. If qi0 is high, then more weight is placed on the parametric model. Secondly, the prior parameter, a, also enters Eq. (15). As discussed above, as a goes to zero less and less weight is placed on the parametric part of the model. As a goes to inﬁnity, the model goes to the purely parametric model of Section 3. Thirdly, the weight placed on any of the other realizations in p(i) is given by Eq. (16). This is the wage density assuming a log-productivity of pj evaluated at the ith observation. Hence, this is a measure of how plausible it is that wi is generated from the wage density calculated using pj. Classical nonparametric kernel algorithms can be interpreted as ﬁtting a density at a point using weighted averages of nearby data points. Note that this is exactly what the Bayesian method is doing. The weights in the local averaging are given by qij which can be interpreted as a measure of how ‘‘close’’ or ‘‘plausible’’ pj is for each j 6¼ i. Fourthly, for ﬁxed a, as N goes to inﬁnity the parametric part of Eq. (13) will receive less and less weight. Hence, asymptotically the approach becomes increasingly nonparametric. Fifthly, Eq. (13) can be interpreted as a smoothed histogram where the smoothing is provided by Gi0. Eqs. (13)–(16) can be used as part of an MCMC algorithm for carrying out nonparametric Bayesian inference. Hence, the previous material is all one needs to know in order to understand and implement Bayesian nonparametric methods. However, in practice, we use a more efﬁcient algorithm outlined in West et al. (1994) (see appendix for more details). 4.2. Measurement Error The derivation of a model with measurement error having a distribution of unknown form proceeds along similar lines to the previous section. In particular, the equilibrium search model with measurement error implies a wage distribution for a ﬁrm given by

Fðwi ji Þ ¼

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1þt p w i i IðBÞ 1 pr t

(17)

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229

We assume ei comes from a distribution, G(ei), with unknown form and model G( ) using a Dirichlet process. Hence, we have wi ji Fðwi ji Þ, i GðÞ Gja DðaG0 Þ where G0 is given by Eq. (9). With these choices (and with p’s replaced by e’s), the conditional priors and posteriors are as in Eqs. (12)–(16) and the same intuitive motivation applies. Further details about Bayesian inference for this model are given in the appendix.

4.3. Interpretation of Bayesian Nonparametric Methods The Bayesian nonparametric approach allows us to ﬁt the wage distribution almost perfectly in the same way as the nonparametric kernel approaches which are increasingly popular in the empirical job search literature (Bontemps et al., 1999, 2000; Nielsen & Rosholm, 1999). However, because the Bayesian approach is based on a well-deﬁned probabilistic model, all the usual tools of Bayesian model comparison and testing can be used. In particular, we can calculate the marginal likelihood for the models described in this section and use them to calculate Bayes factors comparing a nonparametric model to a parametric alternative. As is standard in Bayesian analyses, the marginal likelihood will reﬂect the ﬁt of the model, but will also include a reward for parsimony. Of course, the nonparametric model will ﬁt the data better than corresponding parametric alternative (e.g., the nonparametric model with heterogeneity in productivity will ﬁt better than the parametric model with heterogeneity in productivity). However, the reward for parsimony means that the more complicated nonparametric model may not have the higher marginal likelihood. In essence, Bayesian methods provide us with a formal measure, rooted in probability theory, of whether the added ﬂexibility of the nonparametric model is of sufﬁcient worth. Furthermore, Bayesian methods allow for the comparison of two different nonparametric models. In other words, they allow for a valid statistical comparison of two models, both of which might ﬁt the data almost perfectly. In the context of the empirical job search literature, where there are many different ways of improving data ﬁt (e.g., heterogeneity in productivity or reservation wage, measurement error, etc.), the ability of Bayesian methods

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to allow for formal model comparison is an enormous advantage. Hence, it is useful to explain what a Bayes factor comparing two nonparametric models will be capturing. Traditionally, most authors have focussed on the fact that Bayes factors contain a reward for parsimony (i.e., all else being equal, the Bayes factor supports the more parsimonious model) and a reward for ﬁt (i.e., all else being equal, the Bayes factor supports the model which ﬁts best). However, Bayes factors also reward the model which evinces a higher degree of compatibility between prior and likelihood function.8 If we are comparing two nonparametric models, the ﬁrst two of these rewards will tend to be quite similar and, hence, the Bayes factor will likely be driven by the third. Thus, the choice between two nonparametric models will come down to which one has least conﬂict between prior and data information. We would argue that this is a very sensible basis for model comparison. Remember that a key component of the prior is a distributional assumption about the form of heterogeneity. With heterogeneity in productivity, the base prior, G0, says productivity is log-normally distributed. In the other model, the base prior says measurement error is normally distributed. Furthermore, the values selected for prior hyperparameters can be chosen to make statements like ‘‘heterogeneity in productivity is likely to be fairly small,’’ or ‘‘measurement error will likely have mean zero,’’ etc. A Bayes factor comparing the nonparametric model with heterogeneity in productivity to the one with measurement error will support the model where the data evidence is most consistent with these prior assumptions. So, for instance, if we were to ﬁnd the distribution of productivity to be very different from lognormal, while measurement error was almost normal, then the model with measurement error would be supported. We argue that this is a very sensible basis (perhaps the only basis) for comparison of models which ﬁt perfectly. Furthermore, it formalizes a sensible informal practice. If one were to estimate a nonparametric model and ﬁnd that the results looked odd, this would be informal evidence against the model. The Bayesian approach puts this reasonable practice on a ﬁrm statistical footing.

5. EMPIRICAL RESULTS The data used in this paper are taken from Bowlus et al. (1995) who provide details about the data (see also Kiefer & Neumann, 1993).9 Brieﬂy, the data consists of N ¼ 697 weekly wages for white high school graduates taken from the NLSY and reported in constant 1982 US dollars. All wages are for the ﬁrst job taken after graduation, which lasts for more than 2 weeks and

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involves more than 20 hours per week. Details of the relatively noninformative priors used along with MCMC algorithms for posterior computation are given in the appendix. The mean, median, standard deviation, minimum, and maximum of the wages are 185.38, 175.00, 71.69, 73.92, and 364.72, respectively. Table 1 contains point estimates and standard deviations of all parameters in each of the ﬁve models (i.e., the basic model plus two models with heterogeneity in productivity and two with measurement error). Table 2 contains the logs of the Bayes factors comparing each pair of models. For the models with heterogeneity in productivity, a very large amount of heterogeneity is present. For instance, the point estimates for the parametric model imply a 95% interval for log-productivity of [5.127, 6.161]. If we take the exponential of the upper and lower bounds to obtain a rough idea of

Table 1. Basic

Posterior Means of Parameters (Posterior Standard Deviations in Parentheses). Hetero-Parametric Hetero-Nonparametric Error-Parametric Error-Nonparametric

mp

70.921 (2.878) 4.261 (0.042) 401.81 (0.487) 5.996 (0.001) 365.11 (0.331) –

sp

–

me

–

5.644 (0.019) 0.264 (0.017) –

se

–

–

a

–

–

r g p p h

72.806 (1.115) 4.288 (0.016) –

72.875 (1.042) 4.289 (0.015) –

–

–

–

– 5.643 (0.018) 0.265 (0.017) – – 5,669.8 (1245.8)

71.993 (1.994) 4.276 (0.029) 333.98 (18.66) 5.809 (0.054) 304.87 (16.54) –

78.971 (11.59) 4.359 (0.142) 403.30 (11.82) 5.998 (0.032) 331.19 (18.77) –

–

–

9.781 (6.834) 37.213 (5.073) –

16.61 (6.517) 31.195 (4.116) 28.145 (42.969)

Notes: The headings in columns 2–6 refer to the basic equilibrium search model (Section 2), the parametric model with heterogeneity in productivity (Section 3.1), the nonparametric model with heterogeneity in productivity (Section 4.1), the parametric model with measurement error in wages (Section 3.2), and the nonparametric model with measurement error in wages (Section 4.2), respectively.

232

Table 2.

GARY KOOP

Log Bayes Factors in Favor of Model Listed in First Column. Basic

Basic 0.0 Hetero-parametric 298 Hetero228 nonparametric Error-parametric 3361 Error146 nonparametric

HeteroParametric

0.0 70 3659 444

HeteroNonparametric

ErrorParametric

ErrorNonparametric

0.00 3215

0.00

0.00 3589 374

what this interval implies for productivity we obtain [$168.51, $473.90]. In other words, in order to extend the basic equilibrium search model to obtain a better ﬁt for the wage density, we have to allow for productivity of unskilled workers in different markets to vary from around $150 to $500 per week. For the nonparametric model with heterogeneity in productivity, a is very large relative to N. This indicates that the parametric part of the model is doing quite well at ﬁtting the wage density and the nonparametric part of the model does not add much (see Eqs. (12) and (15)). The model with normal measurement error ﬁnds se to be very large. Interestingly, the estimated upper bound on the wage distribution is less than the highest observed wage. The reason for this anomaly is that the ei’s for the higher wages in the sample are very negative. Remember that Eq. (9) implies that measurement errors are distributed normally, independent of each other and the data. If we calculate the posterior means of the ei’s they strongly seem to violate the latter assumption. In fact, the correlation between the observed wages and the errors is 0U88! That is, to ﬁt the wage density (see Eq. (10)) the model wants to use ei to adjust low wages upward and high wages downward. Hence, ﬁt considerations imply measurement error should be highly correlated with wages. This is at odds with our assumption that measurement error is independent of the wages. The tension between these two imperatives yields a model which ﬁts a bit better than the basic model, but the conﬂict between prior and data information implies that Bayes factors give little support for the model with measurement error (see Table 2). In other words, for measurement error to explain the poor ﬁt of the basic model, it must be the case that high-wage individuals are consistently massively over-reporting their wages while low wage individuals are under-reporting. Since this does not seem reasonable

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(where ‘‘reasonableness’’ is measured by the prior) the Bayes factors massively reject the model. The nonparametric model with measurement error in wages indicates that large departures from the base prior are required in order to ﬁt the data (Remember that the base prior, G0, has measurement error being i.i.d. normal.) One indication of this is that the posterior for a indicates that it is quite small (relative to N). Hence, more weight is being placed on the nonparametric part of the model (see Eq. (15)). These ﬁndings are reinforced in Figs. 1 and 2, which plot the empirical wage density10 along with the wage densities for each of our models evaluated at the posterior means of the parameters. To be precise, for the parametric model with measurement error the wage density is given in Eq. (10). We plug posterior means for r and p into this formula. For each of a grid of values for w, we take the posterior means of me, s2 , and repeatedly simulate e’s and evaluate Eq. (10) at each simulated draw and average the result. This allows us to plot the wage density with e integrated out. A similar strategy is used for the other parametric model. For the nonparametric models we use Eq. (12), evaluated at the posterior mean for a, to integrate out e or p. These ﬁgures conﬁrm that the basic equilibrium search model implies an increasing wage density and, thus, offers a very poor ﬁt for the empirical wage

Fig. 1.

Empirical and Fitted Wage Densities.

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GARY KOOP

Fig. 2.

Empirical and Fitted Wage Densities.

density. In contrast, both of the models with heterogeneity in productivity appear to offer a very good ﬁt (in a smoothed sense). That is, the ﬁtted wage densities for these models look like a nonparametrically kernel smoothed version of the empirical wage density. However, the nonparametric model offers very little improvement in ﬁt over the parametric one. Adding normal measurement error to the equilibrium search model improves ﬁt somewhat, but the ﬁt is not nearly as good as that provided by the models with heterogeneity in productivity. Allowing for measurement error in a nonparametric fashion seems to offer only slight improvements in ﬁt. The reason for the apparent poor performance of the nonparametric measurement error model is partly due to the way we have constructed Fig. 2. Note that the observed wages are not used (i.e., we are integrating out e at each grid point for the wages). But, in order to ﬁt better, the wages and the errors have to be correlated. The Dirichlet process implicitly does this through Eq. (16) of the posterior (i.e., it groups clusters of workers with similar wages as having the same measurement error). However, since we do not use the data in our ﬁgures, we do not use Eq. (16). Intuitively, Figs. 1 and 2 are not well designed to show the posterior ﬁt of the nonparametric models and the apparent lack of a perfect ﬁt (in a smoothed sense) by the nonparametric model in Fig. 2 is to be expected.

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Nevertheless, it is the case that the ﬁt of the nonparametric model with measurement error is not as good as a kernel smoothed nonparametric estimate would provide. To see why the relatively poor ﬁt of this nonparametric model occurred, we investigated what sort of measurement errors would be required for the wage density from the model with measurement error to match perfectly with the empirical wage density. To be precise, we carried out the following experiment. Firstly, we used the raw wage data and constructed a kernel smoothed estimate of the wage density. Secondly, we took Eq. (10) evaluated at the posterior means for p and r, and found the values of the ei’s which would cause Eq. (10) to match exactly with the kernel smoothed density from the ﬁrst step. The resulting values of the ei’s were truly bizarre. For instance, the standard deviation of the ei’s required to ﬁt the wage density perfectly was 519,205. If the data is ordered so that individual 1 has the lowest wage and individual N the highest, then the correlation between ei and ei+1 is 0.99996. It seems that even Dirichlet mixtures of normals have trouble ﬁtting a distribution with such properties when any sort of prior information is allowed for. By letting the priors for a, me, and se become almost completely noninformative, we could obtain a better ﬁt for the nonparametric model. However, such noninformative prior choices would tend to yield Bayes factors which always favored the parametric model. In short, regardless of whether we make reasonable prior choices based on what we expect measurement error to look like (as we have done in this paper), or use a noninformative prior, we will always come to the conclusion that measurement error is not a good way of extending the basic equilibrium search models. The previous discussion can be formalized by examining Table 2. As frequently happens with Bayes factors, results are very strong.11 That is, there is a clear ranking of models in terms of the posterior support they receive: (1) the parametric model with heterogeneity in productivity, (2) the nonparametric model with heterogeneity in productivity, (3) the basic model, (4) the nonparametric model with measurement error, and (5) the parametric model with measurement error. The former two models ﬁt the data very well and the assumption of log-normality for the productivity distribution seems to match the data quite well (i.e., the prior and likelihood are in accordance). Accordingly these two models obtain strong support over the basic equilibrium search model which implies a counterfactual increasing wage density. The models with measurement error do very poorly relatively to the basic model, even though they ﬁt better. The reason for this was outlined above. Brieﬂy, in order to improve ﬁt, the model wants values of the errors which

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are very much at odds with the assumption that errors should be independent of the wages. Hence, if one believes that measurement error should be i.i.d. normal (as the hierarchical prior for the parametric model implies) or departures from i.i.d. normality are not too large (as the prior for the nonparametric model implies), then one should strongly reject measurement error as an explanation for the failure of the equilibrium search model. The nonparametric model with measurement error beats the comparable parametric model due to its improved ﬁt and, more importantly, the fact that its prior is more ﬂexible. Hence, the conﬂict between prior and data information is less for the nonparametric than the parametric model.

6. CONCLUSIONS In this paper, we have developed Bayesian methods for parametric and nonparametric inference in various equilibrium search models. We have argued that a problem with the existing literature is that there are numerous extensions to the basic model which, along with nonparametric methods, can ﬁt the observed wage density perfectly. This opens up the risk that the empirical job search literature will develop in an unsatisfactory manner, where each researcher has his or her own extension on the basic model and all researchers ﬁt the data perfectly. We have argued that Bayesian methods are a logical solution to the problem. Bayesian parametric or nonparametric methods allow for Bayes factors comparing competing models to be calculated. Bayes factors, apart from the usual goodness of ﬁt and reward for parsimony considerations, contain a reward for coherence between data and prior information. Furthermore, the ‘‘priors’’ typically used in the job search literature are hierarchical in nature so that a non-Bayesian would interpret them as part of the likelihood (e.g., we use as a prior that productivity is log-normally distributed, however Koning et al., 1995 treat an identical assumption as part of the likelihood function). In the context of comparing models which all ﬁt well, this reward for coherence between prior and data information can potentially be very important. That is, the researcher is required to specify what he/she thinks are sensible properties for the extension under consideration (e.g., measurement error should be roughly normal, heterogeneity in productivity should not be too large, productivity should increase with training but diminishing returns to training should exist, etc.) and departures from the prior are evidence against the model. This gives a formal basis for the informal strategy of looking at the results one obtains and seeing if they look reasonable.

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An empirical example using the dataset from Bowlus et al. (1995) shows the practicality and usefulness of our approach. Three parametric models (i.e., the basic model and models with heterogeneity in productivity and measurement error) and two nonparametric models (i.e., one with heterogeneity in productivity and one with measurement error) were estimated. Before estimation, we argued that the models with heterogeneity in productivity are much more reasonable than the others. Our empirical results bore out this conjecture. For instance, we were able to provide strong statistical evidence against the models with measurement error – even the nonparametric one which, in theory, could have ﬁt the data perfectly. In essence, it is possible to ﬁt any wage density by allowing for measurement error. However, in the present empirical example, this ‘‘measurement error’’ has to be of such a strange form that it is highly questionable that it truly is measurement error. The example used in the present paper is purely illustrative. That is, we deliberately chose one reasonable extension to the basic model (i.e., heterogeneity in productivity) and one unreasonable one (i.e., measurement error), in order to illustrate our statistical methods. However, as the theoretical job search literature develops new extensions to the basic equilibrium search model, there will undoubtedly be an increasing number of potential models all of which seems reasonable, a priori. The methods developed in this paper provide a statistically rigorous basic for choosing among such competing theories.

NOTES 1. Throughout this paper we use the term ‘‘worker’’ to refer to individuals, regardless of whether they are employed or unemployed. 2. Here, and throughout the paper, by ‘‘perfect’’ ﬁt we mean only in a kernel smoothed sense. 3. The posterior model probability (i.e., the probability that a given model generated the data) is the main tool of Bayesian model comparison. The posterior odds ratio is the ratio of two posterior model probabilities. The posterior odds ratio is also the ratio of the marginal likelihoods for the two models (i.e., the Bayes factor) times the prior odds ratio. 4. It is worth stressing that one reaction to the literature discussed in this paper might be that the researcher should just go out and get better data (e.g., matching employer/employee data such as in Postel-Vinay and Robin, 2002). This is a sensible reaction. The point of the present paper is to argue that, insofar as such data is not available and one is interested in working with the sorts of (widely used) models

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described here, it is good to use a formal statistical approach such as the one adopted here. 5. Formally, t is identiﬁed. However, without data on durations, identiﬁcation only occurs through the bounds of the wage density. In practice, this means identiﬁcation is weak and, accordingly, we simply set t ¼ 2, which is roughly what is typically observed. An earlier version of this paper treated t as an unknown parameter, and all results on model comparison were qualitatively the same to those given in the present version. 6. In the jargon of the literature, we are assuming ‘‘between market’’ rather than ‘‘within market’’ heterogeneity. The latter is much more difﬁcult to deal with theoretically. The interested reader is referred to Bontemps et al. (1999) or Koning et al. (1995) for more discussion of this issue. 7. See Escobar and West (1995) for a discussion (and further citations) that argue that this ‘‘prior’’ plays a role that is similar to the kernel in nonparametric kernel smoothing algorithms. 8. See, for instance, Judge, Grifﬁths, Hill, Lutkepohl, and Lee (1985, pp. 128–131) for a discussion of these issues in the context of the normal linear regression model. 9. The data were obtained from the Journal of Applied Econometrics data archive at http://qed.econ.queensu.ca:80/jae/. Some Gauss code which illustrates how MCMC methods are used in these models is available on my web page: http:// personal.strath.ac.uk/gary.koop/. 10. The empirical wage density is a simple histogram using 20 bins of equal size. 11. Very loosely speaking, log Bayes factors often behave like the difference of log-likelihoods with additional terms reﬂecting the priors. The log-likelihoods of the models used in this paper are extremely different from one another, and the priors give a strong reward for parsimony. These considerations help drive the strong ﬁndings of Table 2.

REFERENCES Antoniak, C. (1974). Mixtures of Dirichlet processes with applications to nonparametric problems. Annals of Statistics, 2, 1152–1174. Bontemps, C., Robin, J., & van den Berg, G. (1999). An empirical equilibrium job search model with search on the job and heterogeneous workers and ﬁrms. International Economic Review, 40, 1039–1074. Bontemps, C., Robin, J., & van den Berg, G. (2000). Equilibrium search with continuous productivity dispersion: Theory and non-parametric estimation. International Economic Review, 41, 305–358. Bowlus, A., Kiefer, N., & Neumann, G. (1995). Estimation of equilibrium wage distributions with heterogeneity. Journal of Applied Econometrics, 10, S133–S152. Burdett, K. (1990). Empirical wage distributions: A new framework for labor market policy analysis. In: J. Hartog, G. Ridder & J. Theeuweis (Eds), Panel data and labor market studies. Amsterdam: North-Holland. Campolieti, M. (1997). Bayesian estimation of discrete duration models. Unpublished doctoral dissertation. University of Toronto, Toronto.

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Chib, S., & Greenberg, E. (1995). Understanding the Metropolis–Hastings algorithm. The American Statistician, 49, 327–335. Escobar, M., & West, M. (1995). Bayesian density estimation using mixtures. Journal of the American Statistical Association, 90, 577–588. Ferguson, T. (1973). A Bayesian analysis of some nonparametric problems. Annals of Statistics, 1, 209–230. Gelfand, A., & Dey, D. (1994). Bayesian model choice: Asymptotics and exact calculations. Journal of the Royal Statistical Society, Series B, 6, 501–514. Geweke, J. (1999). Using simulation methods for Bayesian econometric modelling: Inference, development and communication. Econometric Reviews, 18, 1–74. Jochmann, M., & Leon-Gonzalez, R. (2004). Estimating the demand for health care with panel data: A semiparametric Bayesian approach. Health Economics, 13, 1003–1014. Judge, G., Grifﬁths, W., Hill, R. C., Lutkepohl, H., & Lee, T.-C. (1985). The theory and practice of econometrics (2nd ed.). New York: Wiley. Kiefer, N., & Neumann, G. (1993). Wage dispersion with homogeneity: The empirical equilibrium search model. In: H. Bunzel, P. Jensen & N. Westergard-Nielsen (Eds), Panel data and labor market dynamics. Amsterdam: Elsevier Science Publishers. Koning, P., Ridder, G., & van den Berg, G. (1995). Structural and frictional unemployment in an equilibrium search model with heterogeneity. Journal of Applied Econometrics, 10, S133–S152. Koop, G. (2001). Bayesian inference in models based on equilibrium search theory. Journal of Econometrics, 102, 311–338. Koop, G., & Poirier, D. (2001). Testing for optimality in stationary job search models. The Econometrics Journal, 4, 257–272. Lancaster, T. (1997). Exact structural inference in job-search models. Journal of Business and Economic Statistics, 15, 165–179. Mortensen, D. T. (1990). Equilibrium wage distribution: A synthesis. In: J. Hartog, G. Ridder & J. Theeuweis (Eds), Panel data and labor market studies. Amsterdam: North-Holland. Mortensen, D. T. (1998). Equilibrium unemployment with wage posting: Burdett-Mortensen Meet Pissarides. Working Paper 98-14. Centre for Labor Market and Social Research, University of Aarhus. Nielsen, M., & Rosholm, M. (1999). Wages, training and turnover in a search-matching model, centre for labor market and social research (manuscript). University of Aarhus. Poirier, D. (1995). Intermediate statistics and econometrics. Cambridge: The MIT Press. Postel-Vinay, F., & Robin, J. (2002). Wage dispersion with worker and employer heterogeneity. Econometrica, 70, 2295–2350. Ridder, G., & van den Berg, G. (1997). Empirical equilibrium search models. In: S. Kreps & K. Wallis (Eds), Advances in economics and econometrics: Theory and applications, Proceedings of the Seventh World Congress of the Econometric Society, Cambridge University Press, Cambridge. Ruggiero, M. (1994). Bayesian semiparametric estimation of proportional hazard models. Journal of Econometrics, 62, 277–300. van den Berg, G., & Ridder, G. (1998). An empirical equilibrium search model of the labor market. Econometrica, 66, 1183–1222. West, M., Muller, P., & Escobar, M. (1994). Hierarchical priors and mixture models. In: P. Freeman & A. F. M. Smith (Eds), Aspects of uncertainty. New York: Wiley.

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APPENDIX. BAYESIAN INFERENCE AND COMPUTATION Inference in all models is accomplished by simulating the posterior using Markov Chain Monte Carlo (MCMC) methods. Model comparison is done using Bayes factors (i.e., the ratio of marginal likelihoods for two competing models). For the non-Bayesian reader, deﬁnitions and explanations of all the technical terms used in this appendix are given in Geweke (1999).

The Basic Equilibrium Search Model The notation used in the paper becomes simpler if we work in terms of the logs of the reservation wage, g ¼ ln(r), and productivity, p ¼ ln(p). Let y ¼ (g, p)u indicate the parameters in the model. The likelihood function is given in Eqs. (4) and (5). Given the nonconventional nature of the likelihood, there is no particular need to choose any particular class of priors to facilitate computation. We stress that researchers can use virtually any prior (including noninformative ones) they wish and the methods developed here will apply. In the present paper, we assume p(y) ¼ p(g)p(p), where a priori, p(g) ¼ fN(r0, R0), p(p) ¼ fN(p0, P0). By letting the variance in the normal go to inﬁnity we can get priors which are noninformative relative to the data. Given that the empirical example is largely illustrative and the contribution of the present paper does not relate to prior elicitation, we make relatively noninformative choices for prior hyperparameters. Of course, in a more serious empirical exercise more care should be taken with prior elicitation and a prior sensitivity analysis should be carried out. Noting that we are using weekly wages for recent high school graduates, setting r0 ¼ 5 and p0 ¼ 6 is reasonable, and setting R0 ¼ P0 ¼ 2 makes the prior fairly ﬂat relative to what we expect the data to indicate. Koop (2001) uses a random walk chain Metropolis–Hastings algorithm (Chib & Greenberg, 1995) to carry out Bayesian inference in this model. In the present paper we use a slightly more efﬁcient MCMC algorithm which involves drawing sequentially from the conditional posteriors: p(p|W,g) and p(g|W,p). To be precise, we have pðgjW; pÞ / pðgÞ

1 ðexpðpÞ expðgÞÞN=2

IðBÞ

(A.1)

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To draw from this density we use rejection methods. In particular, we use p(g)I(B) as a source density kernel and use the fact that 1=ðexpðpÞ expðgÞÞN=2 is increasing in g over the interval I(B) to obtain the appropriate bound necessary to calculate the acceptance probability. Next we have pðpjW; gÞ / pðpÞ

1

N Y

1

ðexpðpÞ expðgÞÞN=2 i¼1 ðexpðpÞ wi Þ1=2

IðBÞ

(A.2)

To draw from this density we also use rejection methods. In particular, we use p(p)I(B) Q as a source density1=2kernel and use the fact that ð1=ðexpðpÞ Þ is decreasing in p over the interval I(B) to expðgÞÞN=2 Þ N i¼1 ð1=ðexpðpÞ wi Þ obtain the appropriate bound necessary to calculate the acceptance probability. In order to calculate the marginal likelihood necessary for calculating Bayes factors, we use the modiﬁed harmonic mean approach ﬁrst suggested by Gelfand and Dey (1994), as implemented in Geweke (1999). This involves a truncation parameter labeled p in Geweke (1999, Section 4.3) which we choose to be 0.95. Experimentation indicates that marginal likelihood is not sensitive to choice of p.

Parametric Extensions of the Equilibrium Search Model Heterogeneity in Productivity The model and methods are similar to those outlined for the basic model above and we use the same prior for common parameters. In particular, the posterior conditional for g is identical to Eq. (A.1) except that p and p now vary over individuals and, hence, have a subscript i and I(B) is altered by the fact that h now varies with i. The likelihood function is given by Eqs. (4) and (7). The ﬁrst stage in the hierarchical prior is given by Eqs. (6) (i.e., p(pi) ¼ fN(mp,s2p ) where mp, s2p are unknown parameters to be estimated). The second stage in the hierarchy is given by p(mp) ¼ fN(p0, P0) and pðs2 p Þ ¼ f G ðm0 ; n0 Þ where fG(a, b) denotes the Gamma density with mean a and b degrees of freedom (Poirier, 1995, p. 100). By letting the degrees of freedom in the Gamma go to zero we can obtain a prior which is noninformative relative to the data. Note that we choose the same prior for mp as we did for p in the previous model (although, of course, we could have chosen a different prior). We set m0 ¼ 1.0 and n0 ¼ 1.0, relatively noninformative choices which potentially allow for a huge amount of heterogeneity in productivity.

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An MCMC algorithm can be derived by sequentially drawing from Eq. (A.1) along with p(p|W, g, mp, s2p ), p(mp|W, g, p, s2p ), and pðs2 p jW; g; mp Þ where p now equals (p1, . . . , pN)u. The latter two posterior conditionals simplify to: pðmp jp; s2p Þ ¼ fN ðp1 ; P1 Þ

(A.3)

pðs2 p jp; mp Þ ¼ fG ðm1 ; n1 Þ

(A.4)

and P 2

1 1 2 pi Þ, n1 ¼ n0+N and where P1 1 ¼ PP 0 þ Nsp ; p1 ¼ P1 ðP0 m0 þ sp 2 mp Þ Þ. m1 ¼ n1 =ðn0 m0 þ ðpiQ 2 2 pðpjW; g; mp ; s2p Þ ¼ N i¼1 pðpi jwi ; g; mp ; sp Þ and draws from p(pi|wi, g, mp, sp ) for i ¼ 1, . . . , N can be taken using rejection methods analogous to those used for the basic model. To be precise, we use p(pi) truncated to ensure that piZhi as the source density to generate draws. An acceptance probability can be obtained by noting that p(pi|wi, g, mp, s2p ) is decreasing in pi and has a ﬁnite value at exp(pi) ¼ hi. The marginal likelihood is calculated using the method outlined in Geweke (1999, Section 4.3). However, it is slightly complicated by the fact that the parameter vector includes p which is N-dimensional. To simplify computation, we only include g, mp, s2p as the parameters used to construct the function Geweke (1999) labels f(y ). This choice is simulation consistent and experimentation with artiﬁcial data indicates it to be a good one.

Measurement Error in Wages The likelihood function is given by Eqs. (4) and (10). The prior for g and p is as in the basic model. The ﬁrst stage in the hierarchical prior is given by Eq. (9). The second stage in the hierarchy is given by p(me) ¼ fN(e0, E0) and pðs2 p Þ ¼ f G ðm0 ; n0 Þ. We set e0 ¼ 0, E0 ¼ 10.0, me0 ¼ 10.0, and ne0 ¼ 0.0001. These values indicate that we expect measurement error to have mean zero and be relatively small (i.e., typically less that $20), however we are very uncertain about this prior expectation and hence make very noninformative choices for the other hyperparameters. Our MCMC algorithm involves sequentially drawing from Eqs. (A.1) and (A.2) (with B deﬁned by rowi+eioh) and p(e|W, g, me,s2 ), p(me |W, g, p, s2 ), and pðs2 jW; g; p; m Þ where e ¼ (e1, . . . , eN)u. The latter two posterior conditionals simplify to: pðm j; s2 Þ ¼ fN ðe1 ; E 1 Þ

(A.6)

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and pðs2 j; m Þ ¼ fG ðm1 ; n1 Þ

(A.7)

P 1 1 2 i Þ, ne1 ¼ ne0+N and þ Ns2 where E 1 ; e1 ¼ E 1 ðE 0 e0 þ s 1 ¼ E0 P ði m Þ2 Þ. m1 ¼ v1 =ðv0 m0 þ Q 2 2 pðjW; g; m ; s2 Þ ¼ N i¼1 pði jwi ; g; m ; s Þ and each of p(ei|wi, g, me, s ) can be drawn from using rejection methods analogous to those used for the previous models. We use p(ei) truncated to ensure that rowi+eioh as the source density to generate draws. An acceptance probability can be obtained by noting that p(ei|wi, g, me, s2 ) is increasing in ei and has a ﬁnite value at ei ¼ hwi. The marginal likelihood is calculated as in the previous model with heterogeneity in productivity.

Nonparametric Extensions of the Equilibrium Search Model Heterogeneity in Productivity The setup for this model is as the parametric case except for the treatment of p. Thus, the MCMC algorithm and priors for g, mp, s2p conditional on p are exactly as described above. Similarly the marginal likelihood calculation is (apart from the changes due to the use of a different prior for p) identical to that outlined in the previous section. Hence, we need only describe how to draw from p(p|W, g, mp, s2p ) in order to fully specify our computational methods. We do not do so in any detail since, once methods for calculating Eqs. (14)–(16) are speciﬁed, the remainder of the algorithm is identical to that given in West et al. (1994, pp. 367–369) (or Campolieti, 1997) and the reader is referred there for precise details. Eqs. (14) and (16) can be calculated directly. However, it is not possible to evaluate the integral in Eq. (15) analytically. Hence, it is evaluated using Monte Carlo integration. In particular, we take 1000 draws from g0(pi) (i.e., fN (mp, s2p ) evaluated at the MCMC draws of mp, s2p ), and average the f(wi|pi)’s evaluated at each draw. The nonparametric model has the additional parameter, a, for which we use the prior p(a) ¼ fG(ma, na). By setting ma ¼ N we allocate roughly half the weight to the parametric part of the model (see Eq. (12)), by setting na ¼ 3 we are making the prior relatively noninformative. Note that West et al. (1994) also describes how to draw from the conditional posterior of a.

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Measurement Error The setup for this model is as the parametric model with measurement error except for the treatment of e. Thus, the MCMC algorithm and priors for g, p, me, s2 conditional on e are exactly as described above. The prior for a is the same as for the previous nonparametric model. To draw from p(e|W, p, g, me, s2 ), we use the same method as for the nonparametric model with the material relating to heterogeneity in productivity altered in the obvious way. That is, Eqs. (14)–(16) are calculated using densities from the model with measurement error (rather than the model with heterogeneity in productivity).

DO SUBSIDIES DRIVE PRODUCTIVITY? A CROSS-COUNTRY ANALYSIS OF NORDIC DAIRY FARMS Nadine McCloud and Subal C. Kumbhakar ABSTRACT One of the foremost objectives of the Common Agricultural Policy (CAP) in the European Union (EU) is to increase agricultural productivity through subsidization of farmers. However, little empirical research has been done to examine the effect of subsidies on farm performance and, in particular, the channels through which subsidies affect productivity. Using a Bayesian hierarchical model in which input productivity, efﬁciency change, and technical change depend on subsidies and other factors, including farm location, we analyze empirically how subsidies affect the performance of farms. We use an unbalanced panel from the EU’s Farm Accountancy Data Network on Danish, Finnish, and Swedish dairy farms and partition the data into eight regions. The data set covers the period 1997–2003 and has a total of 6,609 observations. The results suggest that subsidies drive productivity through efﬁciency and input productivities and the magnitudes of these effects differ across regions. In contrast to existing studies, we ﬁnd that subsidies have a positive impact on technical efﬁciency. The contribution of subsidies to Bayesian Econometrics Advances in Econometrics, Volume 23, 245–274 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23008-2

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output is largest for dairy farms in Denmark and Southern, Central, and Northern Sweden.

1. INTRODUCTION AND MOTIVATION Farmers within the European Union (EU) enjoy the beneﬁts of the Common Agricultural Policy (CAP), a system of agricultural subsidies and programs that was established in 1957. The CAP absorbs, on average, about half of the EU’s annual budget which makes this policy the EU’s most expensive budgetary commitment. One of the foremost initial objectives of the CAP is ‘‘to increase productivity, by promoting technical progress and ensuring the optimum use of the factors of production, in particular labor.’’1 The existing empirical literature, however, ﬁnds that subsidies negatively affect productivity (see, e.g., Giannakas, Schoney, & Tzouvelekas, 2001; Rezitis, Tsiboukas, & Tsoukalas, 2003; Iraizoz, Bardaji, & Rapun (2005); Karagiannis & Sarris, 2005; Zhengfei & Oude Lansink, 2006) which suggests a policy prescription of eliminating subsidies to promote productivity, ceteris paribus. These studies use two different modeling approaches both of which may suffer from certain drawbacks. One approach (see, e.g., Zhengfei & Oude Lansink, 2006) allows subsidies to exert a direct inﬂuence on productivity, that is, this approach treats subsidies as traditional inputs such as labor and capital. However, traditional inputs are necessary for production of output while subsidies are not necessary for production of output. Furthermore, a single traditional input can produce some output but subsidies alone cannot produce any output. Thus, the ﬁnding of this modeling approach may be a consequence of model misspeciﬁcation. The other approach (see, e.g., Giannakas et al., 2001; Rezitis et al., 2003; Iraizoz et al., 2005; and Karagiannis & Sarris, 2005) employs a stochastic production frontier approach and allows subsidies to affect productivity only through the mean of technical efﬁciency. The common argument proffered for the effect of subsidies on technical efﬁciency is that subsidies discourage farmers from exerting more effort into their production activities than in the absence of subsidies, and hence reduce productivity. While this modeling approach escapes the aforementioned traditional-input criticism, it does not examine concurrently the impact of subsidies on overall productivity through changes in input productivity, technical efﬁciency, and technical change. Our rationale for considering input productivity as functions of subsidies is guided, in part, by the theoretical prediction of the

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effect of subsidies on traditional inputs. Coupled subsidies distort relative prices of traditional inputs and consequently affect their marginal rates of transformation.2 In the theoretical literature, Hennessy (1998) predicts that in a stochastic environment with producers possessing decreasing absolute risk aversion an increase in coupled or decoupled subsidies can increase producers’ optimal input levels.3 This increase in optimal inputs levels may increase productivity. Also, Roe, Somwaru, and Diao (2002) ﬁnd that under the assumption of imperfect capital markets, in the short- to medium-run subsidies have a positive but small effect on output by increasing capital deepening and employment of labor; however, in the long-run subsidies are not linked to output. With respect to technical change, subsidized producers are less credit constrained and can invest in research and development and advanced technologies thereby achieving technical progress in the long-run. Consequently, restricting subsidies to affect productivity only through technical efﬁciency may also result in an inadequately speciﬁed model that yields invalid inferences on the relation between subsidies and productivity. In this paper, we examine empirically the productivity effects of subsidies through changes in input productivity, efﬁciency change, and technical change, especially when subsidies are coupled to inputs and production environment. Our maintained hypothesis therefore is that subsidies are not traditional inputs but ‘‘facilitating’’ inputs that affect output indirectly by changing the productivity of traditional inputs (technology effect or input productivity), shifting the technology (technological or technical change), and affecting technical efﬁciency. We consider a simple but ﬂexible Bayesian model that enables decomposition of the overall productivity change (i.e., the effect of subsidies on output) into input productivity, technical change, and technical efﬁciency change. We model the input productivity by allowing the production function coefﬁcients to be functions of subsidies and environmental factors such as location. Consequently, subsidies can be complementary or substitutable with the traditional inputs. We also decompose input productivity into input-speciﬁc components. Following the standard production theory we model technical change through time, which in our model is also affected by subsidies. We use a nonfrontier approach to model technical efﬁciency, which also depends on subsidies and location and has a random component to account for latent heterogeneity such as managerial quality. We use an unbalanced data that span the period 1997–2003 and consist of 6,609 farms from three Northern EU countries, Denmark, Sweden, and Finland, which we divide into eight production regions.

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To the best of our knowledge, our work is the ﬁrst Bayesian analysis on the impact of subsidies on productivity. However, the decomposition of productivity into technical efﬁciency, technical change, and input elasticities is not new to the Bayesian literature. Koop, Osiewalski, and Steel (2000) and Koop (2001) respectively study the decomposition of aggregate output growth and sectoral output growth into these three components. In contrast to our nonfrontier approach, Koop et al. (2000) and Koop (2001) use a stochastic frontier approach in their analysis. We note that our idea of subsidies as facilitating inputs is similar but more general than the concept of ‘‘effective-factor correction variable’’ used by Koop et al. (2000). To test empirically our hypothesis, we also consider a traditional model in which subsidies are treated as traditional inputs and therefore impact productivity directly. Using the Deviance Information Criterion (DIC) measure we then investigate whether our ﬂexible model is more consistent with the data than the model that admits subsidies as traditional inputs. The rest of the paper is organized as follows. Section 2 offers different ways of incorporating subsidies into the production function and highlights some of their limitations in modeling the effect of subsidies on productivity. Section 3 contains a description of the farm accountancy data for Denmark, Finland, and Sweden. Section 4 presents and discusses the empirical results. The last section contains a summary of the main ﬁndings and some concluding remarks. We place all of the mathematical details in the Technical Appendix. A GAUSS code for estimating the models is available from the authors upon request.

2. MODELING SUBSIDIES IN PRODUCTION Assume that farmers use a K-variate traditional input x to produce an aggregate output y. The functional relationship between x and y is described by a production function f :RK þ ! Rþ where y ¼ f(x). Suppose the farmers receive subsidies S. The question is how should we incorporate S into the production function. In the preceding section, we discussed several channels through which subsidies can affect productivity. In this section, we introduce several model speciﬁcations to explore which model can identify what channel. In doing so, our objective is to ﬁnd a model that can identify all three channels of productivity growth.

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Suppose the output-subsidy relation can be captured by a production function which is speciﬁed as y ¼ Af ðx; S; t; pÞ

(1)

where t represents the exogenous technology shifter, p the vector of parameters that represents the technology, and A the parameter that represents efﬁciency.4 The implication here is that subsidies are considered as traditional inputs and therefore have a direct effect on output (see, e.g., Zhengfei & Oude Lansink, 2006).5 In Eq. (1) subsidies do not affect output through technical efﬁciency. However, depending on the functional form of f( ), subsidies can affect output indirectly through input productivities and technical change. If we assume the mathematical relation between subsidies and output is y ¼ AðSÞf ðx; t; pÞ

(2)

then in this formulation subsidies are separable from the other inputs and therefore the rate of change in output due to any input xk (elasticity of output with respect to xk) is neutral to subsidies, meaning that the elasticity of none of the other inputs is affected by the level of subsidies. For this speciﬁcation, subsidies affect output only through the efﬁciency function A(S). This is true irrespective of the functional form for A(S) and f( ).6 This speciﬁcation is similar to that employed by Giannakas et al. (2001), Iraizoz et al. (2005), Karagiannis and Sarris (2005), and Rezitis et al. (2003) who model the effect of subsidies on productivity only through technical efﬁciency. A model that is more general than Eqs. (1) and (2) is: y ¼ AðSÞf ðx; S; t; pÞ

(3)

Both Eqs. (1) and (2) can be viewed as special cases of Eq. (3) by imposing appropriate restrictions. Under appropriate functional forms of f( ), Eq. (3) exhibit all three channels of productivity effect. Another possible formulation for the output–subsidy relation is: y ¼ f ðAðSÞx; t; pÞ

(4)

where A(S)x ¼ A1(S)xl, . . . , AK(S)xK and Ak( )W0 is the productivity factor associated with xk, k ¼ 1, . . . , K. In this speciﬁcation, subsidies affect output through input productivity (not necessarily through input elasticities unless the functional form of f( ) is ﬂexible (such as the translog)). Strict positivity on all Ak does not guarantee that subsidies will increase productivity of all inputs. However, if AkW1 then subsidies enhance productivity of input k.

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Hence it is possible to have positive overall productivity even if subsidies reduce the productivity of some inputs.7 For Eq. (4) we see that the productivity effect of subsidies through the technical change channel depends on the functional form of f( ). There are some major limitations of the speciﬁcations in Eqs. (1)–(4). For a log-linear functional form the speciﬁcations in Eqs. (1)–(4) are algebraically equivalent. That is, assuming that f( ) is log linear (Cobb–Douglas) and A( ) is log linear in S, the relationships in Eqs. (1)–(4) become X pk ln xk þ ps ln S þ pT t (5) ln y ¼ p0 þ k

In this speciﬁcation, subsidies are productivity enhancing if the coefﬁcient ps is positive. The pT coefﬁcient represents technical change and the intercept term, p0, captures technical efﬁciency.8 In particular, subsidies in speciﬁcation (5) affect productivity directly just like the traditional inputs. Therefore, if we use speciﬁcations (1)–(4) with a Cobb– Douglas production function (which is equivalent to using Eq. (5)) it is not possible to identify our three proposed channels in the output–subsidy nexus. Since the main purpose of this paper is to examine the three different channels through which subsidies impact productivity, we begin by formalizing the concept of a facilitating input which we will use repeatedly in what follows. Deﬁnition 1. A variable S is a facilitating input in the production of output y if (i) S is not necessary for the production of y, (ii) S alone cannot produce y, and (iii) S affects productivity through at least one channel.9 Note that the ﬁrst two criteria distinguish traditional inputs from facilitating inputs. Given the limitations of Eqs. (1)–(4), we seek another speciﬁcation in which subsidies can be treated as facilitating inputs even for a log-linear production function. In view of this, we specify the production function in the following way: y ¼ AðSÞf ðx; t; pðSÞÞ

(6)

This speciﬁcation accommodates the requirement that subsidies are facilitating inputs and they affect input productivity, technical efﬁciency, and technical change. Also, we can identify all the three channels of transmission even with log-linear speciﬁcation.10 Using a Cobb–Douglas

251

Do Subsidies Drive Productivity?

production function the relationship in Eq. (6) becomes X pk ðSÞ ln xk þ pT ðSÞt ln y ¼ p0 ðSÞ þ

(7)

k

which shows subsidies can affect output through the input coefﬁcients (technology parameters) pk(S), the efﬁciency parameter p0(S), and the time trend parameter pt(S). The speciﬁcation in Eq. (7) is general enough to obtain some of the earlier speciﬁcations as special cases. Using Eq. (7), the output elasticity with respect to S; y;s @ ln y=@ ln S; is @p0 ðSÞ X @pk ðSÞ @pT ðSÞ þ þt (8) ln xk y;s ¼ @ ln S @ ln S @ ln S k and has three components: (i) an efﬁciency component, (ii) a technology component, and (iii) a technological change component in that order.11

2.1. Bayesian Hierarchical Models In this subsection, we elaborate on the econometrics used. We consider ﬂexible models that are based on the general formulations outlined in Eqs. (1) and (6). We begin by ﬁxing notation. The data set has region, farm, and time dimensions which we index respectively as, i, j, and t. The input subscript is k (k ¼ 1, . . . , K). Time (t) and subsidies (S) are separated from the input vector (x). To capture regional differences in productivity, we allow for heterogeneous input-speciﬁc elasticities, technology efﬁciency, and technical change across regions. With these notations our ﬁrst model is: Model I: Subsidies as Traditional Inputs ln yijt ¼ p0;ij þ

K X

pk;ij ln xk;ijt þ ps;ij ln S ijt þ pT;ij t þ uijt

k¼1

p0;ij ¼ b0 þ b1 C 1j þ b2 C 2j þ þ bM1 C M1;j þ vij pk;ij ¼ a0k þ a1k C 1j þ a2k C 2j þ þ aM1;k C M1;j; for k ¼ 1; . . . ; K

(9)

ps;ij ¼ a0s þ a1s C 1j þ a2s C 2j þ þ aM1;s C M1;j pT;ij ¼ a0t þ a1t C 1j þ a2t C 2j þ þ aM1;t CM1;j where Cl, . . . , CM are regional dummies and region M is used as the reference group. Note that in this formulation the efﬁciency parameter p0,ij is

252

NADINE MCCLOUD AND SUBAL C. KUMBHAKAR

assumed to be random and time-invariant. We append a random component to capture heterogeneity from latent productivity differential such as managerial quality and latent time-invariant factors. In fact, our sample is quite heterogeneous in terms of weather conditions and land quality so it seems plausible to allow for latent productivity differential. We do not lump the random productivity component (vij) with the noise terms in the production function (uijt), since our interest is to estimate p0,ij which requires estimation of vij. This is in contrast to a standard random effects model, where the interest commonly lies in estimating the parameters a and p and the vij and uijt terms are collectively treated as the composite error term. The speciﬁcation in Eq. (9) is similar to the standard production function model that accommodates country-speciﬁc intercept and slopes. Our second model is in line with Eq. (7) and is speciﬁed as: Model II: Subsidies as Facilitating Inputs ln yijt ¼ p0;ij þ

K X

pk;ijt ln xk;ijt þ pT;ijt t þ uijt

k¼1

p0;ij ¼ b0 þ b1 C 1j þ b2 C 2j þ þ bM1 C M1;j þ bs ln S ij þ vij pk;ijt ¼ a0k þ a1k C 1j þ a2k C 2j þ þ aM1;k C M1;j þ ask ln S ijt;

(10)

for k ¼ 1; . . . ; K pT;ijt ¼ a0t þ a1t C 1j þ a2t C 2j þ þ aM1;t CM1;j þ aM1; j þ ast ln S ijt: This formulation is more general than the one in Eq. (9). Here the input elasticity coefﬁcients (pk) and technology shifter (pt) are functions of regional dummies as well as subsidies. Technical efﬁciency parameter (p0,ij) is also a function of regional dummies and subsidies (its mean over time), S ij . Clearly, technical efﬁciency is time-invariant and pk,ijt and pT,ijt are time varying. Some informative measures can be extracted from the model in Eq. (10). Deﬁne output elasticity vector with respect to input xk as y;k @ ln y=@ ln xk and as before output elasticity vector with respect to subsidies S as y;s @ ln y=@ ln S. Then y;k;ijt ¼ a0k þ a1k C 1j þ a2k C 2j þ þ aM1;k C M1;j þ ask ln Sijt y;s;ijt ¼ bs þ

K X k¼1

ask ln xk;ijt þ ast t

(11) (12)

253

Do Subsidies Drive Productivity?

For the reference region ey,k;Mjt ¼ a0kþask ln SMjt and for region 1, ey,k;1jt ¼ a0kþa1kþask ln S1jt. Thus, if the subsidies in regions 1 and M are the same, alk represents the difference in the output elasticity of input xk between region 1 (C1) and the reference region (CM). The expression for ey,k shows that subsidies have a second-order effect on productivity of input xk. In particular, since ask ¼ qey,k/q ln S, if askW0 then an increase in subsidies will increase overall productivity and the productivity of input xk. That is, subsidies will have a complementary effect on input xk. Thus, the sign of ask allows us to understand the substitutability and complementarity between the traditional inputs and subsidies.12 Similarly, an increase in subsidies will enhance overall productivity and also increase the contribution of technical efﬁciency to productivity if bs ( ¼ qp0/q ln S), is positive. Also, an increase in subsidies will increase overall productivity and the contribution of technical change to productivity if ast ( ¼ qpT/q ln S) is positive. Aggregate measures such as returns to scale (RTS), technical change (TC) and technical efﬁciency (TE), can also be easily derived from Eq. (10). These are: RTS ijt ¼

X @ ln yijt X X ¼ pk;ijt ¼ y; k;ijt @ ln xk;ijt k2K k2K k2K

(13)

TC ijt ¼ pT;ijt

(14)

TE ij ¼ p0;ij

(15)

Thus RTS represents the output change from proportional changes in all inputs. TE is captured by the p0,ij term. A more formal measure of TE would be TE ¼ expðp0;ij Þ 1 where p0;ij ¼ p0;ij maxi;j ðp0;ij Þ. TC shows the percentage (when multiplied by 100) by which output changes over time, holding everything else unchanged. That is, it indicates shift in the production function over time, ceteris paribus. To investigate empirically whether subsidies may be considered as facilitating or traditional inputs, and also to explicitly account for unobserved factors that may impinge on technical efﬁciency, we compare the ﬁrst and second model speciﬁcations in terms of their consistency with the data. To do so we abstract from the classical mode of estimation and adopt the Bayesian formalism. This is done to avoid the estimation problems and ﬁnite sample inconsistency that are common to the existing classical methods of estimating random effects. Bayesian analysis facilitates estimation of the random effects and provides exact ﬁnite sample results.

254

NADINE MCCLOUD AND SUBAL C. KUMBHAKAR

In addition, economic regularity conditions can easily be imposed in this Bayesian setup. For ease of exposition we discuss the Bayesian model in terms of Model II in Eq. (10). We suppress the logarithm notation and rewrite Eq. (10) as: yijt ¼ p0;ij þ

K X

þxk;ijt ak þ t at þ uijt ¼ p0;i; j þ x~ ijt G þ uijt

(16)

k¼1

p0;ij ¼ zi; j b þ vij where xk;ijt and t are the level 1 covariates interacted with the region dummies and subsidies, each covariate has dimension 1 (Mþ1), and ak ¼ (a0k, a1k, . . . , aM–1,k, as,k)u, for k ¼ 1, . . . , K, at ¼ (a0t, a1t, . . . , aM–1,t, as,t)u and x~ ijt ¼ ½x1;ijt ; x2;ijt ; . . . ; xK;ijt ; t is the 1 (Mþ1) (Kþ1) row vector containing the entire collection of level 1 regressors with G ¼ ða01 ; a02 ; . . . ; a0K ; a0t Þ0 . At level 2, zi;j ¼ ½1; C 1j ; C 2j ; . . . ; C M1;j; S ij is a 1 (Mþ1) time-invariant vector of region dummies and subsidies, and b ¼ (b0, b1, . . . , bM–1, bS)u is the (Mþ1) 1 coefﬁcient vector. We assume iid

iid

that uijt Nð0; s2u Þ and vij Nð0; s2v Þ. us deﬁne Ni as the number ofPdistinct farms in region i, N~ i PLet i1 i ~ ~ r¼1 N r þ 1 for iW1, N 1 1, and N i r¼1 N r . We now consider a matrix formulation of this model. First, we stack the observations so that the fastest running index is time t (t ¼ 1, . . . , Tj), the second fastest running index is the farm index j; j ¼ N~ i ; N~ i þ 1; . . . ; N i , and the slowest running index is the region index i (i ¼ 1, . . . , M). We let Tj be the time span for farm j; thus for our time period 1997–2003, Tjr7 ’j. Then for each farm j in region i we have yi;j ¼ p0;ij iT j þ x~ i;j G þ ui;j , where 0

1 0 1 u~ij1 x~ ij1 By C B x~ C B u~ C B ij2 C B ij2 C B ij2 C B C B C B C yi;j ¼ B .. C; x~ i;j ¼ B .. C; ui;j ¼ B .. C B . C B . C B . C @ A @ A @ A yijT j x~ ijT j u~ijT j yij1

1

0

so that yi, j is a Tj 1 vector, x~ i; j is a Tj (Mþ1) (Kþ1) matrix, ui, j is a Tj 1 vector, and iT j is Tj 1 vector of ones.

255

Do Subsidies Drive Productivity?

If we Ri as the number of panel observations in region i then P denote i T . Also, we let R be the total number of panel observaRi ¼ N j¼N~ i j P Ni R tions thus R ¼ M i¼1 i . Deﬁne fen;i gn¼1 as the collection of standard basis Ni 0 vectors in R . Let E n;i;j en;i iT j be a Tj Ni matrix of zeros and ones with ones only in column n, and denotes the Kronecker product. Furthermore, deﬁne Ei as 0 1 E 1;i;N~ i BE ~ C B 2;i;ðN i þ1Þ C B C Ei B C .. B C . @ A E N i;i;N i

so that Ei is a Ri Ni matrix of zeros and ones. We then write the model for region i as Y i ¼ E i P0i þ X i G þ U i P0i ¼ Z i b þ V i .. .. .. .. . where Z i ¼ ½iN i .C 1 .C2 . .C M1 ..S i is the Ni (Mþ1) matrix of covariates at level 2 of the model, and 0

yi;N~ i

1

0

x~ i;N~ i

1

0

ui;N~ i

1

By ~ C B x~ ~ C Bu ~ C B i;ðN i þ1Þ C B i;ðN i þ1Þ C B i;ðN i þ1Þ C B C B C B C Yi ¼ B C; X i ¼ B C; U i ¼ B C .. .. .. B B B C C C . . . @ A @ A @ A yi;N i x~ i;N i ui;N i 0

vi;N~ i

1

0

S i;N~ i

1

B C Bv ~ C B S i;ðN~ þ1Þ C B i;ðN i þ1Þ C B C i B C C Vi ¼ B C; S i ¼ B .. . B C B C .. . B C @ A @ A vi;N i S i;N i are respectively of dimensions Ri 1, Ri (Mþ1)(Kþ1), Ri 1, Ni 1 and iid Ni 1. Then with the error assumptions we obtain U i N ðRi Þ ð0; s2u I Ri Þ iid and V i N ðN i Þ ð0; s2u I Ni Þ, and the likelihood and population distribution

256

NADINE MCCLOUD AND SUBAL C. KUMBHAKAR

respectively are ind

Y i jP0i; G; s2u N ðRi Þ ðE i P0i þ X i G; s2u I Ri Þ ind

P0i jb; s2v N ðN i Þ ðZ i b; s2v I N i Þ

(17)

(18)

where I N i . and I Ri are the identity matrices in RN i and RRi respectively and N ðRi Þ and N ðN i Þ are multivariate Gaussian distributions. To convert the above model to its Bayesian counterpart, we impose distributional speciﬁcations and independence assumption on the priors b; s2u ; s2v , and G so that: pðb; s2u ; s2v ; GÞ ¼ pðbÞpðs2u Þpðs2v ÞpðGÞ where b N ðMþ1Þ ðb; SÞ

(19)

m r ; 2 2

(20)

n d ;

G s2 v 2 2

(21)

G N ðMþ1ÞðKþ1Þ ðG; OÞ

(22)

s2 u G

and G denotes the gamma distribution, N(Mþ1) and N(Mþ1)(Kþ1) are multivariate Gaussian distributions, and the hyperparameters S; n; d; m; r; G; O are prespeciﬁed constants. Then the model in Eqs. (17) and (18) and the priors in Eqs. (19), (20), (21), (22) constitute a complete Bayesian hierarchical model. To obtain parameter estimates of the hierarchical model we utilize the Gibbs sampler. We simulate the models using 21,000 iterations and for each iteration we generate random draws from the set of posterior conditionals in the order: (a) ½bjY; G; P0i ; s2u ; s2v ; (b) ½GjY; b; P0i ; s2u ; s2v ; 2 2 2 (c) ½P0i jY; b; G; s2u ; s2v ; (d) ½s2 u jY; b; G; P0i ; sv ; (e) ½su jY; b; G; P0i ; su . The values for the hyperparameters and the derivations for these full posterior conditionals are in the appendix. To ensure convergence of the MCMC we carry out different diagnostic checks. In the ‘‘burning in phase’’ we discard the ﬁrst 1,000 iterates to reduce the dependence of the values of the ﬁnal estimate on the starting values of the chain. We plot a separate

257

Do Subsidies Drive Productivity?

autocorrelation function for each set of the posterior iterates to examine the severity of the autocorrelation of each parameter. In the ‘‘thinning phase’’ we then skip every 100 observations reducing the autocorrelations close to zeros. Using the remaining iterates we approximate the posterior mean and standard deviation for each parameter of interest using Monte Carlo estimates. Finally, we use the Savage–Dickey density ratio by Verdinelli and Wasserman (1995) to compute the Bayes factors (BF) for the various hypotheses related to Models I and II. We utilize the DIC by Spiegelhalter, Best, Carlin, and van der Linde (2002) to compare the ﬁt of Models I and II to the data.

3. DATA We use an unbalanced data set from the Farm Accounting Data Network (FADN) that covers individual dairy farms in Denmark, Finland, and Sweden from 1997 to 2003.13 These three countries are quite different in terms of their natural and structural conditions. Denmark has the most favorable climate and land quality for agricultural production and is the most export oriented and competitive. Finland has the least-favorable natural conditions. Sweden is intermediate in that the conditions for agricultural production in Southern Sweden are similar to those in Denmark while the natural conditions in Northern Sweden are similar to those in Finland. There are also differences in the trading beneﬁts enjoyed by these EU economies. Denmark gained accession to the EU in 1973 while Finland and Sweden joined the EU in 1995. Under the CAP, all three economies have been subjected to the same agricultural policy since 1995. In addition, special actions have been implemented to assist farmers in Sweden and Finland to smoothly adjust to their new economic environment with low output prices. These actions include investment aids and a price premium for milk. Given these disparities in production conditions and CAP beneﬁts, we divide the countries into eight regions. These are: Denmark (DK) is one region; three regions are in Sweden (Southern (SS), Central (CS), and Northern (NS) Sweden); and four regions are in Finland (Southern (SF), Central (CF), Western (WF), and Northern (NF) Finland). We select Denmark as the reference region. The total number of observations is 6,609. The total number of distinct dairy farms in the sample is 1,578, of which 647, 441, and 490 are from Denmark, Finland, and Sweden respectively. We use labor, fertilizer, purchased feed, materials, and capital as inputs. Labor is measured in hours

258

NADINE MCCLOUD AND SUBAL C. KUMBHAKAR

of work on the farm and all other inputs are measured in real monetary values. Capital includes costs of machinery and buildings. We report subsidies, excluding investment aids and price support on milk, as a separate direct payment variable in the data. Output is the revenue from sales of milk and other outputs at market prices. We convert monetary values of inputs and outputs to euros using the exchange rates of the national currencies for Denmark and Sweden. Furthermore, we adjust these monetary values for inﬂation by using country-speciﬁc price indices of inputs and outputs.14 Finnish farms are on average the smallest and Danish farms are the largest. In this data set, the average sizes of dairy farms (number of cows) in Denmark, Sweden, and Finland are 84, 36, and 21 respectively. In Table 1 we report the descriptive statistics (mean and standard deviation) of the data. In spite of the size differences of farms across countries, average payments of subsidies are similar except for Northern Sweden which receives the largest payment of subsidies. Farmers in Denmark spend the most on capital, materials and purchase feed while farmers in Eastern Finland spend the least on capital and materials. Labor hours are largest for Southern Sweden followed by Denmark and smallest for Northern Sweden. Farmers in Northern Finland spend the most on fertilizer while farmers in Northern Sweden spend the least on fertilizer. In terms of output, Danish farmers produce the most which is approximately ten times more than farmers in Eastern Finland, the region with the least output. This is quite reasonable since Denmark is the most export-oriented region. Northern Sweden is the most subsidized region and Eastern Finland is the least subsidized. The large standard deviations suggest that inputs, output, and subsidies vary a lot within any given region. Since our main focus is the impact of subsidies on productivity, we look at the distributions of subsidies within and across regions. Fig. 1 shows the plot of total subsidies by region and Fig. 2 shows the plot of subsidies per unit of output (subsidies per each euro of output) also by region. The ﬁgures show variations within and across regions. That is, even after controlling for size differences in terms of output we ﬁnd that the distributions of subsidies differ across regions (although some regions appear more different than others). In addition, Fig. 2 suggests that the distributions of normalized subsidies (subsidies per unit of output) may fall into four groups; at one extreme, Denmark appears to form a group by itself and has the smallest variation in normalized subsidies; at the other extreme, Northern Sweden appears to solely form a group and has the largest variation in normalized subsidies; Central Sweden and Southern Sweden seem to be in one group

Descriptive Statistics (Per Farm).

Country

Finland

Region Variables Output (h) Labor (h) Fertilizer (h) Purchase feed (h) Capital (h) Material (h) Subsidies (h)

SF

EF SD

Mean

SD

Mean

SD

Mean

SD

64,047.69 4,492.741 5,081.632 12,298.59 18,254.12 26,695.76 19,191.16

38,416.07 2,878.016 1,584.153 11,875.75 15,687.56 15,896.74 11,824.15

60,570.66 4,624.093 4,872.485 12,529.03 16,149.76 24,984.35 18,115.03

28,582.26 2,822.304 1,429.608 8,248.635 12,376.5 11,187.06 10,042.36

67,368.05 5,305.944 5,009.353 16,196.27 18,606.59 26,726.53 21,456.46

32,310.88 3,582.036 1,561.821 12,845.66 12,157.84 13,331.87 12,246.55

61,539.32 4,715.297 5,196.239 17,468.13 17,503.02 26,879.43 21,092.3

28,777.6 2,830.338 1,530.016 10,762.28 12,242.5 13,245 14,146.95

Denmark

Region

DK

Output (h) Labor (h) Fertilizer (h) Purchase feed (h) Capital (h) Material (h) Subsidies (h)

NF

Mean

Country

Variables

WF

Do Subsidies Drive Productivity?

Table 1.

Sweden SS

CS

NS

Mean

SD

Mean

SD

Mean

SD

Mean

SD

248,354.1 5,482.042 4,354.637 59,658.59 46,788.27 75,474.4 23,144.07

128,703.1 3,712.306 1,514.928 37,852.78 28,648.08 38,142.61 14,747.95

152,558.2 5,827.902 4,837.15 35,964 38,256.08 49,665.14 19,812.8

144,746.3 6,392.296 2,364.427 36,952.42 37,652.56 43,693.46 24,245.23

115,993.4 3,595.155 4,312.099 30,204.51 30,770.57 37,070.32 18,788.77

99,834.29 2,875.706 2,025.008 31,922.68 23,835.56 25,966.11 16,256.79

78,134.98 3,078.917 3,939.551 23,389.57 29,548.88 38,191.92 40,652.46

66,495.51 3,942.42 1,855.844 22,084.13 25,857.73 31,280.46 32,746.71

259

Note: These values are the regional averages for 1997–2003.

NADINE MCCLOUD AND SUBAL C. KUMBHAKAR

260 1.0

Kernel Density

0.8

0.6

0.4

0.2

0.0

5

6

7

8

9

10

11

12

13

Logarithm of Subsidy Region key (NF);

:

(DK); (SS);

Fig. 1.

(CS);

(SF);

(EF);

(WF);

(NS).

Logarithm of Subsidies.

while Southern, Central, Eastern, and Northern Finland appear to form the intermediate group.

4. EMPIRICAL RESULTS We present empirical results for Model I (which treats subsidies as traditional inputs in Eq. (9)) and Model II (which treats subsidies as facilitating inputs in Eq. (10)). Table 2 shows the posterior means and standard deviations of the level 1 and 2 coefﬁcients associated with Model I. It can be seen from Table 2 (at level 1) that the elasticities of most traditional inputs including subsidies, within any given region, are more than one standard deviation away from zero. This implies that the data may be consistent with regional differences in input elasticities including those associated with subsidies. The posterior mean for output elasticities of

261

Do Subsidies Drive Productivity? 14 12

Kernel Density

10

8 6

4 2 0 0.15

0.30

0.45

0.60

0.75

0.90

1.05

1.20

1.35

Normalized Subsidy Region key (NF);

:

(DK); (SS);

Fig. 2.

(CS);

(SF);

(EF);

(WF);

(NS).

Normalized Subsides.

subsidies, ey,s, is approximately 0.43 for Northern Sweden, 0.15 for Eastern Finland, 0.11 for Southern Finland, and 0.05 for Denmark. Thus for a 10% increase in subsidies, output increases by approximately 4.3% in Northern Sweden, 1.5% for Eastern Finland, 1.1% for Southern Finland, and 0.5% for Denmark. Regional differences in technical change are also borne out by this model. Moreover, technical change for all regions except Denmark is positive. In particular, each posterior mean for technical change is more than two standard deviation away from zero. The posterior estimates of the level 2 coefﬁcients suggest that on average the levels of technical efﬁciency for Southern, Eastern, and Northern Finland and Central Sweden are similar to that of Denmark. However, Southern and Northern Sweden are less technically efﬁcient than Denmark while Western Finland is more technically efﬁcient. The BF for the hypothesis that subsidies have no effect on output, that is, a0s ¼ a1s ¼ ¼ a7s ¼ 0, is 3.025 1041. This suggests that there is an empirical link between subsidies and productivity.

262

NADINE MCCLOUD AND SUBAL C. KUMBHAKAR

Table 2.

Model I – Subsidies as Traditional Inputs. Level 1 Coefﬁcients

Country

Denmark

Region

DK

SF

EF

WF

NF

SS

CS

NS

a0k

a1k

a2k

a3k

a4k

a5k

a6k

a7k

Labor Fertilizer Purchase feed Capital Material

Subsidies

Time

Finland

Sweden

0.0122 0.0576 0.0472 0.0295 0.0247 0.0473 0.0256 0.1395 (0.0057) (0.0167) (0.0169) (0.0169) (0.0141) (0.0104) (0.0121) (0.0137) 0.2515 0.0752 0.0041 0.1289 0.0058 0.0491 0.0948 0.2102 (0.0334) (0.0474) (0.0444) (0.0577) (0.046) (0.0356) (0.0359) (0.05) 0.289 0.0938 0.065 0.2383 0.0821 0.0222 0.0874 0.1672 (0.0179) (0.0255) (0.0258) (0.0246) (0.028) (0.0236) (0.0277) (0.0323) 0.1337 0.0499 0.0436 0.0394 0.0067 0.0591 0.0078 0.3501 (0.0146) (0.0295) (0.0272) (0.0329) (0.0273) (0.0228) (0.0274) (0.0303) 0.3892 0.0883 0.0951 0.0311 0.0446 0.0529 0.0357 0.4115 (0.0289) (0.0504) (0.0449) (0.0513) (0.0472) (0.0315) (0.0417) (0.044) a0s

a1s

a2s

a3s

a4s

0.046 (0.0155)

0.0633 (0.0418)

a0t

a1t

a2t

a3t

a4t

a5t

a6t

a7t

0.0103 (0.0025)

0.0155 (0.0071)

0.0211 (0.0067)

0.0265 (0.0081)

0.022 (0.0059)

0.0125 (0.0049)

0.0126 (0.005)

0.0129 (0.0062)

b5

b6

b7

0.1001 0.0575 (0.0372) (0.0447)

0.0158 (0.0352)

a5s

a6s

0.0164 0.0514 (0.021) (0.0281)

a7s 0.3849 (0.0412)

Level 2 Coefﬁcients

Efﬁciency

b0

b1

b2

b3

0.8186 (0.2537)

0.0721 (0.1617)

0.1871 (0.2211)

0.3046 (0.1876)

b4

0.1636 0.5777 (0.2325) (0.1465)

0.2053 1.2783 (0.1961) (0.288)

Note: These values are posterior mean and standard deviation in ().

Table 3 shows the results of our ﬂexible speciﬁcation, Model II, that accommodates subsidies as facilitating inputs. Three main ﬁndings are worth mentioning. First, the effects of subsidies on labor, fertilizer, purchase feed, capital and materials elasticities are more than four standard deviation away from zero. In addition, subsidies affect capital and material elasticities positively and labor, fertilizer and purchase feed elasticities negatively. To be more precise, and in line with our discussion in the previous section on

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Do Subsidies Drive Productivity?

Table 3.

Model II – Subsidies as Facilitating Inputs. Level 1 Coefﬁcients

Country

Denmark

Sweden

Subsidies

DK

SF

EF

WF

NF

SS

CS

NS

a0k

a1k

a2k

a3k

a4k

a5k

a6k

a7k

a8k

0.0394 (0.0172) 0.0691 (0.0442) 0.1046 (0.0249) 0.0058 (0.0288) 0.0485 (0.043)

0.0354 (0.0172) 0.0009 (0.0438) 0.0941 (0.0281) 0.0152 (0.0265) 0.0293 (0.044)

0.0035 (0.0169) 0.0404 (0.0541) 0.2837 (0.0234) 0.0902 (0.0343) 0.0656 (0.0492)

0.0148 (0.0141) 0.019 (0.0468) 0.0944 (0.0281) 0.0375 (0.0276) 0.0232 (0.0413)

0.0216 (0.0111) 0.0644 (0.0358) 0.002 (0.023) 0.0254 (0.0247) 0.0805 (0.0343)

0.0071 (0.0123) 0.1174 (0.036) 0.0559 (0.0247) 0.0455 (0.0274) 0.0314 (0.037)

0.1505 (0.0139) 0.3036 (0.0449) 0.2639 (0.0317) 0.2658 (0.0299) 0.3057 (0.0429)

0.0464 (0.0046) 0.0372 (0.0093) 0.0459 (0.0069) 0.0512 (0.0089) 0.0624 (0.0114)

a1t

a2t

a3t

a4t

a5t

a6t

a7t

a8t

Labor

0.4897 (0.0477) Fertilizer 0.6027 (0.0951) Purchase 0.743 feed (0.0693) Capital 0.3954 (0.0909) Material 0.2376 (0.115) a0t Time

Finland

0.0264 0.0223 (0.0214) (0.006)

0.0323 0.0234 0.0288 (0.0061) (0.0078) (0.006)

0.013 0.0104 0.0131 (0.0045) (0.0047) (0.006)

0.0018 (0.0022)

Level 2 Coefﬁcients b0 Efﬁciency

b1

b2

b3

1.2135 0.1095 0.1828 0.3382 (0.2215) (0.2014) (0.2305) (0.197)

b4

b5

b6

b7

b8

0.1618 0.5138 0.0589 1.0348 0.038 (0.2107) (0.1338) (0.1375) (0.2749) (0.0273)

Note: These values are posterior mean and standard deviation in ().

the complementarity and substitutability roles of subsidies in Model II, we see that subsidies are strongly substitutable with labor, fertilizer, and purchase feed but complementary with capital and materials. Thus an increase in subsidies results in a reduction in the marginal product of labor (decrease in the number of workers employed by the farm) but an increase in the marginal product of capital. This latter effect suggests that subsidies promote investment in capital or capital deepening. The results are also economically meaningful, a 10% increase in subsidies results in approximately 0.51% reduction in the labor elasticity and 0.46% increase in the capital elasticity. Second, the impact of subsidies on technical change is positive but less than one standard deviation away from zero. Technical change in Denmark appears to be close to zero. Third, the impact of subsidies on efﬁciency is

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NADINE MCCLOUD AND SUBAL C. KUMBHAKAR

positive and more than one standard deviation away from zero. We note that this positive impact of subsidies on technical efﬁciency is in contrast to the empirical ﬁnding of Giannakas et al. (2001), Rezitis et al. (2003), Iraizoz et al. (2005), and Karagiannis and Sarris (2005) who allow subsidies to affect productivity only through the technical efﬁciency channel and ﬁnd a negative impact of subsidies on technical efﬁciency. Also, the results indicate no difference in the efﬁciency levels across Eastern, Northern, and Southern Finland, Central Sweden, and Denmark. In addition, Western Finland is more efﬁcient while Southern Sweden and Northern Sweden are less efﬁcient than Denmark. Overall, the results show that subsidies affect productivity through input elasticities and technical efﬁciency and in different directions. These ﬁndings provide strong empirical support for our hypothesis that subsidies are facilitating inputs. We further analyze the implications of modeling subsidies as facilitating inputs by using the BF to test various hypotheses. We ﬁrst investigate whether the regional differences imposed on the parameters are consistent with the data. For the hypothesis of no difference in efﬁciency levels across regions, that is, b1 ¼ b2 ¼ ¼ b7 ¼ bs ¼ 0, we ﬁnd BF ¼ 0. For the hypothesis of no difference in technical change across regions, that is, a1t ¼ a2t ¼ ¼ a7t ¼ ast ¼ 0, we obtain BF ¼ 8.76 106. Also, for the hypothesis of no difference in input elasticities across regions, we obtain BF ¼ 5.912 10255. In particular, we ﬁnd that the labor elasticities of Western Finland and Central Sweden are similar to that of Denmark, while all other regions have larger labor elasticities. Also, Southern Finland, Eastern Finland, Central Sweden have capital elasticities similar to that of Denmark while Western Finland, Northern Finland and Central Sweden have larger capital elasticities and Southern Sweden has a smaller capital elasticity. Thus, our ﬂexible speciﬁcation appears to ﬁt the data quite well. We now test the validity of our main hypothesis that subsidies affect productivity through three channels – input elasticities, technical efﬁciency and technical change. For the hypothesis that subsidies affect productivity only through technical efﬁciency, that is, as1 ¼ as2 ¼ as3 ¼ as4 ¼ as5 ¼ ast ¼ 0, we ﬁnd that BF ¼ 2.789 10–34. This provides strong empirical evidence that the other channels may play a role in the subsidy-productivity nexus and thus sheds doubt on the subsidy-productivity ﬁnding of Giannakas et al. (2001), Rezitis et al. (2003), Iraizoz et al. (2005), and Karagiannis and Sarris (2005). For the hypothesis that subsidies affect productivity only through technical efﬁciency and technical change, that is, as1 ¼ as2 ¼ as3 ¼ as4 ¼ as5 ¼ 0, we ﬁnd that BF ¼ 6.512 10–36. These two hypotheses suggest that input elasticities are empirically relevant to the

Do Subsidies Drive Productivity?

265

subsidy-productivity nexus. For the hypothesis that input elasticities and technical change are the only two channels, that is, bs ¼ 0, we ﬁnd that BF ¼ 51.26. Finally, for the hypothesis that input elasticities and technical efﬁciency are the only two channels, that is, ast ¼ 0, we ﬁnd that BF ¼ 58.20. These BFs are in favor of input elasticities as important channels through which subsidies drive productivity. Given the posterior mean and standard deviation of the impact of subsidies on technical change, our interpretation of these results is that the data are more consistent with input elasticities and technical efﬁciency as the channels in the link between subsidies and productivity. The regional differences in input elasticities, technical change and efﬁciency suggest that subsidies alone may not be sufﬁcient to make Danish and Swedish farms fully adjusted to their new economic environment, and therefore, other variables may be useful in explaining these regional differences. The results of Models I and II suggest that there is a link between subsidies and productivity in the data. To determine which one of our two speciﬁcations is preferred by the data, we compute the DIC for both models. We ﬁnd that DIC ¼ 29, 016 for Model I and DIC ¼ 28, 815 for Model II. Thus Model II, that accommodates subsidies as facilitating inputs, is preferred by the data. We therefore limit our discussion below to Model II. We now analyze the posterior means of RTS, TC, and TE.

4.1. The Distributions of RTS, TC, and TE Although Model II is not as ﬂexible as the translog, it is ﬂexible enough to render RTS and technical change observation-speciﬁc. Table 4 reports contributions of the various input elasticities along with RTS, TC, and TE. Their corresponding values are mean values by region. For all regions except Northern Sweden, materials contribute the most to RTS while the labor contributes the least to RTS.15 We ﬁnd evidence of increasing RTS for Denmark, Southern and Northern Sweden, and decreasing but close to constant RTS for Southern Finland, Eastern Finland, Western Finland, and Central Sweden. Northern Sweden has the largest value for increasing RTS with purchase feed and materials contributing respectively the most and least to RTS, while Western Finland has the lowest value for decreasing RTS. However, Northern Sweden is the least technically efﬁcient and has achieved no technical progress while Western Finland is the most technically efﬁcient and has showed technical progress. Overall, technical change is mostly positive but small in magnitude. Denmark appears to be

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NADINE MCCLOUD AND SUBAL C. KUMBHAKAR

Table 4.

Model II: Input Elasticities, Returns to Scale, Technology Effect, Technical Change Across Regions.

Country

Denmark

Region

DK

Labor (ey,1)

0.0369 (0.0064) 0.2402 Fertilizer (ey,2) (0.0322) 0.2955 Purchase (0.0166) feed (ey,3) 0.1041 Capital (ey,4) (0.0171) 0.3712 Material (ey,5) (0.03) RTS 1.0479 (0.0454) TE 1.5838 (0.431) TC 0.0089 (0.0027)

Finland

Sweden

SF

EF

WF

NF

SS

0.0763 (0.0161) 0.3093 (0.0431) 0.1909 (0.0188) 0.0983 (0.0243) 0.3227 (0.0409) 0.9974 (0.0491) 1.6933 (0.444) 0.0134 (0.0055)

0.0723 (0.0162) 0.241 (0.0424) 0.2014 (0.0226) 0.1193 (0.0213) 0.3419 (0.0458) 0.976 (0.0576) 1.7666 (0.5283) 0.0234 (0.0057)

0.0334 (0.0157) 0.2805 (0.0557) 0.0118 (0.0168) 0.1942 (0.0314) 0.4368 (0.0482) 0.9568 (0.055) 1.922 (0.4924) 0.0145 (0.0075)

0.0517 (0.0129) 0.2211 (0.0511) 0.2011 (0.0227) 0.1416 (0.0229) 0.348 (0.0417) 0.9635 (0.0644) 1.7457 (0.5792) 0.0199 (0.0056)

0.0585 (0.0092) 0.1758 (0.0316) 0.2975 (0.016) 0.0787 (0.0198) 0.4517 (0.032) 1.0622 (0.0439) 1.07 (0.4209) 0.0041 (0.0035)

CS

NS

0.0298 0.1875 (0.0104) (0.0129) 0.1228 0.5437 (0.0348) (0.0383) 0.3514 0.5594 (0.0188) (0.0269) 0.1496 0.1617 (0.0231) (0.0267) 0.3398 0.0655 (0.0375) (0.0375) 0.9933 1.1944 (0.0514) (0.0398) 1.6427 0.549 (0.482) (0.3551) 0.0015 0.0042 (0.0038) (0.0054)

Note: These values are the posterior means and standard deviation in ( ).

experiencing technical regress. We note that despite the comparative advantage and export-orientation of the farms in Denmark, this region is not superior to the other regions in terms of TC and TE. 4.2. Decomposing the Effect of Subsidies on Productivity We report the individual components, that is, technical efﬁciency, technology and technical change, of the total effect of subsidies on output by region in Table 5. Rather than reporting the output elasticity with respect to subsidies (see Eq. (12)), we report the contribution of subsidies to output, that is qY/ qS, which gives the value of the increase in output for an increase in subsidies by one Euro. To obtain qY/qS, we multiply Eq. (12) by Y/S. The Danish and Swedish farms have relatively larger subsidies contributions than the Finnish farms. In particular, for every h 100 increase in subsidies output increases by h 106.53 for farms in Denmark, h 71.66 for farms in Southern Sweden and h 15.28 for farms in Southern Finland. At the regional level, the variability in total contribution of subsidies is quite large. The efﬁciency effects also differ across regions. We obtain negative technology effects for all Finnish regions,

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Do Subsidies Drive Productivity?

Table 5. Model II: Decomposition of the Marginal Effect of Subsidies into Efﬁciency Effect, Technology Effect, Technical Change Across Regions. Country

Denmark

Region

DK

Technology effect

Finland SF

0.5178 0.0042 (0.1682) (0.0531) Technical change 0.084 0.0204 (0.0971) (0.0236) Efﬁciency effect 0.4636 0.1366 (0.3409) (0.1005) Total 1.0653 0.1528 (0.3473) (0.1017)

Sweden

EF

WF

NF

SS

CS

NS

0.0592 (0.0533) 0.0215 (0.0248) 0.139 (0.1022) 0.1013 (0.1031)

0.037 (0.0505) 0.0197 (0.0228) 0.134 (0.0985) 0.1167 (0.0993)

0.0905 (0.0497) 0.02 (0.0232) 0.132 (0.0971) 0.0615 (0.0972)

0.2534 (0.1355) 0.0706 (0.0815) 0.3926 (0.2887) 0.7166 (0.2904)

0.2242 (0.0896) 0.0461 (0.0533) 0.2624 (0.193) 0.5327 (0.1932)

0.1126 (0.0256) 0.0142 (0.0164) 0.0742 (0.0545) 0.201 (0.0543)

Note: These values are the regional posterior means and standard deviations in ( ) for each component of the marginal effect of subsidies.

although the posterior means of these effects are less than one standard deviation away from zero for Southern and Western Finland. These results require an explanation. There is a threshold effect that is implied by our speciﬁcation for Model II. As a result, regions or farms that do not have a certain level of subsidies have negative input elasticities for some of traditional inputs. Using a truncated multivariate normal as the prior distribution for these elasticities had unfavorable implications for the other parameters of the model and increased the simulation time considerably given the number of observations in the sample. We therefore use the multivariate normal as the prior distribution. For some regions, the technology effects are larger than the efﬁciency effects. In particular, for Denmark and Northern Sweden input elasticities are the main channels through which subsidies affect productivity. For Southern and Central Sweden, technical efﬁciency is the dominant channel in the subsidy–productivity nexus. Overall, regional ranking in decreasing order of the total contribution of subsidies to output is Denmark, Southern, Central and Northern Sweden, Southern, Western, and Eastern Finland, and then ﬁnally Northern Finland.

5. CONCLUSION One of the main purposes of the CAP in the EU is to increase agricultural productivity by subsidizing farmers. To date, however, not much work has

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NADINE MCCLOUD AND SUBAL C. KUMBHAKAR

been done to assess the empirical link between subsidies and productivity in the Nordic countries. This paper investigates empirically the impact of subsidies on agricultural productivity in Denmark, Sweden, and Finland through three channels, input elasticities, technical efﬁciency, and technical change. Our results suggest that subsidies drive productivity through technical efﬁciency and input elasticities and the relative importance of these channels differs across regions. In contrast to the existing studies, we ﬁnd that subsidies have a positive impact on technical efﬁciency. We also ﬁnd that subsidies are substitutable with labor, fertilizer, and purchase feed but complementary with capital and materials. That is, an increase in subsidies will increase overall production by increasing the contribution of capital and materials to production but decreasing the productivity of labor and the other inputs to production. In addition, the contribution of subsidies to output is largest for dairy farms in Denmark and Southern, Central, and Northern Sweden.

NOTES 1. http://www.europarl.europa.eu/factsheets/4_1_1_en.htm 2. A coupled subsidy policy is one that is explicitly related to production decisions. That is, coupled subsidies directly affect the optimal amount of input or output. Otherwise, the subsidy policy is ‘‘decoupled’’; thus the policy does not alter short-run marginal production decisions (see Bezlepkina, Oude Lansink, and Oskam, 2005). 3. Hennessy theorizes that coupled or decoupled subsidies induce a wealth effect and an insurance effect that can alter producers’ decisions. This is because subsidies provide producers with a stable income (the insurance effect) and also increase their wealth (the wealth effect). 4. Here we follow Lau and Yotopoulos (1971) and label A as technical efﬁciency. Note that this is the traditional interpretation which is not equivalent to the deﬁnition of technical efﬁciency in the stochastic frontier literature (Kumbhakar and Lovell, 2000). 5. We note that traditional inputs have direct effects on output. If the functional form of f ( ) permits interactions between inputs, traditional inputs can also have indirect effects on output. 6. In estimating aggregate production functions using time series data, this type of formulation is often used with the efﬁciency parameter being a function of the trend variable (see, Solow, 1956 and Beckmann and Sato, 1969). The present formulation is a variant of this in which A is speciﬁed as a function of S. 7. The formulation is general enough to accommodate various forms of productivity factors. For example, Kumbhakar (2004) uses a formulation in which the Ak( )s are functions of time trend. See Koop et al. (2000) for some other speciﬁcations.

Do Subsidies Drive Productivity?

269

8. Some of these formulations will be different from others if, for example, a translog form is chosen for ln f ( ). In Eq. (2) subsidies are neutral to technical change and inputs in the production technology irrespective of the form of f ( ) chosen. In this case, theoretically, no amount of subsidies can change any of these measures. 9. Subsidies can affect input productivity but not necessarily input elasticity. For example, if f ( ) is log linear in Eqs. (1)–(4), which is the speciﬁcation in Eq. (5), subsidies will affect input productivity but not input elasticities. 10. We emphasize that some of the other speciﬁcations can accommodate subsidies as facilitating inputs (with fewer channels) but cannot identify these channels with a simple log-linear functional form. We could circumvent this problem by using ﬂexible functions such as the translog. The main problem with the ﬂexible functional forms is that they tend to violate properties of the production technology. Furthermore, the parameters cannot be directly interpreted. 11. It is worth pointing out that all these components are related to change in the production function parameters. Thus in a broad sense they all measure ‘‘change in technology.’’ Here we are decomposing ‘‘change in technology’’ into speciﬁc components by focusing on different parameters of the production function. In particular, the ‘‘efﬁciency component’’ is related to a change in the intercept, the ‘‘technology component’’ is related to changes in the input coefﬁcients (elasticities), and ﬁnally the ‘‘technological change component’’ is related to a change in the coefﬁcient on the time trend variable. Note that a change in any of the coefﬁcients can be broadly deﬁned as a shift in the production function which is commonly deﬁned as ‘‘technical change.’’ Thus, here we deviate from the traditional deﬁnition and separate the shift in the production function into several components (such as a shift due to changes in the intercept, input elasticities, and the coefﬁcient associated with the time trend variable). This type of decomposition is not done in the literature. 12. One disadvantage of using the Cobb–Douglas production function is that it precludes an analysis on substitutability and complementarity between the traditional inputs due to the absence of cross-product terms involving these inputs. 13. We thank Timo Sipila¨inen for running these models in his computer. This indirect access to the FADN data was necessary to comply with the restriction that the data may not be transferred to any third-party researchers. The usual disclaimer applies. 14. Since farm-speciﬁc prices are not available and there are not enough variations in prices we cannot use the cost function approach. 15. In particular, material inputs contribute about 35.4%, 32.3%, 35.0%, 45.6%, 36.1%, 42.5%, 34.2%, 5.48% to output respectively in Denmark, Southern Finland, Eastern Finland, Western Finland, Northern Finland, Southern Sweden, Central Sweden, and Northern Sweden; labor contributes about 3.52%, 7.65%, 7.41%, 3.49%, 5.37%, 5.51%, 3.0%, 15.7% respectively.

REFERENCES Beckmann, M. J., & Sato, R. (1969). Aggregate production functions and types of technical progress: A statistical analysis. American Economic Review, 59, 88–101.

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Bezlepkina, I. V., Oude Lansink, A., & Oskam, A. J. (2005). Effects of subsidies in Russian dairy farming. Agricultural Economics, 33, 277–288. Giannakas, K., Schoney, R., & Tzouvelekas, V. (2001). Technical efﬁciency, technological change and output growth of wheat farms in Saskatchewan. Canadian Journal of Agricultural Economics, 49, 135–152. Hennessy, D. A. (1998). The production effect of agricultural income policies under uncertainty. American Journal of Agricultural Economics, 80, 46–57. Iraizoz, B., Bardaji, I., & Rapun, M. (2005). The Spanish beef sector in the 1990s: Impact of the BSE crisis on efﬁciency and proﬁtability. Applied Economics, 37, 473–484. Karagiannis, G., & Sarris, A. (2005). Measuring and explaining scale efﬁciency with the parametric approach: The case of Greek tobacco growers. Agricultural Economics, 33, 441–451. Koop, G. (2001). Cross-sectional patterns of efﬁciency and technical change in manufacturing. International Economic Review, 42, 73–103. Koop, G., Osiewalski, J., & Steel, M. (2000). Modeling the sources of output growth in a panel of countries. Journal of Business and Economic Statistics, 18, 284–299. Kumbhakar, S. C. (2004). Estimation of factor augmenting technical change: The case of US agriculture. Indian Economic Review, 34, 31–53. Kumbhakar, S. C., & Lovell, C. A. K. (2000). Stochastic frontier analysis. Cambridge: Cambridge University Press. Lau, L. J., & Yotopoulos, P. A. (1971). A test for relative efﬁciency and application to Indian agriculture. The American Economic Review, 61, 94–109. Rezitis, A., Tsiboukas, K., & Tsoukalas, S. (2003). Investigation of factors inﬂuencing the technical efﬁciency of agricultural producers participating in farm credit programs: The case of Greece. Journal of Agricultural and Applied Economics, 35, 529–541. Roe, T., Somwaru, A., & Diao, X. (2002). Do direct payments have intertemporal effects on US agriculture? International Food Policy Research Institute, Trade and Macroeconomics Division, Discussion Paper No. 104. http://www.ifpri.org/divs/tmd/dp/papers/ tmdp104.pdf Solow, R. M. (1956). A contribution to the theory of economic growth. The Quarterly Journal of Economics, 70, 65–94. Spiegelhalter, D., Best, N., Carlin, B., & van der Linde, A. (2002). Bayesian measures of model complexity and ﬁt. Journal of the Royal Statistical Society Series B, 64, 583–639. Verdinelli, I., & Wasserman, L. (1995). Computing Bayes factors using a generalization of the Savage–Dickey density ratio. Journal of the American Statistical Association, 90, 614–618. Zhengfei, G., & Oude Lansink, A. (2006). The source of productivity growth in Dutch agriculture: A perspective from ﬁnance. American Journal of Agricultural Economics, 88, 644–656.

TECHNICAL APPENDIX We derive the full conditional posterior distributions for the Bayesian model deﬁned by Eqs. (17), (18), (19), (20), (21), and (22). To obtain the full

271

Do Subsidies Drive Productivity?

conditional posterior distributions for P0i, G, b we make use of the following well-known decompositions of quadratic forms: Lemma A.1. Let a, b, ci, d be random vectors and A, B, C, Di be matrices of conformable dimensions. Suppose a is the vector of interest. Then conditioning on b, ci, d we have the following: a Þ; where a ðC0 CÞ1 C 0 b: (1) ðb CaÞ0 ðb CaÞ / ða a Þ0 ðC0 CÞða P PM M 0 0 0 0 (2) i¼1 ða ci Þ ðDi Di Þða ci Þ / ða ci Þ ð i¼1 Di Di Þða ci Þ; where P P M M 1 0 0 ci ð i¼1 Di Di Þ ð i¼1 Di ci Þ: (3) ða bÞ0 Aða bÞ þ ða dÞ0 Bða dÞ / ða a Þ0 ðA þ BÞða a Þ; where a ðA þ BÞ1 ðAb þ BdÞ: Combining Eqs. (17) and (18) we obtain ½P0i jY; b; G; s2u ; s2v 1 / exp 2 ðY i E i P0i X i GÞ0 ðY i Ei P0i X i GÞ 2su 1 exp 2 ðP0i Z i bÞ0 ðP0i Zi bÞ 2sv

N ðN Þ ðP0i; A i Þ i

(A.l)

ðA:2Þ

where Eq. (A.2) follows from applying Lemma A.l (1) and then (3), and 1 2 0 A i ¼ ðs2 v I N i þ su E i E i Þ ; 0i ¼ ðs2 I N þ s2 E 0 E i Þ1 ðs2 Z i b þ s2 E 0 ðY i X i GÞÞ P v

i

v

i

v

u

i

To see Eq. (A.2), let Y i Y i X i G. Using Lemma A.l (1) and noting our variable of interest and the conditioning set we have, ðY i E i P0i Þ0 ðY i E i P0i Þ / ðP0i P0i Þ0 E 0i E i ðP0i P0i Þ where P0i ðE 0i E i Þ1 E 0i Y i . Then using Lemma A.l (2) we have, 1 1 ðP0i P0i Þ0 E 0i E i ðP0i P0i Þ þ 2 ðP0i Z i bÞ0 ðP0i Z i bÞ 2 su sv 0 2 2 0 0i Þ / ðP0i P0i Þ ðs IN þ s E E i ÞðP0i P u

i

u

i

Hence the result in Eq. (A.2). Combining Eqs. (18) and (19) we obtain ( ) i¼1 1 X 0 2 2 ½bjY; P0i ; G; su ; sv / exp 2 ðP0i Z i bÞ ðP0i Zi bÞ 2sv i

(A.3)

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NADINE MCCLOUD AND SUBAL C. KUMBHAKAR

X1 1 exp ðb b Þ0 ðb bÞ 2 X

N ðMþ1Þ b;

(A.4)

where Eq. (A.4) follows from applying Lemma A.l (1), then (2) and then (3), and ! !1 M M X X 2 0 1 2 0 1 S¼ s Z Zi þ S ; b¼S s Z P0i þ S b v

i

v

i¼1

i

i¼1

Similarly, combining Eqs. (17) and (20) we obtain 2 ½s2 u jY; b; P0i ; G; sv (

) M 1 X 0 exp 2 ðY i E i P0i X i GÞ ðY i E i P0i X i GÞ ðA:5Þ / 2su i¼1 1 2 m =21 ðsu Þ exp 2 r 2ss ðs2u ÞR=2

¼ ðs2u ÞððRþmÞ=2Þ1 ( " #) M X 1 0 exp 2 r þ ðY i E i P0i X i GÞ ðY i E i P0i X i GÞ 2su i¼1 m r

G ; 2 2 where m ¼ R þ m; r ¼ r þ

M X

ðA:6Þ

ðY i E i P0i X i GÞ0 ðY i E i P0i X i GÞ

i¼1

Similarly, combining Eqs. (18) and (21) we ﬁnd n d 2 ; ½s2 v jY; P0i ; b; G; sv G 2 2

!

with n ¼ N M þ n;

d ¼ d þ

M X i¼1

ðP0i Z i bÞ0 ðP0i Z i bÞ

(A.7)

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Do Subsidies Drive Productivity?

Finally using Eqs. (17) and (22) we ﬁnd the full conditional posterior of G to be ½GjY; P0i ; b; s2u ; s2v ( ) M 1 X 0 ðY i E i P0i X i GÞ ðY i E i P0i X i GÞ / exp 2 2su i¼1

ðA:8Þ

1 exp ðG G Þ0 O1 ðG GÞ 2 OÞ

N ðMþ1ÞðKþ1Þ ðG;

ðA:9Þ

where ¼ O

s2 u

M X i¼1

!1 X 0i X i

1

þO

;

G ¼ O

s2 u

M X

! X 0i ðY i

1

E i P0i Þ þ O

G

i¼1

For estimation, we assign values to the parameters of the prior distributions. We consider different sets of hyperparameter values but each set is proper and informative. Economic theory provides us with some guidance to restricting some of these values. For example, input elasticities and returns to scale are non-negative. We assume a0kA{0.2,0.3,0.4,0.5}, a0tA{0.02,0.03,0.04}, astA{0,0.02,0.03,0.04}, askA{0,0.2,0.3,0.4,0.5}, b0A{0.5,1.0,1.5}, bsA{0,0.2,0.3,0.4,0.5}. We deﬁne G ða01 ; a02 ; . . . ; a0K ; a0t Þ0 , with ak ða0k ; a1k ; . . . ; aM1;k ; as;k Þ0 , for k ¼ 1, . . . , K, and at ða0t ; a1t ; . . . ; aM1;t ; as;t Þ0 . Then if we set ak ða0k ; a1k ; . . . ; aM1;k ; as;k Þ0 ¼ ð0:2; 0; . . . ; 0; 0Þ0 for k ¼ 1, . . . , 5, our prior is that farms exhibit constant returns to scale, there are no regional differences in input elasticities and subsidies have no impact on the productivity of all inputs. If we choose a0k ¼ 0.3 for k ¼ 1, . . . 5, then our prior is that farms exhibit increasing returns to scale. For simplicity, we assume a priori that the parameters within b and G are uncorrelated. Nevertheless we choose their prior variances to be diffuse so that the information in the data can contribute more to their posterior correlations. We set S ¼ cs 104 diag(1, 1, . . . , 1,1) and O ¼ cs 0.04 diag(1, 1, . . . , 1,1) 6 6 2 where cs ¼ 1,10. Also, we choose s2 u Gðca 10 ; cb 10 Þ, and sv

6 6 Gðca 10 ; cb 10 Þ with ca, cb ¼ 1,10,100. Here cs, ca, and cb are generic, prespeciﬁed constants that may take different values across priors. We carry out posterior simulation for this model by sequentially drawing from Eqs. (A.2), (A.4), (A.6), (A.7), and (A.9). For the empirical results

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reported, the hyperparameter values are a0k ¼ 0.5 for all k ¼ 1, . . . , 5, a0t ¼ ast ¼ 0.02, b0 ¼ 1.0, bs ¼ 0.5, cs ¼ 1, ca ¼ cb ¼ 1. We ﬁnd that in our sensitivity analysis, the information in the data dominates the information in the prior. That is, the qualitative implications of our ﬁndings are invariant to the choice of prior values. This is expected given the large sample size. For Model I, our sets of hyperparameter values are similar to those we use for Model II.

SEMIPARAMETRIC BAYESIAN ESTIMATION OF RANDOM COEFFICIENTS DISCRETE CHOICE MODELS Sylvie Tchumtchoua and Dipak K. Dey ABSTRACT Heterogeneity in choice models is typically assumed to have a normal distribution in both Bayesian and classical setups. In this paper, we propose a semiparametric Bayesian framework for the analysis of random coefﬁcients discrete choice models that can be applied to both individual as well as aggregate data. Heterogeneity is modeled using a Dirichlet process, which varies with consumers’ characteristics through covariates. We develop a Markov Chain Monte Carlo algorithm for ﬁtting such model, and illustrate the methodology using two different datasets: a household-level panel dataset of peanut butter purchases, and supermarket chain-level data for 31 ready-to-eat breakfast cereal brands.

1. INTRODUCTION Discrete choice models have been widely used in many ﬁelds (e.g., economics, marketing) to model instances where individuals select one Bayesian Econometrics Advances in Econometrics, Volume 23, 275–307 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23009-4

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alternative from a discrete set. In the general setup, a consumer i chooses alternative j from a set of J alternatives if the utility derived from alternative j, ui, j, is the highest, that is, ui;j 4ui;k ; 8k ¼ 1; . . . ; J; kaj. The utility, which is latent, is parameterized as ui;j ¼ x0i;j b þ i;j , where xi,j is a vector of observed characteristics of alternative j, b reﬂects the marginal utility of alternative characteristics (taste parameters), and ei,j is an error term commonly assumed to have an extreme value (0, 1) distribution, giving rise to the multinomial logit model. One objective of the model is to use the estimated tastes parameters to compute elasticities (percent change in the probability of choosing an alternative for a 1% change in one of the observed product characteristics (e.g., price), holding the other product characteristics constant). However, restricting the taste parameters b to be identical across individuals creates the Independence of Irrelevant Alternatives (IIA) problem in the multinomial logit model. For example, an increase in the price for one product implies a redistribution of part of the demand for that product to the other products proportionally to their original market shares and not with respect to their characteristics, as one would expect. This restricts the cross-price elasticities to be proportional to market shares. In order to avoid the IIA problem and estimate more realistic substitution pattern among the different products, heterogeneity across consumers in their tastes for the product characteristics is introduced by allowing the taste parameters b to be individual speciﬁc (bi). Since the true distribution of consumer tastes is not observed, the individual-speciﬁc parameters bi are typically assumed to be drawn from a parametric distribution. Discrete choice models can be estimated using either individual (household) level or aggregate (store, supermarket chain, or market) level data. By individual data we mean consumers and their choices are observed over time. Aggregate-level data consist of total volume (units) sales and dollars sales of a given brand for a store, supermarket chain, or market over time; individual choices leading to these aggregated quantities are not observed. The econometric methodology for the estimation is well documented. For individual-level data see McFadden and Train (2000) for the classical approach, and Yang, Chen, and Allenby (2003) and Rossi, Allenby, and McCulloch (2005) for Bayesian version. For aggregated data, see Berry, Levinsohn, and Pakes (1995) and Nevo (2001) for the classical setting, and Musalem, Bradlow, and Raju (2005, 2006) and Chen and Yang (2006) for the Bayesian paradigm. In both Bayesian and classical models, the distribution of the individualspeciﬁc parameters bi is typically taken to be multivariate normal.

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The distribution of the individual-speciﬁc parameters has important effects on the quantities of interest of the model. For example, in many marketing and economic applications, the individual-speciﬁc parameters are used to compute price elasticities or to predict the demand for established or new products under alternative pricing strategies. In such applications, reliable estimates of the individual-speciﬁc parameters are crucial. The assumption of normality may be too restrictive, since heterogeneity in the population is never known a priori and a normal distribution might not be a good choice; for example, there has been evidence of multimodality in the distribution of taste parameters in marketing studies (e.g., Allenby, Arora, & Rossi, 1998; Kim & Menzefricke, 2004). This warrants a more ﬂexible distribution. There has been some work toward relaxing the normality assumption. Chintagunta, Jain, and Vilcassim (1991) and Kamakura and Russell (1989) used latent class models, which do not capture variation in random coefﬁcients within a latent class. Finite normal mixture models have been used in several studies in the marketing literature (e.g., Allenby et al., 1998; Andrews & Currim, 2003, and references therein). In marketing, for example, the true number of mixing components is essential since many managerial decisions on segmentation, targeting, positioning, and the marketing mix are based on it. However, determining the number of mixing components remains an unresolved issue. Dillon and Kumar (1994, p. 345) argued that ‘‘The challenges that lie ahead are, in our opinion, clear, falling squarely on the development of procedures for identifying the number of support points needed to characterize the components of the mixture distribution under investigation.’’ More recently, Wedel and Kamakura (2000, p. 91) afﬁrmed that ‘‘the problem of identifying the number of segments is still without a satisfactory solution.’’ In a simulation study, Andrews and Currim (2003) showed that most commonly used mixing component retention criteria do not perform well in the context of multinomial choice data. To overcome the difﬁculty of choosing the number of mixing components, Kim and Menzefricke (2004) proposed the Dirichlet process prior due to Ferguson (1973). Basu and Chib (2003) also used the Dirichlet process prior in binary data regression models. However, the relationship between consumer characteristics and the unknown distribution of heterogeneity cannot be assessed using this distribution. Cifarelli and Regazzini (1978) introduced a product of Dirichlet processes that can be used to model dependence when the covariates have a ﬁnite number of levels. In this paper, we propose a model for which heterogeneity is modeled using a nonparametric distribution which depends on consumer’s

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continuous covariates. Instead of assuming a multivariate normal distribution on the individual-speciﬁc parameters (bi), we use a distribution on the space of all possible distributions, and the order-based dependent Dirichlet process prior introduced by Grifﬁn and Steel (2006) is placed on that distribution. Dependence in the Dirichlet process prior is achieved by making the weights in the Sethuraman (1994) representation of the Dirichlet process dependent on consumer’s continuous covariates. An attractive feature of our approach is that unlike the Dirichlet process introduced by Ferguson (1973), the dependent Dirichlet process helps recover a richer variety of heterogeneity distributions while allowing the nonparametric distribution to depend on continuous consumer’s characteristics. We design a Markov Chain Monte Carlo (MCMC) sampler for assessing the model parameters and apply it to a household-level panel dataset of peanut butter purchases and supermarket chain–level data for 31 ready-to-eat breakfast cereal brands. The rest of the paper is organized as follows. Section 2 describes the Mixture of Dependent Dirichlet process models (MDDP). In Section 3, we apply the MDDP model to the discrete choice model with individual data. In Section 4, the model proposed allows estimation with aggregate data. Section 5 contains empirical applications of our methodology. Finally Section 6 presents conclusions. The Matlab code to implement the method introduced in this paper is available on the website http://sylvie.tchumtchoua.googlepages.com/ matlab.

2. MIXTURE OF DEPENDENT DIRICHLET PROCESS MODELS 2.1. The Dirichlet Process The Dirichlet process (Ferguson, 1973) is widely used in Bayesian nonparametric applications. It is deﬁned as follows. Let Y be a probability space, B a s-algebra of subsets of Y, H a probability measure on (Y, B), and M a positive parameter. A random probability measure G on (Y, B) is said to have a Dirichlet process DP (M, H) if for any ﬁnite measurable partition A1, . . . , Ap of the space the vector (G(A1), . . . , G(Ap)) follows a Dirichlet distribution with parameters (M H(A1), . . . , M H(Ap)).

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Using the moments of the Dirichlet distribution, it follows that for AiAB, MHðAi Þ E½GðAi Þ ¼ Pp ¼ HðAi Þ i¼1 MHðAi Þ

(1)

P ð pi¼1 MHðAi Þ MHðAi ÞÞMHðAi Þ HðAi Þð1 HðAi ÞÞ ¼ Var½GðAi Þ ¼ Pp (2) P Mþ1 ð i¼1 MHðAi ÞÞ2 ð pi¼1 MHðAi Þ þ 1Þ The roles of H and M are apparent from Eqs. (1) and (2); H centers the process and is often called the centering distribution or baseline measure. It is a distribution that approximates the true nonparametric shape of G. The scalar M controls the variance of the distribution and is called the precision parameter. It reﬂects our prior beliefs about how similar the nonparametric distribution G is to the base measure H. As Ferguson (1973) established, realizations of G are discrete distributions and thus G is not directly used to model data. Escobar (1994) and MacEachern (1994) deﬁned continuous nonparametric distributions by specifying the DP as prior in a hierarchical framework; the resulting model is referred to as a Mixture of Dirichlet Process (MDP). It arises as follows: Suppose a random vector yi has a parametric distribution indexed by a vector bi, which in turn has a prior distribution with known hyperparameters c0. We have Stage 1: ½ yi jbi f ðbi Þ, Stage 2: ½ bi jc0 f ðc0 Þ, where f( ) is a generic label for a multivariate probability distribution function. The MDP replaces the parametric prior assumption at the second stage with a general distribution G, which in turn has a Dirichlet process prior, leading to the following hierarchical model Stage 1: ½ yi jbi f ðbi Þ, i:i:d:

Stage 2: bi G, Stage 3: G DPðM; HÞ. The above speciﬁcation is a semiparametric speciﬁcation because a fully parametric distribution is given in the ﬁrst stage and a nonparametric distribution is given in the second and third stages. Two representations of the Dirichlet process are frequently used in the literature. One representation widely used for practical sampling purpose is the Polya urn representation (Blackwell & MacQueen, 1973). If we assume i:i:d: that b1 ; . . . ; bn G and G DPðM; HÞ, then Blackwell and MacQueen

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established that Gðbn jb1 ; . . . ; bn1 Þ ¼

n1 X 1 M H d br þ M þ n 1 r¼1 Mþn1

(3)

Using this representation, b1, . . . , bn are sampled as follows. b1 is drawn from the baseline distribution H. The draw of b2 is equal to b1 with probability p1=1/(M+1) and is from the baseline distribution with probability p0=1 p1. The process continues until bn is sampled. Three facts are worth noting about the Polya urn representation. First, the b’s are drawn from a mixture of the baseline distribution and a discrete distribution. Second, if br=b for all r, then b is drawn from the centering distribution with probability one, and therefore the base distribution is the prior. Finally, Probðbr ¼ bs ; rasÞ40, which results in the clustering property of the Dirichlet process (MacEachern, 1994). The n b’s are grouped into k sets, 0okrn, with all observations in a group sharing the same value of b, and observations in different groups have different values of b. Another representation is the stick-breaking prior representation (Sethuraman, 1994; Ishwaran & James, 2001) and is given by 1 X pr dy r (4) G¼ r¼1

where dr is the Dirac measure which places measure 1 on the point t, y1, y2 . . . are i.i.d. realizations of H, and pr ¼ V r Plor ð1 V l Þ where Vr are i.i.d. Beta(1, M). Then yr are referred to as locations, Vr as masses, and pr as the respective weights. By the deﬁnition of the stick-breaking representation, the weights pr ¼ V r Plor ð1 V l Þ tend to be large for small r (recall that the masses Vr are Beta(1, M) random variables), so if r is large, many of the (1Vl) will be multiplied by the weight pr, thus making its value small. 2.2. Introducing Dependence in the Dirichlet Process In many settings, one might be interested in allowing the unknown distribution G as deﬁned above to depend on some covariate W, which could be time, space, or other known covariates. Several papers in the recent literature have extended the Dirichlet process to accommodate this dependence and are all based on the Sethuraman (1994) representation of the DP. MacEachern (1999, 2000) introduced a dependent Dirichlet process (DDP) by replacing either the masses, Vr, or the locations, yr, of the

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stick-breaking representation by stochastic processes. MacEachern, Kottas, and Gelfand (2001) focused on a model where only the locations are stochastic processes. Their model is referred to as ‘‘single p’’ model and has been applied to spatial modeling by Gelfand, Kottas, and MacEachern (2004) and Duan, Guindani, and Gelfand (2007), ANOVA-like models for densities by De Iorio, Muller, Rosner, and MacEachern (2004), and quantile regression by Kottas and Krnjajic (2005). Grifﬁn and Steel (2006) suggested the order-based dependent Dirichlet process (pDDP) that captures nonlinear relationships between the unknown distribution G and covariate W. Dependence is introduced by making the masses, Vr, and the locations, yr, of the stick-breaking representation (4) depending on the covariate W. Speciﬁcally, the elements of the vectors V and y are ranked via an ordering p(W). At each covariate W, we still have the stick-breaking representation (4) (marginally GW is a DP), but the order in which the masses are combined varies over the covariate domain: 1 X d pr ðWÞdypr ðWÞ (5) GW ¼ r¼1

where dk denotes the Dirac measure at k, pr ðWÞ ¼ V pr ðWÞ Plor ð1 V pl ðWÞ Þ P i:i:d: i:i:d: with yk H, V k Betað1; MÞ, and 1 k¼1 pk ðWÞ ¼ 1 a:s. Here p(W) deﬁnes an ordering at the covariate value W and satisﬁes the following condition jW zp1 ðWÞ jojW zp2 ðWÞ jojW zp3 ðWÞ jo where z is the realization of a Poisson process with intensity l. In other words, the ordering p(W) lists the zr in increasing order of absolute distance from W so that the most relevant zr at W are those close to W. An index r that appears ‘‘late’’ in the ordering p(W) (i.e., for which l such that pl(W) ¼ r is high) would have many terms (1Vpl u(W )) multiplied into its weight pr. An inﬁnite number of zr appears over the inﬁnite real line, but only the zr close to the observed covariate value would have signiﬁcant weight. For practical computation, truncation of the point process similar to truncation of the stick-breaking representation is deﬁned. We now turn to the description of how the DP at two distinct covariate values are correlated. As mentioned previously, the marginal distribution of the pDDP at any covariate value follows a DP: GWBDP(M,H). Correlation of two distributions GW 1 and GW 2 depends on the order in which the masses Vr are combined at the covariate values W1 and W2. The intensity parameter l controls how quickly the indexes r change. A large value of l yields more

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densely packed indexes, causing the ordering p(W) to change more quickly from one covariate to another, and consequently the GW will be less correlated. The value of M controls the expected number of indexes with signiﬁcant masses. A large value of M makes more leading terms in the stick-breaking relevant and thus implies more indexes needed to change place in the ordering before the distributions decorrelate. Thus, the intensity parameter l and the precision parameter M control the correlation between the distributions GW 1 and GW 2 . Grifﬁn and Steel deﬁned the explicit expression of the correlation between the distributions GW 1 and GW 2 as: 2ljW 1 W 2 j 2ljW 1 W 2 j exp (6) CorrðGW 1 ; GW 2 Þ ¼ 1 þ Mþ2 Mþ1 where |W1W2| denotes the distance between W1 and W2. Like the Dirichlet process, the pDDP produces discrete realizations. To obtain continuous distributions, the pDDP is imbedded in the hierarchical model as follows: Stage1 : ½yi jbi f ðbi Þ Stage2 : bi GW Stage3 : GW pDDPðM; H; lÞ where H is the baseline distribution, M is the precision parameter, and l is the intensity of the Poisson point process that induces the orderings. In the following two sections, we apply the pDDP model to discrete choice models. The advantage of the pDDP over the ‘‘single p’’ DDP is that it allows dependence to be introduced on both the weights and the atoms. We derive the full conditional distributions and the MCMC sampler for ﬁtting the models. Section 3 presents the case where the discrete choice model is estimated with individual-level data. In Section 4, we extend the model in Section 3 to account for endogeneity and allow estimation with aggregate data.

3. DEPENDENT DIRICHLET PROCESS PRIORS IN DISCRETE CHOICE MODELS WITH INDIVIDUAL-LEVEL DATA 3.1. The Model Assume we have n individuals, each making purchase decisions over T periods, and we observe the choices made by all consumers. In each period,

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each individual chooses one alternative from a set of J alternatives. Deﬁne the following notations: yit=j denotes the event that individual i chooses alternative j at time t, xijt denotes a p-dimensional vector of observed characteristics (price, brand indicator variable, and other product characteristics) of alternative j for individual i in period t, bi denotes the p-dimensional vector of parameters for individual i, eijt represents random variation in consumer choice behavior. The utility individual i derives from choosing alternative j at time t is parameterized as uijt ¼ x0ijt bi þ ijt

(7)

Assuming eijt has an extreme value (0, 1) distribution, the probability that individual i chooses alternative j in period t is given by expðx0ijt bi Þ pijt ¼ Probðyit ¼ jjbi Þ ¼ PJ ; 0 k¼1 expðxikt bi Þ i ¼ 1; . . . ; n;

j ¼ 1; . . . ; J;

(8)

t ¼ 1; . . . ; T

Alternatively, eijt can be assumed to be drawn from a normal distribution, giving rise to the multinomial probit model. However, the model with logit disturbance has the advantage of yielding close form choice probabilities as in Eq. (8) and is easier to implement than the probit model. Moreover, the probit model may not accommodate a large number of products (Chintagunta, 2001). These reasons explain why the logit model is widely used. The likelihood of individual i’s choices over time is then given by pðDi jbi Þ ¼

T Y J Y

D

pijtijt

(9)

t¼1 j¼1

where Dijt=1 if yit=j and 0 otherwise, and yi ¼ ðyit ; . . . ; yiT Þ0 . Our model in Eq. (7) assumes the bi’s are heterogeneous across individuals. We want to model the bi using a nonparametric distribution, while at the same time allowing this distribution to depend on individual characteristics. To accomplish this, we use the mixture of order-based dependent Dirichlet process model described above. The resulting model can be written in hierarchical form as: pðyi jbi Þ ¼

T Y J Y t¼1 j¼1

Probðyit ¼ jjbi ÞDijt

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bi GW GW pDDPðM; H; lÞ There are two properties of the order-based dependent Dirichlet process GW, which give insight into heterogeneity in our model. First, like the Dirichlet process introduced by Ferguson (1973), GW creates clusters of observations in the data. Because there is a positive probability of individuals to share regression parameters, there will be Lrn distinct values of the regression parameters b1, . . . , bn. Second, because GW varies with subject characteristics, the distribution of individuals across the L clusters depends on subject characteristics, and this relationship is not restricted to be linear.

3.2. Prior Distributions for M, H, and l Following Grifﬁn and Steel (2004, 2006), we specify the prior distribution for M as an inverted Beta distribution pðMÞ ¼

nZ0 Gð2ZÞ GðZÞ2

M Z1 ðM þ n0 Þ2Z

where the hyperparameter n0>0, the prior median of M and the prior variance of M (which exists if Z>2) is a decreasing function of Z. Other prior distributions for M have been suggested in the literature; Escobar and West (1995) suggested a gamma distribution whose parameters are elicited by considering the distribution of the number of distinct elements in the ﬁrst n draws from the Dirichlet process. Walker and Mallick (1997) used the formula M=E(o2)/Var(m), where m and o2 are the mean and variance of the unknown distribution. In their inverted Beta distribution, Grifﬁn and Steel interpreted M as a ‘‘prior sample size,’’ because of the form of the Dirichlet process prior predictive distribution derived by Blackwell and MacQueen (1973). The prior distribution for l depends on the precision parameter M, the autocorrelation function, and type of construction used to induce the ordering to vary with the covariate W. Using the permutation construction and assuming a one-dimensional covariate, the distribution of l is 2tn ð2tn l þ 1Þ 2tn l exp pðlÞ ¼ Mþ1 ðM þ 1ÞðM þ 2Þ

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where t is a parameter to be tuned. It is worth mentioning that for more than one covariate, the prior on l has no closed form (see Grifﬁn & Steel, 2006) and can only be approximated numerically. For the centering distribution H, we specify a p-variate normal distribution with unknown mean vector mH and unknown covariance matrix SH, HjmH ; SH ¼ MVNðmH ; SH Þ

3.3. Bayesian Estimation To complete the model speciﬁcation, we assume the following prior distributions for the mean vector and covariance matrix of the baseline distribution H: mH N p ðm0 ; V 0 Þ and SH IW p ðuH0 ; SH0 Þ where Np(m0, V0) denotes a p-dimensional normal distribution with mean vector m0 and covariance matrix V0, and IW p ðuS0 ; S S0 Þ denotes a p-dimensional inverted Wishart distribution with parameters uH0, and SH0; m0, V0, uH0, and SH0 are known. In addition to the parameters {bi}, M, l, SH, and mH, the point process z needs to be sampled. The joint posterior distribution of all model parameters is ! ! n Y T Y J n Y Y Dijt pijt p1 ðbi jGW Þ f ðfbi g; M; l; z; SH ; mH jY; XÞ / i¼1 t¼1 j¼1

i¼1

p2 ðGW jH; M; lÞp3 ðMjlÞp4 ðljM; zÞp5 ðSH ; mH Þ

ð10Þ

where pijt is given in Eq. (8), p1 is the distribution of the regression parameters, p2, the Dirichlet process prior on this distribution, p3, the distribution of the precision parameter that depends on the intensity parameter l, p4, the distribution of the intensity parameter that depends on the point process z and the precision parameter M, and p5, the prior distribution on the parameters of the baseline distribution.

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Deﬁne the n-dimensional vector C such that bi ¼ yCi . The model parameters are estimated via a MCMC algorithm that generates draws from the following sequence and conditional distributions: (1) (2) (3) (4) (5) (6)

Update Update Update Update Update Update

C, y, z, M, l, m and S.

We discuss each of these conditional distributions in turn, but before that we deﬁne some notations. Suppose I={1, . . . , n} is the set of all the n individuals; for a subset B of I, nl (B) represents the number of individuals i in B for which Ci=l and Ql ðBÞ ¼ #fi 2 B such that there exists kol for which pk ðW i Þ ¼ l; where pj ðW i Þ ¼ C i g. That is, Ql (B) is the number of observations for which l appears before Ci in the ordering at Wi. Next we follow the following steps: (1) Generation of C: Propose C according to the following discrete distribution pðC i ¼ ljC l ; M; z; W; DÞ / pðDi jC i ¼ l; C i ; Di ÞpðC i ¼ ljC l ; M; z; WÞ R Q f ðDi jbÞ fjaijCj ¼lg f ðDj jbÞdHðbÞ nl ðI i Þ þ 1 RQ / M þ Ql ðI i Þ þ nl ðI i Þ þ 1 fjaijC j ¼lg f ðDj jbÞdHðbÞ

Y

M þ Qpj ðWÞ ðI i Þ þ 1

jomðlÞ

M þ Qpj ðW Þ ðI i Þ þ npj ðWÞ ðI i Þ þ 1

where pmðlÞ ðWÞ ¼ l. The above expression assumes that clusters are numbered in the order they appear; this implies that for an individual to be allocated to cluster l, it must be true that she is not allocated to clusters appearing before l. Clearly, ðnl ðI i Þ þ 1Þ=ðM þ Ql ðI i Þ þ nl ðI i Þ þ 1Þ is the probability that the individual i is allocated to cluster l given that it can only be allocated to clusters l, l+1, . . . , L, whereas ðM þ Qpj ðWÞ ðI i Þ þ 1Þ= ðM þ Qpj ðWÞ ðI i Þ þ npj ðW Þ ðI i Þ þ 1Þ is the probability that the same individual is not allocated to cluster pj(W). (2) Generation of y: Propose y form the distribution pðyl jC; D; WÞ / Hðyl Þdyl

Y fi:Ci ¼lg

ProbðDi jyl ; :Þ

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A slice sampler (Neal, 2003) can be used to sample from this distribution. Given the draws of C and y, the n-dimensional vector of individualspeciﬁc parameters b are given by bi ¼ yCi . (3) Generation of z: To update the point process z, we use the ‘‘move a current point’’ update in Grifﬁn and Steel. Assume that the current relevant elements of the Poisson process are z ¼ ðz1 ; . . . ; zL Þ. The ‘‘move a current point update’’ consists of choosing at random a point zu and adding to it a random variable with zero mean and a tuning variance. The obtained moved z0u is rejected if it falls outside the truncation region or is accepted with probability ( ) L Y nu ðIÞ þ 1 þ Q0u ðIÞ þ M min 1; n ðIÞ þ 1 þ Q0u ðIÞ þ M u¼1 u (4) Generation of l: The conditional distribution for the intensity parameter (l) depends on the point process z. Sampling l proceeds as follows for a one-dimensional W: For each point of the Poisson process zu, attach a mark mu which is uniformly distributed on (0, 1); Draw a proposed value log l0 Nðlog l; s2l Þ; if l0 ol the points in the data region for which mu 4l0 =l are removed from the point process, otherwise m0u ¼ mu l=l0 ; if l0 4l, a new point process with intensity l0 l is drawn in the data region. (5) Generation of M: Recall that the mass parameter (M) and the ordering process p(W) determine the dependence across the covariate domain, and the number of points in the truncated domain depends on M. To update the value of M, we draw a new point Mu such that log M 0 Nðlog M; s2M Þ, where s2M is chosen to control the overall acceptance rate. If M 0 4M, the truncated region is expanded and the unobserved part of the Poisson process is sampled; If M 0 oM, the truncated region is contracted and points that fall outside the region are removed. If these points have any observations allocated to them, the new point is rejected. Grifﬁn and Steel deﬁne the above move as a reversible jump move where extra points are sampled from the prior distribution. The acceptance rate given by L M 0 pðM 0 jlÞ Y nu þ 1 þ Q0u þ M M pðMjlÞ u¼1 nu þ 1 þ Qu þ M 0

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(6) Generation of m, and S: The full conditional distributions for mH and SH reduce to mH jb ; SH N P ðm ; V Þ and L X SH jb ; mH IW L þ uH ; S H þ ðbk mH Þðbk mH Þ0

!

k¼1

PL P1

P1 1 1 where m ¼ V ðV 1 0 m0 þ l¼1 H bl Þ and V ¼ ðV 0 þ L H Þ ; mH and SH are sampled using direct Gibbs sampling. 3.3.1. Computing Marginal Effects (Elasticities) Recall that probability for consumer i choosing brand j at time t is expðx0ijt bi Þ pijt ¼ Probðyit ¼ jÞ ¼ PJ 0 k¼1 expðxikt bi Þ Assuming consumers do not P make multiple purchases, the market share of brand j at time t is sjt ¼ i sijt =n. Elasticities (percent change in the probability of choosing an alternative for a given change in one of the observed product characteristics xijt,r, holding the other product characteristics constant) are calculated as follows: 8 1 X > > b s ð1 sijt Þxijt;r if l ¼ j > > < nsjt i i;r ijt Zjt;r ¼ 1 X > > b s s x if laj > > : nsjt i i;r ijt ilt ilt;r where bi,r and xijt,r are the lth component of bi and xijt, respectively.

4. DEPENDENT DIRICHLET PROCESS PRIORS IN DISCRETE CHOICE MODELS WITH AGGREGATE DATA AND/OR ENDOGENEITY Very frequently in marketing and economics, the utility model in Eq. (7) includes an unobserved demand shock xjt for each brand j and time t, which is assumed to be correlated with prices, thus creating an endogeneity

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problem. Also discrete choice models are estimated with aggregate (store, chain, or market level) data in some product categories because individual-level data are not available. In this section, we extend the model of Section 3 to account for price endogeneity and allow estimation with aggregate data.

4.1. The Model Assume we observe aggregate market shares, prices, and product characteristics of J brands across T periods of time. We assume the observed market shares are generated by N individuals, each making choice decisions over T periods. The utility that each individual derives from choosing brand j in period t is deﬁned as uijt ¼ x0jt bi ai pjt þ xjt þ ijt

(11)

where xjt and pjt are, respectively, observed product characteristics and price of brand j at time t; they are the same for all consumers; bi and ai represent consumer-speciﬁc tastes for product characteristics. Further xjt represents the effects of variables other than price and observed product characteristics contained in xjt that are not included in the model and that could affect the probability of choosing brand j. It is assumed to be observed by the consumers and the manufacturers, but not by the econometrician. Here eijt represents random variation in consumer choice behavior and is assumed to have an extreme value (0, 1) distribution. The objective is to estimate the parameters bi and ai form the observed aggregate market shares, prices, and product characteristics. Denoting Yi ¼ ðbi ; ai Þ, xt ¼ ðx1t ; . . . ; xJt Þ, the probability that individual i chooses alternative j in period t is given by pijt ¼ Probðyit ¼ jjxt ; Pt ; Yi Þ ¼

expðx0jt bi ai Pjt þ xjt Þ ; J P 0 expðxkt bi ai Pkt þ xkt Þ

(12)

k¼1

i ¼ 1; . . . ; n; j ¼ 1; . . . ; J; t ¼ 1; . . . ; T As previously deﬁned, let Dijt takes a value of 1 if consumer i chooses brand j in period t, and a value of 0 otherwise. We do not observe the individual choices Dijt, but only aggregate share Sjt for each brand in period t. We want to augment observed aggregate shares Sjt with the latent individual

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choices Dijt so that at the aggregate level, the sum of latent individual choices are consistent with the observed shares at each time period (i.e., Pn ¼ Dijt ¼ nS jt ), and at the individual level, augmented choices are i¼1 consistent with utility functions across time periods. For each consumer i, the likelihood of observing choices at purchase occasion 1, . . . , T is pi ¼ ProbðDi jYi ; x; PÞ ¼

T Y J Y

Probðyit ¼ jjxt ; Pt ; Yi ÞDijt

(13)

t¼1 j¼1

Thus, the likelihood for observing choice sequences of all the n consumers, fDi gni¼i , is then given by Prob fDi gni¼i jfYi gni¼1 ; x; P ¼

n Y T Y J Y i¼1 t¼1 j¼1

I Pn i¼1

Dijt ¼nS jt

Probðy ¼ jjx ; Pt ; Yi ÞDijt t it

ð14Þ

where the indicator function ensures that the augmented individual choices Dijt are exactly consistent with the aggregate market shares. There is a potential for correlation between prices and unobserved product characteristics xjt because manufacturers observe the xjt’s and demand for brand j depends on xjt; this makes prices endogenous. We account for endogeneity by using instrumental variables techniques (Villas-Boas & Winer, 1999). We assume Pjt ¼ jwjt þ ojt ; ot MVNð0; So Þ and covðxt ; ot Þ ¼ S ¼

Sx

Sxo

Sox

So

!

where wjt represents a vector of instrumental variables. As before, we want to model the Yi using a nonparametric distribution while at the same time allowing this distribution to depend on consumer characteristics. To accomplish this, we use the order-based dependent Dirichlet process model. The hierarchical form of the model is given by ProbðDit jYi ; xt ; Pt Þ ¼

J Y j¼1

Probð yit ¼ jjxjt ; Pjt ; Yi ÞDijt

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Pt jfxt g; j; fYi g Nðjwt ; S0 Þ xt Nð0; Sx Þ j Nð0; s20j IÞ Yi G W GW pDDPðM; H; lÞ

4.2. Identiﬁcation Since only aggregate data are available, it is important to discuss how the model parameters are identiﬁed by the aggregate data. Identiﬁcation comes from examining the time patterns of the observed aggregate brand shares. The goal of the model is to estimate the distribution of consumer individualspeciﬁc parameters Yi, the covariance between the demand shocks and the prices, S, and the price equation parameter j. By assuming that each Yi is drawn from a distribution that does not have a parametric form but has the order-based dependent Dirichlet process prior, pDDP(M, H, l), with precision parameter M, intensity parameter l, and baseline distribution H assumed to be normally distributed with mean mH and covariance SH, the goal reduces to the estimation of M, mH, SH, l, S, and j from the aggregate brand shares. If each of these parameters induces different behavior of the aggregate brand shares through time, then the model is identiﬁed.1 We discuss each parameter in turn. Recall that the parameters M and l control the correlation of the orderbased dependent Dirichlet process at different values of the covariates. Larger values of M and l cause the marginal Dirichlet processes to decorrelate faster, thus increasing the number of distinct clusters, with consumers having similar covariates sharing the same cluster. More distinct clusters mean there is heterogeneity in consumers’ preferences for product characteristics (price, brand indicators, other product characteristics). For example, one cluster may include consumers that have high income, are loyal to a given brand, and are less price-sensitive, while another cluster is made up of low income, highly price-sensitive consumers. If many consumers are loyal to a given brand, changing the price of that brand

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would not decrease its market share overtime. On the other hand, if few consumers are loyal to that brand, its market share would tend to decline with a price increase. There is also a situation where the negative effect of price due to price sensitivity of some consumers compensates the positive effect due to the behavior of other consumers, thus leaving a less noticeable variation of market shares over time. The price equation parameter j, the off-diagonal blocks, and the lower diagonal block of the covariance matrix S are identiﬁed by the exogenous variations of the instrumental variables over time. The upper diagonal block of S, Sx, represents the covariance matrix of the unobserved demand shocks xjt. Since these demand shocks capture the effect of unobserved demand factors on aggregate demand, a higher value of any of its diagonal element would indicate high volatility of the market share of the corresponding brand. An off-diagonal element Sx(j, j u) measures the similarity of the utilities of brands j and ju over time with respect to demand shocks. Therefore, a high value of Sx(j, j u) implies an identical effect of a demand shock on the shares of brand j and ju, but a different effect on the shares of the remaining brands, thus leading to different market shares patterns over time.

4.3. Bayesian Estimation Lacking observed information on individual choices Dijt, a data augmentation approach (Tanner & Wong, 1987; Albert & Chib, 1993; Chen & Yang, 2006; Musalem et al., 2005) will be used. Instead of integrating out individual choices (Dijt) and individual-level response parameters Yi=(bi, ai) as in the nonlikelihood-based approach (Berry et al., 1995), we treat them as any other unobserved model parameters and use them as conditioning arguments in generating the draws. The prior distribution for mH, SH, and S are assumed to be mH Nðm0 ; V 0 Þ SH IWðuSH 0 ; SSH 0 Þ and S IWðuS0 ; SS0 Þ In the above speciﬁcations, s2j0 , m0, V0, uSH 0 ; S SH 0 , uS0 , and S S0 are known.

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Semiparametric Bayesian Estimation

The joint posterior distribution of all model parameters is f ðfYi g; z; M; l; SH ; S; j; fDijt g; fxt g; PjS; X; w; WÞ ! n Q T Q J Q D p ijt p1 ðfx gjSÞp2 ðfPt gjS; fx gÞ I Pn / i¼1 t¼1 j¼1

n Q

i¼1

Dijt ¼nS jt

ijt

t

t

p3 ðYi jGW Þ p4 ðGW jH; M; lÞp5 ðzÞp6 ðMÞp7 ðlÞp8 ðmH ; j; S; SH Þ

i¼1

(15) where pijt is deﬁned in Eq. (12), and S, P, X, w, and W are matrices of observed market shares, prices, product characteristics, instrumental variables, and consumer characteristics, respectively. The model parameters are estimated via a MCMC algorithm that generates draws from the following sequence and conditional distributions: (1) (2) (3) (4) (5) (6) (7) (8) (9)

Sample Sample Sample Sample Sample Sample Sample Sample Sample

xt, t=1, . . . , T, Dt, t=1, . . . , T, C, y, z, M, l, mH and SH, j and S.

Steps (3)–(8) are the same as in Section 3; therefore here we only discuss steps (1), (2), and (9). (1) Generation of x: The full conditional distribution for xt is given by ! n Y J Y Dijt f ðxt j:Þ / pijt p5 ðfxt gjSÞp6 ðfPt gjS; fxt gÞ i¼1 j¼1

xt is sampled using a random walk Metropolis–Hastings sampling. (2) Generation of Dt: We sample individual choices using a multiple-block Metropolis–Hastings algorithm. Because of the large number of consumers, convergence can be very slow if the single block algorithm is used. We randomly partitioned the set of consumers into b blocks D1t, . . . , Dbt, each of size m. Each block is sequentially updated using the following algorithm: ð0Þ ð0Þ Specify an initial value Dð0Þ t ¼ ðDit ; . . . ; Dbt Þ, Repeat for k=1, . . . , b.

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(i)

Propose a value for the kth block, Dnew kt , conditioned on the current value of the other blocks Dkt from the discrete distribution qk ðDnew kt jDkt Þ ¼

J Y

1 Ok1t OkJt k0t CO m C mOk0t . . . C mOk0t OkJ1;t

I Pm i¼1

j¼1

Dkijt ¼Okjt

Ok1t k0t CO m C mOk0t

kJt where ¼ A!=ðA!ðA aÞ!Þ and . . . CO mOk0t ...OkJ1;t is that satisfy the constraint the total number of combinations of D kijt Pm D ¼ O for all j; O ¼ O O , where O jt jt kjt kjt jt is the integer i¼1 kijt approximation of nSjt and Okjt is the number of consumers in the other blocks that have chosen brand j in period t. To generate a candidate draw Dnew kt from qk, ﬁrst randomly assign Ok0t consumers to the no purchase alternative, then Ok1t consumers among the remaining mOk0t to brand choice 1, and so on until all consumers are allocated. (ii) Calculate the probability of the move ( ) old pðDnew new old kt Þqk ðDkt jDkt Þ ak ðDkt ; Dkt jDkt Þ ¼ min 1; new pðDold kt Þqk ðDkt jDkt Þ Q QJ pDkijt Pm where pðDkt Þ ¼ m ijt i¼1 j¼1 I

C aA

i¼1

Dkijt ¼Okjt

old (iii) Update the kth block with probability ak ðDnew kt ; Dkt jDkt Þ. (3) Generation of j and S: The full conditional distributions for S and j reduce to 0 ! T X xt xt Sj: IW T þ uS0 ; SS0 þ Pt jwt t¼1 Pt jwt

jj: MVNðA; BÞ where A ¼ Cw D ðP f Þ, C ¼ ðL1 þ w0 D1 wÞ1 , 2 D ¼ So Sxo S1 x Sox , and L ¼ sk0 I. 0

1

f ¼ Sxo S1 x x,

Then S and j are sampled using direct Gibbs sampling. 4.3.1. Computing Marginal Effects (Elasticities) Price and advertising elasticities for each chain-period are computed as follows: The conditional probability for consumer i choosing brand j at time t is sijt ¼ Probðyit ¼ jjxt ; Pt ; Yi Þ ¼ PJ

expðx0jt bi ai Pjt þ xjt Þ

k¼1

expðx0kt bi ai Pkt þ xkt Þ

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Semiparametric Bayesian Estimation

Assuming consumers do not make multiple purchases, the market share of brand j at time t is Z 1 X sijt f ðxjt Þdxjt sjt ¼ n i Price elasticities are calculated as follows: 8 p 1X > > ai sijt ð1 sijt Þ > lt > @sjt plt < sjt n i ¼ Zjlt ¼ > plt 1 X @plt sjt > as s > > : sjt n i i ijt ilt

if l ¼ j if laj

5. EMPIRICAL APPLICATIONS 5.1. Discrete Choice Models with Individual-Level Data The model with individual data is estimated with an A.C. Nielson supermarket scanner dataset for peanut butter in the city of Sioux Falls, South Dakota. The objective is to assess the distribution of consumer preferences and investigate how these preferences vary with income (here our covariate W is the income). The data was obtained from the publicly available ERIM database at the University of Chicago Graduate School of Business. We observe consumers and their choices. The number of household is 326 and the total number of purchase is 9,158. There are J=4 brands of peanut butter. The product characteristics include a dummy variable for featured advertising, net price, and three dummy variables for brands 1, 2, and 3. Table 1 summarizes these variables. The following values are chosen for the priors: s2j0 ¼ 100, m0=0, V 0 ¼ SSH 0 ¼ SS0 ¼ 100I, and uSH 0 ¼ uS0 ¼ 2. The MCMC sampler was Table 1.

Descriptive Statistics.

Market share Proportion of observations with feature advertising Average price ($)

Brand 1

Brand 2

Brand 3

Brand 4

24.68 6.86 1.72

29.02 21.15 1.62

12.03 24.43 1.60

34.27 10.60 1.38

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SYLVIE TCHUMTCHOUA AND DIPAK K. DEY 0.16 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0

Fig. 1.

2

4

6

8

10

12

14

16

18

20

Posterior Density of the Precision Parameter M.

run for 15,000 iterations, the ﬁrst 500 being discarded as burn-in period. To assess convergence, we use different starting points for the chain and examine the trace plots of the model parameters (not shown). The nonparametric approach to modeling heterogeneity as described aims at relaxing the unimodality assumption in the distribution of the individualspeciﬁc parameters, and the linearity of the relationship between consumerspeciﬁc parameters and consumer characteristics. Fig. 1 plots the posterior density function of the precision parameter M, which, recall, measures the suitability of a parametric model for the individual-speciﬁc parameters; values close to zero suggests the parametric model is inadequate. From Fig. 1, it appears that most of the values of M are close to 0.5, indicating that the normal centering distribution is very inadequate for the data. Fig. 2 shows the posterior distributions of price sensitivity, advertising intensity, and brand indicators. On the left are displayed the density plot for each parameter, obtained by standard kernel density estimation with window width computed following the recommendation of Silverman (1986). On the right, the relationship between preferences and income is plotted using the Nadaraya–Watson regression estimation method. The density plots reveal that distributions of individual parameters are nonnormal. The conditional density plots further show that the relationship

Semiparametric Bayesian Estimation

Fig. 2.

Density for the Individual-speciﬁc Parameters.

297

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SYLVIE TCHUMTCHOUA AND DIPAK K. DEY

Fig. 2.

Continued.

between individual-speciﬁc parameters and income is nonlinear. It is common knowledge that high-income household are less price-sensitive than low-income households; the conditional density plot for price shows that this is true only for income above $65,000. 5.2. Discrete Choice Models with Aggregate Data The model with aggregate data is applied to a ready-to-eat breakfast cereal dataset. The data were obtained from the Food Marketing Policy Center at the University of Connecticut and is of two types: dollar sales and volume sales measured every four weeks at three supermarket chains in Baltimore, Boston, and Chicago, and household income distribution in each supermarket chains trading areas.

Semiparametric Bayesian Estimation

299

The period of study is January 8, 1996–December 7, 1997. During this period, cereal manufacturers introduced many brands but we focus only on four major brands that were introduced between January 1996 and March 1997, so that each brand is observed for a relatively long time period. These are Kellogg’s Honey Crunch Corn Flakes, General Mill French Toast Crunch, Kellogg’s Cocoa Frosted Flakes, and Post Cranberry Almond Crunch. In addition to the four new brands, the analysis includes 27 established brands. The chain-level share of these established brands varies between 35% and 80% of the total volume of cereal sold at each supermarket chain and quad period. Moreover, these brands are the leading established brands in the four cereal segments: all family, taste-enhanced wholesome, simple health nutrition, and kids’ cereals. The variables used in the analysis include brand’s market share, price, and observed product characteristics (calories, ﬁber, sugar content), and household income. We do not observe consumers and their choices, but only the shares of each cereal brand at each supermarket chain in each period, and the distribution of household income in the trading area of each supermarket chain. Market shares of the brands under consideration are deﬁned by converting the volume sales into servings sold, and dividing by the market size. We assume that each individual has the potential to consume one serving of cereal per day; market size is then computed as the product of the total number of households in the trading area of a supermarket chain and the average household size. The market share of the outside good is deﬁned as the difference between one and the sum of the brands under consideration. Prices are obtained by dividing the dollar sales by the volume sales converted into number of servings. Product characteristics were obtained from cereal boxes and include fat, sugar, and calorie contents. The income variable was obtained by assuming that household income in the trading area of each supermarket chain has a log normal distribution, whose parameters we estimated from the distribution of income. Individual household income is then obtained by drawing a sample of 400 observations from the log normal distribution for each supermarket chain, thus given a total of 1,200 households. Table 2 contains the list of brands included in the analysis as well as the descriptive statistics of price and within-chain market share variables. Within-chain market shares are computed by dividing the volume sales of a given brand by the supermarket chain total volume sales in a given period. Summary statistics for other variables are given in Table 3.

300

Table 2.

SYLVIE TCHUMTCHOUA AND DIPAK K. DEY

Price and Market Share of Brands Included in the Analysis.

Brand

Price ($/Serving)

K Frosted Flakes K Corn Flakes K Frosted Mini Wheat K Raisin Bran K Froot Loops K Rice Krispies K Corn Pop K Special K K Apple Jacks K Crispix K Honey Crunch Corn Flakesa K Cocoa Frosted Flakesa GM Cheerios GM Honey Nuts Cheerios GM Lucky Charms GM Cinnamon Toasted Crunch GM Weathies GM Kix GM Frosted Cheerios GM Total GM Golden Graham GM French Toast Cruncha P Grape Nuts P Raisin Bran P Honey Bunch of Oats P Fruity Pebbles P Honey Comb Post Shredded Wheat P Cranberry Almonds Cruncha Q Cap’n Crunch Q Cap’n Crunch Crunch Berries

Within-chain Market Share (%)

Mean

S.D.

Mean

S.D.

0.482 0.3563 0.8164 0.8076 0.5763 0.6455 0.6200 0.7074 0.6094 0.6629 0.4869 0.5147 0.5700 0.5041 0.6268 0.6241 0.5083 0.7296 0.5188 0.7171 0.6486 0.6232 0.7513 0.7761 0.5133 0.5359 0.5618 0.7733 1.1752 0.4705 0.4576

0.061 0.0590 0.1180 0.1276 0.0955 0.0819 0.0959 0.0862 0.0962 0.0941 0.0816 0.0793 0.0821 0.0555 0.0889 0.0862 0.0740 0.0926 0.0727 0.0783 0.0638 0.1591 0.1446 0.1208 0.0759 0.0756 0.0910 0.1016 0.1621 0.0847 0.0809

4.03 3.92 3.47 3.56 1.91 2.05 1.80 2.02 1.26 1.14 1.42 0.90 3.97 3.14 1.86 1.48 1.27 1.33 1.36 1.16 0.93 0.73 1.82 1.89 1.65 1.03 0.84 1.22 0.67 1.73 1.24

1.99 1.89 1.45 1.73 1.20 0.99 1.09 1.12 0.93 0.59 0.82 0.81 1.48 1.37 1.15 0.77 0.76 0.61 1.09 0.64 0.62 0.72 0.92 1.29 0.98 0.62 0.60 0.71 0.45 1.26 0.94

Source: Authors computation. a New brands.

Table 3.

Calories Fiber Sugar Household income ($)

Sample Statistics.

Mean

Standard

Minimum

Maximum

130.8 1.9677 9.4516 53,761

32.8 1.9754 5.0022 28,117

101 0 0 6,997

220 7.0000 20.0000 216,260

Source: Cereal boxes and samples from the log normal distributions.

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Semiparametric Bayesian Estimation

As instruments for prices we use a set of variables that proxy marginal costs and exogenous variations in prices over time. Over the period covered by our data, in response to low consumption of breakfast cereal, cereal manufacturers slashed cereal prices. To account for these events, we included two indicator variables for April and June 1996. As proxies for marginal production, packages, and distribution costs, we use brand and supermarket chain–indicator variables. Permutation construction is used to induce the ordering to vary with household income. The values n0=1 and Z=0.5 are chosen in the prior of M; the following values are chosen for the other priors: m0=0, V 0 ¼ SSH 0 ¼ SS0 ¼ 100I, and uSH 0 ¼ uS0 ¼ 2. These values are chosen such that the prior variances are very large. The MCMC sampler was run for 20,000 iterations and the last 10,000 iterations were used to obtain parameter estimates. To assess convergence, we use different starting points for the chain and examine the trace plots of the model parameters (not shown). We allowed for heterogeneity in price and cereal characteristics (sugar, ﬁber, and calorie contents) coefﬁcients. Fig. 3 plots the posterior density function of the precision parameter M, which, recall, measures the suitability of a parametric model for the individual-speciﬁc coefﬁcients. From Fig. 3, it appears that most of the values of M are close to 0.1,

2.5

2

1.5

1

0.5

0

0

Fig. 3.

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Posterior Density of the Precision Parameter M.

2

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SYLVIE TCHUMTCHOUA AND DIPAK K. DEY

indicating that the normal centering distribution is very inadequate for the data. Fig. 4 shows the individual-speciﬁc posterior distribution. For each parameter, standard kernel density estimation with window width was computed following the recommendation of Silverman (1986), and the Nadaraya–Watson regression estimation are displayed. Overall, the distributions of consumer preferences are highly non-normal and the relationships between preferences and income are nonlinear. The density plots show that the distribution of price sensitivities, calorie, ﬁber, and sugar preferences are bimodal, thus contrasting the results of Chidmi and Lopez (2007) and Nevo (2001) who assumed a normal distribution for taste parameters. Here, the ﬂexibility of the Dirichlet process that we used to model heterogeneity helps capture multimodality in the distribution

Conditional density Price

Density Price

-2.96

140

-2.962

120

-2.964 100

-2.966

80

-2.968 -2.97

60

-2.972

40

-2.974 20

-2.976

0 -2.985

-2.98

-2.975

-2.97

-2.965

-2.96

-2.955

-2.978

0

Density Calories

0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8 Income

2 x 105

Conditional density Calories

25

0.25

20

0.2 15

0.15 10

0.1

5

0

0

0.05

0.1

0.15

Fig. 4.

0.2

0.25

0.3

0.35

0.05 0

0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8 2 5 Income x 10

Density for the Individual-speciﬁc Parameters.

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Semiparametric Bayesian Estimation Density Fiber

Conditional density Fiber -0.04

12

-0.06

10

-0.08 -0.1

8

-0.12 -0.14

6

-0.16

4

-0.18 -0.2

2

-0.22

0 -0.35

-0.24

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

0

0

0.2 0.4 0.6 0.8

Density Sugar

1 1.2 1.4 1.6 1.8 Income

2 x 105

Conditional density Sugar

10

-0.06

9

-0.08

8

-0.1

7

-0.12

6

-0.14

5 -0.16

4

-0.18

3 2

-0.2

1

-0.22

0 -0.35

-0.24

-0.3

-0.25

-0.2

-0.15

-0.1

-0.05

Fig. 4.

0

0

0.2 0.4 0.6 0.8

1 1.2 1.4 1.6 1.8 Income

2 5

x 10

Continued.

of taste coefﬁcients. The conditional density plots further show that the relationship between tastes parameters and income in nonlinear and highincome households do not have the same preferences as low-income households. Table 4 displays a sample of estimated own and cross-price elasticities. Each entry i, j, where i indexes a row and j a column, represents the percentage change in the market share of brand i for a 1% change in the price of brand j. The values displayed are the median over the 3 supermarket chains and 25 quad-periods considered in the analysis. All own-price elasticities and most cross-price elasticities are larger than those found by Nevo (2001) and Chidmi and Lopez (2007).

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SYLVIE TCHUMTCHOUA AND DIPAK K. DEY

Table 4.

Median Own and Cross-price Elasticities.

Brand 1 Brand 2 Brand 3 Brand 4 Brand 5 Brand 6 Brand 7 Brand 8 Brand 9 Brand 10 Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand Brand

1 1.782 0.063 0.070 0.046 0.095 0.030 0.081 0.037 0.062 2 0.066 1.753 0.049 0.019 0.039 0.115 0.021 0.113 0.016 3 0.116 0.066 1.776 0.091 0.062 0.024 0.035 0.038 0.037 4 0.122 0.041 0.149 1.696 0.063 0.018 0.087 0.023 0.080 5 0.177 0.063 0.070 0.046 1.839 0.030 0.081 0.037 0.062 6 0.072 0.263 0.041 0.018 0.042 1.755 0.023 0.115 0.017 7 0.170 0.049 0.051 0.081 0.090 0.025 1.760 0.028 0.108 8 0.073 0.245 0.051 0.019 0.042 0.109 0.022 1.685 0.017 9 0.168 0.046 0.066 0.100 0.089 0.022 0.149 0.026 1.704 10 0.072 0.263 0.041 0.018 0.042 0.133 0.023 0.114 0.017 11 0.056 0.225 0.070 0.021 0.034 0.083 0.019 0.095 0.015 12 0.127 0.108 0.105 0.029 0.070 0.043 0.033 0.062 0.028 13 0.180 0.054 0.068 0.060 0.096 0.026 0.105 0.032 0.079 14 0.152 0.096 0.069 0.028 0.082 0.044 0.049 0.055 0.037 15 0.066 0.195 0.092 0.022 0.039 0.068 0.020 0.092 0.016 16 0.066 0.259 0.049 0.019 0.039 0.115 0.021 0.113 0.016 17 0.178 0.063 0.070 0.046 0.095 0.030 0.082 0.037 0.062 18 0.073 0.178 0.103 0.023 0.043 0.062 0.021 0.088 0.017 19 0.152 0.096 0.069 0.028 0.082 0.044 0.049 0.055 0.037 20 0.085 0.115 0.186 0.043 0.048 0.038 0.023 0.062 0.021 21 0.120 0.044 0.186 0.243 0.063 0.018 0.067 0.025 0.068 22 0.093 0.199 0.057 0.019 0.052 0.089 0.025 0.101 0.019 23 0.178 0.069 0.048 0.037 0.096 0.037 0.102 0.040 0.066 24 0.167 0.077 0.070 0.035 0.090 0.036 0.063 0.045 0.048 25 0.177 0.063 0.070 0.046 0.095 0.030 0.082 0.037 0.062 26 0.178 0.063 0.070 0.046 0.095 0.030 0.082 0.037 0.062 27 0.056 0.182 0.115 0.028 0.034 0.061 0.019 0.080 0.015 28 0.120 0.086 0.058 0.028 0.096 0.042 0.038 0.053 0.035 29 0.136 0.071 0.060 0.034 0.110 0.035 0.051 0.044 0.047 30 0.120 0.037 0.102 0.127 0.077 0.019 0.083 0.022 0.074 31 0.124 0.038 0.052 0.078 0.101 0.022 0.118 0.027 0.089

0.013 0.041 0.011 0.008 0.013 0.047 0.011 0.039 0.010 1.794 0.032 0.018 0.012 0.018 0.027 0.041 0.013 0.025 0.018 0.016 0.008 0.032 0.016 0.016 0.013 0.013 0.025 0.019 0.016 0.008 0.008

Each entry i, j, where i indexes a row and j a column, represent the median over supermarket chains and time of the percent change in market share of brand i with respect to 1% a change in the price of brand j. The 95% credible intervals are not reported.

6. CONCLUSION In this paper, we have applied a Bayesian semiparametric technique to an important class of models, the random coefﬁcients discrete choice demand models. We speciﬁed a Dirichlet process prior which varies with consumer’s continuous covariates (Grifﬁn & Steel, 2006) for the distribution of consumer heterogeneity. We developed an MCMC algorithm, and illustrate our methodology to estimate the extent of unobserved heterogeneity in

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demand for peanut butter and ready-to-eat breakfast cereal. The empirical results indicate the limitations of the unimodal distribution and the linearity of the relationship between consumer preferences and demographics that are often assumed in modeling consumer heterogeneity.

NOTE 1. In addition to not being restricted to a parametric family, the Dirichlet process has another advantage over a ﬁnite mixture model (e.g., ﬁnite mixture of normals); as a random mixing distribution, it is more parsimonious than a ﬁnite mixture model, which involves a large number of parameters that may not be identiﬁable with aggregate data.

REFERENCES Albert, J. H., & Chib, S. (1993). Bayesian analysis of binary and polychotomous response data. Journal of the American Statistical Association, 88(422), 669–679. Allenby, G. M., Arora, N., & Rossi, P. E. (1998). On the heterogeneity in demand. Journal of Marketing Research, 35, 384–389. Andrews, R. L., & Currim, I. S. (2003). A comparison of segment retention criteria for ﬁnite mixture logit models. Journal of Marketing Research, 40(2), 235–243. Basu, S., & Chib, S. (2003). Marginal likelihood and Bayes factors for Dirichlet process mixture models. Journal of the American Statistical Association, 98(461), 224–235. Berry, S., Levinsohn, J., & Pakes, A. (1995). Automobile prices in market equilibrium. Econometrica, 63, 841–890. Blackwell, D., & MacQueen, J. B. (1973). Ferguson distribution via Polya urn schemes. The Annals of Statistics, 1, 353–355. Chen, Y., & Yang, S. (2006). Estimating disaggregate model using aggregate data via data augmentation of individual choice. Journal of Marketing Research, 4, 596–613. Chidmi, B., & Lopez, R. A. (2007). Brand-supermarket demand for breakfast cereals and retail competition. American Journal of Agricultural Economics, 89(2), 324–337. Chintagunta, P. K. (2001). Endogeneity and heterogeneity in a probit demand model: Estimation using aggregate data. Marketing Science, 20(4), 442–456. Chintagunta, P. K., Jain, D. C., & Vilcassim, N. J. (1991). Investigating heterogeneity in brand preferences in logit models for panel data. Journal of Marketing Research, 28, 417–428. Cifarelli, D. M., & Regazzini, E. (1978). Nonparametric statistical problems under partial exchangeability: The use of associative means. Annali de l’Instituto di Matematica Finianziara dell’ Universita´ di Torino, Series III, 12, 1–36. De Iorio, M., Muller, P., Rosner, G. L., & MacEachern, S. N. (2004). An ANOVA model for dependent random measures. Journal of the American Statistical Association, 99, 205–215.

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Dillon, W. R., & Kumar, A. (1994). Latent structure and other mixture models in marketing: An integrative survey and overview. In: R. P. Bagozzi (Ed.), Advanced methods of marketing research (pp. 295–351). Cambridge: Blackwell Publishers. Duan, J., Guindani, M., & Gelfand, A. (2007). Generalized spatial Dirichlet process models. Biometrika, 94(4), 809–825. Escobar, M. D. (1994). Estimating normal means with a Dirichlet process prior. Journal of the American Statistical Association, 89, 268–277. Escobar, M. D., & West, M. (1995). Bayesian density estimation and inference using mixtures. Journal of the American Statistical Association, 90, 577–588. Ferguson, T. A. (1973). A Bayesian analysis of some nonparametric problems. Annals of Statistics, 1, 209–230. Gelfand, A. E., Kottas, A., & MacEachern, S. N. (2004). Bayesian nonparametric spatial modeling with Dirichlet process mixing. Technical Report, Duke University. Grifﬁn, J. E., & Steel, M. F. J. (2004). Semiparametric Bayesian inference for stochastic frontier models. Journal of Econometrics, 123, 121–152. Grifﬁn, J. E., & Steel, M. F. J. (2006). Order-based dependent Dirichlet processes. Journal of the American Statistical Association, 101, 179–194. Ishwaran, H., & James, L. F. (2001). Gibbs sampling methods for stick-breaking priors. Journal of the American Statistical Association, 96(453), 161–173. Kamakura, W. A., & Russell, G. J. (1989). A probabilistic choice model for market segmentation and elasticity structure. Journal of Marketing Research, 26, 379–390. Kim, J. G., & Menzefricke, U. (2004). Assessing heterogeneity in discrete choice models using a Dirichlet process prior. Review of Marketing Science, 2, 1–41. Kottas, A., & Krnjajic, M. (2005). Bayesian nonparametric modeling in quantile regression. Technical Report, University of California, Santa Cruz. MacEachern, S. N. (1994). Estimating normal means with a conjugate style Dirichlet process prior. Communications in Statistics-Simulations, 23, 727–741. MacEachern, S. N. (1999). Dependent nonparametric processes. In: ASA Proceedings of the section on Bayesian statistical sciences (pp. 50–55). MacEachern, S. N. (2000). Dependent Dirichlet processes. Technical Report, Department of Statistics, Ohio State University, Columbus. MacEachern, S. N., Kottas, A., & Gelfand, A. E. (2001). Spatial nonparametric Bayesian models. Technical Report, ISDS, Duke University. McFadden, D., & Train, K. (2000). Mixed MNL models for discrete response. Journal of Applied Econometrics, 15, 447–470. Musalem, A., Bradlow, E. T., & Raju, J. S. (2005). Who’s got the coupon: Estimating consumer preferences and coupon usage from aggregate data. Technical Report, University of Pennsylvania, Columbus, PA. Musalem, A., Bradlow, E. T., & Raju, J. S. (2006). Bayesian estimation of random coefﬁcients choice models using aggregate data. Technical Report, University of Pennsylvania, PA. Neal, R. M. (2003). Slice sampling. The Annals of Statistics, 31(3), 705–767. Nevo, A. (2001). Measuring market power in the ready to eat cereal industry. Econometrica, 69(2), 307–342. Rossi, P., Allenby, G., & McCulloch, R. (2005). Bayesian statistics and marketing. New York: Wiley. Sethuraman, J. (1994). A constrictive deﬁnition of Dirichlet priors. Statistica Sinica, 639–650. Silverman, B. (1986). Density estimation. London: Chapman and Hall.

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Tanner, M. A., & Wong, W. H. (1987). The calculation of posterior distributions by data augmentation. Journal of the American Statistical Association, 82, 528–550. Villas-Boas, J. M., & Winer, R. S. (1999). Endogeneity in brand choice models. Management Science, 45(10), 1324–1338. Walker, S., & Mallick, B. K. (1997). A note on the scale parameter of the Dirichlet process. Canadian Journal of Statistics, 25, 473–479. Wedel, M., & Kamakura, W. A. (2000). Market segmentation: Conceptual and methodological foundations (2nd ed.). Boston: Kluwer Academic Publishers. Yang, S., Chen, Y., & Allenby, G. M. (2003). Bayesian analysis of simultaneous demand and supply. Quantitative Marketing and Economics, 1, 252–275.

BAYESIAN TWO-STAGE REGRESSION WITH PARAMETRIC HETEROSCEDASTICITY Arto Luoma and Jani Luoto ABSTRACT In this paper, we expand Kleibergen and Zivot’s (2003) Bayesian twostage (B2S) model by allowing for unequal variances. Our choice for modeling heteroscedasticity is a fully Bayesian parametric approach. As an application, we present a cross-country Cobb–Douglas production function estimation.

1. INTRODUCTION After Anderson and Rubin (1949) developed their limited information maximum likelihood (LIML) and Theil (1953) his two-stage least squares (2SLS) technique, instrumental variables (IV) regression became a standard textbook approach in classical econometrics. The development of Bayesian analysis of such models started two decades later, being initiated by Dre`ze (1976); see also, for example, Dre`ze and Morales (1976), Dre`ze and Richard (1983), and Bauwens and van Dijk (1989). Dre`ze’s idea was to equalize the classical and Bayesian analysis of IV models using suitable diffuse priors for

Bayesian Econometrics Advances in Econometrics, Volume 23, 309–328 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23010-0

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the parameters. Unfortunately, his prior ignores important information concerning the near non-identiﬁcation of structural parameters due to weak instruments; see, for example, Kleibergen and Zivot (2003) for discussion. Mainly due to this undesirable property of the Dre`ze prior, recent research on Bayesian analysis of IV models has started to address the abovementioned problem of local non-identiﬁcation; see, for example, Geweke (1996), Kleibergen and van Dijk (1998), and Chao and Phillips (1998, 2002). Following this tradition, Kleibergen and Zivot (2003) developed a new Bayesian two-stage (B2S) approach. In order to mimic classical 2SLS techniques, which essentially handle the problem of local non-identiﬁcation, they constructed a prior for the parameters of the restricted reduced form speciﬁcation and thus functionalized the steps used to obtain the 2SLS estimator. In this paper, we expand Kleibergen and Zivot’s (2003) B2S model by allowing for unequal variances. In classical analysis, modeling heteroscedasticity improves the efﬁciency of estimation and enables the variance estimates to be consistent. Thus, not surprisingly, modeling heteroscedasticity has become standard in classical IV literature; see, for example, White (1982), Cumby, Huizinga, and Obstfeld (1983), and Davidson and MacKinnon (1993). However, there is (to our knowledge) no single Bayesian study of IV models with unequal variances, although from the Bayesian point of view modeling heteroscedasticity should improve the precision of estimates and the quality of predictive inference. The latter follows from the fact that modeling heteroscedasticity allows predictive inferences to be more precise for some units and less so for other. Our choice for modeling heteroscedasticity is a fully Bayesian parametric approach. Speciﬁcally, we assume that varðyi Þ ¼ s2 zy i , where y is the response variable and z a variable explaining the variance. This speciﬁcation requires only one unknown heteroscedasticity parameter (y). Alternatively, we could follow, for example, Geweke (1993) and model heteroscedasticity using a nonparametric approach. This, however, would require estimation of several unknown parameters, which might give rise to identiﬁcation and estimation problems in our relatively complex nonlinear model. To give an empirical illustration of the properties of the heteroscedastic B2S model, we follow Benhabib and Spiegel (1994) and Papageorgiou (2003)1 and construct a simple exercise of aggregate production function estimation as an application. We choose this example, since the problems of endogeneity and heteroscedasticity are well documented in the cross-country growth literature; see, for example, Benhabib and Spiegel (1994) and the surveys of Temple (1999a) and Durlauf, Johnson, and Temple (2005).

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311

The paper is organized as follows: In Section 2 we present a heteroscedastic Bayesian two-stage model (hereafter HB2S model). In Section 3 we give an example of estimating the empirical Cobb–Douglas aggregate production function. Section 4 concludes the paper.

2. THE BAYESIAN TWO-STAGE MODEL WITH PARAMETRIC HETEROSCEDASTICITY Consider the following limited information simultaneous equation model: y1 ¼ Y 2 b þ Zg þ 1

(1)

Y 2 ¼ XP þ ZG þ V 2

(2)

where Y ¼ ð y1 Y 2 Þ is an N m matrix of endogenous variables; Z an N k1 matrix of included exogenous variables; X an N k2 matrix of excluded exogenous variables, that is, instruments; e1 an N 1 vector of errors; and V2 an N (m1) matrix of errors. Vectors b and g contain the structural parameters of interest. The matrices Z and X are assumed to be of full column rank, uncorrelated with e1 and V2, and weakly exogenous for the structural parameter vector b. If the observation vectors yi in the above simultaneous equation model have unequal covariance matrices, they are said to be heteroscedastic. In the following, we will model heteroscedasticity by assuming that the elements e1i of e and the rows V2i of V2 are normally distributed with zero mean and the m m covariance matrix: ! s11 S12 0 (3) Si ¼ varð 1i V 2i Þ ¼ f ðzi ; yÞ S21 S22 where heteroscedasticity is captured by the function f(zi,y), z1, . . . , zN being the known values of some positive-valued variable. Several alternative speciﬁcations of f(zi,y) have been suggested in the literature; see, for example, Judge, Grifﬁths, Hill, and Lee (1985), Greene (1990), Grifﬁths (1999), and Tanizaki and Zhang (2001). Here, we consider the following simple functional form: f ðzi ; yÞ ¼ zy i

(4)

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where y 2 ½0; 1, and the extreme of y ¼ 0 corresponds to homoscedastic errors; see, for example, Greene (1990) and Boscardin and Gelman (1996). If we substitute the reduced form Eq. (2) into the structural form Eq. (1), we get the following nonlinearly restricted reduced form speciﬁcation: y1 ¼ Wd þ v1

(5)

Y 2 ¼ UB þ V 2

(6)

0 where W ¼ ð UB Z Þ, d ¼ ð b g0 Þ0 , U ¼ ð X Z Þ, B ¼ ð P0 G0 Þ0 , and v1 ¼ e1þV2b. Thus, U is an N k matrix, where k ¼ k1þk2, and W is an N (k1þm1) matrix. Denoting ! o11 O12 0 Oi ¼ varð v1i V 2i Þ ¼ f ðzi ; yÞ ; O21 O22

o11:2 ¼ o11 O12 O1 22 O21 ;

f ¼ O1 22 O21

we obtain that e1 ¼ v1V2f is uncorrelated with V2 and var(e1i) ¼ f(zi,y)o11.2. From the reduced form Eq. (5), we can see the possible identiﬁcation problem related to the two-stage approach. The parameter vector d is identiﬁed when W has full column rank, which is equivalent to P having full column rank, and is locally non-identiﬁed when P has a lower rank value. Therefore, the number of instruments has to be at least the number of endogenous regressors, that is, k2Zm1. The model is called just-identiﬁed when k2 ¼ m1 and over-identiﬁed when k2Wm1. In the case of weak instruments (i.e., when the instruments are only weakly correlated with the endogenous regressors), identiﬁcation problems may occur, since P is then close to zero or close to having reduced rank; see, for example, Zivot, Startz, and Nelson (1998) and Shea (1997). Since in the B2S approach suggested by Kleibergen and Zivot (2003) the prior distribution is so constructed that it explicitly incorporates this kind of knowledge, we will choose it as a starting point for our heteroscedasticity-corrected limited information model. Using the restricted form speciﬁcations (5) and (6), we can write the likelihood in the form pðYjX; Z; ZÞ ¼ pðY 2 jX; Z; ZÞpðy1 jX; Z; ZÞ where 0:5 pðy1 jY 2 ; X; Z; ZÞ / o0:5N 11:2 jLj 0 1 expf0:5o1 11:2 ðy1 Wd V 2 fÞ L ðy1 Wd V 2 fÞg

ð7Þ

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313

pðY 2 jX; Z; ZÞ / jO22 j0:5N jLj0:5ðm1Þ 0 1 expf0:5trO1 22 ðY 2 UBÞ L ðY 2 UBÞg

ð8Þ

Here, Z denotes the vector of all parameters and L ¼ diagð f ðz1 ; yÞ; . . . ; f ðzN ; yÞÞ. As a prior distribution, we will use a modiﬁcation of the Jeffreys prior distribution. The Jeffreys prior is deﬁned as pðZÞ / jIðZÞj0:5 , where 2 @ IðZÞ ¼ E log pðYjZÞ @Z@Z0 is the Fisher information matrix for Z. Our modiﬁcations will be as follows: ﬁrst, we calculate the second-order derivative with respect to vec(B) from the logarithm of the conditional density (8) instead of the full log likelihood. Second, we remove the prior dependence between B and d by replacing the corresponding non-diagonal blocks in I(Z) with zero matrices. Then, if we assume that the geometric mean of zi, i ¼ 1, . . . , T, is unity, the joint prior is given by 0:5ð2mþk1 Þ jU 0 L1 Uj0:5ðm1Þ jW 0 L1 Wj0:5 pðZÞ / jO22 j0:5ðmþk1Þ o11:2

(9)

One can normalize the weight variable z by dividing it by its geometric mean. This has two advantages: ﬁrst, one need not adjust the prior distribution, and second, the dispersion parameters o11.2 and O22 have a consistent meaning under different values of y. See also Boscardin and Gelman (1996), who discuss the issue in the context of one-stage regression models. Our derivation of the prior distribution is somewhat arbitrary. However, it yields a prior with desirable properties. The presence of the term jW 0 L1 Wj0:5 in the prior (9) reﬂects the fact that the model is not informative regarding b when P has reduced rank, since this term tends to zero as P tends to a reduced rank value. In the special case, when Z is not in the model and y ¼ 0, our prior reduces to that proposed by Kleibergen and Zivot (2003). The slightly simpler Dre`ze prior jOj0:5ðkþmþ1Þ has the drawback that the marginal posterior of P has a nonintegrable asymptote at P ¼ 0, when the model is just-identiﬁed; see Kleibergen and Zivot (2003) for further discussion on the issue. Multiplying the likelihood function by the joint prior (9) yields, after some tedious algebra, the following conditional and marginal posteriors: 0:5ðmþk1 1Þ jW 0 L1 Wj0:5 pðdjY; f; B; y; o11:2 ; O22 Þ / o11:2 ^ 0 W 0 L1 Wðd dÞg ^ expf0:5o1 ðd dÞ 11:2

ð10Þ

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pðfjY; B; y; o11:2 ; O22 Þ / o0:5ðm1Þ jV 02 L1 MV 2 j0:5 11:2 ^ 0 V 0 L1 MV 2 ðf fÞg ^ expf0:5o1 ðf fÞ 11:2

2

ð11Þ

0 1 pðo11:2 jY; B; y; O22 Þ / o0:5ðNþ2Þ ðv0 L1 MvÞ0:5N expf0:5o1 11:2 v L Mvg 11:2

(12) pðO22 jY; B; yÞ / jO22 j0:5ðNþmþk1Þ jV 02 L1 V 2 j0:5ðNþk1Þ 0 1 expf0:5trO1 22 V 2 L V 2 g

ð13Þ

pðB; yjYÞ / ðv0 L1 MvÞ0:5N jU 0 L1 Uj0:5ðm1Þ jLj0:5m jV 02 L1 MV 2 j0:5 jV 02 L1 V 2 j0:5ðNþk1Þ

ð14Þ

where d^ ¼ ðW 0 L1 WÞ1 W 0 L1 ðy1 V 2 fÞ; ^ ¼ ðV 0 L1 MV 2 Þ1 V 0 L1 My ; f 2

2

1

M ¼ I WðW 0 L1 WÞ1 W 0 L1 ; ^ v ¼ y V 2 f; and V2 ¼ Y 2 UB 1

The distributions given in Eqs. (10) and (11) are multivariate normal, while those in Eqs. (12) and (13) are inverse gamma and inverse Wishart, respectively. The joint marginal posterior for B and y in Eq. (14) does not have a form of any standard p.d.f. Kleibergen and Zivot (2003) discuss some properties of their B2S model and compare it to the original Dre`ze (1976) approach. We brieﬂy review their discussion and make some comparison between our parametric heteroscedasticity-corrected model and their B2S model. 1. As with the Dre`ze and B2S approaches, the posteriors are not invariant to the ordering of the endogenous variables; that is, if y1 is exchanged with some of the variables in Y2, the results do not remain identical. See Dre`ze (1976) for the argument. 2. The mean of the conditional posterior of b in the B2S model is essentially b^ 2SLS . However, this is not true for the HB2S model, since heteroscedasticity correction gives more weight to ‘‘good’’ observations, while b^ 2SLS weighs all observations equally. The difference between the heteroscedastic-corrected estimate of b and b^ 2SLS depends, of course, on the degree of heteroscedasticity.

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315

3. As with the B2S approach, the marginal posterior (14) does not have the nonintegrable asymptote at P ¼ 0 that appears in the Dre`ze approach. The last term in Eq. (14) may be written in the form ^ ðNþk1Þ=2 , where B^ ¼ ðU 0 L1 UÞ1 U 0 L1 Y 2 ^ 0 U 0 L1 UðB BÞj jSþ ðB BÞ 0 1 ^ L ðY 2 U BÞ, ^ and is, for a ﬁxed L, a kernel of a and S ¼ ðY 2 U BÞ matrix-variate Student-t density with N1 degrees of freedom. The other terms are, for a ﬁxed L, bounded from zero and inﬁnity, which implies that the posterior is integrable with respect to B. If the term jW 0 L1 Wj0:5 were not present in the prior, its inverse would appear in the posterior causing an inﬁnite asymptote at the reduced rank values of P. 4. As with the B2S approach (without heteroscedasticity correction), the form of the posterior of B is closely related to the marginal posterior that results from a standard diffuse prior analysis of the reduced form regression of Y2 on U with heteroscedasticity correction.

3. EMPIRICAL EXAMPLE 3.1. Estimated Model To illustrate some of the properties of the HB2S model, we construct a simple exercise of aggregate production function estimation with crosscountry data. We chose this example, since problems of endogeneity and heteroscedasticity are well documented in the cross-country growth literature; see, for example, Benhabib and Spiegel (1994), Papageorgiou (2003), and the surveys of Temple (1999a) and Durlauf et al. (2005). For example, Benhabib and Spiegel (1994) analyzed the biases of coefﬁcient estimates that result from the correlation between the accumulated physical and human capital series and the error term, and found that there is likely to be an upward coefﬁcient bias in the input share of capital and human capital estimates, and a downward bias in estimates of the input share of labor. Our analysis is close to that of Benhabib and Spiegel’s (1994) or Papageorgiou’s (2003). However, we do not separate aggregate labor and human capital stocks; rather we follow Bils and Klenow (2000) and assume that individual human capital stock is related to individuals, years of schooling, and years of experience. This implies that each individual has some degree of human capital and thus aggregate human capital stock should be modeled as Ht ¼ htLt, where ht is average human capital stock per person and Lt is labor force.

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We assume the Romer-type Cobb–Douglas production function Y it ¼ a b Aaþb it K it H Y;it it (see Romer, 1990), where Yit is the output, Ait productivity, Kit physical capital, and HY,it the human capital engaged in ﬁnal-goods production in country i at time t. Taking log differences, we obtain the following equation for long-run growth: Y iT AiT K iT H Y;iT iT ¼ ða þ bÞ log þ a log þ b log þ log log Y i0 Ai0 K i0 H Y;i0 i0 (15) In Eq. (15), we assume that the resource constraint Hit ¼ HA,itþHY,it, where HA,it is the human capital engaged in R&D activities, holds. One problem in estimating Eq. (15) is that we should replace an unobservable log(AT/A0) by some function of observables. Otherwise the estimates of factor shares will be biased; see, for example, Temple (1999a, 1999b). We follow Papageorgiou (2003) and propose the following speciﬁcation for the growth rate of technology: AiT Ai0 A0 ¼ dH A;iT þ mH A;iT 1 (16) Ai0 Ai0 where A 0 is the technology frontier, and d and m are the innovation and imitation parameters, respectively. In Eq. (16), human capital speeds technology growth through innovation and imitation. Using Eq. (16), we can write Eq. (15) in the estimation form max Y iT y ¼ c d i þ ða þ bÞ ðd mÞH A;i0 þ m 0 H A;i0 log Y i0 yi0 K iT H Y;iT þ b log þ uit ð17Þ þ a log K i0 H Y;i0 where di is a vector of deterministic components (constant and dummy variables), and uit a normally distributed error term with zero mean and s11 variance. We follow Benhabib and Spiegel (1994) and Papageorgiou (2003) in assuming that ðY 0 =L0 Þmax =ðY i0 =Li0 Þ ¼ ymax 0 =yi0 approximates A 0 =Ai0 . Since human capital may also speed technology adoption and may be to some extent necessary for technology use, we propose the following two alternative speciﬁcations for technology growth with production aþb a b a b technology: Y it ¼ Aaþb it K it H it it and Y it ¼ Ait K it Lit it ; see, for example, Benhabib and Spiegel (1994) and Bils and Klenow (2000). With similar

Bayesian Two-Stage Regression with Parametric Heteroscedasticity

steps, we obtain the corresponding empirical speciﬁcations max Y iT y ¼ c d i þ ða þ bÞ ðd mÞH i0 þ m 0 H i0 log Y i0 yi0 K iT H iT þ b log þ uit þ a log K i0 H i0 log

max y Y iT ¼ c d i þ ða þ bÞ ðd mÞH i0 þ m 0 H i0 Y i0 yi0 K iT LiT þ b log þ uit þ a log K i0 Li0

317

ð18Þ

ð19Þ

We estimate Eqs. (17–19) using our HB2S model. As a weight variable, we use zi ¼ yi0, where ( ) N Y i0 1X Y i0 log yi0 ¼ exp log Li0 Li0 N i¼1 This corresponds to dividing the output per labor force (Yi0/Li0) by its geometric mean. We use the output per worker in the weight coefﬁcients f(zi,y), i ¼ 1, . . . , N, since we expect countries with higher initial income to have more stable growth paths due to developed institutional structures, which have the ability to reduce the overall risk in society. Alternatively, we could use some institutional indicator. However, since the choice of institutional indicators that approximate the true ‘‘level’’ of institutional quality is somewhat difﬁcult, and far from unique, we decided to abandon this approach.

3.2. Estimation Results The data and choice of instruments are described in Appendix A. To generate a Monte Carlo sample from the joint posterior of y and B, we used a version of the random walk Metropolis algorithm for Markov Chain Monte Carlo (MMCMC). This algorithm uses a multivariate normal distribution for the jump distribution on changes in y and B. Our simulation procedure was as follows: We ﬁrst minimized the negative of the logarithm of the posterior density (14) numerically to obtain the posterior mode and evaluated the Hessian matrix at the minimum. We then used the inverse of the Hessian as an approximation to the posterior covariance matrix of

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(y,vec(B)u)u and scaled it by the factor (2.4)2/d, where d is the number of simulated parameters, to obtain an optimal covariance matrix for the jump distribution; see, for example, Gelman, Carlin, Stern, and Rubin (2004). We then added noise to the posterior mode to obtain overdispersed starting values and simulated three chains of length 100,000. We excluded the ﬁrst half of simulations as a burn-in period in each chain and picked out every tenth draw. The convergence of the chains was checked using Gelman and Rubin’s convergence diagnostic R (also called ‘‘potential scale reduction factor’’); see Gelman and Rubin (1992). The diagnostic values close to 1 indicate approximate convergence and the values smaller than 1.1 are acceptable in most cases. In our case, the diagnostic was estimated as 1.00 for all parameters and all models; the convergence was thus very good. Table B1 in Appendix B shows the simulation results for model (17). After simulating y and B, the other parameter vectors and matrices were simulated from the conditional distributions (10–13). The B2S model could be estimated similarly, except that the covariance matrix of the classical ﬁrst-stage regression (scaled by the factor 2.42/d) could be used as the covariance matrix of the jump distribution.2 We use the posterior mean of deviance, Davg ðYÞ ¼ EfDðY; ZÞjYg, as a measure of model ﬁt. This criterion is called ‘‘average discrepancy’’ by Gelman et al. (2004) who recommend its use for model comparison.3 It is PN estimated as D^ avg ðYÞ ¼ ð1=NÞ i¼1 DðY; Zi Þ, where DðY; ZÞ ¼ 2 ln pðYjZÞ is the deviance and Zi ; i ¼ 1; . . . ; N, are posterior simulations. The average discrepancy is usually greater than DZ^ ðYÞ ¼ DðY; Z^ Þ, where Z^ is a point estimate, such as posterior mode, mean, or median. The difference pD ¼ Davg ðYÞ DZ^ ðYÞ is called the effective number of parameters and is in most cases approximately equal to the number of parameters in nonhierarchical models. The Bayesian equivalent (and also a generalization) of the Akaike information criterion (AIC) is the deviance information (DIC), deﬁned as DIC ¼ DZ^ ðYÞ þ 2pD . The DIC has been suggested as a criterion of model ﬁt when the goal is to select a model with best out-of-sample predictive power; see Spiegelhalter, Best, Carlin, and van der Linde (2002). Fig. 1 displays the residual plots for the ﬁrst- and second-stage regressions, corresponding to Eqs. (7) and (8), respectively. The residuals have been obtained by replacing the unknown parameters by their posterior means and they have been plotted against the normalized initial output y0. The approximate 95% probability belts, based on the normality assumption, are also shown. We see that the ﬁt of the belts seems worse for the log differences of Y and K, since there are no points outside these bands for y0W2. The reason for this is probably that we have only one

319

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2

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3 y0

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Bayesian Two-Stage Regression with Parametric Heteroscedasticity

0

1

2

3 y0

Fig. 1. Residual Plots of the First- and Second-Stage Regressions Corresponding to Eqs. (7) and (8), Respectively. The Dotted Lines are Approximate 95% Probability Intervals, Based on the Normality Assumption. The First Row Gives the Residual Plots against yi0, when log(YT/Y0), log(HY,T/HY,0), and log(KT/K0) of Model (17) are Regressed on the Instrumental Variables. The Second Row Gives the Corresponding Residual Plots for log(YT/Y0), log(HT/H0), and log(KT/K0) when Model (18) is Used. Finally, the Third Row Gives the Residual Plots for log(YT/Y0), log(LT/L0), and log(KT/K0) Corresponding to Model (19). The Residuals are Obtained when the Unknown Parameters are Replaced by their Posterior Means.

heteroscedasticity parameter for all regressed variables and the ﬁt cannot be equally good for all of them. Table 1 shows the estimation results for Eqs. (17–19), obtained using the ordinary least squares (OLS) method and the Bayesian estimation of the B2S and HB2S models. On the basis of the ﬁgures and the posterior

Table 1. Eq. (17)

(aþb)(dm) (aþb)m a b aþb d m y

DummyAfrica DummyLatin D^ avg ðyÞ 2 R

America

Eq. (19)

OLS

B2S

HB2S

OLS

B2S

HB2S

OLS

B2S

HB2S

0.096 (0.048) 0.050a (0.013) 0.432a (0.073) 0.378e (0.137) – – – – – – – – 0.275 (0.064) 0.262 (0.112) 0.187 (0.089) –

0.090 (0.224) 0.067b (0.048) 0.283c (0.107) 0.537d (0.152) 0.820 (0.129) 0.036 (0.269) 0.086b (0.066) – – 0.335 (0.133) 0.348 (0.102) 0.261 (0.104) 378.7

0.077 (0.1496) 0.065 (0.065) 0.339d (0.107) 0.470d (0.147) 0.806 (0.106) 0.031 (0.172) 0.077 (0.089) 0.529 (0.122) 0.305 (0.099) 0.298 (0.105) 0.253 (0.090) 399.0

0.016 (0.0146) 0.002a (0.0005) 0.433a (0.074) 0.365e (0.144) – – – – – – – – 0.255 (0.081) 0.235 (0.106) 0.174 (0.088) –

0.009 (0.035) 0.002 (0.002) 0.293d (0.097) 0.539d (0.150) 0.831 (0.131) 0.012 (0.044) 0.002 (0.002) – – 0.270 (0.150) 0.302 (0.098) 0.244 (0.097) 373.0

0.006 (0.022) 0.001 (0.001) 0.347d (0.103) 0.483d (0.150) 0.829 (0.108) 0.007 (0.028) 0.002 (0.002) 0.519 (0.111) 0.235 (0.109) 0.254 (0.097) 0.237 (0.088) 394.4

0.023 (0.012) 0.002a (0.0004) 0.452a (0.070) 0.408a (0.162) – – – – – – – – 0.351 (0.062) 0.287 (0.121) 0.199 (0.096) –

0.021 (0.034) 0.002 (0.002) 0.313d (0.101) 0.618d (0.185) 0.931 (0.161) 0.022 (0.038) 0.002 (0.0018) – – 0.417 (0.133) 0.387 (0.107) 0.285 (0.105) 417.8

0.017 (0.023) 0.002b (0.001) 0.361d (0.101) 0.547d (0.169) 0.908 (0.136) 0.018 (0.027) 0.002b (0.0017) 0.434 (0.113) 0.376 (0.105) 0.327 (0.107) 0.269 (0.097) 432.1

0.62

–

–

0.62

–

–

0.61

–

–

For the OLS estimates, White’s heteroscedasticity-corrected standard errors are given in parentheses. In the Bayesian models, the posterior means and standard deviations are reported. We do not report the OLS estimates of the imitation and innovation parameters, since exact standard errors are not available for them. a p value of one-sided hypothesis test o0.01. b Parameter W0 with 90–94% probability. c Parameter W0 with 95–98% probability. d Parameter W0 with 99–100% probability. e p value of one-sided hypothesis test o0.05.

ARTO LUOMA AND JANI LUOTO

Constant

Eq. (18)

320

Parameters

Growth Regressions for Eqs. (17–19).

Bayesian Two-Stage Regression with Parametric Heteroscedasticity

321

summaries of the heteroscedasticity parameter y, we see that the data support heteroscedasticity in each model. Heteroscedasticity is especially obvious in the cases of output and physical capital growth, less so in human capital, and only slight in labor growth. When we compare the estimated average discrepancies D^ avg ðYÞ between the B2S and HB2S models, we see that the data lend strong support to the latter. If one would test the signiﬁcance of y using the likelihood ratio test, a signiﬁcant result at the 5% level would correspond to a difference W3 in D^ avg ðYÞ. This follows from the fact that the nested model has one parameter less in this case and from the relation between D^ avg ðYÞ and the number of parameters. It seems that the margin between the estimated average discrepancies of B2S and HB2S depends on the degree of heteroscedasticity, since the gap is around 20 for models (17) and (18), while for model (19) it is about 14 only. Note that model (19) has the lowest heteroscedasticity parameter y due to the small amount of heteroscedasticity in the labor growth series (see the third row in Fig. 1). However, the differences in y are not very signiﬁcant between the models, since they are smaller than the posterior deviation of the parameter. Furthermore, the model deﬁned by Eq. (19) has the smallest estimated average discrepancy (432). However, in the ‘‘economic theory’’ sense, this result does not necessarily indicate that Eq. (19) is more preferable than the other models, since the ﬁrst-stage regression could be more informative in this model, increasing the overall model ﬁt. We also ﬁnd that the IV regression estimates of a are, in general, higher, and the estimates of b lower, than the corresponding OLS estimates.4 Thus, our results conﬁrm the ﬁnding of Benhabib and Spiegel (1994) that there is an upward coefﬁcient bias in the OLS estimates of a and human capital share b [Eqs. (17) and (18)], and a downward bias in the OLS estimates of the labor share parameter b [Eq. (19)]5; see also Griliches and Mairesse (1995) for a discussion of endogeneity of regressors in the aggregate production function approach. Finally, based on the results reported in Table 1, the data are not consistent with the innovation parameter d being positive; see, for example, Benhabib and Spiegel (1994), who obtained similar results in their analyses. On the contrary, there is weak (or moderate) support in the data for the imitation parameter m being positive. Thus, contrary to Papageorgiou (2003), our results slightly favor catch-up progress over country-speciﬁc technological progress as the channel through which accumulation of human capital affects output growth. This is quite sensible, since only about 15% of the countries in our sample have economically meaningful innovation activities; see also Benhabib and Spiegel (1994).

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4. CONCLUSION In this paper, we have presented a relatively straightforward way to model unequal variances in B2S IV regression. We have done this using a fully Bayesian parametric approach. As noted, modeling heteroscedasticity is important also in the Bayesian IV context, since it improves the precision of estimates and the quality of predictive inference. We used a simple production function approach as a tool to provide an empirical illustration of some properties of the heteroscedastic B2S model. On the basis of residual plots and estimated discrepancies between the data and the models, we have shown that the data lend strong support to the use of the HB2S model instead of the homoscedastic B2S model. Because our modeling of heteroscedasticity is relatively limited, we suggest that future research on Bayesian IV regression under unequal variances should focus on multiplicative heteroscedasticity, which is ﬂexible and includes most of the useful formulations for heteroscedasticity as special cases; see, for example, Tanizaki and Zhang (2001).

NOTES 1. See also Barro (1999) and Temple (1999a, 1999b, 2001). 2. The estimation was implemented using R, a statistical computing environment. R is freely available under the General Public Licence at www.R-project.org. The code and data sets are available at http://mtl.uta.ﬁ/codes/HB2S/. 3. Gelman et al. (2004) prefer using discrepancy between data and model to using Bayes factors in model comparisons. They consider Bayes factors to be in most cases irrelevant, since they are used to compute the relative probabilities of the models conditional on one of them being true. 4. See also the OLS results, where the dummies are excluded from the analysis, in Table B2 (Appendix B). Our OLS results are, in general, quite similar to those in previous studies. Speciﬁcally, our estimates for physical capital share a lie between 0.432 and 0.53 and are positive at the 1% signiﬁcance level. The estimates of human capital/labour shares are relatively low and positive at the 5% level. 5. Note that multiplying Lt by ht seems to reduce b more in the IV models than in the ordinary regression model. 6. We have predicted the missing GDP and investment values for some countries in our data. The missing GDP values are predicted using the linear trend model while the investment share (I/GDP) values are predicted using the latest available data points. These countries (and missing years) are Angola (1997–2000), Central Africa (1999–2000), Cyprus (1997–2000), Fiji (2000), Guyana (2000), Haiti (1999–2000), Sierra Leone (1997–2000), and Singapore (1997–2000).

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REFERENCES Anderson, T. W., & Rubin, H. (1949). Estimators of the parameters of a single equation in a complete set of stochastic equations. The Annals of Mathematical Statistics, 21, 570–582. Barro, R. J. (1999). Notes on growth accounting. Journal of Economic Growth, 4, 119–137. Bauwens, L., & van Dijk, H. K. (1989). Bayesian limited information analysis. In: J. J. Gabszewich, J.-F. Richard & L. A. Wolsey (Eds), Economic decision-making: Games, econometrics and optimisation. Amsterdam: North-Holland Publishing Co. Benhabib, J., & Spiegel, M. M. (1994). The role of human capital in economic development evidence from aggregate cross-country data. Journal of Monetary Economics, 34, 143–173. Bils, M., & Klenow, P. J. (2000). Does schooling cause growth? American Economic Review, 90, 1160–1182. Boscardin, J. W., & Gelman, A. (1996). Bayesian computation for parametric models of heteroscedasticity in the linear model. Advances in Econometrics, 11, A87–A110. Chao, J. C., & Phillips, P. C. B. (1998). Posterior distribution in limited information analysis of the simultaneous equations models using the Jeffreys’ prior. Journal of Econometrics, 87, 49–86. Chao, J. C., & Phillips, P. C. B. (2002). Jeffreys’ prior analysis of the simultaneous equations model in the case of nþ1 endogenous variables. Journal of Econometrics, 111, 251–283. Cohen, D., & Soto, M. (2001). Growth and human capital: Good data, good results. OECD Development Centre Technical Papers No. 179. Cumby, R. E., Huizinga, J., & Obstfeld, M. (1983). Two-step two-stage least squares estimation in models with rational expectations. Journal of Econometrics, 21, 333–355. Davidson, R., & MacKinnon, J. G. (1993). Estimation and inference in econometrics (2nd ed.). New York: Oxford University Press. Dre`ze, J. H. (1976). Bayesian limited information analysis of the simultaneous equation model. Econometrica, 44, 1045–1075. Dre`ze, J. H., & Morales, J. A. (1976). Bayesian full information analysis of simultaneous equations. Journal of American Statistical Association, 71, 919–923. Dre`ze, J. H., & Richard, J. F. (1983). Bayesian analysis of simultaneous equation systems. In: Z. Griliches & M. Intriligator (Eds), Handbook of econometrics (Vol. 1). Amsterdam: North-Holland. Durlauf, S. N., Johnson, P. A., & Temple, J. R. W. (2005). Growth econometrics. Handbook of Economic Growth, 1, 555–677. Gallup, J. A., Sachs, J., & Mellinger, A. (1998). Geography and economic development. NBER Working Paper No. 6849. Gelman, A., Carlin, J. B., Stern, H. S., & Rubin, D. B. (2004). Bayesian data analysis (2nd ed.). London: CRC Press. Gelman, A., & Rubin, D. B. (1992). Inference from iterative simulation using multiple sequences. Statistical Science, 7, 457–511. Geweke, J. (1993). Bayesian treatment of the independent student-t linear model. Journal of Applied Econometrics, 8, S19–S40. Geweke, J. (1996). Bayesian reduced rank regression in econometrics. Journal of Econometrics, 75, 121–146. Greene, W. H. (1990). Econometric analysis. New York: Macmillian.

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Grifﬁths, W. E. (1999). Heteroscedasticity. Working Papers in Econometrics and Applied Statistics, No. 104. Department of Econometrics, University of New England. Griliches, Z., & Mairesse, J. (1995). Production functions: The search for identiﬁcation. NBER manuscript. Judge, G. G., Grifﬁths, W. E., Hill, R. C., & Lee, T.-C. (1985). The theory and practice of econometrics. New York: John Wiley and Sons. Kleibergen, F., & van Dijk, H. K. (1998). Bayesian simultaneous equation analysis using reduced rank structures. Econometric Theory, 14, 701–743. Kleibergen, F., & Zivot, E. (2003). Bayesian and classical approaches to instrumental variable regression. Journal of Econometrics, 114, 29–72. Papageorgiou, C. (2003). Distinguishing between the effects of primary and post-primary education on economic growth. Review of Development Economics, 7, 622–635. Romer, P. (1990). Endogenous technological change. Journal of Political Economy, 98, 71–96. Shea, J. (1997). Instrument relevance in multivariate linear models: A simple measure. The Review of Economics and Statistics, 79, 348–352. Spiegelhalter, D. J., Best, N. G., Carlin, B. P., & van der Linde, A. (2002). Bayesian measures of model complexity and ﬁt (with discussion). Journal of the Royal Statistical Society Series B (Statistical Methodology), 64, 583–639. Stock, J. H., & Yogo, M. (2002). Testing for weak instruments in linear IV regression. NBER Technical Working Paper No. 284. Tanizaki, H., & Zhang, X. (2001). Posterior analysis of the multiplicative heteroscedasticity model. Communications in Statistics – Theory and Method, 30, 855–874. Theil, H. (1953). Estimation and simultaneous correlation in complete equation systems. Mimeographed Memorandum of the Central Planning Bureau, The Hague. Temple, J. (1999a). The new growth evidence. Journal of Economic Literature, XXXVII, 112–156. Temple, J. (1999b). A positive effects of human capital on growth. Economic Letters, 65, 131–134. Temple, J. (2001). Generalizations that aren’t? Evidence on education and growth. European Economic Review, 45, 905–918. White, H. (1982). Instrumental variable regression with independent observation. Econometrica, 50, 483–499. Zivot, E., Startz, R., & Nelson, C. R. (1998). Valid conﬁdence intervals and inference in the presence of weak instruments. International Economic Review, 39, 1119–1144.

APPENDIX A. DATA AND INSTRUMENTS Our estimation involves data on 85 countries (see Table B3 in Appendix B). The stock of physical capital is estimated using each country’s investment rates from Penn World Tables 6.1 and perpetual inventory methods. The capital stock in 1960 is estimated using Ki ¼ Ii/(giþdþni), where I denotes the investments, g the growth rate of GDP per worker, d the depreciation rate, and n the growth rate of the population, calculated as the average

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growth rate from 1961 to 1970. The depreciation rate d is assumed to be 0.07.6 In the case of human capital, we follow Bils and Klenow (2000), who approximated the human capital per person using the years of schooling per person and the experience of each age group. Speciﬁcally, we assume that the log of human capital stock of a worker of age a is ln hðaÞ ¼ f ðsÞ þ g1 ða s 6Þ þ g2 ða s 6Þ2

(A.1)

where g1 and g2 are parameters of return to experience, s is average years of schooling, and f ðsÞ ¼ y s1c =ð1 cÞ; c40; y40. Eq. (A.1) is of the same form as that of Bils and Klenow (2000); however, we assume that the inﬂuence of a teacher on human capital is zero. Using Eq. (A.1), we calculate the average human capital stock for all age groups between 20 and 59 in 1970 and 2000 by weighing the human capital of the age group by its proportion of the country’s total population. In Eq. (A.1), we set g1 ¼ 0.0512 and g2 ¼ 0.00071, which corresponds to the average estimates across 52 countries as reported in Bils and Klenow (2000). We set c at 0.28 and set y so that the mean of f 0 ðsÞ ¼ y=sc equals the mean Mincerian returns across 56 countries, which is 0.099; see Bils and Klenow (2000). Finally, HA,it is determined as Hit times the percentage of the population aged 15 or over with some higher education (completeþincomplete). The education series are from Cohen and Soto (2001). The population data are from the International Data Base of the U.S. Census Bureau [Population Division of the International Programs Center (IPC)] and the United Nations population data (1995). The labor stock (L) in each country is obtained from World Development Indicators (2002). The output series have been taken from Penn World Tables 6.1. Since physical and human capitals are accumulated factors, they are endogenous. This causes the simple OLS estimator to be inconsistent. A common means of dealing with the issue of endogeneity is to instrument for endogenous regressors with variables correlated with them but exogenous to them and the regressed variable. Moreover, the validity of an instrument requires that it cannot be a direct growth determinant or correlated with omitted growth determinants; see, for example, Durlauf et al. (2005). Therefore, we instrument the growth rates of aggregate human and physical capital using the distance from the equator (Gallup, Sachs, & Mellinger, 1998) and the following variables in 1970: age dependency ratio (dependents to working-age population), illiteracy rate (%) of people aged 15–24 from

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World Development Indicators (2002), and the level of physical capital per worker. We make the assumption that the distance of a country from the equator, the initial (year 1970) values for age dependency ratio, and youth illiteracy rate are not direct growth determinants; rather they inﬂuence the environment and investment culture, where individuals accumulate physical and human capital. For a more detailed discussion on these topics see, for example, Durlauf et al. (2005). Since one may question the validity of our instruments, we check the consistency of the IV estimators using two speciﬁcation tests. First, Hansen’s test for over-identiﬁcation restrictions is used to see whether the model speciﬁcation is correct and the instruments are uncorrelated with the error process. The second test is for weak instruments. We follow Stock and Yogo (2002), who proposed quantitative deﬁnitions of weak instruments based on the maximum IV estimator bias or the maximum Wald test size distortion. The smallest p value of the Hansen’s test for over-identiﬁcation restrictions for the regression models in this paper is 0.88 and the smallest test statistic of the Stock and Yogo’s test for weak instruments is 9.17. Thus, we can reject the null of weak instruments and cannot reject the null of appropriate instruments at the 5% level. Note that we use classical tests here, since these are readily available and do not demand extra programming effort. Finally, we used African and Latin American country dummies since, based on the above test results, our instruments behave much more appropriately when these dummies are included in the analysis. The reason may be that these dummies approximate some omitted growth determinants that may be correlated with some of our instruments.

Table B1. Parameter

y b11 b12 b13 b14 b15 b16 b17 b18 b19 b21 b22 b23 b24 b25 b26 b27 b28 b29

Summary of the Posterior Simulation of y and B when Model (17) was Used.

Empirical Mean and Standard Deviations, Plus Standard Errors of the Means

Quantiles for Each Variable

Mean

SD

Naiv. SE

T-S SE

2.5%

25%

50%

75%

97.5%

0.5286 0.6697 0.4412 0.0096 0.3303 4.7937 1.1493 0.6680 0.1390 0.0040 0.5981 1.4091 0.0004 0.0326 0.1891 0.1773 0.1368 0.1324 0.0122

0.1166 0.2886 0.4261 0.0036 0.0690 0.8125 0.1901 0.1530 0.2506 0.1059 0.1200 0.1778 0.0015 0.0280 0.3326 0.0767 0.0614 0.1001 0.0422

0.0010 0.0024 0.0035 2.9e05 0.0006 0.0066 0.0016 0.0012 0.0020 0.0009 0.0010 0.0015 1.2e05 0.0002 0.0027 0.0006 0.0005 0.0008 0.0003

0.0024 0.0060 0.0085 6.6e05 0.0014 0.0162 0.0039 0.0030 0.0051 0.0021 0.0026 0.0035 3.1e05 0.0006 0.0068 0.0017 0.0013 0.0020 0.0008

0.3032 1.2414 0.4085 0.0167 0.4674 3.2129 1.5192 0.9682 0.3560 0.2108 0.8356 1.0535 0.0032 0.0212 0.8405 0.3289 0.2558 0.3321 0.0725

0.4485 0.8623 0.1574 0.0120 0.3758 4.2394 1.2758 0.7687 0.0280 0.0760 0.6786 1.2914 0.0014 0.0136 0.4106 0.2286 0.1783 0.1982 0.0158

0.5279 0.6662 0.4451 0.0095 0.3293 4.7897 1.1507 0.6686 0.1406 0.0042 0.5986 1.4107 0.0004 0.0325 0.1909 0.1775 0.1359 0.1314 0.0128

0.6066 0.4807 0.7266 0.0072 0.2846 5.3363 1.0225 0.5667 0.3074 0.0664 0.5172 1.5285 0.0006 0.0516 0.0383 0.1260 0.0952 0.0664 0.0409

0.7612 0.1021 1.2684 0.0025 0.1947 6.3840 0.7689 0.3633 0.6310 0.2024 0.3663 1.7531 0.0026 0.0876 0.4631 0.0269 0.0165 0.0646 0.0946

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APPENDIX B

The Elements of B are Listed by Columns. These Results were Obtained Using the Summary Function of the R Package MCMC pack.

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Table B2.

OLS Results of Growth Regressions for Eqs. (17–19) (Dummies Excluded).

Parameters

Eq. (17)

Eq. (18)

Eq. (19)

Constant

0.197 (0.073) 0.03 (0.054) 0.057a (0.018) 0.231b (0.104) 0.513a (0.060) 0.587

0.159 (0.094) 0.005 (0.017) 0.001b (0.0005) 0.267b (0.124) 0.500a (0.064) 0.592

0.256 (0.068) 0.003 (0.015) 0.001b (0.0004) 0.202b (0.113) 0.534a (0.056) 0.577

85

85

85

(aþb)(dm) (aþb)m b a 2 R OBS

White’s heteroscedasticity-corrected standard errors in parentheses. a p-value of one-sided hypothesis test o0.01. b p-value of one-sided hypothesis test o0.05.

Table B3. Algeria Egypt Jordan Morocco Syria Tunisia Angola Benin Burkina Faso Burundi Cameroon Central African Republic Ethiopia Gabon Ghana Kenya Madagascar

Malawi Mali Mauritius Mozambique Niger Nigeria Senegal Sierra Leone South Africa Tanzania Uganda Zambia Zimbabwe Argentina Bolivia Brazil Chile

Sample of 85 Countries. Colombia Costa Rica Dominican Republic Ecuador El Salvador Guatemala Guyana Haiti Honduras Jamaica Mexico Nicaragua Panama Paraguay Peru Uruguay China

Fiji Indonesia Korea South Malaysia Philippines Thailand Bangladesh India Nepal Australia Austria Belgium Canada Cyprus Denmark Finland France

Germany Greece Ireland Italy Japan The Netherlands New Zealand Portugal Singapore Spain Sweden Switzerland United Kingdom United States Hungary Romania Turkey

BAYESIAN NEAR-BOUNDARY ANALYSIS IN BASIC MACROECONOMIC TIME-SERIES MODELS$ Michiel de Pooter, Francesco Ravazzolo, Rene Segers and Herman K. van Dijk ABSTRACT Several lessons learnt from a Bayesian analysis of basic macroeconomic time-series models are presented for the situation where some model parameters have substantial posterior probability near the boundary of the parameter region. This feature refers to near-instability within dynamic models, to forecasting with near-random walk models and to clustering of $

The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reﬂecting the views of the Board of Governors of the Federal Reserve System or of any other employee of the Federal Reserve System nor the views of Norges Bank (the Central Bank of Norway). This paper contains a number of references to appendices. These appendices contain sampling schemes, conditional and marginal density results, probability density functions, and an overview of models analyzed in the paper. They are available, together with the computer code and data which were used in the empirical applications, on the following website: http:// people.few.eur.nl/hkvandijk/research.htm

Bayesian Econometrics Advances in Econometrics, Volume 23, 331–402 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23011-2

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several economic series in a small number of groups within a data panel. Two canonical models are used: a linear regression model with autocorrelation and a simple variance components model. Several wellknown time-series models like unit root and error correction models and further state space and panel data models are shown to be simple generalizations of these two canonical models for the purpose of posterior inference. A Bayesian model averaging procedure is presented in order to deal with models with substantial probability both near and at the boundary of the parameter region. Analytical, graphical, and empirical results using U.S. macroeconomic data, in particular on GDP growth, are presented.

1. INTRODUCTION Stable economic growth with possibly temporary periodic deviations – better known as business cycles – is one of the most important economic issues for any country. A widely used macroeconomic time series to measure these characteristics of growth and cycles is real gross domestic product (GDP). A commonly used model in this context is the linear autoregressive model with deterministic trend terms. Using such time series and model classes, econometric analysis – Bayesian and non-Bayesian – leads for most industrialized nations to substantial evidence that economic growth evolves according to a trend process that is largely determined by stochastic shocks. Otherwise stated, in the autoregressive models one ﬁnds substantial empirical evidence of a characteristic root that is near the boundary of unity or at this boundary in the parameter region. There exists an enormous literature on this topic; here we only mention the well-known study of Nelson and Plosser (1982) and for empirical evidence on industrialized nations over the past century we refer to van Dijk (2004). There are several other examples in macroeconomics of the existence of substantial posterior probability near the boundary of a parameter region. In business cycle analysis, one may be interested to know how much of the variation in an economic time series is due to the cycle and how much is due to the trend. This issue is relevant in the context of an adequate policy mix for stimulating long-term economic growth and short-term business cycle control. Using a structural time-series model, one may ﬁnd substantial posterior probability of the cyclical component near zero and the relative weights of the trend and cyclical components are then very uncertain. Another example occurs with typical characteristics of ﬁnancial time series

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in stock markets. These series behave close to a random walk model or, otherwise stated, close to a model with a characteristic root close to the boundary of unity. The economic issue of such a process is whether ﬁnancial markets are efﬁcient in the sense that the optimal forecast for future stock market prices is the current price. A fourth example is the club or cluster behavior in panel data models for economic growth in industrialized nations. In this context, convergence of economic growth is studied and the number of clusters may be relatively small. Substantial probability may occur at the boundary of the parameter region of the number of clubs and, as a consequence, large uncertainty exists with respect to the correct number of clubs. To explore the issue of Bayesian near-boundary analysis in basic economic time-series models, one derives their likelihoods and speciﬁes prior information. Our approach with respect to specifying prior information is to start with uniform priors on a large region. The use of such noninformative priors means that we concentrate on the information content in and the shape of the likelihood function. Given our diffuse priors, the posterior distributions of parameters of interest may or may not exist. The latter occurs, in particular, at the boundary of the parameter region, due to nonidentiﬁability of some parameters of interest. We discuss how Information matrix priors or training sample priors may regularize or smooth the shape of irregular posterior distributions. In our analysis, we make use of an interplay of analytical techniques, simulation methods, and graphics. As a simulation technique, we use the Gibbs sampling method. A brief introduction is given in Section 2. We note that graphs in the context of Bayesian analysis are becoming more and more important (see Murrell, 2005). In our analysis, we therefore also place emphasis on presenting results in a graphical way. In this paper, we make use of two classes of canonical models. The ﬁrst class of models is known as the class of single equation dynamic regression models. A ﬁrst contribution of this paper is to show that, for our purpose of a Bayesian near-boundary analysis in the parameter region, basic members of this class of models like unit root models, distributed lag models and error correction models (ECMs) are special cases of the well-known linear regression model with ﬁrst-order autocorrelation in the disturbances. We also indicate that an ECM with near unit root behavior is – for posterior analysis – equivalent to an instrumental variable (IV) regression model with possibly weak instruments (see also Hoogerheide & Van Dijk, 2001 and Hoogerheide & van Dijk, 2008). Interpretation of the models, their posteriors, and the effect of smoothness priors like the Information matrix

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and training sample priors is one aim. A second aim is to illustrate a Bayesian analysis of economic growth in U.S. GDP using different model speciﬁcations. The second class of models deals with variance parameters as parameters of interest. We discuss how the simple regression model with heteroscedasticity can be used as an introduction to the class of hierarchical linear mixed models (HLMM). As a second contribution, we show that the latter model serves as a parent class for time-varying parameter (TVP) models such as state-space models (SSMs) and panel data models. We investigate what happens when the density of one of the variance parameters is located near the zero bound and what happens when the number of components/ groups in a panel is very small. We show that the latter case also leads to a boundary issue. We note that a combination of the ﬁrst and second class of basic models has recently become important in empirical analysis. A third contribution of this paper is to show how Bayesian model averaging over models with substantial posterior probability near and at the boundary leads to better forecasting. That is to say, we do not consider the case where substantial posterior probability is near and/or at the boundary as an econometric inferential problem where model selection is appropriate in order to determine or test whether the economic process is stationary or nonstationary but as a case where model averaging is to be preferred. This may lead to improved forecasting. The results of our analysis may be summarized as ‘‘lessons learnt from models used’’ and the start of a road map for learning Bayesian nearboundary analysis. A summary is presented in Section 7. Some key lessons are: investigate the shape of the likelihood of the parameters of interest; investigate the inﬂuence of smoothness priors in case of substantial nearboundary posterior probability; learn which simulation technique may be used efﬁciently in which situation; apply Bayesian model averaging over models with posterior probability near a boundary and models with substantial probability at the boundary. The topic of this paper should be of interest to Bayesians who make use of basic regression models for economic time series when the focus is on the information content of the likelihood. The topic should interest non-Bayesians who are very knowledgeable about basic econometric models and want to learn how the information in the likelihood function of such models is summarized according to Bayes’ rule. There exists an excellent literature of Bayesian analysis of regression models with autocorrelation, unit root models, distributed lag models, unobserved component models and panel data models. An incomplete list of

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references is given as Chib (1991); Chib and Greenberg (1994, 1995) and Chib and Carlin (1999); Schotman and van Dijk (1991a, 1991b) and Harvey (1989). The purpose of the present paper is to extend this analysis to situations where substantial posterior probability is near or at the boundary of the parameter region. We emphasize that this paper does not result and is not intended to result in a simple message with respect to using one model, one particular prior and one simulation techniques. We do not believe in such simplistic claims but rather in a situation where different priors, different models and simulation algorithms are suitable depending on the problem studied, the data, and the shape of the posterior of the model considered. Our purpose is to investigate a set of models and, next, to explore Bayesian model averaging. The content of this paper is structured as follows. In Section 2, we brieﬂy review some (artiﬁcial) examples of shapes of posterior densities that the researcher may encounter in econometric practice. We also give an introduction to Gibbs sampling. This method is very natural given our derivations of joint, conditional, and marginal posteriors for the linear regression model and this model with possibly autocorrelated disturbances. In Section 3, we present some basic results for the linear regression model that will be used in subsequent sections. In Sections 4 and 5, we present our empirical analysis and present some theoretical results of near-boundary posterior probability for a number of models for economic time series. Section 6 deals with forecasting U.S. GDP using Bayesian model averaging. Section 7 contains a summary of models used and lessons learnt. The Appendices contain some technical details. We note that the technical level of the paper is like that of an introductory graduate econometrics course. Matrix notation is used in order to indicate the common, linear (sub)structure of several models.

2. PRELIMINARIES I: BASICS OF GIBBS SAMPLING AND TYPICAL SHAPES OF POSTERIOR DENSITIES 2.1. Basics of Gibbs Sampling As discussed in, for instance, van Dijk (1999) and Hamilton (2006), the ‘‘simulation revolution in Bayesian econometric inference’’ is to a large extent due to the advent of computers with ever-increasing computational power. This allows researchers to apply alternative Bayesian simulation

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techniques for estimation in which extensive use is made of pseudorandom number generators. One of the most important and widely used simulation methods is Gibbs sampling, developed by Geman and Geman (1984), Tanner and Wong (1987), and Gelfand and Smith (1990). This method has become a popular tool in econometrics for analyzing a wide variety of problems; see for instance Chib and Greenberg (1996) and Geweke (1999). Judging from numerous recent articles in the literature, Gibbs sampling is still gaining more and more momentum. Recent textbooks such as Bauwens, Lubrano, and Richard (1999); Koop (2003); Lancaster (2004); and Geweke (2005) discuss how Gibbs sampling is used in a wide range of econometric models, in particular in models with latent variables. Mixture processes are an important class of latent variable models in econometrics; the most wellknown due to Hamilton (1989). In recent papers by, for instance, Celeux, Hurn, and Robert (2000); Fru¨hwirth-Schnatter (2001); Jasra, Holmes, and Stephens (2005); and Geweke (2007), the issue of convergence of the Gibbs sampler in this class of models is discussed. The posterior distribution in mixture processes may be multimodal and may exhibit ridges often due to near nonidentiﬁcation of parameters. A detailed analysis of this topic is beyond the scope of the present paper but we note and will make use of the distinction by Geweke (2007) between the interpretation of a model and its posterior densities on the one hand and the numerical efﬁciency and convergence of a simulation algorithm on the other hand. One may characterize the Gibbs sampling algorithm as an application of the divide-and-conquer principle.1 First, a K-dimensional parameter vector h is divided into m components y1, y2, . . . , ym, where mrK. Second, for many posterior distributions that are intractable in terms of simulation the lower-dimensional conditional distributions turn out to be remarkably simple and tractable. The Gibbs sampler exploits this feature, as it samples precisely from these conditional distributions. Its usefulness is, for example, demonstrated by Chib and Greenberg (1996) and Smith and Roberts (1993). Since Gibbs sampling is based on the characterization of the joint posterior distribution by means of the complete set of conditional distributions, it follows that a requirement for application of the Gibbs sampler is that the conditional distributions, described by the densities pðyi jhi Þ for i ¼ 1; . . . ; m

(1)

where hi denotes the parameter component vector h without the ith component, can all be sampled from. The Gibbs sampling algorithm starts ð0Þ with the speciﬁcation of an initial set of values: ðyð0Þ 1 ; . . . ; ym Þ and then

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generates a sequence ð2Þ ðJÞ ð1Þ ð2Þ ðJÞ ðyð1Þ 1 ; . . . ; ym Þ; ðy1 ; . . . ; ym Þ; . . . ; ðy1 ; . . . ; ym Þ

(2)

following a process such that yði jÞ is obtained from pðyi jhðij1Þ Þ. Thus, yði jÞ is obtained conditional on the most recent values of the other components. We may summarize the Gibbs sampling algorithm as follows: ð0Þ 1. Specify starting values hð0Þ ¼ ðyð0Þ 1 ; . . . ; ym Þ and set j ¼ 1. 2. Generate (the jth Gibbs step):

yð1jÞ from pðy1 jyð2j1Þ ; . . . ; yðmj1Þ Þ yð2jÞ from pðy2 jyð1jÞ ; yð3j1Þ ; . . . ; yðmj1Þ Þ yð3jÞ from pðy3 jyð1jÞ ; yð2jÞ ; yð4j1Þ ; . . . ; yðmj1Þ Þ .. . jÞ yðmjÞ from pðym jyð1jÞ ; . . . ; yðm1 Þ 3. If joJ, set j ¼ j þ 1, and go back to Step 2. The above algorithm yields a sequence of J realizations hð jÞ ¼ for j ¼ 1, 2, . . . , J, from a Markov chain, converging to the target distribution. We will refer to Step 2 of the algorithm as ‘‘the Gibbs step’’ and for each of the models that we discuss in the subsequent sections we will always indicate what the Gibbs step looks like. Note that the components of h do not necessarily need to be one-dimensional. Generating draws for blocks of parameters where some of the yi components denote a block of parameters is also possible. The Gibbs algorithm is illustrated in Fig. 1 where we show an example path of Gibbs sampled points, when the conditional densities of y1|y2 and y2|y1 are both standard normal and assuming a correlation coefﬁcient of 0.75. The sample path is shown at different stages of the algorithm. The key feature of Gibbs sampling is: ðyð1jÞ ; . . . ; yðmjÞ Þ,

For large enough J the sequence of Gibbs draws, generated from the conditional distributions, is distributed according to the joint and marginal posterior distributions.2 A simple argument for the general case is as follows. Suppose yi and hi have a joint posterior distribution with density p(yi, hi). Thus this posterior should exist, which must be carefully veriﬁed (compare also Sections 3 and 4). Then hi has the marginal posterior distribution with density p(hi).

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(a) 1 substep

(b) 1 step

(c) 2 steps

(d) 10 steps

(e) 100 steps

(f) 1000 steps

Fig. 1. Gibbs Sampling: Example Steps. Notes: Panels (a) through (f ) show consecutive steps of the Gibbs sampler using two conditional posterior densities, p(y1|y2) and p(y2|y1) which are both standard normal with a correlation coefﬁcient of 0.75 together with the 99% highest probability density region. The open circles in ð0Þ panels (a)–(f ) indicate the starting vector ðyð0Þ 1 ; y2 Þ.

Denote by hi and h( j–1), the drawing generated in the ( j1)th step from the marginal posterior with density p(hi). In the jth step of the Gibbs sampling algorithm, yði jÞ is drawn from pðyi jhðij1Þ Þ, which is the density of the conditional distribution of yi given as hðij1Þ The joint density of yði jÞ and hðij1Þ is pðyði jÞ jhðij1Þ Þpðhðij1Þ Þ ¼ pðyði jÞ ; hðij1Þ Þ

(3)

Therefore, ðyði jÞ ; hðij1Þ Þ is distributed according to the joint posterior distribution. So the posterior is an invariant distribution of the Gibbs Markov chain, which is the invariant limiting distribution under the standard assumption of ergodicity. For a more detailed analysis on theoretical properties of the Gibbs sampler, we refer to Smith and Roberts (1993), Tierney (1994), and Geweke (1999).

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Because in practice, it may take some time for the Markov chain to converge, it is common to discard the ﬁrst B draws, where typically B{J. These draws are referred to as the burn-in draws. Consequently, posterior results will be based only on draws h(Bþ1), . . . , h(J) of the generated chain. Furthermore, the sequence of draws sometimes displays some degree of autocorrelation. When autocorrelations are signiﬁcant up to the (h1)th lag, one can consider using only every hth draw and to discard the intermediate draws (h is known as the thinning value).3 An altogether different approach is to generate multiple Markov chains instead of just one chain and to use only the ﬁnal draw from each sequence. Doing so implies that the Gibbs algorithm has to be executed a substantial number of times. When opting for this approach the researcher does not have to worry about which value to choose for h. Although the drawback of this method is that it can be very computationally intensive, it can alternatively help prevent posterior results from being (partially) determined by a particular set of starting values. We show in the next section that randomizing over h(0) can be a worthwhile endeavor when the likelihood displays signs of multimodality. 2.2. Three Typical Shapes of Posterior Densities To illustrate the kinds of shapes that may occur in posterior densities, we work through a number of examples which are based on the model in Gelman and Meng (1991). Suppose that we have a joint posterior density of (y1, y2), which has the following form 1 2 2 2 2 pðy1 ; y2 Þ / exp ay1 y2 þ y1 þ y2 2by1 y2 2c1 y1 2c2 y2 (4) 2 where a, b, c1, and c2 are constants under the restrictions that aZ0 and if a ¼ 0 then |b|o1.4 This Gelman and Meng (1991) class of bivariate distributions has the feature that the random variables y1 and y2 are conditionally normally distributed. In fact, the conditional densities p(y1|y2) and p(y2|y1) can be derived (picked) directly from the right-hand side of Eq. (4) and can be recognized as normal densities: by2 þ c1 1 (5) ; pðy1 jy2 ; a; b; c1 ; c2 Þ N ay22 þ 1 ay22 þ 1 pðy2 jy1 ; a; b; c1 ; c2 Þ N

by1 þ c2 1 ; ay21 þ 1 ay21 þ 1

(6)

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Note that, typically, the joint density of (y1, y2) is not normal. By choosing different parameter conﬁgurations for a, b, c1, and c2, we can construct joint posterior densities with rather different shapes, while the conditional densities remain normal. In the remainder of this section, we consider three types of shapes and we apply the Gibbs sampler to each of these. Although the shapes are all in a way artiﬁcial since they are not based directly on a model and data, doing so will give us some early insights into different shapes of (joint) posterior densities and boundary issues which we discuss in detail in the remainder of this paper. For each of the examples below the jth Gibbs step consists of sequentially drawing from Eqs. (5) and (6): jth Gibbs step for the Gelman–Meng model: Generate yð1jÞ jyð2j1Þ from pðy1 jy2 ; a; b; c1 ; c2 Þ N

byð2j1Þ þ c1

; aðyð2j1Þ Þ2 þ 1 aðyð2j1Þ Þ2 þ 1

Generate yð1jÞ jyð2jÞ from pðy2 jy1 ; a; b; c1 ; c2 Þ N

!

1

byð1jÞ þ c2

1

!

; aðyð1jÞ Þ2 þ 1 aðyð2jÞ Þ2 þ 1

2.2.1. Bell-Shape The ﬁrst parameter conﬁguration that we consider for the joint density in Eq. (4) is the following: [a ¼ b ¼ c1 ¼ c2 ¼ 0] in which case the joint density is given by 1 (7) pðy1 ; y2 Þ / exp y21 þ y22 2 Both the conditional densities and the joint density are standard normal. The latter is depicted in Fig. 2(a). Gibbs sampling simply comes down to obtaining draws by iteratively drawing from standard normal densities.5 A scatter plot of one thousand of such draws is shown in Fig. 2(b). The estimated conditional means and variances are equal to 0 and 1 for both parameters. These are exactly the parameters of the marginal densities which, in this case, we know to be standard normal. In fact, for the chosen parameter conﬁguration, the conditional and marginal densities coincide since the conditional density for y1 does not depend on y2 and vice versa.

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(a) Joint posterior density

(b) Gibbs draws

Fig. 2. Gelman–Meng: Bell-Shape. Notes: Panel (a) shows the Gelman–Meng joint posterior density for y1 and y2 given in Eq. (4) for parameter values [a ¼ b ¼ c1 ¼ c2 ¼ 0] whereas panel (b) shows the scatter plot of 1,000 draws from the Gibbs sampler together with the 99% highest probability density region.

In this particular example, it would therefore obviously not be necessary to use Gibbs sampling. 2.2.2. Ridges The second parameter conﬁguration that we examine is [a ¼ c1 ¼ c2 ¼ 0].6 The joint density is now given by 1 pðy1 ; y2 Þ / exp y21 2by1 y2 þ y22 2 " #" ## " 1 b y1 1 ð8Þ / exp ½ y1 y2 2 b 1 y2 It is apparent from Fig. 3 that the shape of this density depends on the value of b. When b-1, a ridge along the line y1 ¼ y2 appears in the shape of the posterior. The scatter plots of Gibbs draws for this example in Fig. 3 reveal that Gibbs sampling tends to become less efﬁcient in such a case. Ridges may occur in econometric models where the Information matrix tends to become singular, that is when b-1; see the next section for examples. We emphasize that the posterior in Fig. 3(e) and (f) is deﬁned on a bounded region with bounds 75 and þ75. This posterior is constant along the diagonal and it is a continuous function deﬁned on a bounded region and thus a proper density.

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(a) Joint posterior density, b = 0.95

(b) Gibbs draws, b = 0.95

(c) Joint posterior density, b = 0.995

(d) Gibbs draws, b = 0.995

(e) Joint posterior density, b = 1.0

(f) Gibbsdraws, b = 1.0

Fig. 3. Gelman–Meng: Ridges. Notes: Panel (a) shows the Gelman–Meng joint posterior density for y1 and y2 given in Eq. (4) for parameter values [a ¼ c1 ¼ c2 ¼ 0 and b ¼ 0.95] whereas panel (b) shows the scatter plot of 1,000 draws from the Gibbs sampler together with the 99% highest probability density region. Panels (c) and (d) show the same ﬁgures for b ¼ 0.995 and (e) and (f ) for b ¼ 1.0.

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2.2.3. Bimodality The third and ﬁnal conﬁguration we consider is [a ¼ 1, b ¼ 0] and large, but not necessarily equal, values for c1 and c2.7 Here, we select c1 ¼ c2 ¼ 10 which gives 1 pðy1 ; y2 Þ / exp y21 y22 þ y21 þ y22 20y1 20y2 2 " #" # " # 1 0 y1 10 1 2 2 þ y1 y2 ð9Þ / exp ½ y1 10 y2 10 2 0 1 y2 10 At ﬁrst sight the scatter plot of one thousand Gibbs draws, shown in Fig. 4(b), seems perfectly reasonable and posterior means and variances can easily be computed. However, when inspecting the joint density as depicted in Fig. 4(a) we immediately see that the joint density p(y1, y2) is bimodal and that the Gibbs sampler has sampled from one mode but not from the other. Apparently, it tends to get stuck in one of the two modes.8 This is because the modes are too far apart with an insufﬁcient amount of probability mass in between the two modes for the sampler to regularly jump from one to the other. Admittedly, substantially increasing the number of draws will eventually lead to a switch. However, one cannot be certain when this will occur. The scatter plot shows that with a single run, one thousand draws is clearly not enough. However, although not shown here, also one million

(a) Joint posterior density

(b) Gibbs draws

Fig. 4. Gelman–Meng: Bimodality. Notes: Panel (a) shows the Gelman–Meng joint posterior density for y1 and y2 given in Eq. (4) with parameter values [a ¼ 1, b ¼ 0 and c1 ¼ c2 ¼ 10] whereas panel (b) shows the scatter plot of 1,000 draws from the Gibbs sampler together with the 99% highest probability density region.

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draws is still an insufﬁcient number to witness a switch. Therefore, the Gibbs output only provides the researcher with information about a subset of the full domain of p(y1, y2) and posterior results are thus incomplete. One option to try and at least signal the bimodality of the likelihood is to execute the Gibbs sampler several times with widely dispersed initial values. However, we do note that even when doing so the issue of determining how much probability mass is located in each of the modes remains nontrivial. Although the example we discuss here is a rather extreme case, it should be clear that multimodality can result in very slow converge for the Gibbs sampler. Multimodality may occur in reduced rank models when one is close to the boundary of the parameter region. Summarizing, the above examples of a bell-shaped, a ridge-shaped, and a bimodal-shaped density, indicate that it is essential to scrutinize a proposed model and the shape of its posterior distribution before moving on to drawing posterior inference on its parameters through a simulation method. Doing so may not always be straightforward, however, especially in large dimensional spaces.

3. PRELIMINARIES II: JOINT, CONDITIONAL, AND MARGINAL POSTERIOR AND PREDICTIVE DENSITIES FOR THE LINEAR REGRESSION MODEL 3.1. Linear Regression Model We start our model analysis by considering the basic linear regression model where the variation of a dependent variable yt is explained by a set of explanatory variables, as summarized in the (1 K) (row-) vector xt where K is the number of variables in xt (including a constant): yt ¼ x t b þ t ;

t ¼ 1; . . . ; T;

with

t i:i:d N ð0; s2e Þ

(10)

The goal is to draw inference on the (K 1) vector of regression parameters b ¼ ðb1 b2 . . . bK Þ0 9 and the scalar variance parameter s2e . In matrix notation, this model is given by y ¼ Xb þ e;

with

e N ð0; s2e IT Þ

(11)

where y denotes the vector of T time-series observations or crosssectional observations on the dependent variable, y ¼ ðy1 y2 . . . yT Þ0 .

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X ¼ ðx01 x02 . . . x0T Þ0 denotes the matrix of observations on the explanatory variables and IT is an identity matrix of dimension (T T). In the following we provide basic results for the joint, conditional, and marginal posterior densities of the linear regression model in Eq. (11) which are useful for simulation purposes. More details can be found in Zellner (1971), Koop (2003), and Geweke (2005). For an expert reader, we suggest to consider only the summary tables and diagrams in Appendix B. 3.1.1. Joint Density We start by specifying the likelihood for the linear regression model in Eq. (11) as: 1 (12) pð yjX; b; s2e Þ ¼ ð2ps2e ÞT=2 exp 2 ð y XbÞ0 ð y XbÞ 2se Combining the likelihood with a noninformative or Jeffreys’ prior10 pðb; s2e Þ / ðs2e Þ1 gives the joint posterior density pðb; s2e jDÞ

/

ðs2e ÞðTþ2Þ=2

1 0 exp 2 ð y XbÞ ð y XbÞ 2se

(13)

(14)

where we deﬁne D as the data information set, that is D ( y, X). A useful result to facilitate the derivation of the conditional and marginal posterior densities is to rewrite Eq. (14) by completing the squares on b as ^ 0 ð y XbÞ ^ þ ðb bÞ ^ 0 X0 Xðb bÞ ^ ð y XbÞ0 ð y XbÞ ¼ ð y XbÞ

(15)

with b^ ¼ ðX0 XÞ1 X0 y, the OLS estimator of b. One can now rewrite Eq. (14) as i 1 h 0 0 0 2 2 ðTþ2Þ=2 ^ ^ ^ ^ exp 2 ð y XbÞ ðy XbÞ þ ðb bÞ X Xðb bÞ pðb; se jDÞ / ðse Þ 2se (16) The density in Eq. (16) is known as the normal inverted-gamma density of ðb; s2e Þ, see Raiffa and Schlaifer (1961, p. 310) and Zellner (1971, Chapter 3). 3.1.2. Conditional Densities The only part of the posterior in Eq. (16) that is relevant for determining the posterior density of b conditional on a value for s2e is the part that depends ^ only depends on the data D and ^ 0 ð y XbÞ, on b. The ﬁrst part, ð y XbÞ

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does therefore not enter the conditional density of b. From the probability density functions given in Appendix C, we can recognize, for a given value of s2e , a multivariate normal density for b which has mean vector M ¼ b^ and variance matrix S ¼ s2e ½X0 X1 , see Eq. (C-4). Similarly, the conditional density of s2e , for a given parameter vector b, follows from Eq. (C-3) and is inverted gamma with location parameter m ¼ 1=2ð y XbÞ0 ð y XbÞ and n ¼ 1=2ðTÞ degrees of freedom. Summarizing, the conditional posterior densities are

^ s2 ½X0 X1 (17) p bjs2e ; D N b; e 1 1 (18) p s2e jb; D I G ð y XbÞ0 ð y XbÞ; T 2 2 Gibbs sampling for the basic linear regression model consists of iteratively drawing from the conditional densities pðbjs2e ; DÞ and pðs2e jb; DÞ. A scheme of derivations for Gibbs sampler results is presented in the top part of Fig. 5. The jth Gibbs step consists of jth Gibbs step for the linear regression model: j1Þ Generate bð jÞ js2ð from e

^ s2ð j1Þ ½X0 X1 p bjs2e ; D N b; e ð jÞ

Generate s2e jbð jÞ from

2 1 ð jÞ 0 ð jÞ 1 y Xb ; T y Xb p se jb; D I G 2 2

3.1.3. Marginal Densities Ultimately, we are interested in learning about the properties of the marginal densities of b and s2e . In this model, it is straightforward to derive these. Using the results of Appendix C, the marginal posterior densities are given as

^ s^2 ½X0 X1 ; T K (19) pðbjDÞ t b;

1 0

1 pðs2e jDÞ I G y Xb^ y Xb^ ; ðT KÞ (20) 2 2 ^ 0 ð y XbÞ=ðT ^ where s^2 ¼ ð y XbÞ KÞ. A scheme for the derivations for the joint and marginal posterior densities of the linear regression model is given in Fig. B-l. Since in this case one can directly simulate from the marginal

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X X

X

X

X

X

I

X Fig. 5. Sampling Scheme: Posterior and Predictive Results for Gibbs Sampling. Notes: The ﬁgure presents results for Gibbs sampling schemes to obtain posterior and predictive results.

densities without having to rely on the Gibbs sampler to obtain posterior results, we present direct sampling results in Fig. B-2. We emphasize that the derivation of conditional and marginal densities does not change if we were to replace xtb by ryt1 in Eq. (10) using a uniform prior. That is, within a noninformative Bayesian analysis one can go from a static analysis with xtb to a dynamic model: the posterior of regression parameters remains Student-t while in the frequentist world one

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cannot go from a static analysis to a simple dynamic analysis without a change in the properties of the OLS estimators. The same argument also holds for predictive densities which we discuss next. 3.1.4. Predictive Densities Suppose one is interested in constructing forecasts of future values of yt in the linear regression model. A vector of Q future values, y~ ¼ ðyTþ1 yTþ2 . . . yTþQ Þ0 , is then assumed to be generated by the following model: ~ þ e~ y~ ¼ Xb

(21)

~ is a Q K matrix of given values for the independent variables in where X the Q future periods and e~ is a Q 1 vector of future errors which are assumed to be i.i.d. normal with zero mean and variance–covariance matrix s2e IQ . The marginal predictive density for y~ can be derived by integrating the ~ DÞ with respect to b and s2 : joint density p ð~y; b; s2e jX; e Z Z ~ DÞ ¼ ~ D db ds2 pð y~ jX; p y~ ; b; s2e jX; (22) e s2e

b

The joint density is speciﬁed as follows: ~ D ¼ p y~ jb; s2 ; X ~ p b; s2 jD p y~ ; b; s2e jX; e e

(23)

~ is the where pðb; s2e jDÞ is the posterior density in Eq. (14) and pð y~ jb; s2e ; XÞ conditional predictive density of y~ in Eq. (21) which is given as: ~ / s2 1=2Q exp 1 ð y~ XbÞ ~ 0 ð y~ XbÞ ~ (24) p y~ jb; s2e ; X e 2s2e ~ and which is a kernel of a multivariate normal variable with mean Xb covariance matrix s2e IQ . Scheme 1 (of Appendix B) shows a Gibbs sampling scheme for predictive analysis. For each draw of ðb; s2e Þ one can draw y~ from Eq. (24). The draws that are obtained in this way are draws from the predictive density in Eq. (22). The joint density in Eq. (23) is a combination of Eqs. (14) and (24) and becomes: ~ DÞ / ðs2 Þ1=2ðTþQþ2Þ pð y~ ; b; s2e jX; e 1 0 0 ~ ~ exp 2 ðð y~ XbÞ ð y~ XbÞ þ ð y XbÞ ð y XbÞÞ 2se ð25Þ

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The ﬁrst step to analytically obtain the marginal predictive density follows from integrating with respect to s2 which results in: ~ DÞ / ½ð y~ XbÞ ~ 0 ð y~ XbÞ ~ þ ð y XbÞ0 ð y XbÞ1=2ðTþQÞ pð y~ ; bjX; The second step is to complete the squares on b and to integrate with respect to the K elements of b which gives: ^ 0 Hð~y X ^ 1=2ðTþQKÞ ~ DÞ / ½ðT KÞ þ ð y~ X ~ bÞ ~ bÞ pð y~ jX;

(26)

~ 0 XÞ1 X ~ 0 Þ=^s2 . Eq. (26) indicates that y~ has a multivariate where H ¼ ðI þ XðX ^ scale matrix H1, and (TK) degrees ~ b, Student-t distribution with mean X of freedom. By means of Eq. (26), one can draw directly from the predictive density. Schemes 2 and 3, both listed in Appendix B, summarize the derivations of distributions that are needed in a direct sampling and Gibbs sampling scheme. We emphasize again that in a Bayesian noninformative framework all these derivations carry over directly to a dynamic model with lagged endogenous variables.

4. SINGLE EQUATION DYNAMIC REGRESSION MODELS For this class of models, the near-boundary issue refers to near-instability of dynamic models. This is an important boundary issue in the sense that it has substantial implications for efﬁcient forecasting. The purpose of this section is threefold. We start with derivations of posteriors of parameters of interest for different dynamic model speciﬁcations (and the construction of corresponding Gibbs samplers). We show here the uniformity of the derivations for different model structures. We also discuss interpretation of the deterministic terms in autoregressive models with a focus on the issue of near-boundary analysis. The key feature in this context is: under what conditions do the dynamic economic processes under consideration return to a deterministic mean or trend and/or when does there exist a random walk or stochastic trend? Is there a substantial probability mass in the stationary region and/or on the boundary of a random walk or stochastic trend model? These are boundary issues that have important implications for forecasting. Finally, we present empirical illustrations using some major U.S. macroeconomic and ﬁnancial series.

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4.1. Posterior Analysis and Gibbs Samplers 4.1.1. Linear Regression with Autocorrelation We are now ready to analyze the extension of the model in Eq. (10) by allowing the error terms to have ﬁrst-order autocorrelation.11 That is: y t ¼ xt b þ n t

t ¼ 1; . . . ; T

nt ¼ rnt1 þ t ; with t i:i:d N ð0; s2e Þ

(27) (28)

where r is the parameter that determines the strength of the autocorrelation. For expository purposes with respect to the derivation of the conditional and marginal posterior densities we distinguish between two cases: one where the domain of r is not restricted and one where it is. We emphasize that for economic purposes the domain of this parameter is in most cases restricted to the interval 1rrr1. We note that later we will distinguish between the cases where r is 1 and where r is in the bounded interval of (0,1). The domain for the remaining parameters is given by NoboN and 0os2e oN. When r ¼ 0, the autocorrelation model coincides with the basic linear regression model since nt reduces to a white noise series. As we will see later, difﬁculties occur when there is a constant term and r has substantial posterior probability mass at the edges of its domain. By substituting Eq. (28) in Eq. (27) and rewriting the resulting expression in matrix notation, we obtain y ry1 ¼ Xb X1 br þ e;

with

e i:i:d N ð0; s2e IT Þ

(29)

where y1 and X1 denote the one-period lagged values of y and X. This reformulation shows that the autocorrelation model is nonlinear in its parameters b and r. This problem of inference on a product (or ratio) of parameters is a classic issue. A detailed analysis is, however, beyond the scope of the present paper. For an early example see Press (1969) and we refer to Fieller (1954) and van Dijk (2003) for more references. Although this issue of nonlinearity hampers parameter estimation and inference when using frequentist estimation approaches, obtaining posterior results using Gibbs sampling is straightforward as we will show below, in the case where r is unrestricted and no such deterministic terms as a constant or trend occur in the equation. We ﬁrst turn to deriving the joint, conditional, and marginal densities. It will become apparent that the autocorrelation model serves as a template for several other well-known econometric models.

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4.1.1.1. Joint, Posterior, and Marginal Densities. The combination of the likelihood of the autocorrelation model with the noninformative prior in Eq. (13) and a uniform prior on r on a large region, and, further, assuming that the initial observations are ﬁxed nonrandom quantities, gives the following joint posterior density, pðb; r; s2e jDÞ / ðs2e Þ1=2ðTþ2Þ 1 exp 2 ð y ry1 Xb þ X1 brÞ0 2se ð y ry1 Xb þ X1 brÞ

ð30Þ

where D once again represents the known data ( y, X). In case the domain of the parameter r is bounded, we make use of an indicator function I(b, r) which is 1 on the domain speciﬁed (which is usually (NoboN), (1rrr1) and 0 elsewhere). Thus, we obtain a truncated posterior density deﬁned on the region indicated. We now derive the expression for the conditional densities pðbjr; s2e ; DÞ, pðrjb; s2e ; DÞ, and pðs2e jb; r; DÞ and the marginal densities pðbjDÞ and pðrjDÞ. For analytical convenience, we start with the derivations for the case where r is not restricted. To facilitate the derivation of the conditional densities, it is useful to rewrite the model in Eq. (29) in two different ways. In each case, we condition on one of the two types of regression coefﬁcients. First, we rewrite Eq. (29) conditional on values of r, ( n y ¼ yn ðrÞ y ry1 (31) yn ¼ Xn b þ e where Xn ¼ Xn ðrÞ X rX 1 Second, we rewrite Eq. (29) conditional on values of b which then becomes ( y~ ¼ y~ ðbÞ y Xb (32) y~ ¼ r~y1 þ e where y~ 1 ¼ y~ 1 ðbÞ y1 X1 b To derive the conditional density for b we use Eq. (31) to rewrite the joint posterior density. Doing so gives us the joint density of the basic linear regression model again so we can reuse all our earlier derivations. It, therefore, follows immediately that the conditional density for b is n multivariate normal with mean m ¼ b^ ðXn 0 Xn Þ1 Xn 0 yn and variance matrix S ¼ Sb s2e ðXn 0 Xn Þ1 . Similarly, using Eq. (32) we obtain the result that the conditional density for the unrestricted parameter r is normal with

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mean m ¼ r^ ð y~ 01 y~ 1 Þ1 y~ 01 y~ and variance s2 ¼ s2r s2e ð y~ 01 y~ 1 Þ1 . The conditional density for s2e is again inverted gamma with parameter m ¼ 1=2e0 e 1=2ð y ry1 Xb þ X1 brÞ0 ð y ry1 Xb þ X1 brÞ and n ¼ 1=2ðTÞ degrees of freedom. Summarizing, we have ^ Sb Þ pðbjr; s2e ; DÞ N ðb; ^ s2r Þ pðrjb; s2e ; DÞ N ðr; 1 1 pðs2e jb; r; DÞ I G e0 e; T 2 2 Whereas in the basic regression model Gibbs sampling was unnecessary because the marginal densities could be derived analytically, here we do not have analytical results and therefore we need Gibbs sampling. This is due to the fact that the marginal densities of b, r, and s2e are not a member of any known class of densities. We show this as follows. After integrating out s2e from the joint density we get pðb; rjDÞ / ½ð y ry1 Xb þ X1 brÞ0 ð y ry1 Xb þ X1 brÞT=2 ð33Þ We can rewrite this joint density in two different ways T=2 ^ 0 y~ 01 y~ 1 ðr rÞ ^ pðb; rjDÞ / y~ 0 My~ 1 y~ þ ðr rÞ T=2 pðb; rjDÞ / y0 n MXn yn þ ðb bn Þ0 X0 n Xn ðb bn Þ

(34) (35)

where My~ 1 and MXn are idempotent residual maker matrices of y~ 1 and X, respectively.12 Integrating out r from Eq. (34) and b from Eq. (35) gives the marginal densities ðT1Þ=2 pðbjDÞ / ð y XbÞ0 My1 X1 b ð y XbÞ 1=2 ð y1 X1 bÞ0 ð y1 X1 bÞ ð36Þ ðTKÞ=2 pðrjDÞ / ð y ry1 Þ0 MXrX1 ð y ry1 Þ 1=2 ðX rX1 Þ0 ðX rX1 Þ

ð37Þ

In case the parameter r is restricted to the interval [1, 1] we proceed as follows. Eqs. (30), (34), and (35) are now changed with inclusion of the indicator function I(b, r). The right-hand side of Eq. (36) now contains ^ r Þ where F ^ r Þ Fðð1 rÞ=s the function c(b) given as cðbÞ ¼ Fðð1 rÞ=s

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353

stands for the standard normal distribution function. The conditional normal density of r given b, s2e , and D is in this case a truncated normal density and the right-hand side of Eq. (37) is now changed with the inclusion of the indicator function I(r) which is deﬁned as 1 on the interval [1, 1] and 0 elsewhere. Both densities in Eqs. (36) and (37) – and their truncated variants – do not belong to a known class of density functions which means that we need Gibbs sampling to obtain posterior results. Despite the fact that the marginal densities of b and r cannot be determined analytically, applying the Gibbs sampler is, however, a straightforward exercise, conditional upon the fact that all variables in the data matrix X have some nontrivial data variability. 4.1.1.2. Fisher Information Matrix. The Fisher Information matrix can provide information as to whether problems are likely to occur when r approaches the edges of its domain, in the sense that the joint posterior density becomes improper. The Fisher Information matrix is deﬁned as minus the expectation of the matrix of second-order derivatives of the log likelihood with respect to the parameter vector h ¼ ðb; r; s2e ), that is I ¼ E½ðd2 ln LðhjDÞÞ=dhdh0 . For the linear model with autocorrelation the Information matrix is given by13 2 2 3 d ln L d2 ln L d2 ln L 6 7 drdb0 drds2e 7 6 dr2 6 7 6 2 7 6 d ln L d2 ln L d2 ln L 7 7 I ¼ E 6 6 dbdr dbdb0 dbds2e 7 6 7 6 7 6 d2 ln L d2 ln L d2 ln L 7 4 5 s2e dr ds2e db0 ds4e 2 3 T 0 0 7 6 1 r2 6 7 6 7 0 6 7 ðX rX1 Þ ðX rX1 Þ 6 0 7 0 ¼6 ð38Þ 7 s2e 6 7 6 7 6 T 7 4 0 5 0 2s4e The inverse of the Information matrix shows that even when |r| ¼ 1 none of the variances ‘‘explode.’’ In the next sections, we will see that this not always

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needs to be the case. More general, under the condition that all variables in X have some variability, there are no issues in terms of impropriety of the joint posterior density when r reaches the edge of its domain. 4.1.1.3. Gibbs Sampling for the Unrestricted Case of r. jth Gibbs step for the linear regression model with autocorrelation: ð j1Þ ð j1Þ Generate bð jÞ jrð j1Þ ; s2e from pðbjr; s2e ; DÞ N ðb^n ; Sðbj1Þ Þ ð j1Þ ð jÞ j1Þ Generate rð jÞ jbð jÞ ; s2e from pðrjb; s2e ; DÞ N ðr^ ; s2ð Þ r 0 ð jÞ ð jÞ 2 ð jÞ 2 ð jÞ Generate se jb ; r from pðse b; r; DÞ I Gð1=2ðe eð jÞ Þ; 1=2ðTÞÞ

We can see from the conditional densities given earlier that the Gibbs sampler has no difﬁculties with the nonlinearities in the likelihood. This is due to the fact that conditionally on one regression parameter, the model for the other regression parameter is the basic linear regression model as shown in Eqs. (31) and (32). In fact, the joint posterior density of r and any element of b, or the other way around, resembles the density shown in Fig. 2(a). Therefore, the Gibbs sampler is a very convenient tool for drawing inference on the parameters in these types of models. We distinguish between a Gibbs step when there is no truncation for r (then all draws are accepted) and the case of a truncated domain for r. In the latter case, a simple solution for the Gibbs step is to ignore drawings outside the bounded region (1oro1). A more efﬁcient algorithm has been developed by Geweke (1991, 1996). 4.1.2. Distributed Lag Models: Koyck Model A further extension of the basic linear regression model is the univariate distributed lag model.14 This model has proven to be one of the workhorses of econometric modeling practice since it offers the econometrician a straightforward tool to investigate the dependence of a variable on its own history or on the history of exogenous explanatory variables. Here we focus in particular on the well-known Koyck model, which is popular in for example marketing econometrics to investigate the dynamic link between sales and advertising. The general distributed lag model has, in principle, an inﬁnite number of parameters. Koyck (1954) proposed a model speciﬁcation in which the lag parameters are a geometric series, governed by a single unknown parameter. The resulting model is known as the geometric distributed lag model or simply as the Koyck model. Below, we discuss the

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boundary issue that can occur in this model which results in a parameter (near) nonidentiﬁcation issue. The Koyck model is given by yt ¼ bwt þ vt wt ¼ ð1 rÞ

t ¼ 1; . . . ; T 1 X

ri xti

(39) (40)

i¼0

nt ¼ rnt1 þ t ; with t i:i:d: N ð0; s2e Þ

(41)

where we allow for ﬁrst-order autocorrelation in the error term. It is assumed that 0rrr1, NoboN, and 0os2e oN. Note that the effect of lagged values of the (here single) explanatory variable xt is determined solely by r and that this parameter is assumed to be equal to the ﬁrst-order autocorrelation parameter. In marketing econometrics, the parameter r is usually referred to as the ‘‘retention’’ parameter. We assume that nt is serially correlated. One may also assume that nt is i.i.d. Then the transformed model has an MA(1) error. Another closely related model that also gives a boundary problem is the so-called partial adjustment model.15 This model is given as ynt ¼ bwt þ nt

(42)

yt yt1 ¼ ð1 rÞðynt yt1 Þ þ t

(43)

where yt is observed but ynt unobserved. For the Koyck model, substituting Eq. (41) in Eq. (39) gives a similar type of expression as we found for the linear model with autocorrelation. In particular, in matrix notation one obtains y ry1 ¼ bðw rw1 Þ þ e Eq. (40) puts additional structure on the term wrw1. More speciﬁcally, it holds that wrw1 ¼ (1r)x which gives y ¼ ry1 þ bð1 rÞx þ e; with e N ð0; s2e IT Þ

(44)

The result in Eq. (44) shows that the Koyck model is nested in the autocorrelation model and that therefore all earlier derivations hold here as well. The main difference, however, is that contrary to the autocorrelation model, the speciﬁc structure that is placed on the exogenous variable will result in a boundary issue when r is near 1. We can understand why this is

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the case by realizing that b will be near nonidentiﬁcation for values of r close to 1. This means that y effectively becomes a random walk and that exogenous variables no longer have any inﬂuence on y. When r ¼ 1, then b is not identiﬁed and the model reduces to a random walk. We will analyze the joint, conditional, and marginal densities to give insights in the consequences of the nonidentiﬁcation of b when applying the Gibbs sampler. 4.1.2.1. Joint, Posterior, and Marginal Densities. Derivations for the joint and conditional densities are very similar to before. Therefore, we only report the joint and conditional densities for the case of the bounded domain of r. The joint density, after integrating out s2e is speciﬁed as T=2 Iðb; rÞ pðb; rjDÞ / ðy ry1 bð1 rÞxÞ0 ðy ry1 bð1 rÞxÞ where I(b,r) is an indicator function which is 1 on the region bounded by 0 r r r 1, NoboN, and 0 elsewhere. The conditional densities – given that r is an element of the interval (0, 1) – are given as pðbjr; s2e ; DÞ N ðbn ; s2b Þ ^ s2r Þ pðrjb; s2e ; DÞ T N ðr; 1 1 pðs2e jb; r; DÞ I G e0 e; T 2 2 where T N indicates a truncated normal density. The parameters in the conditional densities are speciﬁed as 1 bn ¼ ðxn 0 xn Þ1 xn 0 yn ¼ ð1 rÞ2 x0 x x0 ð y ry1 Þ

(45)

1 s2b ¼ s2e ðxn 0 xn Þ1 ¼ s2e ð1 rÞ2 x0 x

(46)

and 1 r^ ¼ ð y~0 1 y~ 1 Þ1 y~0 1 y~ ¼ ðy1 bxÞ0 ðy1 bxÞ ðy1 bxÞ0 ð y bxÞ (47) 1 s2r ¼ s2e ð y~ 01 y~ 1 Þ1 ¼ s2e ð y1 bxÞ0 ð y1 bxÞ

(48)

and eue ¼ (yry1b(1r)x)u( yry1b(1r)x). The density for r is truncated to the unit interval which is indicated by the density notation T N . At ﬁrst sight, it may seem straightforward to apply the Gibbs sampler to the Koyck model. However, upon closer inspection of the conditional

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density parameters it becomes clear that a problem can occur for values of r close to 1. Suppose that a value near 1 is drawn for r. The conditional variance of b given this draw will be close to inﬁnity, see Eq. (46), which means that any large value is likely to be drawn for b. If the next draw for b is indeed large then the conditional variance of r goes to zero, see Eq. (48). As a result the next draw for r is again going to be close to 1, see Eq. (47). This means that the Gibbs Markov chain will converge very slowly. Convergence is not achieved for the case r ¼ 1 since this is an absorbing state of the Markov chain. The extent of this problem depends on how much probability mass there actually exists close to r ¼ 1 and at r ¼ 1. When r ¼ 1, it follows directly from the joint posterior p(b, r|D) that p(b|r ¼ 1, D) is constant. Thus when r ¼ 1 the conditional density of b is uniform on the interval (NoboN) and as a consequence it is improper. The conditional density of r is just the value of the truncated normal in the point r ¼ 1. The economic issue is that we cannot learn (draw inference) on the parameter when r ¼ 1, which is basically very natural for a random walk model. To understand the behavior of the Gibbs sampler further we examine the marginal densities in detail. Given 0oro1, the marginal densities for b and r are as follows: ðT1Þ=2 1=2 ðy1 bxÞ0 ðy1 bxÞ cðbÞ pðbjDÞ / ðy bxÞ0 My1 bx ðy bxÞ (49) ðT1Þ=2 0 1=2 ½x x ð1 rÞ1 pðrjDÞ / ðy ry1 Þ0 Mð1rÞx ðy ry1 Þ

(50)

where c(b) is similar as in the previous section but now such that r is deﬁned on the bounded interval (0, 1). Focusing on the density for r, we can recognize it to be a Student-t type density, but with an additional factor (1r)1. It is exactly this factor that is causing the behavior of the Gibbs sampler. The reason is that the joint density p(b, r|D) is improper at r ¼ 1 for NoboN. Graphically, this means that the joint density has a ‘‘wall,’’ similar to the ridge that is depicted in Fig. 3(e). The marginal density for r will tend to inﬁnity when r tends to 1. To reiterate what we said before, the extent of the problem – given the speciﬁcation of the model – depends on the data at hand. If the likelihood assigns virtually no probability mass to the region close to r ¼ 1 then the marginal for b will be virtually indistinguishable from a Student-t density.

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Furthermore, the marginal density for r will still tend to inﬁnitely close to r ¼ 1, but if this event happens to be far out in the tail of the distribution then this should not pose a serious problem. We shall show an example of this data feature in the empirical analysis relating to U.S. inﬂation and growth of real GDP. If, on the other hand, substantial probability mass is near r ¼ 1 then measures should be taken to prevent the Gibbs sampler from reaching that part of the domain of r or, alternatively, to try and regularize the likelihood. Choosing an appropriate prior density can do the trick. 4.1.2.2. Fisher Information Matrix. Analyzing the Information matrix gives similar insights in the irregularity in the joint density close to and equal to r ¼ 1 and furthermore, it provides us with a direction for a possible solution to tackle this irregularity. The Information matrix follows directly from Eq. (38) by substituting in XrX1 ¼ (1r)x. Therefore 3 2 T 0 0 7 6 1 r2 7 6 7 6 2 0 7 6 ð1 rÞ x x 7 6 0 0 (51) I ¼6 2 7 se 7 6 7 6 T 5 4 0 0 2s4e The Information matrix again shows that when r is close to 1, the variance of r is zero (the inverse of the ﬁrst diagonal element) whereas the variance of b goes to inﬁnity (the inverse of the second diagonal element). When r ¼ 1, then the determinant of the Information matrix is zero. 4.1.2.3. Gibbs Sampling when 0oro1. The Gibbs jth step is given by jth Gibbs step for the distributed lag model: ð j1Þ

Generate bð jÞ jrð j1Þ ; s2e from pðbjy; x; r; s2e Þ N ðbðj1Þ ; sb2ðj1Þ Þ ð j1Þ ð jÞ Generate rð jÞ jb ; s2e from pðrjy; x; b; s2e Þ T N ðr^ ð jÞ ; sr2ðj1Þ Þ 0 ð jÞ ð jÞ Generate s2e jb ; rð jÞ from pðs2e y; x; b; rÞ I Gð1=2ðeðjÞ eð jÞ Þ; 1=2ðTÞÞ When r is 1 it follows that Gibbs sampling is inappropriate. 4.1.2.4. Potential Solutions: Truncation of Parameter Region, Information Matrix Prior, or Training Sample Prior. In order to apply the Gibbs sampler without serious convergence problems something should be done

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about the irregularity in the joint density close to r ¼ 1. A number of potential solutions have been proposed in the literature to circumvent this problem (see Schotman & van Dijk, 1991a; Kleibergen & van Dijk, 1994, 1998). Here we only brieﬂy touch upon the several options in order to just give a ﬂavor of how to tackle the impropriety of the likelihood. One can distinguish three solution approaches: (i) truncation of the parameter space, (ii) regularization by choosing a prior that sufﬁciently smoothes out the posterior, and (iii) use of a training sample to specify a weakly informative prior for b. In terms of applying the ﬁrst solution, one can truncate the domain of r and check whether there is probability mass near 1. Imposing an upper bound can be achieved by selecting for example a local uniform prior. The goal would be to only allow draws for r that are at least Z away from 1 with ZW0 to prevent a wall in the joint posterior density. Choosing a speciﬁc value for Z would necessarily be a subjective choice. However, once a value for Z is agreed upon one can apply the Gibbs sampler. Alternatively, one can use a Metropolis–Hastings type step in which only draws that fall below 1Z are accepted. For an example of this method e.g. see Geman and Reynolds (1992). As for the second solution, one can try and regularize the likelihood in the neighborhood of r ¼ 1 such that it becomes a proper density. This can be achieved by using a prior that is chosen in such the way that it eliminates the factor (1r)1. From the Information matrix in Eq. (51) we can construct the following Jeffreys’ type prior for b given r and s2e ,16 pðbjp; s2e Þ /

ð1 rÞ2 s2e

for 0oro1

(52)

Deriving the joint and marginal densities with this prior will show that it eliminates the factor (1r)1 from the marginal density of r. What happens is that the marginal density for r is now integrable everywhere except for r ¼ 1 which in turn has a zero probability of occurring. The third solution is an alternative way of regularizing the posterior density. One can use a training sample17 to specify a weakly informative prior for b. Schotman and van Dijk (1991a) specify the following prior s2e 2 for 0oro1 (53) pðbjr; se Þ / N y0 ; ð1 rÞ2 where y0 is the initial value of the time series for y. The intuition behind this prior is that as r approaches 1 it becomes increasingly difﬁcult to learn

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about b from the data since the unconditional mean of y, given as (1r)b, does not exist for r at 1. The prior is stronger for smaller values of r but approaches an uninformative prior for r-1. It is derived from the unconditional distribution of y0 under the assumption of normality. The effect of this normal prior on the joint posterior density is that it eliminates the pronounced wall feature in the joint density. We will see an example of this approach when we discuss the unit root model. We conclude that for solutions (ii) and (iii) one has to – in most cases – replace the simple Gibbs procedures by other Monte Carlo integration methods. This is a topic outside the scope of the present paper. Further solutions, which we do not discuss here in the detail, are to reparameterize the model in such a way that the Gibbs sampler can be used without any problems for the reformulated model. However, one still has to translate the posterior results back to the original model. Without imposing some sort of prior, similar problems will still occur only now at a different stage in the analysis. For examples of reparametrization see for instance Gilks, Richardson, and Spiegelhalter (2000). Finally, modiﬁed versions of the Gibbs sampler such as the collapsed Gibbs sampler (see Liu, 1994), where some parameters can be temporarily ignored when running the Gibbs sampler (in this case r) can be useful in this context as well. In the empirical application in Section 5 we only use the truncation prior approach. 4.1.3. Autoregressive Models and Error Correction Models with Deterministic Components We present the issue of near-boundary analysis in the context of an autoregressive model with deterministic components. The simplest example is a ﬁrst-order autoregressive model with an additive constant, given as yt ¼ c þ ryt1 þ t

(54)

This model can be respeciﬁed as an ECM around a constant mean using a restriction on c. We start with rewriting Eq. (54) as yt ¼ mð1 rÞ þ ryt1 þ t

(55)

where c is now restricted as c ¼ m(1–r). We can rewrite the latter equation as a mean reversion model, see Schotman and van Dijk (1991a, 1991b), Dyt ¼ ðr 1Þðyt1 mÞ þ t

(56)

Here one can see the expected ‘‘return to the long-term unconditional mean (m) of the series’’ when 0oro1. That is, when yt1 is greater than m and

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0oro1 then the conditional expected change in yt, given previous observations, is negative while in the opposite case the expected change in positive. Furthermore, when r tends to 1 then in the ECM speciﬁcation (55) one has a smooth transition from stationarity to a random walk model. In other words one approaches the boundary in a continuous way. On the contrary in Eq. (54) one has a transition from stationarity to a random walk with drift: one hits the boundary with a ‘‘jump.’’ The models are much farther apart then the ones in the ECM setup. Note that the constant term c in Eq. (54) does not have a direct interpretation in terms of being the mean of the process while in Eq. (55) the constant m is the long-term unconditional mean of the series given 0oro1. Similar as for the Koyck model, imposing this particular ECM structure introduces a boundary issue when there is substantial posterior probability near r ¼ 1. In the ECM model for yt, the interpretation of m depends on whether the series y is stationary (ro1) or whether it has a unit root (i.e., r ¼ 1). In the latter case, the mean of y does not exist and m is thus nonidentiﬁed. Therefore, even when y is a weakly stationary process, any value for m along the real line is likely to be drawn in the Gibbs sampler when r is sampled close to 1. This will not only make it very difﬁcult to pinpoint the posterior mean of m but it also causes the sequence of draws for r to have difﬁculties moving away from r ¼ 1. Of course, r close to 1 can be an indication that one should model ﬁrst differences of y instead of y itself which will circumvent the entire issue altogether. However, for series such as interest rate levels there is no clear economic interpretation why these should be I(1) processes and one is left with dealing with the boundary issue nonetheless. For series that are near unit root, substantial probability mass will lie close to r ¼ 1 and at r ¼ 1 so that the impropriety of the joint posterior poses a serious issue. As an example we depict the joint density for the unit root model for a series of monthly data on the 10-year U.S. Treasury Bond yield in Fig. 6(a). A time-series plot of this series is given in Fig. 7(b). Fig. 6(a) clearly shows the pronounced wall feature close to and at r ¼ 1. In order to resolve the impropriety of the joint density a local uniform prior or truncation of the domain for r could be used. Another possibility would be to use a regularizing prior like the Schotman and van Dijk (1991a) prior. The joint density that results from combining the data likelihood with this particular prior is shown in Fig. 6(b). The joint density no longer has a wall feature close to r ¼ 1 although it still ﬂattens out somewhat near the edge of the domain. We note that this posterior may also be interpreted as the exact likelihood including the initial observation. For details see Schotman and van Dijk (1991a).

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(a) Uniform prior

(b) Schotman and Van Dijk (1991a) prior

Fig. 6. Joint Posterior Density in the Unit Root Model. Notes: Panel (a) shows the joint posterior density p(r, m|y) when we use a uniform prior as in Eq. (13) whereas panel (b) shows the same posterior density however now with the prior proposed by Schotman and van Dijk (1991a) as given in Eq. (53). In both panels, we use the endof-month 10-year U.S. Treasury Bond constant maturity yield for the period January 1960–July 2007 as the data vector y.

We emphasize that the autoregressive model with an additive constant, Eq. (54), can be treated like the linear regression model of Section 3. Direct sampling or a simple Gibbs procedure is possible. The model with the ECM interpretation can be written as in the autoregressive form of Sections 4.1.1 and 4.1.2 and deriving the corresponding Gibbs sampling formulas is left to the interested reader. We also refer to that subsection for the convergence issues of the Gibbs sampler. Next, we treat the autoregressive model with additive linear trend. We start with a distributed lag model of order two, yt ¼ c þ bt þ r1 yt1 þ r2 yt2 þ t ; with t i:i:d: N ð0; s2e Þ

(57)

where t captures a linear increasing trend. We can rewrite this model as an ECM as follows. Consider ð1 r1 L r2 L2 Þðyt m dtÞ ¼ t ; with t i:i:d: N ð0; s2e Þ

(58)

using c ¼ m(1r1r2)þd(r1þ2r2) and b ¼ (1r1r2)d and where L is the lag operator; Lyt ¼ yt1. Applying this operator to Eq. (58) we obtain yt ¼ ð1 r1 r2 Þm þ dðt r1 ðt 1Þ r2 ðt 2ÞÞ þ r1 yt1 þ r2 yt2 þ t (59)

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(a) inflation

(b)10-year Treasury Bond yield

(c) Dividend Yield

Fig. 7. Macroeconomic and Financial Series. Notes: Shown in this ﬁgure are in panel (a) quarterly changes in log U.S. price levels (CPI), panel (b) end-of-month levels of the U.S. 10-year Treasury Bond constant maturity yield, and panel (c) monthly dividend yield on the Standard & Poor’s 500 Index.

This equation can be rewritten further as Dyt ¼ d þ ðr1 þ r2 1Þðyt1 m dðt 1ÞÞ r2 ðDyt1 dÞ þ t

(60)

which shows that yt is mean-reverting towards a linear trend when r1þr2o1, otherwise we have a random walk with drift (if r1þr2 ¼ 1). This is our ECM model with linear trend. Similar as in Section 4.1.1, the derivation of the conditional densities can be simpliﬁed if we rewrite Eq. (60) by conditioning on one of the two types of regression coefﬁcients. Like for the linear model with autocorrelation, the idea is that given r ¼ [r1, r2]u one has a linear model in b ¼ [m, d]u whereas given b one has a linear model in r.

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First, we rewrite (58) conditional on values for r: ynt ¼ xnt b þ t ( n yt ¼ ynt ðrÞ yt r1 yt1 r2 yt2 where xnt ¼ xnt ðrÞ ½1 r1 r2 ; t r1 ðt 1Þ r2 ðt 2Þ

(61)

y~t ¼ r~yt þ t ( y~t ¼ y~t ðbÞ y m dt where y~ t ¼ y~ t ðbÞ ½yt1 m dðt 1Þ; yt2 m dðt 2Þ

(62)

Posterior densities and predictive densities can now be derived in the same fashion as for the linear and autoregressive models from Sections 3 and 4.1.2. We note that we use the restriction (r1þr2)o1 in the Gibbs sampling scheme, since in the unrestricted case the posterior is improper. We can summarize this section by stating that what we did was to make a distinction between additive deterministic terms in autoregressive models and ‘‘interpretable’’ deterministic terms in ECMs. The interpretation of mean or trend reversion is important in economics. In addition, the resulting forecasts from these models can be quite different. Given our interest in near-boundary analysis, the smooth transition to the boundary (from stationarity to unit roots) is relevant for model comparison, model averaging and forecasting. This is a well-known topic in the literature, see Sims and Uhlig (1991) and Schotman and van Dijk (1991a, 1991b, 1993) among others. In this paper we do, however, not focus on computing posterior model probabilities for model selection for the choice or test for stationarity and unit root cases. Our interest in these models is primarily from a forecasting perspective.

4.2. Illustrative Empirical Analysis Using Macroeconomic Series 4.2.1. Possible Unit Root Models in Inﬂation, Interest Rates, and Dividend Yield Before we apply the models discussed in this section to analyze our main macroeconomic series of interest, U.S. GDP growth, we apply the autoregressive model of the previous paragraph to three time series which are of important economic relevance: U.S. inﬂation, the 10-year U.S. Treasury Bond yield and the Standard & Poor’s 500 Index Dividend Yield.

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Fig. 7 shows time-series plots of each of the series. We note that this section is by no means meant to be a full attempt at modeling these series empirically, it is purely for illustrative purposes. We aim at analyzing the posterior mean, a unit root and, next, how to deal with the latter in a model averaging procedure for forecasting purposes. The ﬁrst series we analyze is inﬂation. We collected quarterly U.S. CPI ﬁgures from the Federal Reserve of Philadelphia database. We then construct inﬂation, pt, as the quarterly differences in log price levels, pt ¼ ln(CPIt)ln(CPIt1). The data sample runs from 1984:Q1 to 2006:Q3, for a total of 91 observations. The model we estimate is speciﬁed as in Eq. (55): pt m ¼ rðpt1 mÞ þ t ; with t i:i:d: N ð0; s2e Þ

(63)

This speciﬁcation allows us to analyze ﬁrst-order autocorrelation in inﬂation growth. Fig. 8 and Table 1 show posterior results based on 10,000 draws from the Gibbs sampler.18 The ﬁrst column of Table 1 shows that ﬁrst-order autocorrelation seems to be an important feature of inﬂation as it cannot be rejected at any reasonable level of credibility. We note again that the posterior is improper but that the Gibbs sampler does not detect this (i.e., it does not reach the absorbing state in our ﬁnite set of random drawings) since we are so far away from the boundary. A simple truncation of the region for r seems a practical solution in this case. Fig. 8(b) conﬁrms that the value of r ¼ 1 will only occur with extremely low probability. The second series we consider is the U.S. 10-year Treasury Bond constant maturity yield. Data for this series were collected from the FRED database and the sample spans the period January 1960 to July 2007 for a total of 571 monthly observations. The model we apply to analyze unit root behavior in the 10-year yield yt is again the ECM model in Eq. (55). Learning about the value of r is crucial for several reasons. For example, r determines whether long-run forecasts will converge to a long-term mean (ro1) or whether they will display explosive behavior (rW1). We analyze three speciﬁcations of the ECM model. In the ﬁrst application, we use a noninformative prior on r and we refer to this approach as TBU. We note that with this case of a ﬂat prior on a large region it can be shown, using the results of Sections 4.1.1 and 4.1.2, that the posterior of (m,r) is improper. Yet a naive application of the Gibbs sampler will produce results. We emphasize that this is not a limitation/fault of the Gibbs sampler. It is simply the consequence of a poor search of the shape of the posterior and it is simply a wrong application of

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Gibbs. However, this may occur and has occurred in practice. We use TBR to indicate the second approach for which we estimate the model under stationarity by truncating the parameter space to ro1. Finally, in the third approach we consider model Eq. (54) and impose a unit root (r ¼ 1) and analyze a random walk with drift model, TBRW. Fig. 9 and Table 1 show that the improper posterior density of r, which has an asymptote at 1, has substantial probability mass close to 1 and even some mass beyond 1. These results suggest that the 10-year Treasury yield may be nonstationary which is difﬁcult to interpret economically, however. The second application therefore uses a truncation prior to restrict r to be lower than 1. Panel (b) in Fig. 9 shows the posterior density of r. Posterior densities for the other model parameters look similar in both cases. To assess the restrictiveness or importance of restricting r to be smaller than 1 we perform a small forecasting exercise.19 In particular, we make 3, 6,

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Table 1. Posterior Results for Inﬂation, Treasury Bond, and Dividend Yield Series. Parameters

CPI

TBU

TBR

TBRW

DYU

DYR

DYRW

– – – 0.0005 – – 0.0002 – – – (0.0122) – – (0.0006) 0.7706 7N 6.9739 – 7N 0.2480 – (0.0675) – (1.1956) – – (0.0586) – 0.4487 0.9992 0.9895 1 0.9993 0.9893 1 (0.0994) (0.0025) (0.0014) – (0.0025) (0.0019) – 0.1083 0.0856 0.0854 0.0855 0.0177a 0.0176a 0.0177a (0.0166) (0.0051) (0.0051) (0.0050) (0.0012) (0.0011) (0.0011)

c m r s2e Predictions RMSPE – 3 months RMSPE – 6 months RMSPE – 12 months

0.3031 0.3184 0.3441

– – –

0.6922 0.9569 1.4099

0.7015 0.9844 1.4914

– – –

0.0213 0.0307 0.0454

0.0211 0.0303 0.0443

Notes: The table presents posterior means, posterior standard deviations (in between parentheses), and the root mean square prediction error (RMSPE) for models on U.S. quarterly inﬂation growth (CPI), U.S. 10-year Treasury Bond constant maturity yield (TB) and Standard & Poor’s 500 Index dividend yield (DY). The subscripts for TB and DY distinguish between using an unconstrained parameter space (TBU, DYU), imposing stationarity with ro1 (TBR, DYR) and imposing a random walk with drift (TBRW, DYRW). a Indicates that the number has been multiplied by a factor 100. The RMSPE results are for 3, 6, and 12 months ahead predictions for the sample 1985:Q1–2007:Q2.

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Fig. 9. Posterior Density Histograms for 10-Year Treasury Bond. Notes: Shown in this ﬁgure are posterior density histograms for the model in Eq. (55) applied to the 10-year U.S. Treasury Bond yield. Panel (a) shows the density of r with an uninformative prior whereas panel (b) shows the density of r using a truncation prior to ensure that ro1. The densities are based on 10,000 draws from the Gibbs sampler.

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and 12 months ahead forecasts (with the forecast taken to be the mean of the predictive density) starting in January 1985. We construct forecasts using an expanding window approach. In particular, we use the sample from January 1960 to December 1984 to construct the forecast for January 1985. We then expand the estimation sample to include the realized January 1985 yield value in order to construct the forecast for February 1985 and we keep expanding the sample in the same way until the ﬁnal forecast, July 2007, which is based on the estimation sample January 1960 to June 2007. The bottom rows of Table 1 report results. We assess forecasting accuracy by means of the root mean squared prediction error (RMSPE). The results seem to indicate that imposing the truncation is relevant since the RMSPE is reduced for all horizons. The third and ﬁnal series that we examine is the dividend yield on the Standard and Poor’s 500 Index which is a commonly used predictor variable for forecasting stock returns (see Keim & Stambaugh, 1986; Campbell & Shiller, 1988; Fama & French, 1988). We use monthly data from January 1966 to December 2006 (492 observations). Fig. 7(c) shows that the dividend yield has a quite different pattern in the initial 20 years compared to the 1990s and beyond which can induce unit root type behavior. Whether or not the dividend yield displays unit root-type behavior is a widely discussed topic in the literature (see Cochrane, 2008). As in the previous example we analyze the model with an uninformative prior (indicated by DYU), a truncation prior (DYR), and imposing a random walk with drift structure (DYRW). Results are similar to those for the 10-year yield and are shown in Fig. 10. 8000

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Fig. 10. Posterior Density Histograms for Dividend Yield. Notes: Shown in this ﬁgure are posterior density histograms for the model in Eq. (55) applied to the Standard & Poor’s 500 dividend yield. Panel (a) shows the density of r with an uninformative prior whereas panel (b) shows the density of r using a truncation prior to ensure that ro1. The densities are based on 10,000 draws from the Gibbs sampler.

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A substantial part of the improper posterior of r (with asymptote at 1) lies again beyond 1. However, restricting r to be lower than 1 now actually slightly worsens forecast accuracy for all horizons. The random walk with drift speciﬁcation gives the most accurate forecasts albeit that the differences are very small. The results in Table 1 indicate that the data does not provide a clear answer as to whether the dividend yield series is stationary or nonstationary. As mentioned earlier the results of this section are for illustrative purposes only. A more detailed analysis for the Treasury Bond and dividend yield series should involve Bayesian diagnostic checking on the (mis-)appropriateness of using Gibbs sampling results from improper posteriors and using a misspeciﬁed model like the autoregressive model with a possible unit root in a forecasting exercise. Further, Fig. 7(b) and (c) suggest a time-varying trend and this data feature should be modeled. However, a detailed Bayesian misspeciﬁcation and/or empirical analysis for these series is beyond the scope of the present paper and is therefore left to the interested reader. 4.2.2. U.S. Real GDP Growth We develop our empirical analysis mainly on one key macroeconomic series: U.S. GDP growth. We apply the models from the previous paragraphs (as well as those in Section 5.1.2) to model this series to illustrate boundary issues and to show how to approach these. We collected real GDP (seasonally adjusted) ﬁgures from the U.S. Department of Commerce, Bureau of Economic Analysis. Fig. 11(a) plots the log quarterly (a)

(b)

Fig. 11. U.S. GDP. Notes: Shown in this ﬁgure are in panel (a) quarterly log levels of real U.S. GDP whereas panel (b) shows the quarterly GDP growth rate (in % terms). The sample is 1960:Q1–2007:Q2.

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GDP level for our sample 1960:Q1 to 2007:Q2 (190 observations) and shows that GDP has followed an upward sloping pattern but with ﬂuctuations around this trend. The quarterly growth rate of log GDP, gt ¼ ln GDPtln GDPt1, shown in Fig. 11(b), underlines these ﬂuctuations with periods of positive changes followed by periods of negative changes, clearly indicating business cycles. The sample average growth rate is positive but with a high level of variation which is mainly due to different cycles (for more details we refer to Harvey, Trimbur, & van Dijk, 2007). We apply the various linear models we discussed earlier to model and assess empirical facts on stochastic shocks over the full sample so as to assess these models’ suitability in a out-of-sample forecasting exercise. In the forecast exercise we use an initial in-sample period from 1960:Q1 to 1992:Q2 to obtain initial parameter estimates and we forecast the GDP growth ﬁgure for 1992:Q3. We then expand the estimation sample with the value in 1992:Q3, reestimating the parameters, and we forecast the next value for 1992:Q4. We continue this procedure up to the last value and we end up with a total of 60 forecasts. Also here we remark that this section is not meant as a fullﬂedged empirical analysis of U.S. GDP. We mainly want to analyze and compare the various linear models for illustrative purposes. It is also for this reason that we ignore the ‘‘great moderation’’ that occurred in ﬂuctuations of real GDP since 1985 in our analysis, see Stock and Watson (2002). 4.2.2.1. Random Walk with Drift for GDP Growth. The ﬁrst model that we put forth to describe U.S. GDP growth is the univariate constant model in Eq. (10): gt ¼ c þ t ; with t i:i:d: N ð0; s2e Þ

(64)

We refer to this model by the label ‘‘random walk with drift’’ in tables and graphs. This model allows us to infer the average growth rate of GDP although with the drawback that other ﬂuctuations cannot be explained. Section 3 provides details on how to compute posterior densities and predictive densities using the Gibbs sampler. We note that in this case direct sampling is also a good strategy for efﬁcient computation. Fig. 12 shows the posterior densities pðcjs2e ; DÞ and pðs2e jc; DÞ and clearly reveals the conditional normal and conditional inverted gamma densities. Also shown is the joint posterior density. Table 2 and Fig. 13, which summarize posterior results, show that the growth rate is on average positive, well distributed around the mean value of 3.25% (in annual terms) with a small variance. However, the model explains relatively little as the residual variance is quite close to the unconditional

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Fig. 12. Joint and Conditional Posterior Densities. Notes: Panel (a) shows the joint posterior of c and s2e whereas panels (b) and (c) show the conditional posterior density of c for given values of s2e and the data D; pðcjs2e ; DÞ and the conditional density of s2e for given values of c and the data; pðcjs2e ; DÞ, respectively. We apply the linear regression model in Eq. (64) to quarterly U.S. GDP growth.

variance of GDP growth which equals 0.71%. In addition, the posterior standard deviation of s2e is quite substantial as well. Moving to Table 3 which contains the MSPE results of our forecast exercise, we do ﬁnd that the model has higher forecasting power than the benchmark random walk (RW) model. The RMSPE of the constant mean model is lower for all horizons compared to the random walk. 4.2.2.2. Simple Error Correction Model for GDP Growth. The second speciﬁcation that we propose is the ECM in Eq. (55). Allowing for

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Table 2. Parameters

m c d b r1 r2 s2e

Posterior Results for U.S. GDP. Growth

Levels

RW þ drift

Simple ECM

ECM

Trend

– – 3.2500a (0.2460)a – – – – – – – – 0.7214 (0.0749)

3.2516a (0.3308)a – – – – – – 0.2719 (0.0715) – – 0.6750 (0.0712)

16.8940 (3.8574) – – 0.0087 (0.0157) – – 1.1624 (0.0901) 0.2596 (0.0715) 0.6711 (0.0704)

– – 1.5417a (0.5512)a – – 0.0004 (0.0001) 1.2246 (0.0706) 0.2727 (0.0701) 0.6490b (0.0682)

Notes: The table presents posterior means and posterior standard deviations (in between parentheses) for models for log real U.S. GDP level/growth. The considered models are (i) the random walk with drift model (for GDP growth), (ii) the simple ECM model in Eq. (55) (for GDP growth), (iii) the ECM model in Eq. (60) (for GDP growth), and (iv) the trend model in Eq. (57) (for GDP level). Note that the interpretation of the parameter m is different in Eqs. (55) and (60). In the simple ECM, m represents the unconditional mean of the series whereas for the ECM in Eq. (60) it represents the constant of the linear trend line. The difference in interpretation also applies to c. For the random walk with drift model, it represents the drift whereas for the trend model it is the constant of the trend line. a Indicates that the number has been annualized (multiplied by a factor of 4). b Indicates a multiplication factor of 104.

autocorrelation in GDP growth may be beneﬁcial. The model is given by gt m ¼ rðgt1 mÞ þ t ; with t i:i:d: N ð0; s2e Þ

(65)

and we refer to it as ‘‘simple ECM.’’ Section 4 gives estimation details for this model. Posterior results are shown in the second column of Table 2. It can be seen for m that whereas its posterior mean is again positive and very close to the posterior mean of c in the previous model, the explanatory power of the model increases. The posterior mean of r is 0.27 and Fig. 14 shows that nearly all probability mass is to the right of 0. The latter observation provides evidence of the presence of autocorrelation in growth. Here we see again an example where the posterior is improper. However,

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Table 3. Forecast Accuracy Results for U.S. GDP Growth. RW

[1] RMSPE – 1Q 0.6198 RMSPE – 2Q 0.5462 RMSPE – 4Q 0.6118

RW þ Drift Simple Trend ECM ECM [2] 0.4738 0.5299 0.5257

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[2], [3] [4], [5], [6] [2]–[6]

0.4761 0.4713 0.4724 0.4726 0.4700 0.5178 0.5050 0.5220 0.5318 0.4742 0.5251 0.5054 0.5202 0.5223 0.4888

0.4802 0.4991 0.5131

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Notes: The table presents RMSPE results for U.S. GDP growth for various models. The models used are: [1], the random walk model; [2], the random walk with drift model; [3], the ECM model in Eq. (55); [4], the model with additive trend Eq. (57); [5], the error correction model in Eq. (60); [6], the time-varying parameter model Eqs. (91) and (92). Three Bayesian model averaging (BMA) schemes are: (i) averaging over the random walk with drift and simple ECM model, (ii) averaging over the trend, ECM, and time-varying parameter model, and (iii) averaging over all models. The out-of-sample period is 1992:Q3–2007:Q2 (60 forecasts) and an expanding window approach is used for each model. For the BMA models we use the initial 30 forecasts from 1985:Q1 to 1992:Q2 as training period.

the data keep the parameter r away from 1. We use as potential solution a truncated density where the truncation is a binding constraint in practice. Finally, the density of s2e has both a lower posterior mean and posterior variance which suggests that this model explains more variation in GDP growth than the random walk with drift model. Similarly, allowing for

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0.3

0.4

0.5

0.6

(a)

0

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

(b) 2

Fig. 14. GDP: Simple ECM for Growth. Notes: Shown in this ﬁgure are posterior density histograms for the model in Eq. (65). Panel (a) shows the density for r and (b) for s2e . The densities are based on 10,000 draws from the Gibbs sampler.

ﬁrst-order autocorrelation in growth improves forecast accuracy as shown in Table 3 although the improvement is only marginal (and only for longer horizons). A possible explanation is that both models have a nearly identical posterior mean for m. 4.2.2.3. Autoregressive Model with Additive Linear Trend for GDP Level. In addition to using models for GDP growth, we also analyze models that draw inference on the level of GDP. In particular we consider an autoregressive distributed lag model for GDP level. Within these types of models one attempts to explain the level of GDP by using the information in lagged GDP levels and a linearly increasing variable. The model we consider here is that of Eq. (57) and is given as: ln GDPt ¼ c þ bt þ r1 ln GDPt1 þ r2 ln GDPt2 þ t ; with t i:i:d: N ð0; s2e Þ

(66)

which we refer to as ‘‘trend.’’ The results in Section 3 can be used to draw inference on the parameters in this model. Table 2 shows that the constant (or long-term growth rate) is positive. However, its value is substantially lower than when we model the growth rate directly and cannot be directly interpreted as being the average growth rate. Part of the long-term growth rate is now being explained by the trend (long-term trend) which, as we may expect, has a positive posterior mean and a small standard deviation. The behavior of the lag parameters of past GDP levels is also interesting to examine. The ﬁrst lag has a distribution which lies almost entirely beyond 1, implying compounding GDP growth.

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Bayesian Near-Boundary Analysis 700

1400

600

1200

500

1000

400

800

300

600

200

400

100

200

0 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02

(a) 1 + 2 in Eq. (66)

1.04

0

0.82

0.84

0.86

0.88

0.90

0.92

0.94

0.96

(b) 1 + 2 in Eq. (60)

Fig. 15. GDP: Autoregressive Model with Linear Trend and ECM. Notes: Shown in this ﬁgure is the posterior density histograms of the persistence parameter r1þr2 for the trend model Eq. (66) in panel (a) and the error correction model (ECM) Eq. (60) in panel (b). The densities are based on 10,000 draws from the Gibbs sampler.

However, the parameter of the second lag is always negative, which captures the cycles in GDP. The residual variance is very small compared to the previous two models. Panel (a) of Fig. 15 shows the posterior density histogram of the persistence in log GDP, as measured by r1þr2. We note that we do not truncate the parameter space of r1 and/or r2 here. We emphasize that even with a ﬂat prior on a large region, the posteriors of the parameters in Eq. (63) are proper. Whereas this unrestricted model does well in terms of in-sample ﬁt, the fact that there is some probability mass for (r1þr2)W1 (as shown in Fig. 15) implies that the model can potentially display explosive behavior when forecasting future values. In fact, Table 3 shows that the forecast accuracy of this model is high, but not superior to that of the random walk with drift model. 4.2.2.4. Error Correction Model for GDP Level. As a ﬁnal model for GDP level we analyze the ECM given in Eq. (60). Panel (b) in Fig. 15 shows the persistence in GDP. Here we do truncate the parameter space such that (r1þr2)o0.95. Not doing so results in r1 and r2 often adding up to values close to 1 in the Gibbs sampling algorithm which makes inference on the remaining parameters m and d difﬁcult. The ECM with truncation provides accurate forecasts, even though these are marginally worse than the forecasts of the simple ECM and the trend models. Because the forecast performance of all models is very similar it is unclear whether one should decide to model the level of GDP or its growth rate

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directly to forecast GDP growth. In Section 6 we propose a simple procedure, that of model averaging, to alleviate this decision problem. As we show in Table 3, the model averaging of a random walk with drift and a simple ECM gives the most accurate forecasts. 4.3. Remark on Equivalence Between Boundary Problems in Dynamic Regression Models and Instrumental Variable Models The ﬁnal class of models that we discuss in the current section is the class of multivariate models. The issues involved here are similar to those surrounding univariate unit root models, that is nonidentiﬁability of parameters. This results in the Information matrix being singular, or alternatively, in the Hessian having a reduced rank. This reduced rank problem can occur in several well-known models, such as for example cointegration models, vector autoregressive (VAR) and simultaneous equation models (SEM) which in turn are closely linked to IV models. To show which role nonidentiﬁability plays in these models we give an example by means of a just identiﬁed IV model and in particular we focus on the incomplete simultaneous equation model (INSEM). Our analysis, which is necessarily brief, is based on van Dijk (2003) and Hoogerheide, Kaashoek, and van Dijk (2007) and we refer to these studies for a more in-depth analysis. Consider the INSEM model as it is speciﬁed in Zellner, Bauwens, and van Dijk (1988)20 y ¼ xb þ e

(67)

x ¼ zp þ m

(68) "

½e

0

0

m Nð½ 0 0 ; RÞ; with R ¼

s2e

se;m

se;m

s2m

# (69)

with y, x, and z all having dimensions (T 1) and b and p being scalar parameters. In this model, Eq. (67) is to be interpreted as the structural relation of interest, x is an endogenous variable and z is the (weakly exogenous) instrument. Similarly, b is the structural parameter of interest and |p| measures thep quality ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ of the instrument. Furthermore, the correlation parameter r ¼ se;m = s2e s2m measures the degree of endogeneity of x in the equation for y. Eqs. (67)–(69) comprise the structural form of the INSEM. By substituting Eq. (68) in Eq. (67) we can derive the reduced form which is

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given by y ¼ zpb þ n

(70)

x ¼ zp þ v

(71)

0

with n ¼ eþnb and ½e m as in Eq. (69). We can interpret the reduced form model as a multivariate regression model, which is nonlinear in the parameters b and r as in Eq. (44). As was the case in the unit root model this nonlinearity can lead to a nonidentiﬁability problem. In particular, when we assume a ﬂat prior on the parameters b and p, then when p ¼ 0 (this is usually labeled as the case of no identiﬁcation or case of irrelevant instruments) the joint posterior density is improper because it is ﬂat and nonzero in the direction of b. In fact, the joint density looks very similar to that in Fig. 6(a) in the sense that it has a wall at p ¼ 0. Therefore, b is not identiﬁed when p ¼ 0 whereas it is for any p 6¼ 0. In a multivariate setting where y, x, and z are all matrices and b and p are matrices as well, the identiﬁcation problem of (part of the elements) of b occurs when p ¼ 0 or when p is of reduced rank. The above problem is known as local nonidentiﬁcation and is discussed in detail in Kleibergen and van Dijk (1998). As a result of the local nonidentiﬁcation problem, the marginal density of p is nonintegrable because of inﬁnite probability mass near p ¼ 0 (see Kleibergen & van Dijk, 1998). Whether or not the impropriety of the joint density will be revealed in the output from the Gibbs sampler is unclear. Slow convergence of the Gibbs sampler due to the fact that p ¼ 0 is acting as an absorbing state could be an indication. Examples of bimodal posterior densities on bounded intervals are given in Hoogerheide et al. (2007). A possible solution to circumvent the local nonidentiﬁcation problem in INSEM models would again be the speciﬁcation of sensible prior densities. However, it can be an arduous task to ﬁnd conjugate priors, mainly since these will have to curtail multiple parameters all at the same time.

5. VARIANCE COMPONENTS AND STATE-SPACE MODELS We now switch our attention to drawing inference on variance parameters instead of regression parameters. In particular, we focus on situations when a variance component tends toward the zero bound or when a degrees of freedom restriction may be violated or an identiﬁcation problem arises.

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We do so by again analyzing a canonical type of model, the so-called HLMM. This model is a variance components model, that is, the relative importance of several variances is the object of study. A second feature of this canonical model is the presence of unobserved components. The starting-point of our analysis will be a basic speciﬁcation of the HLMM. This model serves as a parent model for extensions such as a SSM and a panel data model, which we discuss subsequently. 5.1. Heteroscedasticity and Hierarchical Linear Mixed Models Before we specify the basic setup of the HLMM we ﬁrst discuss two preliminary models, focusing on variances of disturbances. The models serve to identify the following two issues: (i) a degrees of freedom bound which refers to a sufﬁcient number of observations or a sufﬁcient number of cross-sectional groups and (ii) an identiﬁcation issue or labeling issue with respect to the different variance components. 5.1.1. Linear Regression Model with a Small Number of Observations or Naive Heteroscedasticity 5.1.1.1. T ¼ Small, Degrees of Freedom Bound. In Section 3, we analyzed the basic linear regression model. Now we revisit this model which we simplify using xt ¼ 1 and b ¼ m yt ¼ m þ t ;

t ¼ 1; . . . ; T; with t N ð0; s2e Þ

(72)

We emphasize that the number of observations T may refer to the number of observation in a time series and to individuals or groups of individuals in a cross-section. For notational convenience, we use the same symbol T here for time-series and cross-section observations. If we use a uniform prior on both m and s2e , pðm; s2e Þ / 1

(73)

then, from the results of Table B-1 in Appendix B we can derive the marginal densities of m and s2e . Note, however, that because the prior is now that of Eq. (73) instead of Eq. (13), we have two degrees of freedom less. The marginal densities are given by ðT 3ÞT ^ ;T 3 pðmjyÞ t m; s2 1 1 0 2 ^ ðy it mÞ; ^ ðT 3Þ pðse jyÞ I G ðy iT mÞ 2 2

Bayesian Near-Boundary Analysis

379

with y the full vector of observations; y ¼ ½y1 . . . yT 0 ; m^ ¼ 1=Tði0T yÞ, and s2 ¼ y0 MiT y. The degrees of freedom for the general linear regression model with the prior in Eq. (73) is TK2. For the model in Eq. (72), we have K ¼ 1. From the parameters of the marginal densities and the conditions given in Appendix C, it is clear that in order for these Student-t and inverted gamma densities to exist one needs more than three observations, that is TW3. Analogous results can be derived for the existence of higher order moments. For illustration, Fig. 16 shows that the right tail of an inverted gamma density tends to zero at a rate that is too small when the number of degrees of freedom is too small. For instance, for T ¼ 4 the zeroth moment exists but the higher moments do not.21 The conditional densities, using a uniform prior, are given by s2 ^ e p mjy; s2e N m; T 2 1 1 p se jy; m IG ðy iT mÞ0 ðy it mÞ; ðT 2Þ 2 2 Only focusing on these conditional densities shows that T ¼ 3 is a already sufﬁcient for the Gibbs sampler to function properly. However, it follows from our analysis that the marginal densities for m and s2e do not exist. Thus, we have a simple case where the Gibbs sampler can be applied as a simulation method, whereas the joint and marginal densities do not exist

Fig. 16. Inverted Gamma Density. Notes: The graph shows the inverted gamma density function, as speciﬁed in Eq. (C-3), for m ¼ 10 and for a varying number of degrees of freedom, n.

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(similar as the ridge example in Section 2.2), see also the discussion in Koop (2003). Therefore, the generated Gibbs sequences cannot be interpreted. We emphasize that for the usual number of time-series observations this degrees of freedom restriction is obviously of no signiﬁcance. However, for the case of the number of groups in a panel it may become restrictive. In Section 5.5, we give an example using a panel data model. 5.1.1.2. Naive Heteroscedasticity and a Degrees of Freedom Bound. Consider model (72) in which each observation yt, for t ¼ 1, . . . , T, is allowed to have its own variance parameter s2t . When we use a uniform prior on m as well as on each of the s2t components then the posterior of ðm; s21 ; . . . ; s2T Þ is given as: T Y 1 2 2 2 2 1=2 (74) ðst Þ exp 2 ðyt mÞ pðm; s1 ; . . . ; sT jyÞ / 2st t¼1 The posterior is unbounded since for some t, one may have yt ¼ m and/or s2t ¼ 0, then pðm; s21 ; . . . ; s2T jyÞ !N. One solution would be to partition the observations into groups, assuming that per group the variance is constant whereas it is allowed to be different across groups. Each partition needs to be chosen in such a way that it contains a sufﬁcient number of observations. Our main point, although trivial as it may seem, is that the degrees of freedom restriction implies that one needs multiple observations to draw inference on variance components. This becomes particularly relevant in dynamic panels with groups of observations, see Section 5.5. We note that Geweke (1993) uses a weakly informative inverted gamma density which makes the posterior more regular. The linear regression model with naive heteroscedasticity serves as introduction to the class of HLMM, since the latter class is not so wellknown in economics. 5.1.2. Hierarchical Linear Mixed Model An example of a canonical model with multiple variance components is the class of HLMM. Instead of T variances, one has two variance components with the additional issue of two stochastic processes. We introduce this class through the following hierarchical model with two variance components yt ¼ mt þ t ; with t N ð0; s2e Þ

for t ¼ 1; . . . ; T

mt ¼ y þ Zt ; with Zt N ð0; s2g Þ and E½t Zs ¼ 0

(75) (76)

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Bayesian Near-Boundary Analysis

with l ¼ (m1, . . . , mT)u a vector containing the time-varying mean of y and y the mean of the distribution of mt which, for any t, is normal with variance s2g . This model serves as a parent model for more elaborate models such as SSMs or panel data models. Before moving on to introducing and discussing these models, we analyze the base model by distinguishing between two cases. The ﬁrst case assumes that s2e is known with T being small whereas the second case is the opposite: s2e is assumed unknown and T is large. Each case helps to gain a better understanding of the existence conditions for joint, conditional, and marginal posterior distributions of the HLMM class of models. Note that unless stated otherwise, we assume a uniform prior for each of the variance components. 5.1.2.1. s2e ¼ 1 and T Small: A Degrees of Freedom Bound. Because s2e is given and equal to 1, the only unknown variance component is s2g . The requirement on a minimum number of degrees of freedom as discussed in the previous paragraph is of importance here. Sensible posterior results can only be obtained when there is a sufﬁcient number of observations. The conditional densities pðyjs2g Þ and pðs2g jyÞ can be derived by substituting Eq. (76) in Eq. (75) yt ¼ y þ t þ Zt ; with t N ð0; s2e ¼ 1Þ; Zt N ð0; s2g Þ

(77)

The joint posterior density is

p y; s2g js2e ¼ 1; y / s2g þ 1

1=2ðT1Þ

"

1 ð y iT yÞ0 ð y iT yÞ exp 2ðs2g þ 1Þ

#

(78) n

By using the transformation of random variables: s2g s2g þ 1, one is back in the situation of the beginning of Section 5.1. A degrees of freedom bound is necessary for obtaining existence of posterior moments. Gibbs sampling is possible but not sensible when the number of observations (or the number of groups) is less than or equal to 3. This is essentially the same result as is more formally derived by Hobert and Casella (1996). We refer the interested reader to that reference for details but we emphasize here that substitution of Eq. (76) in Eq. (75) yields a simpliﬁed derivation of the Hobert and Casella result on the degrees of freedom restriction. As before, the Gibbs sampler may seem to work in this model even when the marginal posterior densities for y and s2g do not exist, see Hobert and Casella (1996) for an example and discussion.22

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5.1.2.2. s2e Unknown and T Large: An Identiﬁcation or Labeling Issue. By taking T large enough, one no longer needs to worry about the marginal posterior densities possibly being nonexistent. However, making the ﬁrst variance component, s2e , unknown as well introduces a new issue. More speciﬁcally, one now has to deal with an identiﬁcation or labeling issue in the sense that it is not possible to distinguish the two variance components from each other. Why this is the case can be made clear as follows. Note ﬁrst that since T is assumed to be large enough, the marginal densities of s2e and s2g will exist. However, respecifying the model in Eq. (77) to y ¼ iT y þ e þ g; with e N ð0; s2e IT Þ and g N ð0; s2g IT Þ

(79)

yields that the unconditional mean and variance of y are given by E½y ¼ iT y and V½ y ¼ ðs2e þ s2g ÞIT . Also, the joint posterior density which, after integrating out y, is given by " # ^ 0 ðy iT yÞ ^ 1 ðy iT yÞ 2 2 2 2 1=2ðT1Þ exp pðsg ; se jyÞ / ðsg þ se Þ (80) 2 s2g þ s2e with y^ ¼ y being the average of y. Clearly, only the total variance is identiﬁed, not the individual components. Furthermore, the roles of s2e and s2g are interchangeable (this explains the use of the term labeling issue). This holds true for any value of the signal-to-noise (SN) ratio which is deﬁned as SN ¼ s2g =s2e . Fig. 17 shows the joint density for SN ratios of 1 and 0.5. Panels (a) and (b) show that irrespective of the SN ratio the joint density is perfectly symmetrical. It is also clear from the ﬁgure that the joint density will always have a ridge. Note that everywhere along this ridge the sum of the variance components is the same. This becomes evident by ﬁrst deﬁning n ¼ e þ g and s2n ¼ s2e þ s2g and then recognizing the resulting model as the basic linear regression model which only has a single variance component. The model in Eqs. (75) and (76) basically splits up this single component into two components which explains the ridge. However, because this ridge is on a bounded domain the joint density is nevertheless integrable.23 The Gibbs sampler can, therefore, be used to obtain posterior results. The joint posterior density of ðs2e ; s2g Þ is given in Eq. (80). The conditional density of s2e , given some draw or value for s2g is given by " # ^ 0 ð y iyÞ ^ 1 ð y iyÞ 2 2 2 2 2 1=2ðT1Þ pðse jsg ¼ s g ; yÞ / ðs g þ se Þ exp (81) 2 s 2g þ s2e

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Bayesian Near-Boundary Analysis

(a) SN = 1

(b) SN = 0.5

s2e

s2g

Fig. 17. Joint Posterior Density of and with a Uniform Prior. Notes: Panels (a) and (b) show the joint density in Eq. (80) with a signal-to-noise (SN) ratio of 1 and 0.5, respectively. For both panels y was simulated from Eqs. (75) and (76) with y ¼ 1 and for panel (a) s2e ¼ s2g ¼ 1 whereas for panel (b) s2e ¼ 2; s2g ¼ 1 was used.

and in a similar way one obtains pðs2g js2e ¼ s 2e ; yÞ. Thus we are back in the situation of Section 5.1.1. However, now these conditional posterior densities are identical and the role of the two variance components is interchangeable. The dynamic processes in Eqs. (75) and (76) have an identical structure. The result is an identiﬁcation or labeling issue since it is possible to relabel s2e as s2g and everything remains the same. A further problem arises when instead of a uniform prior, an independent Jeffreys’ prior for s2e and s2g is used, pðy; s2e ; s2g Þ / ð1=s2e Þð1=s2g Þ, in which case the joint density becomes 2

0

3 !1=2ðT1Þ ^ ^ y i y i y y T T 1 1 1 6 1 7 exp4 pðs2g ; s2e jyÞ / 2 2 5 (82) s2g þ s2e se sg s2g þ s2e 2 Fig. 18 shows that the Jeffreys’-type prior causes the joint density to shoot off to inﬁnity for either s2e ! 0 or s2g ! 0.24 Therefore, the joint posterior is now improper and the Gibbs sampler will not converge.25 The Jeffreys’ Information matrix prior in this case is given as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 1=2 2 2 (83) pðsg ; se Þ / cðs2g ; s2e ; TÞ 1 1 where cðs2g ; s2e ; TÞ ¼ 1=2ðT þ 3Þðs2g þ s2e Þ2 . Clearly the determinant in the right-hand side of Eq. (83) is zero. This is in a certain sense obvious since the gradients of the log likelihood are the same for s2e and s2g and the ridge in

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(a) SN = 1

(b) SN = 0.5

Fig. 18. Joint Posterior Density of s2e and s2g with a Jeffreys’-Type Prior. Notes: Panels (a) and (b) show the joint density in Eq. (82) with a SN ratio of 1 and 0.5, respectively. For both panels y was simulated from Eqs. (75) and (76) with y ¼ 1 and for panel (a) s2e ¼ s2g ¼ 1 whereas for panel (b) s2e ¼ 2 and s2g ¼ 1 was used.

Fig. 17 shows the constancy of the likelihood function. We note that in our evaluation of the likelihood we marginalized in an analytical way (by substituting the state Eq. (76) into the measurement Eq. (75)) with respect to the unobserved component mt and thus we make use of the EM method to evaluate the Information matrix. In Hobert and Casella (1996), Theorem 1, a number of conditions are stated that ensure propriety of the posterior density in HLMM models. Our derivation is a simpliﬁed version which we achieve by substituting the state equation into the measurement equation. Summarizing, our analysis indicates the following. A uniform prior yields a proper posterior on a bounded region for ðs2g ; s2e Þ, compare also Gelman (2006). However, there exists an identiﬁcation or labeling issue for s2g and s2e (see Fru¨hwirth-Schnatter, 2001). A weakly informative prior like an (independent) Jeffreys’-type prior is not appropriate and leads to improper posteriors. A natural conjugate informative prior has the possible disadvantage of cutting away posterior probability mass observed near zero. 5.1.2.3. Solutions: Structural Time-Series or Cross-Sectional Information. A number of solutions exist to prevent the problems presented in cases (i) and (ii). For case (i) increasing the number of observations or groups beyond three and assuming constant variance for the observations or groups will prevent the degrees of freedom problem. To solve the identiﬁcation issue of case (ii) one can proceed in a number of ways. One possibility of dealing with this problem is to impose an identiﬁability constraint on the variance components,

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for example, s2e 4s2g . Imposing this constraint in the Gibbs sampler aids in classifying the Gibbs draws to either of the variance components. However, it should be noted that ‘‘identiﬁcation’’ is only coming from the constraint and not in any way from the data. We note that a smoothness prior like the Jeffreys’-type prior is also not a solution. Another possibility is to extend the basic HLMM in such a way that one can distinguish s2e from s2g . Two possible directions can be taken here. The ﬁrst direction is to change the dynamics of m by changing the speciﬁcation of the model in Eqs. (75) and (76) to that of a SSM. The variance components can then be identiﬁed from the imposed additional model structure. The second direction is to use a second source of information. Including additional information via more dependent variables in a panel data model enables one to identify s2g from the cross-sectional observations. We discuss both types of models in the following sections.

5.2. State-Space Model: A Random Walk for mt Starting from the HLMM in the previous paragraph we can specify a SSM by introducing time-series dynamics for the latent variable. Specifying a random walk process for the state variable mt gives yt ¼ mt þ t ; with t N ð0; s2e Þ and mt ¼ mt1 þ Zt ; with Zt N ð0; s2g Þ and

t ¼ 1; . . . ; T E½t Zs ¼ 0

(84) (85)

This model, which is generally known as the local level model or TVP model (see Harvey, 1989) is a basic speciﬁcation of a SSM and has been studied extensively in the literature (see Koop & van Dijk, 2000). More elaborate SSMs are easily obtained by including explanatory variables in the measurement Eq. (84) and state Eq. (85), see Hamilton (1994) or Kim and Nelson (1999) for an overview. The main tool for drawing inference in SSMs is the Kalman ﬁlter. This recursive procedure computes the optimal estimate of the unobserved state vector l given the data y and given values for the remaining parameters, see Kim and Nelson (1999) for more details. Popular algorithms for drawing Bayesian inference in SSMs are given in Carter and Kohn (1994), Fru¨hwirth-Schnatter (1994), De Jong and Shephard (1995), and Durbin and Koopman (2001). The speciﬁcation in Eq. (85) implies that mt is a random walk process which follows from recursively substituting mt1, mt2, etc. Due to the

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additional structure of the SSM one can now distinguish s2e from s2g and therefore identify both variance components. We explain the Gibbs sampler by solving Eq. (85) in a recursive way which yields t X Zi þ t (86) y t ¼ m0 þ i¼1

For simplicity, we assume that the initial value m0 equals zero so that we can obtain the posterior density of ðs2g ; s2e Þ as 1 pðs2g ; s2e jy; m0 ¼ 0Þ / js2g V þ s2e IT j1=2 exp y0 ðs2g V þ s2e IT Þ1 y (87) 2 where V ¼ CCu and C is the so-called random walk generating matrix which is deﬁned as 2 3 1 0 0 6 .. 7 61 1 .7 6 7 C¼6. . . 7 6 .. .. .. 07 4 5 1 1 1 Fig. 19 depicts the joint posterior density of s2e and s2g for 100 data points generated for the case where the ‘‘true’’ s2e is chosen as 4 and the ‘‘true’’ s2g is chosen as 1. In a similar way as indicated before, we can derive the conditional posterior densities of s2g js2e ¼ s 2e and s2e js2g ¼ s 2g . Both conditional densities are again inverted gamma-type which are translated by a constant. It follows that Gibbs sampling is now proper because the variance components are distinguishable. Details are omitted in order to save space. Figs. 20 and 21 show the conditional densities for some draws of the variance parameters. 5.3. State-Space Model: Gibbs Sampling with Explanatory Variables We explain the Gibbs step in a SSM by means of a model that is slightly more complicated than the above local level model: yt ¼ xt bt þ t ; with t N ð0; s2e Þ and t ¼ 1; . . . ; T

(88)

bt ¼ bt1 þ gt ; with gt N ð0; Rg Þ and E½s gk;t ¼ 0

(89)

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Bayesian Near-Boundary Analysis

Fig. 19. Joint Posterior Density of s2e and s2g . Notes: The graph shows the joint density Eq. (87) for 100 data points generated for the case where the ‘‘true’’ s2e is chosen as 4 and the ‘‘true’’ s2g is chosen as 1.

2

2

(a) 2

(b) 2

Fig. 20. Conditional Posterior Densities of s2e and s2g . Notes: Panels (a) and (b) show the conditional posterior densities for s2e and s2g for 100 data points generated for the case where the ‘‘true’’ s2e is chosen as 4 and the ‘‘true’’ s2g is chosen as 1.

with xt a (1 K) vector of explanatory variables, bt the (K 1) state vector with individual elements bk,t for k ¼ l, . . . , K, and RZ is a (K K) diagonal covariance matrix with diagonal elements s2g;k for k ¼ 1, . . . , K. We use this model in an empirical illustration below. It is convenient to ﬁrst factorize

388

1.0

MICHIEL DE POOTER ET AL. 2

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.0

0.0

2

(b) 2

(a) 2

Fig. 21. Marginal Densities of s2e and s2g . Notes: Panels (a) and (b) show the marginal densities for s2e and s2g for 100 data points generated for the case where the ‘‘true’’ s2e is chosen as 4 and the ‘‘true’’ s2g is chosen as 1.

the likelihood when deriving the Gibbs conditional densities. From the hierarchical structure of the model it follows that pð yjb; s2e ; RZ Þ ¼ pð yjb; s2e ÞpðbjRZ Þ

(90)

where b is the T K matrix of latent states. Furthermore, we use bk to denote the kth column of b and bt to denote the tth row of b. The density pðbjs2Z Þ has to be factorized further down to its individual elements p(bk,t|bk,t1). It is straightforward to show that the Gibbs step in this case is given by26 jth Gibbs step for the SSM with explanatory variables: ð j1Þ

Generate bð jÞ js2e

; RðZj1Þ from pðbjy; s2e ; RZ Þ KFS

ð jÞ

Generate s2e jbð jÞ ; RðZj1Þ from

0

1 2 1 ð jÞ0 ð jÞ0 y Xb y Xb p se jy; b; RZ I G ; ðT 2Þ 2 2 ð jÞ

ð jÞ

Generate s2g;k jbð jÞ ; s2 from

0

1 1 ð jÞ jÞ jÞ bk bð1;k bðkjÞ bð1;k p s2g;k jy; b; s2e I G ; ðT 2Þ 2 2 where KFS represents the Kalman ﬁlter sampler using one of the abovementioned algorithms.

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5.4. Empirical Application: U.S. GDP Growth In our empirical analysis to infer and forecast U.S. GDP growth, we also use the local level/TVP model given in Eqs. (84) and (85). Specifying a random walk for the state variable mt gives the TVP model27 gt ¼ ct þ t ; with t N ð0; s2e Þ and t ¼ 1; . . . ; T

(91)

ct ¼ ct1 þ Zt ; with Zt N ð0; s2g Þ and E½t Zs ¼ 0

(92)

This model can be interpreted as capturing time-varying growth. As a special case of this model we have the random walk with drift model when there is no time-variation in ct. When we apply the TVP model to our U.S. GDP growth series we obtain that the posterior mean of s2e equals 0.7030 compared to 0.7214 for the random walk model with drift. This suggests that allowing for GDP growth to vary over time does indeed improve in-sample ﬁt. However, the increased in-sample explanatory power of the TVP model does not help in out-ofsample forecasting as shown in Table 3.

5.5. Panel Data Model The attractive feature of panel data models is that by using time-series observations in addition to cross-sectional information, one can control for time-varying and cross-section speciﬁc variables as well as account for unobserved heterogeneity. The cross-sectional information results from including multiple dependent variables in the model. By grouping dependent variables that are hypothesized to have similar characteristics one can then proceed to identify the parameters for each group. Extensive discussions on panel data models can be found in recent textbooks by Baltagi (2001), Arellano (2002), and Hsiao (2003), among others. As an example of panel data models, we discuss the following random effects model in which we allow for only a single group. yi;t ¼ mi þ i;t ; with i;t N ð0; s2e Þ and t ¼ 1; . . . ; T; i ¼ 1; . . . ; N mi ¼ y þ Zi ; with Zi N ð0; s2g Þ

(93) (94)

where l ¼ ðm1 m2 . . . mN Þ0 . The double subscript on y reﬂects that one now has observations across time as well as across groups. The model allows for

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differences in mean, mi, across individuals by modeling these as random draws for a normal distribution with mean y and variance s2g . As before, the vector l, which contrary to the SSM is now constant over time but varies across groups, consists of latent variables and can be sampled alongside the other parameters in the Gibbs sampler. Note that inference on s2g is based on the cross-sectional observations whereas for s2e variation across the crosssection as well as over time is utilized. Therefore, by including data on multiple individuals, the identiﬁcation issues for the variance components do not exist. However, inference is only possible if a group consists of a sufﬁcient number of individuals. Otherwise a degrees of freedom issue emerges. Throughout this section, we assume a uniform prior on the parameters. 5.5.1. Gibbs Sampling As for the SSM, the likelihood for a random effects panel model can be factorized as pðYjl; y; s2e ; s2g Þ / pðYjl; s2e Þpðljy; s2g Þ The matrix Y contains the observations on all individuals for all time periods. We denote the time-series observations on the ith individual by yi (column i of Y) and the observations on all individuals at time t by the vector yt (the tth row of Y). Furthermore, deﬁne the overall sum of squares as E 0 E ¼ ½vecðYÞ ðIN iT Þm0 ½vecðYÞ ðIN iT Þm where vec( ) is the operator that stacks the columns of Y into a single vector of dimensions TN 1, is the Kronecker product, and IN is a (N N) identity matrix. Given these deﬁnitions, the Gibbs step can be shown to be jth Gibbs step for the panel data model: Generate mði jÞ jyð j1Þ ; s2e

ð j1Þ

; s2g

ð j1Þ

from

pðmi jY; y; s2e ; s2g Þ N

Generate yð jÞ jmð jÞ ; s2e

ð j1Þ

j1Þ ; s2ð from g

pðyjY; m; s2e ; s2g Þ

Mi;

s2e s2e

ð j1Þ

ð j1Þ

s2g

ðj1

þ Ts2g

!

Þ

ð j1Þ

1 1 ð j1Þ iN mð jÞ ; s2g N N N

391

Bayesian Near-Boundary Analysis ð jÞ

Generate s2e jmð jÞ ; yð jÞ ; s2g

ð j1Þ

pðs2e jY; m; y; s2g Þ ð jÞ

Generate s2g jmð jÞ ; yð jÞ ; s2e

from 1 ð jÞ0 ð jÞ 1 I G E E ; ðTN 2Þ 2 2

ð jÞ

from 1 ð jÞ 1 ð jÞ 0 ð jÞ ð jÞ 2 2 pðsg jY; m; y; se Þ I G ðm iN y Þ ðm iN y Þ; ðN 2Þ 2 2

where Mi, for i ¼ 1, . . . , N, is deﬁned as ð j1Þ

Mi ¼

s2g s2g

ð j1Þ

0 t ð j1Þ iT y þ

þ ð1=TÞs2e

s2e Ts2g

ð j1Þ

ð j1Þ ð j1Þ

þ s2e

yði jÞ

(95)

The expression in Eq. (95) shows that draws for mi are based on a weighted average of the information in the cross-section (through yði jÞ ) and the information in the time series (through zt) and that the weights are determined by the two variance components. See also Gelfand, Hills, Racine-Poon, and Smith (1990) for more details. 5.5.2. Empirical Application: Cross-Country GDP Growth We use the Gibbs sampler to analyze the random effects model for a panel of OECD annual real per capita GDP growth rates (in %). The dataset consists of 17 industrialized countries which include Australia, Canada, New Zealand, Japan, the USA, and 12 Western European countries, for the period 1900–2000. It should be noted that the setup of the panel model that we consider here is very limited. For example, we assume that growth rates are independent across countries and that there is no autocorrelation in growth rates. Nevertheless, it may serve as a good starting-point from which to consider more elaborate models. Table 4 shows posterior results for the full panel (ﬁnal column) that includes all individual countries (as a single group). In the table, we only report posterior standard deviations for Australia since those for the other countries are qualitatively similar. The mean growth rate y of the 17 countries is estimated at 1.90%. Interestingly, some part of the variation in the data is due to cross-country differences in growth, which is reﬂected by the estimate of s2g . The Scandinavian countries seem to have experienced the highest average growth rates over the twentieth century, as well as Italy and

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Table 4. Posterior Results for the Random Effects Panel Data Model. Country

Parameters y^ s^ 2e s^ 2g

Australia Austria Belgium Canada Denmark Finland France Germany Italy Japan Netherlands New Zealand Norway Sweden Switzerland UK USA

m^ 1 m^ 2 m^ 3 m^ 4 m^ 5 m^ 6 m^ 7 m^ 8 m^ 9 m^ 10 m^ 11 m^ 12 m^ 13 m^ 14 m^ 15 m^ 16 m^ 17

N¼3

N¼4

N¼5

N¼6

N ¼ 10

N ¼ 17

1.292a (0.562) 50.716 (4.251) 4.279 (35.062) 1.525a (0.633) 1.765b 1.610a

1.426b (0.506) 45.286 (3.246) 2.219 (7.272) 1.563b (0.587) 1.785b 1.642b 1.883b

1.542b (0.449) 39.833 (2.568) 1.444 (2.215) 1.589b (0.543) 1.811b 1.669b 1.906b 1.922b

1.667b (0.407) 37.138 (2.182) 1.154 (1.420) 1.629b (0.522) 1.842b 1.706b 1.938b 1.953b 2.224b

1.882b (0.311) 47.215 (2.146) 0.697 (0.532) 1.731b (0.533) 1.907b 1.795b 1.980b 1.989b 2.210b 1.932b 1.831b 2.151b 2.464b

1.903b (0.208) 38.042 (1.321) 0.415 (0.215) 1.752b (0.448) 1.908b 1.808b 1.976b 1.987b 2.185b 1.937b 1.841b 2.133b 2.417b 1.846b 1.588b 2.271b 1.966b 1.873b 1.677b 1.923b

Notes: The table shows posterior means and posterior standard deviations (in between parentheses) for the random effects panel model Eqs. (93) and (94) when applied to the full panel (N ¼ 17), and several subsets (N ¼ 3, 4, 5, 6, 10) of annual real per capita percentage GDP growth rates for OECD countries. The sample period is 1900–2000 with GDP levels for 1900–1949 obtained from Maddison (1995) whereas those for 1950–1998 were obtained from Maddison (2001). For 1999 and 2000, the data were obtained from the GGDC Total Economy Database, http://www.ggdc.net. All the levels are measured in 1990 U.S. dollars converted at Geary–Khamis purchasing power parities, see Maddison (1995) for a full description. We applied a log transformation to remove the exponential trend in GDP levels across time. Posterior results are based on 100,000 draws after a burn-in of B ¼ 10,000 draws and selecting every h ¼ 10th draw. a Indicates that zero is contained in the 99% but not in the 90 and 95% highest posterior density (HPD) region. b Indicates that zero is not contained in the 99% HPD region.

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Japan, due to their postwar growth spurt. The Australian, New Zealand, and the UK economies witnessed comparatively low growth. Apart from including all the countries we also estimated the model with fewer countries.28 These results, which are shown in the ﬁrst ﬁve columns of Table 4 corroborate the analytical result from Section 5.1 which for a panel model translates to a minimum required number of individuals in a group. The results for N ¼ 3 show that, compared to the results for larger N, the posterior mean and standard deviation for s2g are very large. Especially, the standard deviation of 35.602 seems to indicate that the second moment does not exist. In fact, we know that with N ¼ 3 neither posterior mean nor posterior standard deviation indeed exists. Including at least one additional country helps to identify the mean but still not the variance of s2g . From N ¼ 6 onwards the variance seems to be more reasonable, although the values are still comparatively large. We reemphasize that this panel data model is used for illustrative purposes only. For a more detailed analysis of cross-country growth analysis over a long period we refer to, for example, Barro (1991), Sala-i-Martin (1994), Quah (1997), and Zellner and Palm (2004).

6. FORECASTING U.S. GDP GROWTH USING MODEL AVERAGING Since the seminal article of Bates and Granger (1969) several papers have shown that combinations of forecasts can outperform individual forecasts in terms of symmetric loss functions. For example, Stock and Watson (2004) ﬁnd that using forecast combinations to predict output growth in seven countries generally perform better than forecasts based on single models. Marcellino (2004) has extended this analysis to a large European dataset and reaches broadly the same conclusion. Bayesian approaches have been widely used to construct forecast combinations, see Leamer (1978), Hodges (1987), Draper (1995), Min and Zellner (1993), and Strachan and van Dijk (2007). In general, one derives the posterior probability for any individual model and combines these. The predictive density accounts then for model uncertainty by averaging over the probabilities of individual models. Sala-i-Martin, Doppelhoffer, and Miller (2004) propose a Bayesian model averaging method based on frequentist estimates of individual models and apply their method to long-term GDP growth. Ravazzolo, van Dijk, and Verbeek (2007) extend this approach to a complete Bayesian estimation of a linear regression framework to combine individual models.

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Suppose we are considering J predictive models to forecast GDP growth. The predictive density of gTþ1 ¼ ln GDPTþ1ln GDPT given the data up to time T, D ¼ ( yT, XT), is computed by inferring the following linear regression g^Tþ1 ¼ c þ

J X

bj g^j;Tþ1 þ Tþ1 ;

with

Tþ1 i:i:d: N ð0; s2e Þ

(96)

j¼1

where g^i;Tþ1 is the forecast given by model j ¼ 1, . . . , J, which can be chosen as the mean of the conditional predictive density for model mj given D; p(gTþ1|D, mj), for j ¼ 1, . . . , J. We use this Bayesian averaging scheme to combine the growth forecasts from the models analyzed in Sections 4.2.2 and 5.4. We estimate the model in Eq. (96) and compute forecasts by applying uninformative priors and using the derivations in Section 3. For point estimates and forecasts we use the posterior mean.29 The set of models we discussed earlier can be grouped in two classes: (i) speciﬁcations that model GDP growth with a constant intercept as the random walk plus drift and the simple ECM and (ii) speciﬁcations that model GDP growth with trends as the additive linear trend model for levels, the ECM and the local level model. We showed in Table 3 that the ﬁve models provide similar out-of-sample results and a practitioner may therefore ﬁnd it difﬁcult to choose which individual model to use. As an alternative, we propose applying the Bayesian averaging approach using three schemes. In the ﬁrst scheme, we average models which only have constant terms (m,c), in the second scheme we average models with trends, and as third and ﬁnal scheme we average over all the ﬁve models. In the empirical application, we compute additional forecasts from individual models for the period 1985:Q1–1992:Q2, for a total of 30 observations, and we use these to compute the ﬁrst forecast for 1992:Q3 for the averaging schemes in Eq. (96). We use an expanding window to update the model averaging coefﬁcients. The results of the three schemes as presented in the ﬁnal columns of Table 3. The table provides evidence that model averaging is an appealing strategy for forecasting. All the three schemes produce MSPEs which are similar to the best individual model. The ﬁrst scheme in fact even outperforms the best individual model which is the additive linear trend model for levels. The combination of models with only constant terms gives the most accurate forecasts even if the two individual models used in this strategy are less precise than the other models. A possible explanation is that having to determine only two averaging weights results in smaller estimation error than in the other two schemes, where three and ﬁve weights need to be

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derived, respectively. For the ﬁrst scheme, we ﬁnd that the estimated weights of the random walk with drift and the simple ECM model do show quite some variation over time. A promising extension over the averaging scheme in Eq. (96) would therefore be to make the weights time varying; however, we consider this to be beyond the scope of this paper. To summarize, Table 3 indicates that model averaging is a safe and accurate strategy, but that its success is likely to depend on how it is implemented.

7. FINAL REMARKS: MODELS USED AND LESSONS LEARNT Using a set of basic economic time-series models, focusing on dynamic processes, and time-varying structures, we presented the results of a Bayesian analysis for the case where substantial posterior probability is near and/or at the boundary of the parameter region. As canonical models, we considered the dynamic regression model with autocorrelation in the disturbances and the HLMM in a variance components setup. We have indicated how several empirically relevant model structures are simple extensions of these two canonical ones. A summary of models used and their key features is shown in Table A-1. The results of our Bayesian analysis may be summarized as ‘‘lessons learnt.’’ We describe these as follows: Single equation dynamic regression models and no boundary issues: When the model and data information are such that boundary issues do not occur even under diffuse priors, then posteriors are – at least – locally proper and basic simulation techniques like Gibbs sampling can successfully be used for computing posterior and predictive results. The role of deterministic trend terms should be carefully evaluated. Single equation dynamic regression models and the presence of boundary issues: In many macroeconomic processes, the information in the data is weak and the mass of the likelihood function may be close to the boundary of the parameter region. Examples are nearly nonstationary processes or nearly nonidentiﬁed processes as exhibited by inﬂation, interest rates, dividend yield, and GDP processes. The single equation dynamic regression model serves in this case as a workhorse for unit root models, distributed lag models, and ECMs. A ﬂat prior leads to improper posteriors. Regularization or smoothness priors like the Information matrix prior may be fruitfully used to obtain proper posteriors. Gibbs sampling may fail and the use of more indirect sampling methods like importance sampling and/or Metropolis–Hastings is to be recommended.

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Time-varying parameter models and the presence of boundary issues: A simple regression model with a time-varying variance explains the structure of HLMM. The latter class serves as workhorse for SSMs and panel data models. Boundary issues occur due to a lack of degrees of freedom or nonidentiﬁcation of the variance component. The degrees of freedom problem may occur in panel data problems when a small number of groups is considered. Here we concentrate on right tail behavior of the posterior density. When there exists a substantial probability near the zero variance bound, uniform priors are still recommended, since conditional conjugate priors will cut off relevant data mass near zero, see also the recommendation by Gelman (2006). Informative dynamic structure in time series is used in SSMs and a sufﬁcient number of units are used in the cross-sectional models to regularize the shape of the posterior. This is shown theoretically and empirically. Given these ‘‘lessons,’’ a practical path for an empirical economic researcher is to investigate the shape of the posterior distribution of the parameters of interest and to classify this shape in two categories. As long as this shape is approximately elliptical and much probability mass is in the interior of the parameter region, then applying Gibbs sampling is straightforward and yields accurate results. When the posterior distribution has strong nonelliptical contours and substantial mass is at the boundary of the parameter region then warning signals for the researcher should appear. It depends on the speciﬁcation of the model and the information in the data in which situation a researcher is located. A second advice for the empirical researcher is to apply Bayesian model averaging in cases where substantial posterior probability is at the boundary of the parameter region, compare Section 6. Some directions on how to continue further in nonstandard cases are left for future research. One could think of a reparameterization of the model, the use of subjective informative priors, and the use of predictive priors, see Geweke (2005).

NOTES 1. We are necessarily brief in our explanation of the Gibbs sampler. See Casella and George (1992) or Hoogerheide, van Dijk, and van Oest (2008) for a more elaborate discussion. 2. Strictly speaking, sensitivity to initial conditions persists, but it becomes negligible if the sequence of Gibbs draws mixes sufﬁciently well. 3. The current consensus in the literature, however, seems to be to always include the information of all draws, even when these are correlated.

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4. These restrictions are to insure that the joint density in (4) is integrable and therefore a proper probability density function. 5. For all three examples in this section, we used a burn-in period of B ¼ 10,000 draws and we set the thinning value h ¼ 10. 6. Note that this parameter vector violates the earlier stated parameter restrictions when bZ1. 7. See also Hoogerheide et al. (2007) for a further analysis of the three types of shapes (elliptical shapes, bimodality, and ridges) for posterior densities in the IV model. 8. Which of the two modes the Gibbs sampler gets stuck in depends on the initial ð0Þ values ðyð0Þ 1 ; y2 Þ. 9. A simple case is when xt1 and b ¼ m, in which case one only estimates the mean and variance of y. 10. A noninformative prior for the regression parameters can simply be speciﬁed as p(b)p1. For a variance parameter, a noninformative prior comes down to p(s2)p(s2)1 which follows from specifying a uniform prior for the logarithm of s2, see Box and Tiao (1973), Chapter 1 for more details. If one has prior information, it is strictly advisable to include this in the analysis (see the discussion in Lancaster (2004) and Geweke (2005)). Specifying conjugate priors is, however, not always an easy task, especially when one is faced with a large dimensional parameter region. Here we focus on noninformative priors since we are concerned with what we can learn about the model parameters through the data likelihood. 11. For a more general discussion on Bayesian inference in dynamic econometric models, we refer to Chib (1993) and Chib and Greenberg (1994). 12. The general residual maker matrix is given as MA ¼ IT Að A0 A Þ1 A0 . 13. We note here that we focus on long-term expectations which implies that E[ y] ¼ E[ yl] ¼ Xb for lW0. In reality, T is ﬁnite and therefore (small) sample means should be considered. For expositional purposes, however, we focus solely on longterm expectations; see Kleibergen and van Dijk (1994) for a ﬁnite sample analysis. 14. For an extensive overview of distributed lag models (see Griliches, 1967). 15. We are indebted to William Grifﬁths for pointing this out. 16. In general the Jeffreys’ Information matrix prior is proportional to the square root of the determinant of the information matrix of the considered model. For our purposes, however, we use a somewhat stronger prior because we need (1r)2 instead of (1r)1 to regularize the likelihood. For more details and an advanced analysis on similar Jeffreys’ priors we refer to Kleibergen and van Dijk (1994, 1998). 17. For details on training samples we refer to O’Hagan (1994). 18. The Gibbs sampler is applied with a burn-in period of 4,000 draws and a thinning value of two. 19. Because of the posterior being improper, we do not report forecast results for TBU and DYU in Table 1. 20. The reason this model is called just identiﬁed is because there is only a single instrument, z. 21. A Jeffreys’ type prior, pðs2e Þ / 1=s2e , increases the number of degrees of freedom with 1. As a result, densities now exist for TW1. 22. Note that Hobert and Casella (1996) assume an independent Jeffreys’ prior as a result of which the inverted gamma density of s2g has one degree of freedom since their sample consists of T ¼ 2 groups/observations.

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23. The density shown in Fig. 3(e) on the other hand has a ridge on the domain [0, N) [0, N) which makes it nonintegrable on this domain. For expository purposes, we restricted the domain in this ﬁgure and therefore for the ﬁgure the posterior is a proper density. 24. Although Fig. 18 is similar in shape as Fig. 4 the two ﬁgures have a very different interpretation. Whereas Fig. 4 shows a density that has two well-deﬁned modes (albeit far apart) the density in Fig. 18 is only well behaved in the domain (d, N) (d, N) for a d that is sufﬁciently far away from zero. The latter density goes to inﬁnity when either of the variance components tends to zero. 25. For an early Bayesian paper in this ﬁeld we refer to Grifﬁths, Drynan, and Prakash (1979). 26. If one allows for correlation between the errors in the transition equation one would have to generate draws for Rg from an inverted Wishart density which is given in for example Poirier (1995). 27. We also analyzed a more general formulation of the state equation by estimating ct ¼ a0 þ a1ct1 þ Zt but this resulted in posterior densities which were very tightly centered around 0 (for a0) and 1 (for a1) so we settled for the random walk speciﬁcation. 28. We selected countries according to their alphabetical ordering in the full panel. Although this is somewhat arbitrary we expect results using a random selection of countries to be similar. 29. An alternative approach, often associated with the general deﬁnition of Bayesian model averaging, is to restrict c ¼ 0 and to approximate bj with the posterior probability based on the marginal likelihood for model j. We refer to Ravazzolo et al. (2007) for a comparison of these and alternative methods.

ACKNOWLEDGMENTS This paper is a substantial revision and extension of De Pooter, Segers, and van Dijk (2006). We are very grateful to participants of the 3rd World Conference on Computational Statistics & Data Analysis (Cyprus, 2005), the International Conference on Computing in Economics and Finance (Geneva, 2007), and the Advances in Econometrics Conference (Louisiana, 2007), for their helpful comments on earlier versions of the paper and we are in particular indebted to William Grifﬁths, Lennart Hoogerheide, and Arnold Zellner for many useful comments. Herman van Dijk gratefully acknowledges ﬁnancial support from the Netherlands Organization for Scientiﬁc Research (NWO).

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FORECASTING IN VECTOR AUTOREGRESSIONS WITH MANY PREDICTORS Dimitris Korobilis ABSTRACT This paper addresses the issue of improving the forecasting performance of vector autoregressions (VARs) when the set of available predictors is inconveniently large to handle with methods and diagnostics used in traditional small-scale models. First, available information from a large dataset is summarized into a considerably smaller set of variables through factors estimated using standard principal components. However, even in the case of reducing the dimension of the data the true number of factors may still be large. For that reason I introduce in my analysis simple and efﬁcient Bayesian model selection methods. Model estimation and selection of predictors is carried out automatically through a stochastic search variable selection (SSVS) algorithm which requires minimal input by the user. I apply these methods to forecast 8 main U.S. macroeconomic variables using 124 potential predictors. I ﬁnd improved out-of-sample ﬁt in high-dimensional speciﬁcations that would otherwise suffer from the proliferation of parameters.

Bayesian Econometrics Advances in Econometrics, Volume 23, 403–431 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23012-4

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1. INTRODUCTION It is common practice today to collect observations on many variables that potentially help explain economic variables of interest such as inﬂation and unemployment. Technological progress has allowed the collection, storage, and exchange of huge amounts of information without much effort and cost. In turn, this has signiﬁcantly affected recent macroeconomic modeling techniques. Current academic research is focused on ﬁnding solutions on how to efﬁciently handle large amounts of information with, for example, Stock and Watson (2002) using 215 predictors to forecast 8 major macroeconomic variables for the U.S. economy. Bernanke and Boivin (2003), among others, argue that this is also the case nowadays in central banks, where it is customary for researchers and decision makers to monitor hundreds of subsidiary variables during the decision-making process. These reasons justify the current trend in applied modeling with large datasets. The modern econometrician has tools adequate enough to successfully extract information from hundreds of predictor variables and compute more accurate forecasts than ever before. It is noteworthy that these tools mainly do not rely on economic theory in an explicit way; rather they are statistical and consequently atheoretical methods that are used to cover the unfortunate gap between theoretical models and their empirical validation. Within the sum of all possible options, two methods in particular have recently gained ground: dimension reduction and model averaging. Among many others, Bernanke, Boivin, and Eliasz (2005), Favero, Marcellino, and Neglia (2005), Gianone, Reichlin, and Sala (2004), Stock and Watson (2002, 2005a, 2005b), and Koop and Potter (2004) show how forecasts can be improved over univariate or multivariate autoregressions, using either dynamic factors or Bayesian model averaging (BMA), or both techniques, when a rich dataset is in hand. In this paper I examine empirically the merit of using factors extracted from a large set of explanatory variables and at the same time implementing BMA/selection in the context of macroeconomic vector autoregressions (VARs). While factor methods have already been examined thoroughly in multivariate models, the challenging task of model averaging/selection is implemented with a stochastic search variable selection algorithm (henceforth SSVS) proposed by George and McCulloch (1993, 1997) and George, Sun, and Ni (2008). The proposed approach is ﬂexible as its output can easily be used for selection of a single best model or model averaging. The SSVS adds to a recent and expanding literature on different approaches to BMA in VARs

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(Strachan & van Dijk, 2007; Andersson & Karlsson, 2008). The innovation of the speciﬁc prior formulation is that it is more appropriate for VAR models compared to previous model selection priors used in multivariate regressions (Brown, Vannucci, & Fearn, 1998, 2002; Brown et al., 1998, 2002). That is because each right-hand side variable is allowed to enter in all, some, or none of the VAR equations, and not only in all or none of them. The additional advantages come from the fact that this class of restriction search algorithms is extremely simple to use and automated. Furthermore, certain versions of these algorithms can incorporate variable selection when the number of predictors is larger than the number of time series observations. The following section deﬁnes the Bayesian VAR model when many variables are available. Within this ‘‘large model approach’’ the large number of variables is replaced with a small number of factors and several aspects of this approach are discussed. In Section 3, the stochastic restriction search is introduced as a means of efﬁciently selecting a subset of macroeconomic variables or factors that should be restricted from the VAR speciﬁcation, based only on the information in the data. The prior speciﬁcation necessary for model selection is analyzed, as well as the interpretation of model selection probabilities as a special case of BMA. Section 4 outlines the setting of the empirical section (data, forecasting models, prior hyperparameters, and comparison statistics), and the results of the forecasting performance of various VAR speciﬁcations. Section 5 concludes the paper with a summary and thoughts for further extension of the basic framework presented in this paper.

2. THE MODEL Let yt be m 1 vector of variables of interest (that we want to forecast) observed for t ¼ 1; . . . ; T. Unlike previous univariate studies (Stock & Watson, 2002; Koop & Potter, 2004), mW1 and I deﬁne a forecasting model for y using a general VAR representation: y0tþ1 ¼

p1 X

y0ti ai þ w0t c0 þ 0tþ1

(1)

i¼0

where the parameter matrices ai and c0 are of dimensions (m m) and (N m) respectively, yti ; i ¼ 0; . . . ; p1 are lagged values of the endogenous variable y, wt is a (N 1) vector containing current and lagged values of some exogenous predictor variables, and for simplicity assume iid Gaussian

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errors t Nð0; SÞ. This model can be estimated both by OLS and Bayesian methods, provided that the total number of explanatory variables will not exceed the total number of time series observations T. I propose to adopt a Bayesian setting which allows for a uniﬁed treatment of this model in high dimensions. For a review of the VAR under standard prior speciﬁcations and different sampling methods, the reader is referred to Kadiyala and Karlsson (1997). Assume you have available observations xt ¼ ðx1t ; . . . ; xnt Þ0 on some macroeconomic quantities, where n is large (in the order of hundreds). A popular and simple method to incorporate into an econometric model all the information inherent in a large set of variables, is to reduce their dimension into a lower-dimensional vector of k n latent factors and insert these in the VAR model as explanatory variables: x0t ¼ lf 0t þ u0t y0tþ1 ¼

p1 X i¼0

y0ti ai þ

p2 X

f 0tj bj þ 0tþ1

(2)

(3)

j¼0

where ft is a k 1 vector of unobserved factors, l the matrix of factor loadings, and ut are independent normal errors, ut Nð0; WÞ. In Eq. (3) the same assumptions hold as in Eq. (1), with the only difference that now N ¼ k ð p2 þ 1Þ. For simplicity xt is demeaned which is equivalent to imposing a constant term in the factor equation, equal to the sample mean (which in this model coincides both with the MLE of the constant or the mode of its posterior under a diffuse prior). The factors ft are unobserved quantities and usually it is assumed that they follow a normal distribution with diagonal covariance matrix. One more convention in the factor model literature is to impose the covariance matrix of the innovations, W, to be also diagonal so that Eq. (2) reduces to n independent equations. Estimation methods vary from principal component analysis (PCA) to full likelihoodbased approaches. The ultimate goal of using the factor model is to obtain the factor scores ft as a valid reduced representation of the manifest vector xt, so that factor identiﬁability issues play no actual role here and will not be further discussed. In terms of the general forecasting VAR model in Eq. (1), I replace the predictors wt with the principal components estimates of the factors 0 0 F^ t ¼ ½ f^t0; f^t1 ; . . . ; f^tp 0 , i.e., as if they were observed data. Note that this 2 speciﬁcation is slightly different from the Bayesian dynamic factor model (or factor-augmented VAR) used in Bernanke et al. (2005). From their point of

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view the dynamic factor model (DFM) is treated as a state-space model, which has the advantage of a probably more efﬁcient one-step estimation of the factors (i.e., along with the parameters of the model) through the Kalman ﬁlter algorithm. But this comes at a huge computational cost which makes the application of this model prohibitive in the recursive forecasting setting adopted in this study. After all, Stock and Watson (2005a) implemented a large-scale forecasting exercise involving DFMs where they compare several frequentist, full Bayes, and empirical Bayes approaches. However, having in mind the challenge of preserving parsimony in previous dynamic factor speciﬁcations,1 I provide the interested reader with an accompanying technical document (and MATLAB code) with instructions on how to implement factor and lag order selection on the DFM. This involves generalizing the restriction search algorithm in the loadings matrix, by way of specifying which variable loads on each factor. The factors replace the original variables in order to allow richer dynamics. The factors are allowed to have up to p2+1 lags, while if the original variables xt ¼ ðx1t ; . . . ; xnt Þ0 were included as predictors then – for a typical macroeconomic dataset with monthly observations on many variables – a degrees of freedom problem would occur if more than one or two lags were assumed. However, when the number of lagged factors is unknown, estimating traditional information criteria or Bayes factors routinely used in small VAR models is not a convenient solution. For N ¼ k ð p2 þ 1Þ larger than 20 the number of all possible models will tend virtually to inﬁnity so that pairwise comparison is practically infeasible using an AIC/BIC-type criterion or prior predictive (marginal) densities and Bayes factors. A reasonable proposed solution from a Bayesian point of view is to use shrinkage subjective priors. For example, the Minnesota prior imposes restrictions on parameters which correspond to higher order lags of y, whereas the prior weight (i.e., the prior mean) for the parameter on the ﬁrst own lag in each of the m equations is equal to one, and zero on the ﬁrst lag of the rest (m1) dependent variables. While this approach will work well in VARs which include only lags of the dependent variables, it is difﬁcult to adopt this approach in the models examined here. This happens because there is no theoretical or empirical justiﬁcation for constructing a subjective prior on predictor variables. Introducing any kind of subjective prior information in this model is not an easy task. These priors may not be speciﬁed concretely because of the lack of prior information regarding joint distributions or the large amount of models involved in the analysis. In that respect, subjective prior beliefs require a huge amount of input from the researcher. It is unrealistic to

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assume that uncertainty about the true model speciﬁcation can be described meaningfully using ones’ own beliefs; hence prior elicitation should be based mainly on economic theory. The problem with this approach is that in many cases economic theory has empirically proven to be bad guidance in proposing relevant predictors. Stock and Watson (2003) argue that this is the case when forecasting inﬂation: ‘‘the literature does suggest [ . . . ] variables with the clearest theoretical justiﬁcation for use as predictors often have scant empirical predictive content.’’ The discussion so far has focused on the ‘‘large –n’’ case, avoiding mentioning anything about how small or large the dimension m of the dependent variable y should be. Although macroeconomic VARs typically contain as dependent variables three or four fundamental quantities that describe the economy, when forecasting, the actual number of variables of interest can grow large. A decision maker would be interested to forecast future values of many series, like production, employment/unemployment, short- and long-term interest rates, consumer and producer price inﬂation, exchange rates, and many other nominal or real quantities. This is easily handled with the model selection algorithm which is the focus of the next section. The methods described below apply to large VARs in a general sense, that is (i) when the number of predictors n-N and the number of dependent variables m is small, (ii) when m-N and n is small, or (iii) when both m, n-N, although the empirical application is centered upon the ﬁrst case.

3. BAYESIAN MODEL SELECTION AND AVERAGING As was mentioned in the introductory section, when the number of candidate models is too large to enumerate, posterior sampling methods are necessary for the computation of marginal likelihoods for model comparison. Stochastic search algorithms that base on a Markov chain on model space identify regions of high posterior probability and can be used for model selection or to obtain posterior weighted estimates for model averaging. When applied to small models, these algorithms have the ability to search the entire model space, while in large settings only more plausible models are visited. An indicator (zero/one) variable g, epitomizes the core of Bayesian model selection using stochastic search techniques. Let us deﬁne the vector c ¼ ðg1 ; . . . ; gs Þ as the complete set of indicators, where s is the maximum number of parameters in the model. Then we can proceed by deﬁning a prior pðcÞ which combined with the likelihood marginalized over c,2 pðdatajcÞ, will give zero or one value (for each gi, i ¼ 1; . . . ; s) from the updated (based on

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data) posterior distribution pðcjdataÞ. This posterior distribution pðcjdataÞ entails all the necessary information for model selection and averaging. Imposing the vector of parameters, say h ¼ ðy1 ; y2 ; . . . ; ys Þ, to have a structure conditional on the values of c has the implication that when gi takes the value 1, the respective parameter yi will be estimated according to its unrestricted posterior density. Otherwise when a parameter is restricted not to enter the ‘‘true’’ model its posterior mean value is imposed to be zero or approximately zero, which means that (in linear models) the respective variable is excluded. There are many ways to implement this general strategy and many alternative methods exist which involve several prior speciﬁcations. An analytical review of model averaging and selection is offered in Hoeting, Madigan, Raftery, and Volinsky (1998) and Chipman, George, and McCulloch (2001). A computationally fast restriction search is described in this section which is based on the SSVS algorithm of George and McCulloch (1993, 1997). Deﬁne zt ¼ ½ y0t ; y0t1 ; . . . ; y0tp1 ; w0t 0 , then the VAR model in familiar matrix form is obtained by stacking the row vectors yt+1, zt, and et for t ¼ 1; . . . ; T: y ¼ zu þ ; Nð0; SÞ

(4)

where y ¼ ½y02 ; . . . ; y0Tþ1 0 ; z ¼ ½z01 ; . . . ; z0T 0 ; u ¼ ½a0 ; . . . ; ap1 ; c0 , and ¼ ½01 ; . . . ; 0T 0 . Note that when forecasts are projected h-steps ahead, y is the matrix y ¼ ½y1þh ; . . . ; yTþh 0 (see next section for a deﬁnition). Let nf be the total number of elements in f ¼ vecðuÞ ¼ ðf1 ; f2 ; ; fnf Þ0 . From these elements the m, in total, constants are always included in the models (left unrestricted) and admit a typical normal prior ðfc Þ N fc ; nI m where fc is the block of f containing the constants. Let fk be the vector of the nf m parameters that are subject to restriction search and let c ¼ ðg1 ; g2 ; . . . ; gnf m Þ be a vector of 0–1 variables. Then, impose a mixture prior for each of the nf m elements fki , i ¼ 1; . . . ; nf m, conditional on each gi k fi jgi 1 gi N 0; t20i þ gi N 0; t21i (5) This is a mixture of two normals prior for the parameters that we want to ﬁnd restrictions. The hyperparameters t0i , t1i , are selected in such a way that t0i is small (or even zero) and t1i is large. Subsequently each parameter fi is restricted with zero prior mean and very small (or zero) prior variance when

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gi ¼ 0, while for gi ¼ 1 has a large (locally uninformative) prior and in that respect is left unrestricted. This prior is easily combined with the standard normal prior (Kadiyala & Karlsson, 1997) when we want to leave some parameter elements unrestricted (like constants, trends or the ﬁrst lagged value of each of the dependent variables). In that case there will be nf x elements with prior density as the one in Eq. (5), with the rest x in total, unrestricted elements admitting a typical normal prior-posterior density. It would not make sense to deﬁne the gi’s if these were deﬁned subjectively and not updated by the information in the data. Hence a Bernoulli prior on these variables is placed, which updated by the likelihood pðdata; cÞ will result in a Bernoulli posterior pðcjdataÞ of 0–1 values that index each mixture component in Eq. (5). The elements of the vector c are independent Bernoulli pi 2 ð0; 1Þ random variables with prior density: Y g PðcÞ ¼ pi i ð1 pi Þð1gi Þ ; i ¼ 1; 2; . . . ; nf m (6) Such prior reduces computational costs and leads to a posterior density which is easy to derive. In this case pðgi ¼ 1Þ ¼ pi ¼ 1 pðgi ¼ 0Þ so that pi determines the prior probability that fki is large enough and should be left unrestricted. By selecting pi o1=2, models with an unreasonably large number of parameters are downweighted in order to highlight the signiﬁcance of parsimonious models. The special case where pi ¼ 1=28i ¼ 1; . . . ; nf m, is equivalent to a constant uniform prior PðcÞ 1=2nf . This prior is uninformative in the sense that it favors each parameter equally; see Section 4.2 in this paper and the discussion in Chipman et al. (2001). The hierarchical mixture prior described above is straightforward to interpret and can be applied virtually to any model for which a normal prior can be speciﬁed3 as the conjugate prior that leads to easy derivation of the underlying posterior. A different version of the SSVS is used in Brown et al. (1998) for a multivariate regression model used to predict three variables using 160 predictors. Following the suggestions of George and McCulloch (1997) and Smith and Kohn (1996) they set in Eq. (5) t0i ¼ 0 and t1i ¼ g ðz0g zg Þ1 . This prior implies that the ﬁrst component of the mixture is a Dirac delta function, i.e., a function that puts point mass at zero and hence whenever gi ¼ 0; fki will be exactly zero. The second component is Zellner’s g-prior speciﬁcation and suggestions for setting uninformative values of g (although in a univariate context) are given in Fernandez, Ley, and Steel (2001). An updated and computationally more efﬁcient version of this prior speciﬁcation appears in Brown et al. (2002), where more variables than

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observations can be handled. The shortcoming of this approach is that it will not treat each equation in the VAR individually, but instead will choose the variables in z that are more probable in all equations together. Put simply, if z is only lags of the dependent variable, the latter approach will choose the same number of lags in each equation, while the approach proposed here restricts each element of the vector fk individually, and hence each explanatory variable may enter in only some of the m VAR equations. Nevertheless, the Brown et al. (2002) implementation of the SSVS algorithm is a valuable complement to the one used here, and undoubtedly a useful tool in empirical analysis with focus on prediction. Smith and Kohn (2002) extend the stochastic search for parameter restrictions to the covariance matrix of longitudinal data. George et al. (2008) apply their idea to the covariance matrix of structural VARs: motivated by the fact that identifying restrictions on the covariance are usually imposed on the elements of a reparametrization of R, they focus on restricting the elements of the m m upper triangular matrix W satisfying: 0

R ¼ W1 W1 They then derive a mixture of normals prior, as in Eq. (5), for the nondiagonal elements of W, while the diagonal is integrated out with a gamma prior. Matrix W has the form: 2 3 c11 c12 . . . c1m 6 7 6 0 c22 . . . c2m 7 6 7 W¼6 . .. .. .. 7 6 .. 7 . . . 4 5 0 0 . . . cmm so let W ¼ ðc11 ; . . . ; cpp Þ0 and g ¼ ðg0 2 ; . . . ; g0 m Þ0 ¼ ðc12 ; c13 ; c23 ; . . . ; cm1;m Þ0 be the vectors of the diagonal and upper diagonal elements respectively, where gj ¼ ðc1j ; . . . ; cj1; j Þ0 for j ¼ 2; . . . ; m. Similarly to Eq. (5), let xj ¼ ðo1j ; . . . ; oj1; j Þ0 be a vector of 01 variables and each element of g j has prior ðZij joij Þ ð1 oij ÞN 0; k20ij þ oij N 0; k21ij ; for i ¼ 1; . . . ; j 1 (7) As in the case of the vector c, assume that the elements of x ¼ ðx0 2 ; . . . ; x m Þ0 are independent Bernoulli qij 2 ð0; 1Þ random variables so that YY o PðxÞ ¼ qij ij ð1 qij Þð1oij Þ ; i ¼ 1; 2; . . . ; m j ¼ 1; . . . ; m 1 (8) 0

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For i ¼ 1; . . . ; m, cii has a gamma prior density cii Gammaðai ; bi Þ;

for

cii 40

(9)

For more information on these priors the reader is referred to the analytical calculations of George et al. (2008) where it is shown that ﬁnding restrictions on the covariance matrix based solely on the data provides an attractive alternative to identifying restrictions imposed in structural VARs. It should be clear from the prior speciﬁcation that the SSVS is an intuitive extension of the Bayesian conjugate (normal – inverse Wishart) prior. In the empirical application I adopt a fast sampling scheme (see Section 4.2) to draw from the posteriors of c and x, which makes computation feasible in multivariate models. The parameter posteriors are given in detail in Appendix A (Technical Appendix). Although selection of prior hyperparameters seems to be fairly automatic in this setting, prior elicitation is an important factor in model selection.

4. EMPIRICAL APPLICATION 4.1. Data I use the Stock and Watson (2005b) dataset which is an updated version of the Stock and Watson (2002) dataset that is widely used in empirical applications. This version consists of 132 monthly variables pertaining to the US economy measured from 1960:01 to 2003:12. The 132 predictors can be grouped in 14 categories: real output and income; employment and hours; real retail, manufacturing, and trade sales; consumption; housing starts and sales; real inventories; orders; stock prices; exchange rates; interest rates and spreads; money and credit quantity aggregates; price indexes; average hourly earnings; and miscellaneous. The data were transformed to eliminate trends and nonstationarities. All the data and transformations are summarized in Appendix B.

4.2. Selection of Prior Hyperparameters Implementation of Bayesian model selection requires all the priors to be proper, as the ones described in Section 3. Noninformative improper priors are not suitable to calculate Bayes factors and posterior model probabilities.

Forecasting in Vector Autoregressions with Many Predictors

413

Even though there are certain methods which overcome this difﬁculty (BIC approximations, intrinsic, or fractional Bayes factors), the standard practice in the Bayesian model selection literature is to use only proper priors. This does not necessarily mean that noninformative proper priors cannot be speciﬁed. It is easy to choose the hyperparameters in such a way that all the priors are locally noninformative. Selection of t0i, t1i and k0i, k1i can be made along the guidelines of Chipman et al. (2001, p. 86). For a given threshold xi, higher posterior weighting can be allocated to those c values for which jfi j4di when gi ¼ 1, if t0i, t1i satisfy t1i =t0i ¼ x2i log 1 t0i t1 1i A similar argument can be made for the choice of k0i and k1i. Alternatives for a more objective selection of these hyperparameters exist, but at the cost of a substantial increase in computational calculations. The ﬁrst one is to use empirical Bayes criteria in the spirit of George and Foster (2000), while a fully Bayes approach would require to place an inverted-Gamma hyperprior on each t0i, t1i and k0i, k1i. Selection based on the formula above is a simple task which can be easily implemented in large models. George et al. (2008) argue that even if the restriction search algorithm is not effective in selecting the correct restrictions on u, the results can still be used to obtain improved forecasts. The only source of difﬁculty may arise in eliciting the hyperparameters of the Bernoulli random variables c (similarly x). The prior structure that appears in Eq. (6) (similarly in Eq. (8)) is an ‘‘independence prior,’’ in the sense that each element of c (x) is assumed to be independent of the rest. This simpliﬁcation makes it difﬁcult to account for similarities or differences between models when the correlation between the explanatory variables is high. While priors that ‘‘dilute’’ probability across neighborhoods of similar models (Chipman et al., 2001; Yuan & Lin, 2005) are able to correct this shortcoming, it is preferable to use an orthogonal transformation of the variables in z, by applying a singular value decomposition. This allows exploring the model space in considerably less iterations, which subsequently decreases the computational cost in multivariate models. Hence, in the forecasting exercise, I apply the restriction search to the model yTþh ¼ GT mh þ Tþh where G=zH are orthogonal variables and m ¼ H 1 f; see Koop and Potter (2004). This approach will speed up computations, even though orthogonality

414

DIMITRIS KOROBILIS

does not lead to posterior independence of elements of c. The default choice pi ¼ 1=2 in Eq. (6) and qij ¼ 1=2 in Eq. (8) may result in a uniform prior, but this would not be a noninformative prior about model size. A rule of thumb is that if the researcher anticipates many (few) restrictions on the model then the choice should be pi ; qij o1=2 ( pi ; qij 41=2). Prior sensitivity analysis using real and simulated data showed that pi ¼ qij ¼ 1=2 is able to identify restrictions quite well and hence is left as the default reasonable choice. Following the suggestions of George et al. (2008) and George and McCulloch (1997), I adopt a fast sampling scheme for c and x, which requires to set t0i and k0ij small, but different than zero. According to the preceding discussion in this subsection and the absence of prior beliefs about speciﬁc parameters I set t0i ¼ t0 ¼ 0:01, t1i ¼ t1 ¼ 70 for all i ¼ 1; . . . ; nf m, and k0ij ¼ k0 ¼ 0:01; k1ij ¼ k1 ¼ 30 for all j ¼ 2; . . . ; m and i ¼ 1; . . . ; j 1. For the intercept term, the typical normal prior has mean fc ¼ 1 and variance v=100. A default noninformative choice for the parameters of the Gamma density is ai ; bi ¼ 0:01. 4.3. Implementation of Bayesian Model Averaging/Selection At this point, as it is practically impossible to summarize model selection results from the recursive forecasting exercise, I summarize the average posterior probability of some of the variables in the dataset without extracting factors, i.e., replacing wt with xt ¼ ðx1t ; . . . ; xnt Þ0 in speciﬁcation (1), and using the full sample of observations from 1960:1 to 2003:12. I consider a New Keynesian VAR with three variables (unemployment, consumer price index,4 and federal funds rate) regressed on a constant, 14 autoregressive lags and the remaining 129 variables in the dataset used as predictors. This gives a total of 129+13 3=168 right-hand side variables (excluding the constant) to choose from in each equation. The horizon chosen in this illustration is h=12. The unemployment and interest rate are transformed to stationarity by taking ﬁrst differences. The consumer price index is transformed by taking the second difference of the logarithm. A parameter should either be included or excluded, hence the number of all possible models is 2168 in each VAR equation and 2168 3=5.2e+151 in total. The BMA posterior probabilities are computed for each parameter i ¼ 1; . . . ; nf as: Eðgi jyÞ ¼

S X s¼1

pðgi jyÞ

Forecasting in Vector Autoregressions with Many Predictors

415

where S is the number of after burn-in draws from the posterior pðgi jyÞ. This says that the average probability is actually the proportion of models drawn by the Gibbs sampler that contain the corresponding variable. Tables C1 and C2 summarize the results for those predictor variables and own lags, respectively, that have the highest probabilities. Variables which had average posterior probability less than 0.5 in all of the three equations are not included at all in the tables. Each element in these tables is the BMA posterior probability and can be interpreted simply as the probability that the corresponding right-hand side variable should be included. For this speciﬁc application the variables are not orthogonalized in order to retain the interpretation of the probabilities as the amount of belief that the respective variable is included in the model. The results are based on 150,000 iterations with a burnin period of 50,000 leaving 100,000 draws to evaluate the posterior of c. Elicitation of prior hyperparameters is based on the values described in Section 4.2. Note that the probabilities for W are 0.52, 1, and 1 for each of the upper diagonal elements c12, c13, and c23 respectively. Once all these probabilities are available, it is straightforward to interpret them. This output can be used to implement BMA if all variables contribute to the ﬁnal forecast according to their probability, no matter how high or low this probability is. Looking for example at Table C1, the spread of the 10-year interest rate from the federal funds rate variable will contribute to the ﬁnal forecast of the unemployment rate, the consumer price index, and the interest rate in 100, 86.1, and 100% of the occasions (models visited by the sampler), respectively. In contrast the same output can be used to select the best single model. Barbieri and Berger (2004) show that in the context of Bayesian model selection the optimal model is the median probability model. According to this result, only the variables which have average probability larger than 0.5 in each equation will be unrestricted. These probabilities are presented in Tables C1 and C2. Hence, in this ‘‘best’’ model, the 1, 5, and 10-year interest rate spreads should be included in all three equations, while capacity utilization should enter only the unemployment equation. The results presented in Table C1 are also subject to economic interpretation. Space restrictions, however, do not allow further analysis in this study. Structural interpretation is not the main focus, but forecast improvement is. This is an issue examined in the following section. 4.4. Forecasting in Large VAR Models The ﬁrst estimation period is set to 1960:1 and a simulated real-time forecasting of ytþh is done from 1983:1 through 2003:12-h, for horizons h=1, 6, and 12.

416

DIMITRIS KOROBILIS

Each VAR model has eight dependent variables of interest (with their sort mnemonic from the dataset in parentheses): Personal Income (A0M052), Industrial Production (IPS10), Employment Rate (CES002), Unemployment Rate (LHUR), 3-month Treasury Bill Rate (FYGM3), Producer Price Index (PWFSA), Consumer Price Index (PUNEW), and PCE Deﬂator (GMDC). This leaves a total of 124 variables to explore their predictive content. As mentioned earlier, all the variables are transformed to stationarity, a fact that implies a speciﬁc transformation of the variable ytþh proper for forecasting. Let vit denote the untransformed value of yit for each of the eight monthly dependent variables i, then yitþh ¼ ð1200=hÞ lnðvitþh =vit Þ for i ¼ ðA0M052; IPS10; CES002Þ, yitþh ¼ vitþh vit for i ¼ ðLHUR; FYGM3Þ, and yitþh ¼ ð1200=hÞflnðvitþh =vit Þ hD lnðvit Þg for i ¼ ðPWFSA; PUNEW; GMDCÞ. The principal components are estimated from the 124 variables in the dataset using the same sample period as the VAR. Several multivariate forecasting exercises in the literature (cf. Stock & Watson, 2002) focus on ﬁnding the best performing model. In contrast, here the main challenge is to improve forecasts when the number of predictors grows large and the researcher has no prior information about which is the correct model size. Thus, the maximum potential number of factors and lags is deliberately set to large, ‘‘uninformative’’ values. In particular, 10 principal components (k=10) are extracted from the factor model in Eq. (2), while the VAR in Eq. (3) contains a constant, 13 autoregressive lags ( p1=12), and 13 lagged factors ( p2=12). This gives a maximum of 221 (plus the constant, which is unrestricted) potential predictors of each of the 8 dependent variables. For the purpose of the empirical application forecasts are computed from: (i) VAR with SSVS and model averaging, (ii) VAR with SSVS and model selection, and (iii) VAR estimated using OLS with selection of predictors with the Bayesian information criterion (which has a larger penalty for less parsimonious models relative to the Akaike information criterion, and is a rough approximation to the Bayes factors). The predictors in the latter method are orthogonalized and the total number of possible models considered is equal to the maximum number of right-hand side variables and subsequently selection of the best model is implemented in a ﬁnite number of calculations. An appropriate common way to quantify out-of-sample forecasting performance is to compute the root mean square forecast error (RMSFE) statistic for each forecast horizon h: vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u2003:12h 2 u X h y i;tþh yi;tþh; j RMSFEij ¼ t t¼1982:12

where y i;tþh is the realized (observed) value of y at time t+h for the ith series and yi;tþh; j is the mean of the posterior predictive density at time t+h,

Forecasting in Vector Autoregressions with Many Predictors

417

for the ith series, from the jth forecasting model. The RMSFE of each model is reported relative to the RMSFE of a benchmark VAR with a constant and seven lags of the dependent variables, estimated with OLS: rRMSFEhij ¼

RMSFEhij RMSFEhiVARð7Þ

This VAR(7) model is not chosen because of its higher forecasting ability compared to other alternatives. Following the standard convention in the literature an AR(2) model would be a better candidate to serve as the benchmark model. But note that the VAR(7) is nested to the VAR with factors, which will give a better picture of whether the restrictions found by the SSVS are actually the ones that will lead to reduced RMSFE statistics, compared to a more parsimonious alternative. The forecasting performance of the models based on the relative RMSFE for horizons h=1, 6, 12, is summarized in Table C3. These are the averaged values of the RMSFEs over the forecast period, 1983:1 through 2003:12-h. The results are encouraging about the application of the restriction search algorithm in large models. In most occasions the BMA and Bayesian model selection give improved results compared to the BIC selection. Note that the improvement is not only due to the fact that the models of interest contain more predictors than the benchmark model. It is noteworthy that in some occasions only lags of the dependent variable are selected from the restriction search, while for most samples the number of important lagged factors, for each dependent variable, is not more than ﬁve. This is supported by the fact that the average RMSFE (results not reported here) of the large VAR with factors but without selection of predictors (i.e., a heavily overparametrized model) is, as expected, extremely high relative to the VAR(7). An important feature of the restriction search algorithm applied to the speciﬁc VAR is that the forecasts from Bayesian model selection are better than the forecasts from BMA. The practical difference of the two approaches is that BMA shrinks the posterior means of the parameter with low probability toward zero, while Bayesian model selection imposes that these parameters (with probability less than 0.5) will be exactly zero.

5. CONCLUSIONS This paper addresses the forecasting performance of Bayesian VAR models with many predictors using a ﬂexible prior structure which leads to output that can be used for model selection and model averaging. For eight U.S. monthly macroeconomic variables of interest forecasting accuracy is

418

DIMITRIS KOROBILIS

improved over least squares estimation and selection of predictors using the Bayesian information criterion. Without arguing that the choice of prior hyperparameters was the best possible and done with a strict ‘‘objective’’ criterion (like in other BMA applications, see Fernandez et al., 2001), the gains from the standard automated choices are appreciable. As already mentioned, there are many proposals in the Bayesian literature for efﬁcient elicitation of prior hyperparameters for model selection and some of them were discussed in the paper. Nevertheless, the merit of the SSVS for VAR models lies in its simplicity and intuitive interpretation. With regard to other macroeconometric speciﬁcations, the ﬂexibility of the restriction search algorithm suggests many interesting extensions. Firstly, note that it is straightforward to adopt it in general piecewise-linear multivariate regressions that allow for thresholds, Markov switching or structural breaks; an interesting area for future research. Secondly, I only considered the case where the number of dependent variables, m, is small and the number of predictors grows large. But as already mentioned the restriction search algorithm may also be used when the number of dependent variables grows large. Banbura, Giannone, and Reichlin (2007) examine this case using shrinkage priors and ﬁnd huge gains from this large VAR speciﬁcation. Lastly, an interesting direction for future research would be the empirical application of the restriction search algorithm in the Bayesian dynamic factor model. This approach will probably improve forecasting performance and impulse response analysis in DFMs that lack parsimony (cf. Bernanke et al., 2005 and Stock & Watson, 2005b).

NOTES 1. Bernanke et al. (2005) use 13 lags in their factor-augmented VAR in which case they had to estimate more than 200 parameters (plus the factor loadings). 2. This is the part of the full likelihood function that involves only the parameter vector c as a random variable. 3. For example, recently Tadese et al. (2005) applied this idea to variable selection in a clustering mixture model. 4. Actually the New Keynesian VAR will include the consumer price index (CPI) annual inﬂation and not the level of CPI, but for simplicity the variable is used as it is provided from the dataset.

ACKNOWLEDGMENTS The author would like to thank Gary Koop, an anonymous referee and participants and discussants at several workshops and conferences for

Forecasting in Vector Autoregressions with Many Predictors

419

helpful comments. Part of this research was supported by the Department of Economics, University of Strathclyde, which I gratefully acknowledge.

REFERENCES Andersson, M. K., & Karlsson, S. (2008). Bayesian forecast combination for VAR models. In: S. Chib, W. Grifﬁths, G. Koop & D. Terrell (Eds), Bayesian Econometrics (pp. 501–524). Advances in Econometrics, Vol. 23. Oxford: Elsevier. Banbura, M., Giannone, D., & Reichlin, L. (2007). Bayesian VARs with large panels. London: CEPR. Barbieri, M. M., & Berger, J. O. (2004). Optimal predictive model selection. The Annals of Statistics, 32, 870–897. Bernanke, B. S., & Boivin, J. (2003). Monetary policy in a data-rich environment. Journal of Monetary Economics, 50, 525–546. Bernanke, B. S., Boivin, J., & Eliasz, P. (2005). Measuring the effects of monetary policy: A factor-augmented vector autoregressive (FAVAR) approach. Quarterly Journal of Economics, 120, 387–422. Brown, P. J., Vannucci, M., & Fearn, T. (1998). Multivariate Bayesian variable selection and prediction. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 60, 627–641. Brown, P. J., Vannucci, M., & Fearn, T. (2002). Bayes model averaging with selection of regressors. Journal of the Royal Statistical Society. Series B (Statistical Methodology), 64, 519–536. Chipman, H., George, E. I., & McCulloch, R. E. (2001). The practical implementation of Bayesian model selection. In: P. Lahiri (Ed.), Model selection (pp. 67–116). Lecture Notes – Monograph Series, Vol. 38. Beachwood, OH: Institute of Mathematical Statistics. Favero, C. A., Marcellino, M., & Neglia, F. (2005). Principal components at work: The empirical analysis of monetary policy with large datasets. Journal of Applied Econometrics, 20, 603–620. Fernandez, C., Ley, E., & Steel, M. (2001). Benchmark priors for Bayesian model averaging. Journal of Econometrics, 100, 381–427. George, E. I., & Foster, D. P. (2000). Calibration and empirical Bayes variable selection. Biometrika, 87, 731–747. George, E. I., & McCulloch, R. E. (1993). Variable selection via Gibbs sampling. Journal of the American Statistical Association, 88, 881–889. George, E. I., & McCulloch, R. E. (1997). Approaches to Bayesian variable selection. Statistica Sinica, 7, 339–379. George, E. I., Sun, D., & Ni, S. (2008). Bayesian stochastic search for VAR model restrictions. Journal of Econometrics, 142, 553–580. Gianone, D., Reichlin, L., & Sala, L. (2004). Monetary policy in real time (and comments). In: M. Gertler & K. Rogoff (Eds), NBER Macroeconomics Annual 2004 (pp. 161–224). Cambridge: The MIT Press. Hoeting, J., Madigan, D., Raftery, A., & Volinsky, C. (1998). Bayesian model averaging: A tutorial. Statistical Science, 14, 382–417. Kadiyala, K. R., & Karlsson, S. (1997). Numerical methods for estimation and inference in Bayesian VAR-models. Journal of Applied Econometrics, 12, 99–132.

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Koop, G., & Potter, S. (2004). Forecasting in dynamic factor models using Bayesian model averaging. Econometrics Journal, 7, 550–565. Smith, M., & Kohn, R. (1996). Nonparametric regression using Bayesian variable section. Journal of Econometrics, 75, 317–343. Smith, M., & Kohn, R. (2002). Parsimonious covariance matrix estimation for longitudinal data. Journal of the American Statistical Association, 97, 1141–1153. Stock, J. H., & Watson, M. W. (2002). Macroeconomic forecasting using diffusion indexes. Journal of Business and Economic Statistics, 20, 147–162. Stock, J. H., & Watson, M. W. (2003). Forecasting output and inﬂation: The role of asset prices. Journal of Economic Literature, 41, 788–829. Stock, J. H., & Watson, M. W. (2005a). Forecasting with many predictors. Unpublished Manuscript. Princeton University, Princeton, NJ (prepared for The handbook of economic forecasting). Stock, J. H., & Watson, M. W. (2005b). Implications of factor models for VAR analysis. Unpublished Manuscript. Princeton University, Princeton, NJ. Strachan, R. W., & van Dijk, H. K. (2007). Bayesian model averaging in vector autoregressive processes with an investigation of stability of the US great ratios and risk of a liquidity trap in the USA, UK and Japan. Econometric Institute Report EI 2007-11. Erasmus University Rotterdam, the Netherlands. Yuan, M., & Lin, Y. (2005). Efﬁcient empirical Bayes variable selection and estimation in linear models. Journal of the American Statistical Association, 100, 1215–1225.

APPENDIX A. TECHNICAL APPENDIX – A GIBBS SAMPLER FOR VAR RESTRICTION SEARCH The priors described in Section 3 combined with the likelihood function of a VAR model, will allow us to draw the full conditional distributions. The likelihood of the VAR model y ¼ zu þ , Nð0; SÞ, where R1=WWu, is:

1 Lðyju; WÞ / jWjT exp tr W0 ðy zuÞ0 ðy zuÞW 2

1 ^ 0 WW0 ðz0 zÞ ðu uÞ ^ ¼ jWjT exp ðu uÞ 2

1 ^ 0 ðy zuÞ ^ tr WW0 ðy zuÞ 2 ^ is the MLE of u. This form of the likelihood function allows to where u derive the posterior of u. In order to derive the posterior of the elements of W ﬁrst rewrite the likelihood function in convenient form. Deﬁne SðuÞ ¼ ðy zuÞ0 ðy zuÞ and write SðuÞ ¼ ðsij Þ. For j ¼ 2; . . . ; m deﬁne the (m1) vectors sj ¼ ðs1j ; . . . ; sj1; j Þ0 containing the upper diagonal elements of S(u), and the (m1) matrices Sj containing the upper left j j submatrix of S(u). Deﬁne also v1 ¼ s11 and vi ¼ jSi j=jSi1 j ¼ sii s0i S1 i1 si for i ¼ 2; . . . ; m.

421

Forecasting in Vector Autoregressions with Many Predictors

The likelihood function takes the following form Lðyju; WÞ

( )# m m 0 X 1 X 2 1 1 c vi þ Zj þ cjj Sj1 sj S j1 Zj þ cjj S j1 sj / ðcii Þ exp 2 i¼1 ii i¼1 j¼2 m Y

"

T

Let ( D ¼ diagðh1 ; h2 ; . . . ; hnf Þ with hi ¼

t0i ;

if gi ¼ 0

t1i ;

if gi ¼ 1

; for i ¼ 1; . . . ; nf m: (

Similarly, deﬁne Dj ¼ diagðh1j ; . . . ; hj1; j Þ where hij ¼

k0ij ; k1ij ;

if oij ¼ 0 , for if oij ¼ 1

i ¼ 1; . . . ; j 1 and j ¼ 2; . . . ; m. Then we can rewrite Eqs. (5) and (7) as ðfk jcÞ Nð0; DDÞ iid

ðgj jxj Þ N j1 ð0; Dj Dj Þ;

for j ¼ 2; . . . ; m

respectively. Denote the combined prior of fc and fk as f Nðf0 ; V 0 Þ. Given starting values, model parameters are drawn from their conditionals for r ¼ 1; . . . ; R iterations. 1. Draw ðwðrÞ juðr1Þ ; cðr1Þ ; xðr1Þ ; yÞ from the gamma distribution 1 2 ðcii ju; c; x; yÞ Gamma ai þ T; Bi 2 where 8 1 > > if i ¼ 1 < b1 þ s11 ; 2 Bi ¼ 1 > > : bi þ sii s0i ½S i1 þ ðDi Di Þ1 1 si ; if i ¼ 2; . . . ; m 2 2. Draw ðgðrÞ jwðrÞ ; uðr1Þ ; cðr1Þ ; xðr1Þ ; yÞ from the normal distribution ðgj jw; u; c; x; yÞ N j1 ðlj ; Dj Þ where

1 lj ¼ cjj S j1 þ ðDi Di Þ1 si

1 Dj ¼ S j1 þ ðDi Di Þ1

422

DIMITRIS KOROBILIS

3. Draw ðxðrÞ jgðrÞ ; CðrÞ ; uðr1Þ ; cðr1Þ ; xðr1Þ ; yÞ from Bernoulli distribution uij1 oij Bernoulli uij1 þ uij2 where for i ¼ 1 ; . . . ; j 1 and j ¼ 2; . . . ; m: uij1

uij2

! c2ij 1 ¼ exp 2 qij k0ij 2k0ij

! c2ij 1 ¼ exp 2 ð1 qij Þ k1ij 2k1ij

4. Draw ðfðr1Þ jcðr1Þ ; RðrÞ ; xðr1Þ ; yÞ from the normal distribution, where f ¼ vecðuÞ, and RðsÞ is computed from wðsÞ and gðsÞ . ðfjc; g; x; w; yÞ N nf ðl; DÞ where o

1 n ^ þ ðV 0 Þ1 f l ¼ ðWW0 Þ ðz0 zÞ þ ðV 0 Þ1 ððWW0 Þ ðz0 zÞÞf 0

1 D ¼ ðWW0 Þ ðz0 zÞ þ ðV 0 Þ1 where f^ are the elements of the ‘‘stacked’’ matrix of MLE coefﬁcients, ^ ¼ vecððz0 zÞ1 z0 yÞ. that is f^ ¼ vecðuÞ 5. Draw ðcðrÞ jwðrÞ ; uðrÞ ; xðrÞ ; yÞ from Bernoulli distribution ui1 gi Bernoulli ui1 þ ui2 where for i ¼ 1; . . . ; nf : ui1 ¼

1 f2 exp i2 pi t0i 2t0i

1 f2i ui2 ¼ exp 2 ð1 pi Þ t1i 2t1i

Forecasting in Vector Autoregressions with Many Predictors

423

APPENDIX B. DESCRIPTION OF DATA This table lists the 132 variables in the dataset used. The third column indexes the respective transformation applied to each of the variables to ensure stationarity (at least approximately). Let vt and xt be the untransformed value and transformed values respectively, then there are ﬁve cases: (1) lv: xt ¼ vt (level), (2) ln: xt ¼ logðvt Þ (logarithm), (3) Dlv: xt ¼ vt vt1 (ﬁrst difference), (4) Dln: xt ¼ lnðzt =zt1 Þ (growth rate), and (5) D2ln: xt ¼ Dðlnðvt =vt1 ÞÞ (second difference of the logarithm). This table is from Stock and Watson (2005b) and the reader should seek there the original source of the data. No.

Mnemonic

Trans.

1.

A0M052

Dln

2.

A0M051

Dln

3. 4.

A0M224_R A0M057

Dln Dln

5.

A0M059

Dln

6. 7.

IPS10 IPS11

Dln Dln

8. 9.

IPS299 IPS12

Dln Dln

10.

IPS13

Dln

11.

IPS18

Dln

12.

IPS25

Dln

13. 14.

IPS32 IPS34

Dln Dln

15.

IPS38

Dln

Description Personal income (ar, bil. chain 2000 $) Personal income less transfer payments (ar, bil. chain 2000 $) Real consumption (A0M224/GMDC) Manufacturing and trade sales (mil. chain 1996 $) Sales of retail stores (mil. chain 2000 $) Industrial production index-total index Industrial production index – products, total Industrial production index – ﬁnal products Industrial production index – consumer goods Industrial production index – durable consumer goods Industrial production index – nondurable consumer goods Industrial production index – business equipment Industrial production index – materials Industrial production index – durable goods materials Industrial production index – nondurable goods materials

424

DIMITRIS KOROBILIS

APPENDIX B. (Continued ) No.

Mnemonic

Trans.

16.

IPS43

Dln

17.

IPS307

Dln

18. 19. 20. 21.

IPS306 PMP A0M082 LHEL

Dln lv Dlv Dlv

22.

LHELX

Dlv

23.

LHEM

Dlv

24.

LHNAG

Dlv

25.

LHUR

Dlv

26.

LHU680

Dlv

27.

LHU5

Dln

28.

LHU14

Dln

29.

LHU15

Dln

30.

LHU26

Dln

31.

LHU27

Dln

32.

A0M005

Dln

33.

CES002

Dln

34.

CES003

Dln

35.

CES006

Dln

Description Industrial production index – manufacturing Industrial production index – residential utilities Industrial production index – fuels NAPM production index (percent) Capacity utilization (mfg) Index of help-wanted advertising in newspapers (1967=100;sa) Employment: ratio; help-wanted ads/ no. unemployed clf Civilian labor force: employed, total (thous.) Civilian labor force: employed, nonagricultural industries (thous.) Unemployment rate: all workers, 16 years & over (%) Unemployment by duration: average (mean) duration in weeks Unemployment by duration: persons unemployed less than 5 wks (thous.) Unemployment by duration: persons unemployed 5 to 14 wks (thous.) Unemployment by duration: persons unemployed 15 wks+(thous.) Unemployment by duration: persons unemployed 15 to 26 wks (thous.) Unemployment by duration: persons unemployed 27 wks+(thous.) Average weekly initial claims, unemployment insurance (thous.) Employees on nonfarm payrolls – total private Employees on nonfarm payrolls – goods-producing Employees on nonfarm payrolls – mining

Forecasting in Vector Autoregressions with Many Predictors

425

APPENDIX B. (Continued ) No.

Mnemonic

Trans.

36.

CES011

Dln

37.

CES015

Dln

38.

CES017

Dln

39.

CES033

Dln

40.

CES046

Dln

41.

CES048

Dln

42.

CES049

Dln

43.

CES053

Dln

44.

CES088

Dln

45.

CES140

Dln

46.

A0M048

Dln

47.

CES151

lv

48.

CES155

Dlv

49.

AOM001

lv

50. 51.

PMEMP HSFR

lv ln

52.

HSNE

ln

53.

HSMW

ln

54. 55.

HSSOU HSWST

ln ln

Description Employees on nonfarm payrolls – construction Employees on nonfarm payrolls – manufacturing Employees on nonfarm payrolls – durable goods Employees on nonfarm payrolls – nondurable goods Employees on nonfarm payrolls – service-providing Employees on nonfarm payrolls – trade, transportation, and utilities Employees on nonfarm payrolls – wholesale trade Employees on nonfarm payrolls – retail trade Employees on nonfarm payrolls – ﬁnancial activities Employees on nonfarm payrolls – government Employee hours in nonagricultural establishments (ar, bil. hours) Average weekly hours of production or nonsupervisory workers on private nonfarm payrolls Average weekly hours of production or nonsupervisory workers on private nonfarm payrolls Average weekly hours: manufacturing (hours) NAPM employment index (percent) Housing starts: nonfarm (1947–58); total farm & nonfarm (1959) (thous.) Housing starts: Northeast (thousands of units) Housing starts: Midwest (thousands of units) Housing starts: South (thousands of units) Housing starts: West (thousands of units)

426

DIMITRIS KOROBILIS

APPENDIX B. (Continued ) No.

Mnemonic

Trans.

56.

HSBR

ln

57.

HSBNE

ln

58.

HSBMW

ln

59.

HSBSOU

ln

60.

HSBWST

ln

61. 62. 63.

PMI PMNO PMDEL

lv lv lv

64. 65.

PMNV A0M008

lv Dln

66.

A0M007

Dln

67.

A0M027

Dln

68.

A1M092

Dln

69.

A0M070

Dln

70.

A0M077

Dlv

71. 72. 73. 74. 75.

FM1 FM2 FM3 FM2DQ FMFBA

D2ln D2ln D2ln Dln D2ln

76.

FMRRA

D2ln

77.

FMRNBA

D2ln

78.

FCLNQ

D2ln

Description Housing authorized: total new priv housing units (thousands) Houses authorized by build. permits: Northeast (thousands of units) Houses authorized by build. permits: Midwest (thousands of units) Houses authorized by build. permits: South (thousands of units) Houses authorized by build. permits: West (thousands of units) Purchasing managers’ index (sa) NAPM new orders index (percent) NAPM vendor deliveries index (percent) NAPM inventories index (percent) Mfrs’ new orders, consumer goods and materials (bil. chain 1982 $) Mfrs’ new orders, durable goods industries (bil. chain 2000 $) Mfrs’ new orders, nondefense capital goods (mil. chain 1982 $) Mfrs’ unﬁlled orders, durable goods indus. (bil. chain 2000 $) Manufacturing and trade inventories (bil. chain 2000 $) Ratio, mfg. and trade inventories to sales (based on chain 2000 $) Money stock: M1 (bil$, sa) Money stock: M2 (bil$, sa) Money stock: M3 (bil$, sa) Money supply – M2 in 1996 dollars (bci) Monetary base, adjusted for reserve requirement changes(mil$, sa) Depository inst. reserves: total, adjusted for reserve req changes (mil$, sa) Depository inst. reserves: non-borrowed, adj reserve req changes (mil$, sa) Commercial & industrial loans outstanding in 1996 dollars (bci)

Forecasting in Vector Autoregressions with Many Predictors

427

APPENDIX B. (Continued ) No.

Mnemonic

Trans.

79.

FCLBMC

lv

80.

CCINRV

D2ln

81.

A0M095

Dlv

82.

FSPCOM

Dln

83.

FSPIN

Dln

84.

FSDXP

Dlv

85.

FSPXE

Dln

86.

FYFF

Dlv

87. 88.

CP90 FYGM3

Dlv Dlv

89.

FYGM6

Dlv

90.

FYGT1

Dlv

91.

FYGT5

Dlv

92.

FYGT10

Dlv

93.

FYAAAC

Dlv

94.

FYBAAC

Dlv

SCP90 SFYGM3 SFYGM6 SFYGT1 SFYGT5 SFYGT10 SFYAAAC SFYBAAC

lv lv lv lv lv lv lv lv

95. 96. 97. 98. 99. 100. 101. 102.

Description Wkly rp lg com’l banks:net change com’l & indus loans (bil$,saar) Consumer credit outstanding – non-revolving Ratio, consumer installment credit to personal income (pct.) S&P’s common stock price index: composite (1941–43=10) S&P’s common stock price index: industrials (1941–43=10) S&P’s composite common stock: dividend yield (% per annum) S&P’s composite common stock: priceearnings ratio (%) Interest rate: Federal funds (effective) (% per annum) Commercial paper rate Interest rate: U.S. Treasury bills, sec market, 3-mo. (% per annum) Interest rate: U.S. Treasury bills, sec market, 6-mo. (% per annum) Interest rate: U.S. Treasury const maturities, 1-yr. (% per annum) Interest rate: U.S. Treasury const maturities, 5-yr. (% per annum) Interest rate: U.S. Treasury const maturities, 10-yr. (% per annum) Bond yield: Moody’s AAA corporate (% per annum) Bond yield: Moody’s BAA corporate (% per annum) CP90 – FYFF (spread) FYGM3 – FYFF (spread) FYGM6 – FYFF (spread) FYGT1 – FYFF (spread) FYGT5 – FYFF (spread) FYGT10 – FYFF (spread) FYAAAC – FYFF (spread) FYBAAC – FYFF (spread)

428

DIMITRIS KOROBILIS

APPENDIX B. (Continued ) No.

Mnemonic

Trans.

103.

EXRUS

Dln

104.

EXRSW

Dln

105.

EXRJAN

Dln

106.

EXRUK

Dln

107.

EXRCAN

Dln

108.

PWFSA

D2ln

109.

PWFCSA

D2ln

110.

PWIMSA

D2ln

111.

PWCMSA

D2ln

112.

PSCCOM

D2ln

113.

PSM99Q

D2ln

114. 115. 116.

PMCP PUNEW PU83

lv D2ln D2ln

117.

PU84

D2ln

118. 119.

PU85 PUC

D2ln D2ln

120. 121. 122.

PUCD PUS PUXF

D2ln D2ln D2ln

123.

PUXHS

D2ln

124.

PUXM

D2ln

Description United States; effective exchange rate (merm) (index no.) Foreign exchange rate: Switzerland (Swiss franc per U.S.$) Foreign exchange rate: Japan (yen per U.S.$) Foreign exchange rate: United Kingdom (cents per pound) Foreign exchange rate: Canada (Canadian $ per U.S.$) Producer price index: ﬁnished goods (82=100,sa) Producer price index: ﬁnished consumer goods (82=100,sa) Producer price index: intermed mat. supplies & components (82=100,sa) Producer price index: crude materials (82=100,sa) Spot market price index: bls & crb: all commodities (1967=100) Index of sensitive materials prices (1990=100)(bci-99a) NAPM commodity prices index (percent) CPI-u: all items (82–84=100,sa) CPI-u: apparel & upkeep (82–84=100,sa) CPI-u: transportation (82–84=100,sa) CPI-u: medical care (82–84=100,sa) CPI-u: commodities (82–84=100,sa) CPI-u: durables (82–84=100,sa) CPI-u: services (82–84=100,sa) CPI-u: all items less food (82–84=100,sa) CPI-u: all items less shelter (82–84=100,sa) CPI-u: all items less medical care (82–84=100,sa)

429

Forecasting in Vector Autoregressions with Many Predictors

APPENDIX B. (Continued ) No.

Mnemonic

Trans.

Description PCE, impl price deﬂator (1987=100) PCE, impl price deﬂator: Durables (1987=100) PCE, impl price deﬂator: Nondurables (1996=100) PCE, impl price deﬂator: Services (1987=100) Average hourly earnings of production or nonsupervisory workers on private nonfarm payrolls: goods Average hourly earnings of production or nonsupervisory workers on private nonfarm payrolls: construction Average hourly earnings of production or nonsupervisory workers on private nonfarm payrolls: manufacturing U. of Michigan index of consumer expectations (bcd-83)

125. 126.

GMDC GMDCD

D2ln D2ln

127.

GMDCN

D2ln

128.

GMDCS

D2ln

129.

CES275

D2ln

130.

CES277

D2ln

131.

CES278

D2ln

132.

HHSNTN

Dlv

APPENDIX C Table C1.

Average Posterior Probabilities of Explanatory Variables.

Explanatory Variable at Time t

Personal income IP index – Final products IP index – Manufacturing Capacity Utilization Employment ratio Civilian labor force: Total employed Employees on nonfarm payrolls – Total private Employees on nonfarm payrolls – Manufacturing Employees on nonfarm payrolls – Serviceproviding

Dependent Variable ut+12

cpit+12

rt+12

0.141 0.251 0.593 1 0.011 0.428 0.811

0.001 0.003 0.016 0.124 0.002 0.003 0.018

0.949 0.564 0.170 0.032 0.992 0.652 0.317

0.50

0.014

0.33

1

0.023

0.826

430

DIMITRIS KOROBILIS

APPENDIX C. (Continued ) Explanatory Variable at Time t

Employees on nfm prl – Trade, transportation, and utilities Employees on nonfarm payrolls – Wholesale trade Employees on nonfarm payrolls – Financial activities Average weekly hours of production Housing starts: Total Housing authorized: Total Houses authorized by building permits: Northeast Houses authorized by building permits: Midwest Houses authorized by building permits: South Houses authorized by building permits: West Consumer installment credit to Personal income (ratio) S&P’S common stock price index: Composite S&P’s composite common stock: Dividend yield Commercial paper rate (spread from Fed Funds Rate) 3-month interest rate (spread from FFR) 6-month interest rate (spread from FFR) 1-year interest rate (spread from FFR) 5-year interest rate (spread from FFR) 10-year interest rate (spread from FFR) Bond yield: Moody’s BAA corporate (spread from FFR) NAPM commodity prices index CPI-U: Durables CPI-U: All items less shelter

Dependent Variable ut+12

cpit+12

rt+12

0.878

0.003

0.682

0.296

0.003

1.000

0.687

0.008

0.697

0.001 0.879 1 1

0.082 0.001 0.001 0.105

0.941 0.040 1 0.003

1

0.025

0.018

1

0.001

0.006

1 0.013

0.000 0.001

1 1

0.962 0.092

0.132 0.001

0.004 0.937

0.028

0.7452

0.851

0.002 0.005 0.941 1 1 0.001

0.087 0.002 0.752 0.982 0.861 0.000

1 1 0.992 1 1 0.978

0.001 0.172 0.246

0.867 0.002 0.006

0.857 0.543 0.692

Notes: Highest average posterior probabilities of inclusion among 129 explanatory variables in a VAR in unemployment (ut+h, unemployment rate: all workers, 16 years and over), consumer price index (cpit+h, cpi-u: all items), and interest rate (rt+h, interest rate: effective federal funds rate) for h=12. Variables with probability of inclusion less than 0.5 in all three equations are not reported in the table. Reported probabilities that are larger than 0.5 are in bold.

431

Forecasting in Vector Autoregressions with Many Predictors

Table C2.

Posterior Probabilities of Autoregressive Lags.

Dependent Variable

Most Important Lags (ProbabilityW0.5)

Model Average Posterior Probability

ut+12

rt7 rt13 Own lags from 1 to 7 (cpit to cpit6): cpit7 rt6

0.56 0.74 1

cpit+12

rt+12

0.83 1

Notes: Highest average posterior probabilities of inclusion of autoregressive lags in a VAR in unemployment (ut+h, unemployment rate: all workers, 16 years and over), consumer price index (cpit+h, cpiu: all items), and interest rate (rt+h, interest rate: effective federal funds rate) for h=12.

Table C3.

Forecast Comparison.

RMSFE Relative to VAR(7) with No Factors PI

PPI

CPI

PCED

BVAR with factors (Bayesian model averaging) h=1 0.94 1.00 0.90 0.96 1.08 h=6 1.06 0.96 0.93 0.94 0.95 h=12 0.97 0.92 0.99 1.02 0.98

0.88 0.92 0.92

0.95 1.05 0.95

1.09 0.94 0.96

BVAR with factors (model selection) h=1 0.86 0.98 0.87 0.96 h=6 0.90 0.97 0.85 0.92 h=12 0.87 0.99 0.91 0.98

1.06 0.94 0.89

0.91 0.94 0.87

0.93 0.98 0.99

0.91 0.93 0.96

VAR with h=1 h=6 h=12

1.22 1.12 0.99

0.99 0.97 0.90

1.01 1.06 1.10

0.97 0.94 0.95

factors 0.92 0.93 0.97

IP

EMP

UR

(BIC selection) 0.99 0.94 0.99 0.97 0.94 0.94 1.04 0.98 1.05

TBILL

Notes: The variables of interest are: PI: Personal Income (A0M052), IP: Industrial Production (IPS10), EMP: Employment Rate (CES002), UR: Unemployment Rate (LHUR), TBILL: 3-month Treasury Bill Rate (FYGM3), PPI: Producer Price Index (PWFSA), CPI: Consumer Price Index (PUNEW), and PCED: PCE Deﬂator (GMDC).

BAYESIAN INFERENCE IN A COINTEGRATING PANEL DATA MODEL$ Gary Koop, Roberto Leon-Gonzalez and Rodney Strachan ABSTRACT This paper develops methods of Bayesian inference in a cointegrating panel data model. This model involves each cross-sectional unit having a vector error correction representation. It is ﬂexible in the sense that different cross-sectional units can have different cointegration ranks and cointegration spaces. Furthermore, the parameters that characterize short-run dynamics and deterministic components are allowed to vary over cross-sectional units. In addition to a noninformative prior, we introduce an informative prior which allows for information about the likely location of the cointegration space and about the degree of similarity in coefﬁcients in different cross-sectional units. A collapsed Gibbs sampling algorithm is developed which allows for efﬁcient posterior inference. Our methods are illustrated using real and artiﬁcial data.

$

Gary Koop, Roberto Leon-Gonzalez, and Rodney Strachan are Fellows of the Rimini Centre for Economic Analysis.

Bayesian Econometrics Advances in Econometrics, Volume 23, 433–469 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23013-6

433

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GARY KOOP ET AL.

1. INTRODUCTION The growing availability of panel data with large T dimension (i.e., where the number of time series observations is large) has stimulated a growth in research, both empirical and theoretical, which discusses time series issues in panel data models. Of particular interest are issues relating to nonstationarity and cointegration. In this paper, we develop a Bayesian approach to the analysis of cointegration in panels. We use a modeling framework which allows for great ﬂexibility in the way heterogeneity across cross-sectional units is incorporated. In particular, we allow for both cointegrating vectors and ranks to vary over N. Our use of Bayesian methods allows for the cointegrating ranks to be treated as random variables. Thus, our methods can either be used to select a particular model with speciﬁed cointegrating ranks or to average across different cointegrating ranks. We also consider restricted models of interest (e.g., where all cross-sectional units have the same cointegrating rank). The use of Bayesian methods requires elicitation of a prior. We develop two priors, a noninformative and an informative one. The latter allows for the incorporation of prior beliefs that the same cointegrating relationship exists for all cross-sectional units. Furthermore, it allows for what we call ‘‘soft homogeneity’’ restrictions (i.e., that comparable parameters in different cross-sectional units are likely to be similar to one another). We derive efﬁcient methods of posterior analysis in our class of models and illustrate our methods using artiﬁcial data and an application involving a monetary exchange rate model (see Groen, 2000; Groen & Kleibergen, 2003). The importance of this area of research is evidenced by the increasing tendency for researchers to employ panels of nonstationary processes in empirical studies in macroeconomics and international economics. For instance, the survey paper by Baltagi and Kao (2000) identiﬁes many areas of application, including purchasing power parity (PPP), growth convergence, and international R&D spillovers. To give one example which illustrates the issues which can be addressed through the use of panel data consider Jacobson, Lyhagen, Larsson, and Nesse´n (2002). These authors use a multivariate panel cointegration model and demonstrate that, although strong purchasing power parity restrictions are rejected, the location of the cointegrating space is similar for all countries considered. This provides some evidence in support of PPP. In terms of the frequentist econometric literature, there has been a range of methods proposed to obtain inference relating to cointegration in panel data models. Among many others, we note that residual-based, LM and

Bayesian Inference in a Cointegrating Panel Data Model

435

likelihood-based tests have been proposed by Kao (1999), McCoskey and Kao (1998), Pedroni (2004), Larsson, Lyhagen, and Lo¨thgren (2001), and Groen and Kleibergen (2003). The estimation methods used in these papers vary from OLS through maximum likelihood and generalized method of moments. The extent of this literature prevents us giving even a reasonable coverage here and so we refer the reader to the surveys by Phillips and Moon (2000) and Baltagi and Kao (2000). While the bulk of the work to date has used frequentist methods, there have been a number of Bayesian approaches to obtain inference in dynamic panels. Pesaran, Hsiao, and Tahmiscioglu (1999) present frequentist and Bayesian approaches to estimating the mean of the cross-sectional distribution for the coefﬁcients in a dynamic panel data model. They show that the Bayesian approach performs reasonably well in ﬁnite samples, and is even preferable to some consistent estimators when ﬁnite sampling performance is considered. Although they impose a stability condition, thus precluding discussion of issues relating to unit roots and cointegration, this assumption could be relaxed (see also Hsiao & Pesaran, 2004). Li (1999) investigates PPP by considering support for symmetry and proportionality restrictions in the PPP relationship. She allows for stationary AR(1) errors in the relationship between log exchange rates and prices. Interestingly, while this paper does not explicitly consider cointegration, with one small change the model of Li could – using a triangular setup as proposed by Phillips (1991) – be easily extended to allow investigation of whether or not cointegration between log exchange rates and prices occurs. We are aware of only one paper explicitly proposing a Bayesian approach to estimation of a cointegrating system in panel data models. Carmeci (2005) presents a state space model which implies cointegration by directly modeling the common stochastic trends. Under the assumption that the cointegrating rank is known and assumed equal in every cross-sectional unit, the author develops Bayesian methods for estimation. We are not aware of any paper that presents a fully Bayesian method of inference on cointegration in panels, when the cointegrating rank is unknown and may differ across cross-sectional units. The present paper attempts to address this gap in the literature. The remainder of the paper is organized as follows. Section 2 introduces the model and describes the elements of Bayesian analysis: likelihood, priors, and methods of posterior simulation. Section 3 illustrates our methods using artiﬁcial data and Section 4 demonstrates the ﬂexibility of inference in the application used in Groen and Kleibergen (2003), which involves an interesting set of restrictions implied by economic theory. Section 5 concludes the paper.

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GARY KOOP ET AL.

2. THE MODELS In a standard time series framework, cointegration is typically investigated using a vector error correction model (VECM). To establish notation, to investigate cointegrating relationships involving an n-vector, yt, we write the VECM for t ¼ 1, . . . , T as: Dyt ¼ Pyt1 þ

l X

Gh Dyth þ Fd t þ t

(1)

h¼1

where the n n matrix P ¼ abu, where a and b are n r full rank matrices and dt denotes deterministic terms.1 The value of r determines the number of cointegrating relationships. et is a normal mean zero error with positive deﬁnite covariance matrix. Before extending Eq. (1) to the panel data case, it is important to digress brieﬂy to motivate an important issue in Bayesian analysis of cointegrated models. The VECM suffers from both local and global identiﬁcation problems. The local identiﬁcation problem occurs since, if a ¼ 0, b does not appear in the likelihood function. The global identiﬁcation problem can be seen by noting that P ¼ abu and P ¼ aGG1bu are identical for any nonsingular G. This indeterminacy is commonly surmounted by imposing the so-called linear normalization where b ¼ [Ir Bu]u. However, there are some serious drawbacks to this linear normalization (see Strachan & Inder, 2004; Strachan & van Dijk, 2004). Researchers in this ﬁeld (see Strachan & Inder, 2004; Strachan & van Dijk, 2004; Villani, 2005, 2006) point out that it is only the cointegration space that is identiﬁed (not particular cointegrating vectors) and that, for most purposes (including prior elicitation), it is preferable to think in terms of the cointegration space. Accordingly, we introduce notation for the space spanned by b, p ¼ spðbÞ. We can generalize Eq. (1) to the panel data case by including i subscripts to denote the cross-sectional unit which we refer to as the ‘‘individual’’ hereafter (where i ¼ 1, . . . , N). That is, yi,t is an n vector2 of observations on the dependent variables for individual i at time t and the panel VECM is written as: Dyi;t ¼ Pi yi;t1 þ

li X

Gi;h Dyi;th þ Fi d i;t þ i;t

(2)

h¼1

where now Pi ¼ ai b0i where ai and bi are n ri full rank matrices. Our model allows for the number of cointegrating relationships to vary across

Bayesian Inference in a Cointegrating Panel Data Model

437

individuals and thus, we extend our previous notation such that the cointegration spaces are now pi ¼ spðbi Þ. The covariance matrices for vectors ei,t are assumed to be Eði;t 0j;s Þ

¼

(P

for t ¼ s

0

for tas

ij

(3)

In other words, we are assuming the errors to be uncorrelated over time, but correlated across equations for a given individual and correlated across individuals. Note that the last assumption differs from much of the previous literature. For instance, Larsson et al. (2001) use a more restrictive model assuming Eði;t 0j;s Þ ¼ 0 if i 6¼ j for all t and s. Although allowing for a correlation between errors for different individuals is not usually done with microeconomic survey data, with macroeconomic panels where the ‘‘individuals’’ are countries such a correlation is potentially important. We are therefore following the more general model of Groen and Kleibergen (2003) which does allow for such a correlation. Note also that our model is more ﬂexible than the one of Groen and Kleibergen (2003) in that we relax the assumption of a common cointegrating rank. There are many features of Eq. (2) that the researcher might be interested in. For each individual, we would naturally be interested in the dimension of the cointegrating space, ri, and whether ri ¼ r for all i. Other questions of interest relate to the cointegrating spaces, pi ¼ spðbi Þ. A restricted version of Eq. (2) would have the same cointegrating relationships (i.e., the same ri and bi) for every individual and, thus, pi ¼ p. Alternatively, if different individuals have different numbers of cointegrating vectors, then we might be interested in whether all of the cointegrating spaces lie within some more general one. That is, if rirr for i ¼ 1, . . . , N and p is a cointegration space with dimension r, then we might be interested in investigating whether pi p for i ¼ 1, . . . , N. As a simple illustration of how these questions might arise, consider the balanced growth hypothesis in the real business cycle model presented by King, Plosser, Stock, and Watson (1991). Assume yi,t ¼ (ci,t, ai,t, gi,t)u where ci,t is log consumption for country i, ai,t the log investment for that country, and gi,t the log income. If the elements of the vector yi,t are I(1) and are cointegrated then 0orio3. If there are two cointegrating relationships (ri ¼ 2) and the logs of the great ratios of consumption to income and investment to income are stable such that ci,tgi,t and ai,tgi,t are I (0) in

438

GARY KOOP ET AL.

every country, then the cointegrating space, pi , is 0 1 1 0 B C 1 A pi ¼ p ¼ sp@ 0 1 1 In an empirical analysis using panel data, it would be of interest to investigate this restriction (i.e., whether two cointegrating vectors exist for each country and whether their values are consistent with the great ratios). However, it is possible that some countries might only have one cointegrating relationship, so that ri ¼ 2 for some countries and ri ¼ 1 for others. In this case, investigating whether pi ¼ p for all i ¼ 1, . . . , N is not reasonable. Instead, the researcher may be interested in investigating whether the cointegrating relationships either involve the great ratios individually (for ri ¼ 2) or involve a linear combination of the (logs of ) the great ratios. In terms of our notation, this involves investigating whether pi p for i ¼ 1, . . . , N. In most empirical applications, the cointegrating spaces will be of most interest and, hence, the researcher will be most interested in a set of models deﬁned by restrictions on these. However, it is also common for the set of models to be broadened by considering different forms of the deterministic processes, di,t, and the number of lags li and it might be desirable to allow these to vary across individuals. Thus, in empirical work, the researcher might want to consider a very wide range of models indeed. However, in order to focus on the central issues relating to cointegration, we will assume a common lag length for all individuals (i.e., li ¼ l for all i) and common deterministic process (i.e., di,t ¼ dt for all i) and develop methods of inference for ri and pi . 2.1. The Likelihood Function In this section we show two representations of the likelihood, involving different parameterizations, which we draw on in our discussion of posterior simulation. Note that the matrix of long-run multipliers can be written as: 0 0 bi a0i ¼ ½bi ki ½ai k1 i bi Ai

(4)

where bi is restricted to be semi-orthogonal (for reasons described in the next section) and ki is positive deﬁnite and deﬁned so that Ai is semiorthogonal. Here we have used bi ¼ bi ki and ai ¼ Ai k0i . There are many

Bayesian Inference in a Cointegrating Panel Data Model

439

choices for ki which satisfy these properties, but a convenient one we will use here is: 0

ki ¼ ða0i ai Þ1=2 ¼ ðbi bi Þ1=2

(5)

For reasons explained below, our posterior simulator will involve switching between the parameterizations in Eq. (4). To establish notation, we collect the n n blocks Sij into the Nn Nn matrix S ¼ {Sij}. Collecting the (n 1) vectors ei,t into (T n) matrices ei ¼ (ei,1, . . . , ei,T)u, then collecting these matrices into the (T Nn) matrix e ¼ (e1, . . . , eN), we obtain e ¼ vec (e) being the vector of errors. This vector has covariance matrix Eðee0 Þ ¼ V e ¼ ðS I T Þ

(6)

The density of the errors, a key building block in forming the likelihood for this model, is then 1 1 jSjT=2 exp e0 S1 I T e ¼ jSjT=2 exp trS1 0 2 2 We next give two representations for e that will prove useful in developing a sampling scheme for the parameters. We rewrite Eq. (2) by deﬁning zi;t ¼ b0i yi;t1 , the 1 (kþri) vector X i;t ¼ ðz0i;t ; Dy0i;t1 ; . . . ; Dy0i;tl ; d 0t Þ, where k is the number of deterministic terms plus n times the number of lags (assumed to be common to all individuals and, hence, we have not included an i subscript), and the (kþri) n matrix Bi ¼ (ai,Ci,1, . . . , Ci,l, Ui)u and, thus, Dy0i;t ¼ X i;t Bi þ 0i;t

(7)

If we stack the vectors in Eq. (7) over t as Dyi ¼ (Dyi,1, . . . , Dyi,T)u and X i ¼ ðX 0i;1 ; . . . ; X 0i;T Þ0 then, we can write Dyi ¼ XiBiþei. Vectorizing this equation gives us the form vecðDyi Þ ¼ ðI n X i ÞvecðBi Þ þ ei or yi ¼ xi bi þ ei where yi ¼ vecP (Dyi), xi ¼ (InXi), bi ¼ vec(Bi), and ei ¼ vec(ei) such that Eðei e0j Þ ¼ ij I T . We collect the vectors yi and bi into the vectors y ¼ ðy01 ; . . . ; y0N Þ0 and bP¼ ðb01 ; . . . ; b0N Þ0 , and deﬁne the matrix x as the TNn Nnðk þ rÞ (where r ¼ N i¼1 ri =N) block diagonal matrix with diagonal equal to (x1, . . . , xN). Using these deﬁnitions, we can express the full system of

440

GARY KOOP ET AL.

equations as yxb ¼ e. The likelihood can now be expressed as 1 0 1 T=2 Lðb; S; bÞ ¼ jSj exp ðy xbÞ V e ðy xbÞ 2 0 i 1h 2 T=2 1 ^ ^ exp s þ b b V bb ¼ j Sj 2

ð8Þ

1 0 1 ^ 0 1 where s2 ¼ yu MVy, MV ¼ V 1 e V e xVx V e , b ¼ Vx V e y, Ve ¼ (SIT), 0 1 1 and V ¼ ðx V e xÞ . Thus from our ﬁrst representation of the likelihood, we can see that the form of the posterior for b conditional upon the bi and S is normal if the (conditional) prior for b is ﬂat or normal. Our next representation of the likelihood demonstrates that we can obtain a normal form for the cointegrating vectors (conditional on the other parameters of the model). That is, the conditional posterior density of the vector bb ¼ ðb0b ;1 ; . . . ; b0b ;N Þ0 , where bb ;i ¼ vecðbi Þ, can be shown to be normal. To do this let us again rewrite Eq. (2) but this time deﬁne the 1 k vector wi;t ¼ ðDy0i;t1 ; . . . ; Dy0i;tl ; d 0t Þ, and the k n matrix Ci ¼ (Gi,1, . . . , Gi,l, Fi)u and, thus

Dy0i;t ¼ y0i;t1 bi A0i þ wi;t C i þ 0i;t

(9)

If we stack the vectors over t as Dyi ¼ (Dyi,1, . . . , Dyi,T)u, yi,1 ¼ (yi,0, . . . , yi,T1)u, and wi ¼ ðw0i;1 ; . . . ; w0i;T Þ0 , then we can write 0 Dyi ¼ yi;1 bi Ai þ wi C i þ i . Vectorizing this equation we obtain vecðDyi wi C i Þ ¼ ðAi yi;1 Þvecðbi Þ þ vecði Þ or y^i ¼ x^ i bb ;i þ ei where y^i ¼ vecðDyi wi Ci Þ and x^ i ¼ ðAi yi;1 Þ. Now stack the vectors y^i into y^ ¼ ðy^01 ; . . . ; y^0N Þ0 and deﬁne x^ as the TNn Nnr block diagonal matrix with diagonal equal to ðx^ 1 ; . . . ; x^ N Þ so we can express the system of ^ b ¼ e. The likelihood can now be expressed as equations as y^ xb 1 0 T=2 1 ^ b Þ V e ðy^ xb ^ b Þ Lðb; S; bÞ ¼ jSj exp ðy^ xb 2 1 2 0 T=2 1 ^ ^ exp ½sb þ ðbb bb Þ V b ðbb bb Þ ð10Þ ¼ j Sj 2 0 1 1 ^ ^ V ^ M V b ¼ V 1 ^ b x^ 0 V 1 ^ and where s2b ¼ y^0 M V b y, e V e xV e , bb ¼ V b x e y, 0 1 1 ^ . This representation of the likelihood shows that the form V b ¼ ðx^ V e xÞ

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of the posterior for bb (conditional upon the Ci and S) is normal if the (conditional) prior for bb is normal.

2.2. Priors In this section, we describe two classes of priors which may be useful for empirical research. The ﬁrst of these is a noninformative prior, suitable for reference or benchmark purposes. The second is an informative prior which contains what we call ‘‘soft homogeneity’’ restrictions. That is, in many cases economic theory suggests that the cointegration space should be the same for different individuals and of a particular form. In an empirical analysis, the researcher might not want to impose this sort of homogeneity in a strong sense, but, through the use of priors, we can do so in a soft sense. That is, rather than setting parameters to have the same values for all individuals, we specify common informative priors that favor parameter values which are similar for different individuals. This is likely to be of particular beneﬁt since our model contains many parameters and, thus, issues relating to possible overparameterization and efﬁciency of estimation are likely to be important. Before describing the priors, we should highlight some important issues that have arisen in Bayesian analyses of cointegration in nonpanel data contexts. As discussed previously, the VECM suffers from a lack of identiﬁcation (both locally and globally). A large literature has grown which develops priors which attempt to surmount the problems this causes (see the survey paper by Koop, Le´on-Gonzalez, & Strachan, 2005; Koop, Strachan, van Dijk, & Villani, 2006). We will not recreate the arguments of this literature in detail here. Sufﬁce it to note that it is unsatisfactory to use some apparently sensible approaches. For instance, at ﬁrst sight it seems sensible just to use a standard prior (e.g., a ﬂat prior or a normal one) on B after imposing the linear normalization b ¼ [Ir Bu]u. As discussed in Kleibergen and van Dijk (1994), the local nonidentiﬁcation of the model causes problems for this sort of Bayesian approach. The issue here is that when a has reduced rank (e.g., a ¼ 0) the conditional posterior distribution for B|a is equal to its prior (i.e., since B does not enter the likelihood function at the point a ¼ 0 there is no data-based learning about B and, thus, its posterior equals its prior at this point). If the prior for B|a ¼ 0 is improper (as it is in the common ‘‘noninformative’’ case), then the posterior will also be improper. Formally, Kleibergen and van Dijk (1994) associate the local nonidentiﬁcation problem with

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nonexistence of posterior moments and nonintegrability of the posterior (under a common noninformative prior). Kleibergen and van Dijk (1998) additionally point out that local nonidentiﬁcation implies an absorbing state in a Gibbs sampler, thereby violating the convergence conditions for the sampler. With regards to global identiﬁcation, Strachan and Inder (2004) show how the use of linear identifying restrictions places a restriction on the estimable region of the cointegrating space. This paper also provides an extensive discussion of further problems associated with the use of linear identifying restrictions. Strachan and van Dijk (2004) show that a ﬂat and apparently ‘‘noninformative’’ prior on B in the linear normalization favors regions of the cointegration space near where the linear normalization is invalid. Hence, the linear normalization is used under the assumption that it is valid while at the same time the prior says that the normalization is likely to be invalid. Coming out of this literature is the strong message that prior elicitation should be made directly off the cointegration space itself (which is all that is identiﬁed). Several papers, including Strachan (2003), Strachan and Inder (2004), and Villani (2005, 2006) propose various approaches which involve such a focus. In this paper, we extend the general framework outlined in Strachan (2003) and Strachan and Inder (2004) to our panel cointegration model. The advantages of this approach are that they allow us to avoid identiﬁcation restrictions that may restrict the estimable cointegration space, allow for priors which are, in a sense, noninformative (but are proper and, hence, allow for calculation of posterior odds ratios), and offer a convenient framework for incorporating prior information (if the researcher wishes to incorporate it). To brieﬂy motivate this approach (in the nonpanel case), note that a cointegrating space p is an r-dimensional hyperplane in a n-dimensional space and its relation to the cointegrating vectors b is that these vectors lie in and thereby identify that plane. Consider the case where n ¼ 2 and a single cointegration vector exists which is parameterized in polar coordinates b ¼ (cos y sin y)u, where yA[p/2, p/2). It is only y which determines the cointegration space and, thus, as shown in Strachan and Inder (2004), the length of b can be restricted to be unity for identiﬁcation. Crucially, and in contrast to the linear normalization, this identifying restriction does not restrict the estimable cointegration space or distort the weight on the space of the parameter of interest, p. A natural candidate for a noninformative distribution on p is the uniform distribution on y, the parameter governing the direction of b and therefore p.

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To extend these intuitive concepts to general n and r, some additional deﬁnitions are required. Our aim is to provide a rigorous deﬁnition of the intuitive idea of assigning equal prior probability to every possible cointegration space of dimension r. As described in Strachan and Inder (2004), having b being semi-orthogonal such that bub ¼ Ir identiﬁes the cointegrating vectors without placing any restrictions on the cointegrating space. The set of all n r semi-orthogonal matrices is called the Stiefel manifold Vn;r . The Stiefel manifold is a compact space and admits a uniform distribution. In the case where r ¼ 1, one might conceptualize the collection of directions of all n-dimensional unit vectors, b 2 Vn;1 , as describing an n-dimensional unit sphere centered at the origin. Thus, we may visualize a uniform distribution on the n-dimensional unit sphere as characterizing a uniform distribution on Vn;1 . For rW1, we can think of each vector in b as describing a unit sphere with the additional restriction that the vectors are all orthogonal to each other. The Grassman manifold Gn;r is the abstract space of all possible r-dimensional planes of Rn . The cointegration space is an element of the Grassman manifold, that is p 2 Gn;r . In the VECM only the space spanned by the columns of b is identiﬁed, such that we only have information on p ¼ spðbÞ 2 Gn;r . A uniform prior for the cointegration space is therefore given by the uniform distribution on Gn;r . For calculating posterior odds ratios, proper priors are required to avoid Bartlett’s paradox (see Bartlett, 1957). But, since b now has a compact support, the prior over the cointegration space is proper. Formally, this approach uses the natural relationship between the Grassman manifold and the Stiefel manifold and the development of measures on these spaces presented in James (1954). In particular, a key result is that the uniform distribution on Vn;r induces a uniform distribution on Gn;r (see James, 1954; Strachan & Inder, 2004). Thus, it is possible to work with the semiorthogonal matrices, i.e., b 2 Vn;r , and adjust all integrals to account for the fact that Vn;r , is a larger space than Gn;r . In this paper, we have only sketched out the basic ideas relating to prior elicitation in the cointegration models, and refer the reader to the literature we cite above for further details. Sufﬁce it to note here that, in this paper, we extend these ideas to work with the panel cointegration model. 2.2.1. A Noninformative Prior Let bb ¼ ðb0b;1 ; . . . ; b0b;N Þ0 , where bb,,i ¼ vec(bi), contain all the parameters which determine the cointegration spaces. The remaining parameters of the model are S and b, where b is deﬁned between Eqs. (7) and (8). Noting that,

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conditional upon bb, the model reduces to a linear one (see Eq. (7)), a plausible candidate is the standard noninformative prior for multivariate linear models: (11) p b; S; bb / jSjðNnþ1Þ=2 where we add the additional restriction, arising from our wish to be noninformative about the cointegrating space and have an identifying restriction which does not limit that space, that bi is semi-orthogonal such that b0i bi ¼ I ri . Note, however, that although the marginal prior for bb is proper, the joint prior for all the parameters is improper. The impropriety relating to the prior for S is not a problem, since it is common to all models.3 However, a proper prior would be required for the remaining parameters should we wish to calculate posterior odds ratios comparing different cointegrating ranks. That is, if we wish to estimate a single model for speciﬁed values for ri (and speciﬁed values for l and dt) the prior given in Eq. (11) will be appropriate. However, if we are wishing to compare this single model to another with different values for ri (and/or different values for l and dt), then an informative prior for b would be required. It is to such an informative prior to which we turn. However, it is worth noting in passing that a researcher who is interested in model comparison, but would prefer to avoid informative priors, could use information criteria to approximate marginal likelihoods or could adopt a training sample approach. That is, Eq. (11) could be used as a noninformative prior which is then combined with a training sample (e.g., the initial 10% of the data) to yield a ‘‘posterior.’’ This ‘‘posterior’’ can then be used as an informative prior in a posterior analysis involving the rest of the data. See O’Hagan (1995) for a discussion of such an approach. 2.2.2. An Informative Prior (Including Soft Homogeneity Restrictions) In many cases the researcher may wish to specify an informative prior on the cointegrating space. For instance, in our previous example arising from King et al. (1991), the researcher may wish to elicit a prior which implies that the cointegrating space lies in the region implied by the great ratios. Alternatively, the researcher may wish to elicit a prior which implies that the cointegration spaces (or other parameters) for different individuals are similar. We refer to the latter as soft homogeneity restrictions. In addition, in order to avoid the issues relating to Bartlett’s paradox discussed in the previous section, the researcher may wish to elicit an informative prior for b.

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Here we describe an approach to prior elicitation which incorporates these aspects. Some aspects of our prior are best motivated in the context of our posterior simulation algorithm. Hence, we digress brieﬂy to informally discuss computation. Posterior computation is greatly complicated by the fact that bi is semi-orthogonal which precludes use of the simple Gibbs sampling methods described, e.g., in Geweke (1996). For the nonpanel cointegration model, Koop et al. (2005) develop an efﬁcient method of posterior simulation based on the idea of a collapsed Gibbs sampler developed in Liu (1994) and Liu, Wong, and Kong (1994). To give some preliminary intuition for this, consider the relationships deﬁned in Eq. (4). For prior elicitation or posterior computation, we may consider either (bi, ai) or (bi ; Ai ). Crucially, in the ﬁrst of these parameterizations, bi is semiorthogonal while ai is unrestricted, whereas in the second it is bi which is unrestricted whereas Ai is semi-orthogonal. In the next section we develop a collapsed Gibbs sampler which alternates between these two parameterizations. Arguments made in Liu (1994) and Liu et al. (1994) prove that this will be more efﬁcient than a Gibbs sampler which works only with (bi, ai) or (bi ; Ai ). Even more crucially, with the priors developed in this section, the collapsed Gibbs sampler will only involve draws from the normal distribution (and inverted Wishart4 for S), enormously simplifying the computational burden. We now turn to our informative prior and begin by discussing b and S. Typically, these parameters will be of less importance in an empirical exercise than the prior on the cointegrating space. For S we maintain the noninformative prior given in Eq. (11), although an inverted-Wishart form could also easily be handled. For b we assume:5 1 (12) b N 0; V n where n is a scalar which controls the degrees of informativeness or precision of the prior. Note that n can be interpreted as a shrinkage parameter and, thus, Eq. (12) shares some similarities with shrinkage priors commonly used in the VAR literature (see Litterman, 1986). Note, however, that we treat n as a parameter (rather than a hyperparameter selected subjectively by the researcher). Now consider the prior covariance matrix (up to the scalar shrinkage parameter) V in Eq. (12). Of course, any choice for V can be made. Here we motivate a particular form for the elements of V which relate to ai or,

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equivalently, Ai. Considering the relationships in Eq. (4) and surrounding discussion, it makes sense, analogous to our noninformative prior for the semi-orthogonal bi, to assume that the n ri semi-orthogonal matrices (Ai, . . . , AN) are a priori independent and that: pðAi ; . . . ; AN jt; v; bb Þ / 1

(13)

as this implies a uniform but proper density for each of subspaces deﬁned by the Ai for i ¼ 1, . . . , N. Given the relationships in Eq. (4) we can derive a prior for (bi ; Ai ) from a prior for (bi, ai) or vice versa. The prior (13), along with the prior for bi (to be deﬁned shortly), implies that 1 1 (14) b Þ I vecðai Þjt; bi ; n N 0; ðb0i P1 n i t n and, thus, that the diagonal blocks of V that correspond to ai are equal to 1 I n (where Pt will be deﬁned shortly). The remaining elements ðb0i P1 t bi Þ of V, corresponding to the parameters (C1, . . . , CN), can be speciﬁed using either informative or noninformative choices and will be further discussed below. For the cointegration spaces, pi (and therefore for bi) it is often desirable to have a prior which allows for a common location across individuals. If an economist believes a parameter is likely to have a particular value, she will often place more prior mass around this likely point. To extend this idea from parameters to spaces, some new ideas are required. To provide some intuition, consider the case where we have a prior belief that the space of bi should be approximately the space of H where H is semi-orthogonal and is of the same dimension as bi (we will extend this to allow H to have a different number of columns from bi below). To obtain the semi-orthogonal matrix H the researcher can ﬁrst specify the matrix Hg containing desired coefﬁcient values and then use the transformation H ¼ Hg (Hgu Hg)1/2. The matrix H constructed in this way will span the same space as Hg but will be semi-orthogonal. For instance, if, motivated by King et al. (1991), we wanted a prior reﬂecting the fact that the great ratios are probably cointegrating relationships, we would set: 1 0 1 0 C B 1 A Hg ¼ @ 0 1 1

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Hg is not semi-orthogonal but H ¼ Hg(Hgu Hg)1/2 will be (and will span the same space). A dogmatic prior would be obtained by setting bi ¼ H which places all of the prior mass for pi at pH ¼ spðHÞ. Strachan and Inder (2004) develop an informative, but nondogmatic prior, for the cointegration space and we adopt a similar approach here. Intuitively, we want a prior which says the cointegration spaces, pi , are likely to be close to pH ¼ spðHÞ and, thus, farthest from pH ? ¼ spðH ? Þ where H> is the orthogonal complement of H. The pi s are weighted averages of pH and pH ? and we can elicit a prior about these weights. One way to motivate our informative prior is through its implications for bi . To this end, suppose we have an n ri matrix Zi with all elements being i.i.d. N(0, n1). A standard result tells us that the space of Zi will be uniformly distributed over the Grassman manifold. If we simply set bi ¼ Z i and use this as a prior for bi then it would be noninformative over the cointegrating space. To get a dogmatic informative prior for bi (and, thus, the cointegrating space), we can project Zi into pH . Another standard result in matrix algebra says this projection is given by bi ¼ HH 0 Z i . At the opposite extreme, we could project Zi into pH ? as bi ¼ H ? H 0? Z i if we wanted a cointegration space as far away from pH as possible. A nondogmatic informative prior can be introduced by introducing the random variable Z (with distribution centered at 0) which centers the prior over pH , but attaches weight to other spaces as: bi ¼ HH 0 Z i þ ZH ? H 0? Z i ¼ PZ Z i where PZ ¼ HHuþZH>Hu>. Using the properties of the normal distribution, it follows that

vecðbi Þ Z; n

1 0 N 0; I ri PZ PZ n

But, given the structure of PZ, it follows that PZ PZ0 ¼ HH 0 þ Z2 H ? H 0? ¼ PZ2 . Thus, Z enters the prior only through Z2 and, accordingly, we introduce the notation t ¼ Z2 and use the following conditional normal prior for bi : vecðbi Þjt; n

1 N 0; I ri Pt n

(15)

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or, equivalently,

1

bb t; neN 0; V b n

(16)

where V b ¼ diagðI ri Pt Þ.6 This prior will more strongly weight the cointegrating space toward H the closer t is to zero. At t ¼ 1, this prior is uniform over the Grassman manifold (since Pt ¼ 1 ¼ In) and tW1 implies more weight toward the space of H>. Therefore, it is sensible to either truncate the distribution of t to the region (0,1] or to choose the hyperparameters in the prior for t so that tW1 is a very unlikely event. Note that our informal motivation implicitly assumed H to be of the same dimension as b. However, if we deﬁne HAVs,n to be a known n s (sZri) matrix and H>AVns,n its orthogonal complement, then our prior expresses the belief that the cointegration space pi is likely to be included in the higher dimensional space pH .7 For any p(t) and p(n), we can write the joint prior for bi and (n, t) as ( ) Nnr=2 N 2p nX 0 Nðnrr2 Þ=2 pðnÞpðtÞ t exp trbi Pt1 bi n 2 i¼1 ( ) Nnr=2 N 2p nX 0 Nðnrr2 Þ=2 0 ¼ pðnÞpðtÞ t exp trbi HH bi n 2 i¼1 ( ) N n X 0 0 trbi H ? H ? bi ð17Þ exp 2t i¼1 where r2 ¼ ð

PN

2 i¼1 ri Þ=N.

In our empirical work, we select for p(n) the form: n Gðmn ; nn nN r Þ

(18)

where Gðmn ; nn nN r Þ denotes the gamma distribution with mean mn and degrees of freedom nn nN r. Note that the degrees of freedom depend on nN r. This arises out of our wish to have the prior pðnjbi ¼ 0Þ the same for every model we consider in our model comparison exercise. Such a condition is necessary for using the Savage–Dickey density ratio as we do below. For brevity, we will not provide details, but it turns out that if p(n) has the form given in Eq. (18) then the resulting prior for n satisﬁes the (reasonable and commonly used) conditions for the Savage–Dickey density ratio to be used.

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Using the transformations bi ¼ bi ki and ki k0i ¼ $ with change of measure ðdbi Þ ¼ 2ri j$jðnri 1Þ=2 (d$) (dbi), and using Eq. (18) to integrate out n, we can obtain YN 2 (19) pðt; bb Þ ¼ pðtÞtNðnrr Þ=2 i jb0i Pt1 bi jn=2 cr Q Q 2 ri where cr ¼ 2N rpNðr r Þ=4Nnr=2 N i j G½ðn þ 1 jÞ=2. Since the cointegrating space pi is identiﬁed given a value for bi , the expression in Eqs. (17) or (19) can be regarded as the joint prior for pi ; t conditional upon ri. From the form in Eq. (17), a convenient form of prior for t1 that suggests itself is gamma t1 Gðmt ; nt Þ possibly truncated to [l,N) to ensure to1. Alternatively, we could choose values, as we do in our application, such mt ¼ 5 and nt ¼ 15 which will ensure P(to1)E1. In the truncated case the normalizing constant cr in Eq. (19) is adjusted by dividing by P(t1Wl). Note that, if we use appropriate common values for mt and nt for every individual, we will ensure that each pi has its prior mass near to pH ¼ spðHÞ. This is an example of what we refer to as a soft homogeneity restriction. That is, we are not restricting, a priori, each individual to have the same cointegration space, but we are expressing the view that different individuals are likely to have similar cointegration spaces. In general, such soft homogeneity restrictions can be imposed in two ways with this prior. First, priors (such as the prior for t) can be the same or can share common locations. Second, we can choose V deﬁned in Eq. (12) to have a structure which implies correlation between the same parameters for different individuals. Here we brieﬂy describe one strategy for specifying V. The Nnðk þ r Þ Nnðk þ r Þ matrix V can be partitioned into nðk þ ri Þ nðk þ ri Þ blocks on the diagonal V ii , which can be chosen to have various forms (see Eq. (14) for the form relating to the a0i s). On the off-diagonals, it would often make sense for the nðk þ ri Þ nðk þ rj Þ matrices V ij to have zeros in the rows and columns relating to the a0i s. Thus, no a priori correlation8 is assumed between the a0i s. However, it will usually be sensible to assume that vec(Ci) and vec(Cj) are positively correlated with one another, a priori. This can be done by specifying the nk nk matrix of prior covariances between the elements of vec(Ci) and vec(Cj) to be equal to rb I nk , where 0orb o1. This completes our speciﬁcation of an informative prior which has three key properties: (i) it allows for prior information about the likely location of the cointegration space to be incorporated; (ii) it allows for prior

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information about the degree of similarity in coefﬁcients across individuals (which we refer to as soft homogeneity restrictions); (iii) it contains a parameter n which allows for shrinkage of coefﬁcients on short-run dynamics and deterministic terms.

2.3. Posterior Computation Using the priors speciﬁed above and the likelihood in Eqs. (8) and (10), we can derive various posterior conditional densities of use in our posterior simulation algorithm. Using standard results (e.g., Bauwens et al., 1999, P Chapter 9), the conditional posterior of can be conﬁrmed to be inverted Wishart with degrees of freedom parameter T and scale matrix eue, where e is deﬁned just before Eq. (6). Similar standard results P can be used to obtain the posterior distribution for b conditional upon ( , bb) which is normal with 1 bb and covariance V b ¼ ½V 1 þ nV 1 1 . mean bb ¼ VV The ﬁnal block in a standard Gibbs sampler would involve the cointegrating vectors, bb ¼ ðb0b;1 ; . . . ; b0b ; NÞ0 , where bb,i ¼ vec(bi). Because of the semi-orthogonality of bi, this posterior conditional is difﬁcult to draw from directly. However, the conditional posterior distribution of bb turns out to be normal (we remind the reader that bb ¼ ðb0b ;1 ; . . . ; b0b ;N Þ0 , where bb ;i ¼ vecðbi Þ). To be precise, recalling the deﬁnition of Ci before Eq. (9), 0 0 0 and deﬁning ci ¼ (vec(Ai)u, vec(Ci)u)u and P c ¼ ðc1 ; . . . ; cN Þ , the posterior distribution for bb conditional upon ( , c) is normal with mean bb ¼ 1 1 1 ^ V b V 1 . b bb and covariance V b ¼ ½V b þ nV bP In the process of drawing the parameters ( , b, bb ), we need to draw n and t1. The conditional posterior distribution for n is gamma with mean h i1 m n ¼ n n ðnn nN rÞ=mn þ b0 V 1 b and degrees of freedom n n ¼ Nnk þ nn . The conditional distribution for of freedom n t ¼ nt þ Nðnr r2 Þ and mean t1 is gamma with PN degrees 0 m t ¼ n t ½ nt =mt þ n i¼1 trbi H ? H 0? bi 1 . From these conditional distributions we summarize the following sampling scheme using a collapsed Gibbs sampling method: P P 1. Initialize (b, , bb, n, t) ¼ (b(0), (0), b (0), n(0), t(0)). P PN 0 2. Draw |b, bb, n, t from IW ð i¼1 i i ; TÞ. P 3. Draw b| , bb, v, t from N ðbb ; V b Þ. 4. Calculate Ai ¼ ða0i ai Þ1=2 ai and create c.

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P 5. Draw bb jc, , v, t from Nðbb ; V b Þ. 0 6. Decompose each bi as bi ¼ bi ki using ki ¼ ðbi bi Þ1=2 and bi ¼ bi k1 . Aiki. Construct ai ¼P , t from Gðm n ; n n Þ. 7. Draw n|b, bb, P 8. Draw t1|b, bb, , v from Gðm t ; n t Þ. 9. Repeat steps 2–8 for a suitable number of replications. Note that, in this sampler, the transformations involving the long run multipliers are based on Eq. (5). To see why these steps sufﬁce P to set up a posterior simulator, we ﬁrst show that, conditional on (n, t, ), steps 3–6 deﬁne a collapsed Gibbs sampler (Liu, 1994). To show this, note from Eq. (4) that ai can be decomposed into (Ai, ki), and that therefore the draw of b in step 3 is a draw of (c, k1, . . . , kN). Similarly, bi can be decomposed . . , kN). into (bi, ki) and a draw of bb in step 5 is a draw of (bb, ki, .P Therefore, dropping for simplicity the conditioning arguments (v, t, ), the value of c obtained in step 3 is a draw from c|bb, that is obtained marginally on (k1, . . . , kN). Similarly, the value of bb obtained in step 5 is a draw from bb|c (i.e., obtained again marginally on (k1, . . . , kN)). Therefore, steps 3–6 deﬁne the collapsed Gibbs sampler suggested by Liu (1994) and Liu et al. (1994), who show that this algorithm is more efﬁcient than a standard Gibbs sampling algorithm (i.e., one which simply draws from the conditional posteriors of b and bb). Finally, we extend the collapsed Gibbs sampler with steps that generate (k1, . . . , kN), S, v, and t from their corresponding conditional posterior densities and it is trivial to show that the posterior density continues to be the invariant distribution of the sampler. For a more detailed explanation of this algorithm in the context of a standard (nonpanel) cointegration model see Koop et al. (2005). We will usually be interested in comparing different models nested within the general model deﬁned above. For instance, we might wish to compare the unrestricted model with one where the same cointegrating rank holds for all individuals. We also might wish to calculate the posterior for ri for i ¼ 1, . . . N. The Savage–Dickey density ratio (see Verdinelli & Wasserman, 1995) proves to be a simple and efﬁcient way of doing so. That is, it allows us to compute the Bayes factor comparing every model to a base model (e.g., the model where cointegration does not occur for any individual). This information can then be used to compare any two models, build up the posterior for ri for i ¼ 1, . . . N, do Bayesian model averaging, or select a single model. To compute the Bayes factor for the model Mr with a particular set of cointegrating ranks: r ¼ (r1, r2, . . . , rN) against model M0

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with r ¼ (0, . . . , 0), we note that the restricted case occurs when ai ¼ 0. As ai and Pi have the same singular values (which determine the rank of a matrix, e.g., Golub & van Loan, 1996), Pi ¼ 0 occurs if and only if ai ¼ 0. If we deﬁne a ¼ (vec(a1))u, . . . , vec(aN)u)u, we can use the conditional posterior distribution and (marginal) prior for a to compute the Savage–Dickey density ratio (SDDR): B0;r ¼

pðajM r; yÞja¼0 pðajM r Þja¼0

(20)

Thus we can use output from our Gibbs sampler and the prior to estimate the required ratio: P PðmÞ ðmÞ ðmÞ ðmÞ 1=M M ; C 1 ; . . . ; CðmÞ ; v ; bb ; yÞja¼0 m¼1 pðajM r ; N ;t ^ (21) B0;r ¼ pðajM r Þja¼0 where m ¼ 1, . . . , M denote the (post burn-in) Gibbs sampler replications and (m) superscripts denote the replications themselves (or, as below, functions of these replications). We begin by deriving the analytical expression for p(a|Mr)|a ¼ 0. Using the properties of the gamma distribution and the MACG distribution (Chikuse, 1990), it can be shown that the marginal prior for a evaluated at a ¼ 0 is !Nðnrr2 Þ=2 2nt GððNðnr r2 Þ þ nt Þ=2Þ pðajM r Þja¼0 ¼ mt Gðnt =2Þ !Nnr=2 Gðnn =2Þ nn Nnr mn p Gððnn NnrÞ=2Þ This expression gives us the denominator of the SDDR. The numerator of the SDDR is the marginal posterior for a evaluated at zero. Using the fact that the posterior for b conditional upon (S, bb) is Nðbb ; V b Þ, it follows that a is Nðba ; V a Þ, where ba and V a are given by the elements of bb and V b that correspond to a. Therefore, the Gibbs sampler can be used to estimate the numerator of the SDDR as: M ð2pÞNnr=2 X 1 ðmÞ0 ðmÞ1 ðmÞ ðmÞ 1=2 jV a j exp ba V a ba M 2 m¼1 There are other restricted versions of our general model in which the researcher may be interested. The appendix describes how variants on the methods described above can be used to calculate Bayes factors relating to

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these models. Here we just list the restrictions of interest. Firstly, in practice it is often the case that there is interest in testing overidentifying restrictions of the form pi pH for a subset of the countries i ¼ 1, . . . , N. This restriction can be imposed by writing bi ¼ Hji, where ji 2 V ri ;s is an unknown s ri full rank matrix. Our empirical example in the next section shows how such a restriction can arise. Secondly, we would also like to obtain the probability that all countries have the same unknown cointegrating space p ¼ spðbÞ. Finally, the appendix also describes how to calculate the probability that spðbÞ spðHÞ in the case in which all countries share the same unknown b.

3. ILLUSTRATION USING SIMULATED DATA This section uses simulated data to illustrate the properties of the proposed methodology and its robustness to the speciﬁcation of the prior. Instead of a conventional Monte Carlo experiment, we draw on ideas outlined in Selke, Bayarri, and Berger (2001) to develop a simulation experiment which, as we explain below, better reveals the performance of our approach. We consider seven data generating processes (DGPs) and one prior speciﬁcation: H g ¼ ð 1 1 Þ0 ; nt ¼ 15; mt ¼ 5; nv ¼ 42; mn ¼ 21; and rb ¼ 0:4 (we remind the reader that H ¼ H g ðH g0 H g Þ1=2 ). Except for H, this is the same prior that we use in the empirical application in the next section. We consider N ¼ n ¼ 2, T ¼ 85,9 l ¼ 0, an intercept in all models (dt ¼ 1) and, in each DGP, we ﬁx the error covariance matrix equal to the value used by Groen and Kleibergen (2003) in their Monte Carlo experiment: 1 0:8 0:70 0:60 and Sij ¼ with iaj Sii ¼ 0:8 1 0:60 0:85 We assume that there are only four possible models: M1: (r1 ¼ r2 ¼ 0), M2: (r1 ¼ 0, r2 ¼ 1), M3: (r1 ¼ 1, r2 ¼ 0), and M4: (r1 ¼ r2 ¼ 1). In a conventional Monte Carlo experiment draws from a DGP would involve simply drawing from a single model (with parameters set to particular values). This is consistent with the hypothesis testing ideas underlying frequentist econometrics (e.g., the idea of null hypothesis and the importance allocated to frequentist concepts such as the size of a test). However, as argued in Selke et al. (2001) and Berger and Selke (1987), the ideas underlying Bayesian model comparison are very different. Accordingly, following their arguments, in our simulation experiment we repeatedly

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draw datasets from different distributions. In particular, we set up distributions over our model space and parameter spaces and draw from these. For each draw of a model and parameter values, we then draw an artiﬁcial dataset. All our DGPs involve the same distributions over the model space and, accordingly, each of our seven DGPs arise from different distributions over the parameters. Note that these distributions have the same functional form as our priors, but the hyperparameters selected to create our DGPs do not have to coincide with the prior hyperparameters we use to estimate our models. To be precise, in each of our DGPs data is drawn from each model with probability 1/4, which is equal to the prior probability of each model. Conditional on model Mi, the parameters are drawn from distributions that are of the same form as the prior, but with different hyperparameters. In all cases we use rb ¼ 0:4. The speciﬁcation of the remaining hyperparameter values for each of these distributions is given in Table 1. Note that DGP1 involves the same informative distribution over the cointegrating space as we use in our prior, but the remaining DGPs are less informative. For the remaining parameters, we have a wide variety of speciﬁcations. The speciﬁcations in each DGP imply we draw Ai (deﬁned in Eq. (4)) from a uniform distribution on the Stiefel manifold. For DGP2 to DGP7 we ﬁx t equal to 1, which implies that bi is also drawn from a uniform distribution. This contrasts with the prior we use for pi ¼ spðbi Þ, which gives more weight to the space deﬁned by H. In addition, DGP3 to DGP7 vary in the expected value of ki. Higher values of n1 imply higher expected values for ki and therefore higher expected values for the singular values of Pi. Note that there is 95% prior probability that n1 lies in the interval Table 1. Speciﬁcation of the (Hyper) Parameters for the Distributions from which the Parameters are Drawn in the Simulation Experiment. DGP

DGP1 DGP2 DGP3 DGP4 DGP5 DGP6 DGP7

Parameter (Brief Description of the Distribution) pi

b

H g ¼ ð 1 1 Þ; nt ¼ 15; mt ¼ 5 t ¼ 1 (uniform) t ¼ 1 (uniform) t ¼ 1 (uniform) t ¼ 1 (uniform) t ¼ 1 (uniform) t ¼ 1 (uniform)

nn ¼ 42; mn ¼ 21 nn ¼ 42; mn ¼ 21 n1 ¼ 0.05 (n1 equal to its prior mean) n1 ¼ 0.2 (large var(ai)) n1 ¼ 0.5 (very large var(ai)) n1 ¼ 0.02 (small var(ai)) n1 ¼ 0.002 (very small var(ai))

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(0.032, 0.077). Therefore, DGP4 and DGP5 specify a value of n1 that is large compared to the prior information, whereas DGP6 and DGP7 specify a value that is small. For each DGP, 2500 artiﬁcial datasets were generated. For each dataset, the posterior probability of each model (i.e., each rank combination) was calculated. In order to analyze the performance of posterior probabilities in this setup, let us deﬁne the following concepts (see Selke et al., 2001, for the development of these concepts). Let Ci(0.5) be the set of datasets in which model Mi had posterior probability higher than 0.5. Assume that a model is selected whenever its posterior probability is higher than 0.5 and let Ri(0.5) be deﬁned as an error rate that gives the proportion of the samples in Ci(0.5) that were not generated from model Mi. To motivate why these are interesting metrics, we digress brieﬂy to provide a bit of the theory from Selke et al. (2001). Consider the ideal case where the distribution used to generate the datasets is the same as the prior. For this case, suppose Mi is chosen whenever its posterior model probability (pi) is equal to a particular value pi . From the deﬁnition of posterior model probability, the error rate that results (i.e., the proportion of samples that were classiﬁed as Mi but were in fact generated from another model) is equal to 1 pi . Thus, posterior model probabilities, unlike p-values, are constructed to reﬂect true error rates (see also Berger & Selke, 1987, for discussion). However, it is unlikely that we will ever simulate a dataset that results in posterior probability of Mi being exactly pi so this approach is hard to implement. Therefore, one possibility would be to accept those draws with posterior model probability lying in the interval (pi ; pi þ ), where e is a small number. This is what Selke et al. (2001) do. Alternatively, a simple rule of thumb such as ‘‘select M i if pi 4pi ’’ can be used (as we have done with pi ¼ 0:5) and the average value of pi ðpi Þ among the datasets in Ci (0.5) can be reported and the previous reasoning implies this will also be informative about the error rate Ri(0.5). In particular, if the number of datasets is large and are generated from the prior, Ri(0.5) will be equal to 1 pi . Table 2 shows the values of pi ; Ri ð0:5Þ, and the number of datasets in Ci(0.5). Overall, the strategy of choosing Mi when piW0.5 seems to work very well, selecting the correct model much of the time. Recall that DGP1 draws all model parameters, except for S, from the prior. Not surprisingly, therefore, Table 2 shows that for DGP1, Ri(0.5) is very close to ð1 pi Þ for every i ¼ 1, . . . , 4. These two quantities are still close for every i for DGP2 and DGP3, which indicates that posterior model probabilities are still a reliable measure of error when the prior of bi is misspeciﬁed and/or n1 is ﬁxed to a particular value instead of being random. When n1 ¼ 0.2

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Table 2. Error Rates (Ri(0.5)), One Minus the Average Posterior Probabilities ð1 pi Þ and Number of Samples in Ci(0.5) ðN~ i Þ for Each DGP.

DGP1

DGP2

DGP3

DGP4

DGP5

DGP6

DGP7

Ri ð0:5Þ 1 pi N~ i Ri ð0:5Þ 1 pi N~ i Ri ð0:5Þ 1 pi N~ i Ri ð0:5Þ 1 pi N~ i Ri ð0:5Þ 1 pi N~ i Ri ð0:5Þ 1 pi N~ i Ri ð0:5Þ 1 pi N~ i

M1

M2

M3

M4

0.07 0.06 842 0.05 0.05 844 0.04 0.05 830 0.01 0.07 855 0.004 0.079 765 0.11 0.06 840 0.36 0.10 868

0.05 0.05 527 0.03 0.04 541 0.05 0.04 545 0.03 0.04 540 0.105 0.067 531 0.06 0.05 558 0.22 0.10 514

0.06 0.05 545 0.04 0.05 519 0.03 0.04 518 0.07 0.05 531 0.107 0.061 542 0.07 0.05 498 0.24 0.10 542

0.03 0.02 330 0.03 0.02 344 0.03 0.02 354 0.09 0.03 307 0.255 0.046 392 0.02 0.03 341 0.05 0.07 289

(DGP4), which is far outside the prior 95% credible interval of (0.032, 0.077), ð1 pi Þ is still close to Ri(0.5) for every i. Similarly, when n1 ¼ 0.02 (DGP6), which is small compared to prior information, posterior model probabilities continue to be a reliable measure of error for every i. However, when n1 ¼ 0.5 (DGP5), posterior model probabilities are not reliable when model M4 is chosen (ð1 pi ÞoRi ð0:5Þ), although they still seem to be reasonable when models M1 to M3 are selected. Something similar, but in the opposite direction, happens when n1 is very small (DGP7). In this case, the posterior model probability is only a reliable measure of error when M4 is chosen. Thus, the simulation illustrates that posterior model probabilities, unlike p-values (Selke et al., 2001), are reliable measures of error unless the prior for n1 is seriously misspeciﬁed. Therefore, some amount of careful prior elicitation for n1 is desirable to avoid such misspeciﬁcation. In our case, for

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example, it should be noted that DGP5 tends to generate very explosive processes whenever ri ¼ 1, resulting in data that would be extremely unreasonable (at least for standard applications with macroeconomic data such as the one considered in the next section). For example, it can be shown that DGP5 implies that about 45% of the datasets would have (|y1,t|W1000) for every t ¼ 1, . . . , T when r1 ¼ r2 ¼ 1, which is not sensible for macroeconomic data such as that which we use in our application. Table 3 shows other measures that illustrate the performance of Bayesian model selection in this context. For each DGP and model from which the Table 3.

DGP1

DGP2

DGP3

DGP4

DGP5

DGP6

DGP7

M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4 M1 M2 M3 M4

Two Summaries (%i and Pi) for Each DGP.

%1

%2

%3

%4

P1

P2

P3

P4

0.99 0.05 0.05 0.03 0.99 0.04 0.03 0.01 0.98 0.03 0.03 0.00 0.95 0.01 0.01 0.01 0.85 0.00 0.00 0.00 0.98 0.07 0.09 0.03 0.98 0.24 0.23 0.10

0.00 0.94 0.00 0.07 0.00 0.95 0.00 0.03 0.01 0.96 0.00 0.04 0.02 0.96 0.00 0.01 0.07 0.93 0.00 0.00 0.01 0.92 0.01 0.07 0.01 0.73 0.02 0.19

0.01 0.01 0.94 0.07 0.00 0.00 0.95 0.04 0.00 0.00 0.96 0.04 0.03 0.00 0.97 0.04 0.06 0.00 0.91 0.01 0.01 0.00 0.90 0.08 0.01 0.02 0.74 0.21

0.00 0.01 0.01 0.84 0.00 0.01 0.01 0.92 0.00 0.01 0.01 0.92 0.00 0.03 0.02 0.94 0.02 0.06 0.08 0.99 0.00 0.01 0.01 0.82 0.00 0.01 0.01 0.49

0.94 0.05 0.05 0.02 0.94 0.04 0.04 0.01 0.94 0.03 0.03 0.00 0.90 0.01 0.01 0.01 0.79 0.00 0.00 0.00 0.94 0.07 0.08 0.03 0.91 0.21 0.22 0.09

0.03 0.91 0.00 0.06 0.03 0.92 0.00 0.03 0.03 0.93 0.00 0.04 0.05 0.94 0.00 0.01 0.08 0.90 0.00 0.00 0.03 0.89 0.01 0.07 0.04 0.70 0.03 0.19

0.03 0.01 0.91 0.07 0.03 0.00 0.92 0.05 0.03 0.00 0.93 0.04 0.05 0.00 0.94 0.04 0.09 0.00 0.88 0.01 0.03 0.01 0.87 0.08 0.04 0.03 0.70 0.20

0.00 0.03 0.04 0.84 0.00 0.03 0.04 0.91 0.00 0.04 0.03 0.91 0.00 0.05 0.06 0.94 0.03 0.10 0.11 0.99 0.00 0.04 0.04 0.82 0.00 0.05 0.05 0.51

Notes: %i is the percentage of times that Mi has largest posterior model probability. Pi is the average posterior model probability of Mi. The bold numbers indicate cases where the model being reported (i.e., with %i or Pi) is the model generating the data.

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data was generated, it gives the proportion of times (denoted %i) that model Mi had the largest posterior probability. In addition, it shows the average posterior model probability (denoted Pi) of Mi for each DGP and each generating model. Note that the proportion of times that the correct model has largest posterior model probability is almost always near or above 90%, and that on average posterior model probabilities are accordingly large. The exception is DGP7, where the detection rate of the true model is lower, as are average posterior model probabilities. This is to be expected, as lower values of n1 mean that data generated when rW 0 will be more similar to data generated when r ¼ 0, and hence model selection becomes more difﬁcult and we see slightly larger values of %1 and P1.

4. EMPIRICAL WORK In this section we investigate support for the monetary model of the exchange rate commonly employed in international ﬁnance. We focus upon the speciﬁcation proposed by Groen (2000) which implies a particular testable relationship among the following variables: ei,t, the log of the exchange rate for country i at time t; mi,t, the log of the ratio of the quantity of domestic to foreign money supply; and xi,t, the log of the relative real income. Groen (2000) shows that in a long-run model for bilateral exchange rates, the theory implies the relation ei;t b1 mi;t b2 xi;t ¼ b0 þ zi;t will be stationary (i.e., zi,t should be an I (0) process) with b1 ¼ 1 and b2o0. If the variables in the vector yi,t ¼ (ei,t, mi,t, xi,t) are I(1), this model implies they cointegrate with a particular cointegrating space. The data are quarterly and consist of U.S. dollar exchange rates and the log ratio of money (m) and income (x) for France (i ¼ 1), Germany (i ¼ 2), and the United Kingdom (i ¼ 3) to the U.S. equivalents. The data runs from the ﬁrst quarter of 1973 to the last quarter of 1994. The data were those used in Groen and Kleibergen (2003) and are described in detail in Groen (2000). We have chosen this application because the economic model implies a varied and clear set of testable restrictions on the cointegrating space. That is, we have a requirement that the cointegrating rank be one for all countries, a linear restriction on b1, as well as an inequality restriction upon b2. We note that it is often the case that the economic model of interest implies such a set of joint restrictions, some of which are linear and some are

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nonlinear. In such a case, classical inference usually proceeds with a mixture of sequential testing and informal inference to gather evidence for or against the model, with no single statistic with known power to indicate the degree of support in favor of the model. Therefore, the classical work of Groen, which tested sequentially the rank restriction and the other restrictions, provided only informal evidence about the degree of support for the model. An advantage of using the Bayesian approach is that we are able to provide a formal summary of the evidence for the model via posterior model probabilities. We are also able to assess the evidence, if desired, for components of the model. For example, we may be interested in whether the variables cointegrate or whether the cointegrating ranks are common to all countries, or whether the b’s are common across all countries. Within the speciﬁcation of the statistical model we use, the monetary exchange rate model implies that ri ¼ 1 for each country and that the cointegrating spaces are restricted. In particular, if we deﬁne the orthogonal matrix H as: 2 3 1 p ﬃﬃ ﬃ 0 6 2 7 6 7 6 7 1 H¼6 7 6 pﬃﬃﬃ 0 7 4 5 2 0 1 and introduce the semi-orthogonal vector " # j1;i ji ¼ j2;i we can write these restrictions as: 2 3 1 p ﬃﬃ ﬃ j 6 2 1;i 7 6 7 j 6 7 1 bi ¼ Hji ¼ 6 ; with 2;i 40 7 6 pﬃﬃﬃ j1;i 7 j1;i 4 5 2 j2;i

(22)

Note that, for ri ¼ 1, this set of restrictions does not actually require that all panels share the same cointegrating space, since the inequality restriction allows a different value of f2,i for each panel, provided that it has the same sign as f1,i. However, all of these spaces will be subspaces of the space deﬁned by H. Other restrictions of more general interest in the cointegrating

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panel data model are equality of the ranks for all panels (ri ¼ r for all i), and equality of the cointegrating spaces for all panels, pi ¼ p such that ri ¼ r and bi ¼ b for all i. We compute posterior probabilities distributions for the cointegrating ranks from both unrestricted and restricted models. We consider two types of restrictions. The ﬁrst imposes the same unknown cointegrating space: bi ¼ b and ri ¼ r for all i. The second restricts the cointegrating space of at least one country, such that sp (bi)Dsp (H) for some i with ri ¼ 1, 2.10 This makes a total of 221 models. Following Groen and Kleibergen (2003), all models include an intercept and three seasonal dummies and we ﬁx the number of lags equal to three. As in the artiﬁcial data experiment in the previous section, we choose our prior hyperparameters as: nn ¼ 42, mn ¼ 21, nt ¼ 15, mt ¼ 5, and rb ¼ 0:4. We use 15,000 replications of the sampling algorithm presented in Section 2.3. For the sake of comparison, we also calculate the Bayesian information criterion (BIC) for each of these models.11 Recall we let i ¼ 1, 2, 3 correspond to France, Germany, and UK, respectively. The BIC selects the model with (r1 ¼ r2 ¼ r3 ¼ 0) as the best model, followed by the model with (r1 ¼ 1, r2 ¼ r3 ¼ 0) and no other restrictions. If posterior model probabilities are calculated using the BIC approximation, these two models would get 90% of the probability. However, the actual posterior model probabilities calculated using our approach are spread over a wide range of models: no less than 28 models would be required to contain 98% of the probability. Table 4 presents the details of the ﬁve most likely models, which get 71.4% of the probability mass. All these models assign rank equal to one to France and Germany and Table 4.

Posterior Probabilities for the Five Most Likely Models.

r1

r2

r3

o1

o2

o3

E

Prob.

1 1 1 1 1

1 1 1 1 1

0 0 0 2 1

1 1 0 1 1

1 0 1 1 1

0 0 0 1 1

0 0 0 0 0

0.35 0.12 0.10 0.10 0.05

Notes: The ﬁrst three columns indicate the rank of each country in a particular model. i ¼ 1, 2, 3 corresponds to France, Germany, and UK, respectively. In the following three columns oi takes value 1 when the restriction sp(bi)Dsp(H) is imposed and 0 otherwise. E takes value 1 if the restriction sp(b1) ¼ sp(b2) ¼ sp(b3) is imposed and zero otherwise. The last column indicates the probability of each model.

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restrict sp (bi)Dsp(H) in at least one country. In particular, the model with ri ¼ 1 and sp(bi)Dsp(H) for every i, which gives support to the monetary exchange model, has a non negligible probability that is equal to 0.05. Conditional on this model, Pr(f2i/f1i W 0 for i ¼ 1, 2, 3) ¼ 0.12,12 which means that the probability of all the restrictions implied by the monetary exchange model holding in every country is 0.12 0.05 ¼ 0.006. The probability of many other restrictions of interest can be evaluated by simply adding up the posterior model probabilities of models in which the restriction is true. For example, Pr(r1 ¼ r2 ¼ r3) ¼ 0.09, Pr(sp(b1) ¼ sp(b2) ¼ sp(b3)) ¼ 0.004, Pr(r2 ¼ 1) ¼ 0.86, Pr(sp(b1)Dsp(H), r1W0) ¼ 0.79. Finally, the probability that (sp(bi)Csp(H), ri ¼ 1) for at least one country is 0.94, which again gives support to the monetary exchange model holding in at least one country.

5. CONCLUSION In this paper, we have discussed Bayesian inference in cointegrated panel data models. We adopt a very general speciﬁcation where each individual is characterized by its own VECM. Special cases of this model allow for individuals to have common cointegrating rank and/or common cointegrating spaces. We develop a noninformative prior as well as an informative prior which allows for sensible priors on the cointegration spaces. The latter prior also allows for prior information about the degree of common structure across individuals to be used. Efﬁcient posterior simulation is carried out using a collapsed Gibbs sampler. While we consider this a useful start to employing Bayesian methods in this area of models, there are a number of directions for future development. For instance, in a PPP study, Li (1999) argues that estimating relationships of interest individually for each country results in overly noisy estimates. On the other hand, imposing strict homogeneity by assuming these relationships are the same for all countries tends to be overly severe due to the differences in macroeconomic policies in each country. Such severe restrictions are often rejected. Li suggests specifying an unknown hierarchical prior and conducting inference upon the distribution from which the parameters for the PPP relations come, not upon the actual PPP parameters themselves. In this paper we have assumed that the cointegrating spaces came from a common known prior distribution and investigated support for common cointegrating spaces (pi ¼ p for all i). To adopt the Li approach, a hierarchical prior could be placed upon the prior distribution for the

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cointegrating spaces, rather than assuming a known prior distribution. That is, a prior could be placed upon pH in Section 2.2.2. Further, while we have provided a method of conducting inference upon a class of models, we have only alluded to the conduct of policy advice. The application of this class of models to policy via, say, forecasts or cross impulse responses could provide useful information on, for example, the forms of international linkages. Finally, Bayesian model averaging over combinations of pi , li, and di could be used to provide inference using an even wider set of models.

NOTES 1. The exact form of the deterministic terms is not crucial to our derivations so we leave these unspeciﬁed. See Johansen (1995, pp. 81–84) for a commonly used set of choices. 2. It is not complicated to allow for yi,t to be of dimension ni, but we will assume ni ¼ n, for simplicity. Similarly it is straightforward to extend our results to the unbalanced panel case. 3. When calculating posterior odds ratios, it is common to make use of improper, noninformative priors over parameters which are common to all models (see Kass & Raftery, 1995). Ferna´ndez, Ley, and Steel (2001) employ such a prior for an error variance in a model averaging exercise. As they point out, the prior in Eq. (11) is invariant to scale transformations and, although it is not strictly Jeffreys’ prior, it is that part of Jeffreys’ prior related to S and widely accepted as a noninformative prior for S. 4. See Bauwens, Lubrano, and Richard (1999, p. 305) for a deﬁnition of the inverted Wishart distribution. 5. In the following material, we use notation where lower bars (e.g., as in V) denote prior hyperparameters which must be selected by the researcher. 6. Note that b and bb share elements in common (i.e., ki) and therefore, the prior speciﬁcation on b has implications on the prior of bb This is the reason why the shrinkage parameter, n, appears in Eq. (16). Note that n does not affect the marginal prior for cointegrating spaces. 7. If we have the case that son1, then we will have models with rWs and the above prior distribution is not applicable. In this case, in the absence of economic theory to guide us, we would assume a uniform prior distribution for pi . 8. If the cointegrating relations are exactly identiﬁed, all individuals share the same cointegrating rank and the same cointegrating relationship holds for all equations, then it would make sense to assume the adjustment coefﬁcients (ai) are a priori correlated. However, without these restrictions, it does not make sense to assume that the columns of ai and aj will be correlated. 9. These 85 observations were the last 85 observations of 135. That is, the initial 50 were discarded. 10. If ri ¼ 1, then pi pH , while if ri ¼ 2, then pi ¼ pH .

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11. In order to calculate the penalty for the number of parameters in the BIC, we count the parameters in the semi-orthogonal but otherwise unrestricted bi matrix as nri r2i which is the dimension of the Grassman manifold Gn;ri deﬁned above (Strachan & Inder, 2004). Similarly, when bi is restricted such that pi pH , we ﬁx the penalty corresponding to the semi-orthogonal but otherwise unrestricted fi matrix to be 2ri r2i . We use our algorithm to search for the maximum value of the actual likelihood by using 1000 draws from a modiﬁed posterior density. This modiﬁcation increases the accuracy of the obtained maximum likelihood values and consists in analyzing the posterior that results when the sample size is increased by a factor of 600 and the additional data is just a replication of the real data. Therefore, the maximum value of the log likelihood function in this modiﬁed dataset is 600 times the value of the log likelihood in our real data. And most importantly, the dispersion of the posterior around the mode will be much smaller and therefore the accuracy of the maximized likelihood will be much larger. 12. This probability was approximated by the proportion of draws from the posterior of this model in which the restriction was veriﬁed. 13. Further motivation is given in Strachan and Inder (2004). Strachan and van Dijk (2004) show how this speciﬁcation can be implemented in a macroeconomic example.

ACKNOWLEDGMENTS The authors would like to thank the participants at the 6th Annual Advances in Econometrics Conference at Louisiana State University, Econometric Society Australasian Meeting, RBNZ-CAMA Conference Macroeconometrics and Model Uncertainty at the Reserve Bank of New Zealand, and the Workshop in Bayesian Econometric Methodology at the Sveriges Riksbank Stockholm, Sweden. We would also like to thank participants at seminars at the Chicago Graduate School of Business, the University of Queensland, and the University of Leicester.

REFERENCES Baltagi, B., & Kao, C. (2000). Nonstationary panels, cointegration in panels and dynamic panels: A survey. Advances in Econometrics, 15, 7–51. Bartlett, M. (1957). A comment on D. V. Lindley’s statistical paradox. Biometrica, 44, 533–534. Bauwens, L., Lubrano, M., & Richard, J.-F. (1999). Bayesian inference in dynamic econometric models. Oxford: Oxford University Press. Berger, J., & Selke, T. (1987). Testing of a point null hypothesis: The irreconcilability of signiﬁcance levels and evidence (with discussion). Journal of the American Statistical Association, 82, 112–139.

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Carmeci, G. (2005). A Bayesian state space approach to cointegration in panel data models. Working Paper available at http://www.cide.info/conf/papers/1128.pdf Chikuse, Y. (1990). The matrix angular central Gaussian distribution. Journal of Multivariate Analysis, 33, 265–274. Ferna´ndez, C., Ley, E., & Steel, M. F. J. (2001). Benchmark priors for Bayesian model averaging. Journal of Econometrics, 100, 381–427. Geweke, J. (1996). Bayesian reduced rank regression in econometrics. Journal of Econometrics, 75, 121–146. Golub, G. H., & van Loan, A. (1996). Matrix computations (3rd ed.). Baltimore, MD: John Hopkins University Press. Groen, J. (2000). The monetary exchange rate model as a long-run phenomenon. Journal of International Economics, 52, 299–319. Groen, J., & Kleibergen, F. (2003). Likelihood-based cointegration analysis in panels of vector error-correction models. Journal of Business and Economic Statistics, 21, 295–318. Hsiao, C., & Pesaran, M. H. (2004). Random coefﬁcient panel data models. Working Paper available at http://www.econ.cam.ac.uk/faculty/pesaran/rcm17june04.pdf Jacobson, T., Lyhagen, J., Larsson, R., & Nesse´n, M. (2002). Inﬂation, exchange rates and PPP in a multivariate panel cointegration model. Working Paper available at http:// www.riksbank.se/upload/7405/wp_145.pdf James, A. T. (1954). Normal multivariate analysis and the orthogonal group. Annals of Mathematical Statistics, 25, 40–75. Johansen, S. (1995). Likelihood-based inference in cointegrated vector autoregressive models. Oxford: Oxford University Press. Kao, C. (1999). Spurious regression and residual-based tests for cointegration in panel data. Journal of Econometrics, 90, 1–44. Kass, R., & Raftery, A. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795. King, R., Plosser, C., Stock, J., & Watson, M. (1991). Stochastic trends and economic ﬂuctuations. American Economic Review, 81, 819–840. Kleibergen, F., & van Dijk, H. (1994). On the shape of the likelihood/posterior in cointegration models. Econometric Theory, 10, 514–551. Kleibergen, F., & van Dijk, H. (1998). Bayesian simultaneous equations analysis using reduced rank structures. Econometric Theory, 14, 701–743. Koop, G., Le´on-Gonzalez, R., & Strachan, R. (2005). Efﬁcient posterior simulation for cointegrated models with priors on the cointegration space. Forthcoming in Econometric Reviews. Koop, G., Strachan, R., van Dijk, H. K., & Villani, M. (2006). Bayesian approaches to cointegration. In: T. C. Mills & K. Patterson (Eds), The Palgrave handbook of econometrics (Vol. 1, pp. 871–898). Econometric theory. Basingstoke: Palgrave-Macmillan. Larsson, R., Lyhagen, J., & Lo¨thgren, M. (2001). Likelihood-based cointegration tests in heterogeneous panels. Econometrics Journal, 4, 109–142. Li, K. (1999). Testing symmetry and proportionality in PPP: A panel data approach. Journal of Business and Economic Statistics, 17, 409–418. Litterman, R. (1986). Forecasting with Bayesian vector autoregressions: Five years of experience. Journal of Business and Economic Statistics, 4, 25–38. Liu, J. S. (1994). The collapsed Gibbs sampler with applications to a gene regulation problem. Journal of the American Statistical Association, 89, 958–966.

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Liu, J. S., Wong, W., & Kong, A. (1994). Covariance structure of the Gibbs sampler with applications to comparisons of estimators and augmentation schemes. Biometrika, 81, 27–40. McCoskey, S., & Kao, C. (1998). A residual-based test of the null of cointegration in panel data. Econometric Reviews, 17, 57–84. O’Hagan, A. (1995). Fractional Bayes factors for model comparison. Journal of the Royal Statistical Society, Series B, B 57, 99–138. Pedroni, P. (2004). Panel cointegration: Asymptotic and ﬁnite sample properties of pooled time series tests with an application to the purchasing power parity hypothesis. Econometric Theory, 20, 597–625. Pesaran, M. H., Hsiao, C., & Tahmiscioglu, K. (1999). Bayes estimation of short-run coefﬁcients in dynamic panel data models. In: C. Hsiao, K. Lahiri, L.-F. Lee & M. H. Pesaran (Eds), Analysis of panels and limited dependent variables: A volume in honour of G. S. Maddala (pp. 268–296). Cambridge: Cambridge University Press. Phillips, P. C. B. (1991). Optimal inference in cointegrated systems. Econometrica, 59, 283–306. Phillips, P. C. B., & Moon, H. (2000). Nonstationary panel data analysis: An overview of some recent developments. Econometric Reviews, 19, 263–286. Selke, T., Bayarri, M. J., & Berger, J. (2001). Calibration of P-values for testing precise null hypotheses. The American Statistician, 55, 62–71. Strachan, R. (2003). Valid Bayesian estimation of the cointegrating error correction model. Journal of Business and Economic Statistics, 21, 185–195. Strachan, R., & Inder, B. (2004). Bayesian analysis of the error correction model. Journal of Econometrics, 123, 307–325. Strachan, R., & van Dijk, H. (2004). Valuing structure, model uncertainty and model averaging in vector autoregressive processes. Econometric Institute Report EI 2004-23, Erasmus University Rotterdam. Verdinelli, I., & Wasserman, L. (1995). Computing Bayes factors using a generalization of the Savage–Dickey density ratio. Journal of the American Statistical Association, 90, 614–618. Villani, M. (2005). Bayesian reference analysis of cointegration. Econometric Theory, 21, 326–357. Villani, M. (2006). Bayesian point estimation of the cointegration space. Journal of Econometrics 134, 645–664.

APPENDIX. OVERIDENTIFYING RESTRICTIONS AND OTHER RESTRICTED MODELS Probability of Overidentifying Restrictions In practice it is often the case that there is interest in testing the overidentifying restriction pi pH for a subset of countries i ¼ 1, . . . , N. Our empirical example showed how such a restriction can arise. This restriction can be imposed by writing bi ¼ Hji, where H is a n s

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semi-orthogonal matrix and ji 2 V ri; s an unknown s ri full rank matrix, with sZri. This speciﬁcation13 requires a prior for ji and in this paper we use a uniform prior for ji on V ri ;s , although an informative MACG prior (Chikuse, 1990) could also be used. Within this framework, it is also possible to use the SDDR to obtain the Bayes factor, which in this case is deﬁned as the probability of the restricted (overidentiﬁed) model over the probability of the unrestricted model. Note that one can write bi ¼ Hji þ H ? li , where ji ¼ H 0 bi is a s ri, full rank matrix and li ¼ H 0? bi a (ns) ri full rank matrix. Therefore, the overidentifying restriction can be imposed by setting li ¼ 0. The numerator in the SDDR can be easily estimated by noting that (l1, . . . , lN) is a linear transformation of bb and therefore the conditional posterior of (l1, . . . , lN) given (c, S, v, t) is a normal density (Bauwens et al., 1999). In particular, let H~ be a block diagonal matrix with diagonal blocks equal to: ðI r1 H ? ; . . . ; I rN H ? Þ. The conditional posterior density of l ¼ 0 0 ðvecðl1 Þ0 ; . . . ; vecðlN Þ0 Þ0 is normal with variance H~ V b H~ and mean H~ bb , where V b and bb were deﬁned in Section 2.3. The numerator of the SDDR for this restriction is estimated as:

M

ð2pÞðnsÞN r=2 X 1 ðmÞ0 0 ðmÞ 1 ~ 0 ðmÞ

~ 0 ðmÞ ~ 1=2 H b b exp bb H~ H~ V b H~

H V b H

M 2 m¼1 P where N r ¼ N i¼1 ri . The denominator, which is given by the prior of l evaluated at zero, is equal to:

pðlÞjl¼0 ¼

!ðnsÞN r=2 2nt ðnv nN rÞ mt mn p Gððnt þ ðn sÞN rÞ=2ÞGððn sÞN r þ ðnv nN rÞ=2Þ Gðnt =2ÞGððnv nN rÞ=2Þ

Estimation with Equal Cointegrating Spaces: {sp(bi) ¼ sp(b), i ¼ 1, . . . , N}. We would also like to obtain the probability that all countries have the same cointegrating space. For this purpose, we ﬁrst discuss how to set up a posterior simulator for the model that restricts all cointegrating spaces to

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be the same. Let b be an n r semi-orthogonal matrix that represents the cointegrating space common to all countries. We rewrite the matrix of long-run multipliers as: ~ a0 ba0i ¼ bi DD1 a0i b~ i where D is a r r, symmetric positive deﬁnite matrix. We stress that unlike ki , which was deﬁned as one of the components of the polar decomposition of ai (Golub & van Loan, 1996, p. 149), the matrix D is not identiﬁed. However, the introduction of D facilitates posterior computations because neither b~ nor a~ i are subject to restrictions. Our strategy is to specify a proper prior on D and to use a simple Gibbs sampling algorithm. We note that this strategy could also be used for the case in which cointegrating spaces differ between countries, but stress that it would be less efﬁcient than the ‘‘k-algorithm’’ that we specify in Section 2.3 above for two reasons. Firstly, the k-algorithm (implicitly) integrates out the parameter D, and it thereby achieves a comparative advantage (Liu, 1994). Secondly, the k-algorithm draws bb and c marginally on (k1, . . . , kN), which is likely to result in smaller autocorrelations in the Markov chain. We specify the same prior on the cointegrating space as in Section 2.2.2. ~ and assume that b ~ follows a priori a N(0, n1 IrPt) We deﬁne bb~ ¼ vecðbÞ b where Pt is deﬁned as above. Note that this implicitly speciﬁes a proper prior for D and that the marginal prior for b continues to be the same as the one presented in Section 2.2.2. As a prior for vecð~ai Þ we choose a N(0, n1Inr). Note that this prior speciﬁcation implies that the prior mean of ai is zero and the variance–covariance matrix of vec(ai) conditional on 1 I n Þ Therefore, the prior mean and variance for (n, b, t) is ðn1 ðb0 P1 t bÞ (a1, . . . , aN) are the same as in Section 2.2.2. 0 0 Let us deﬁne b~ ¼ ðb~1 ; . . . ; b~N Þ0 , where b~i ¼ vecðð~ai ; Gi;1 ; . . . ; Gi;l ; Fi Þ0 Þ. ~ where V~ is speciﬁed in the same way as The prior for b~ is a Nð0; n1 VÞ, V except for the diagonal blocks corresponding to a~ i , which are now equal to the identity matrix. ~ let x€ i ¼ ð~ai y Þ, In order to obtain the posterior conditional of b, i;1 0 0 0 0 1 1 1 ^ € , bb~ ¼ V b~ xV € e y, ~ and V b~ ¼ n1 I r Pt x€ ¼ ðx€ 1 ; . . . ; x€ N Þ , V b~ ¼ ðx€ V e xÞ The conditional posterior of b~ given (b, S, t, n) is normal with covariance 1 1 ^ and mean bb~ ¼ V b~ V b1 V b~ ¼ ½V b1 ~ þ V b~ ~ bb~ . ~ ~ The conditional posterior of b given (b, S, t, n) is a normal with mean bb~ and variance V b~. The expressions for bb~ and V b~ are obtained in the same way as bb and variance V b respectively in Section 2.3, except we replace all bi ~ Similarly the posterior for n has the same form as in everywhere with b.

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~ such that n has a Section 2.3, except we replace b with b~ and V with V, 0 1 1 ~ and gamma posterior distribution with mean n v ½ðnn NnrÞ=mn þ b~ V~ b degrees of freedom n n ¼ Nnk þ nn . The conditional distribution for t1 is gamma with degrees of freedom n t ¼ nt þ ðn rÞr and mean 0 ~ 1 . m t ¼ n t ½ nt =mt þ n1 trb~ H ? H 0? b Therefore, a Gibbs sampling algorithm is deﬁned by simply sampling ~ S; t; nÞ, Sjðb; ~ b ~ ; t; nÞ, tjðb; ~ b ~ ; S; nÞ, and ~ ~ ; S; t; nÞ, b ~ jðb; iteratively from bjðb b b b b ~ b ~ ; t; SÞ. A sample from the posterior of (b, b, S, t, n, D) can be njðb; b obtained using the following transformations: 0

~ 1=2 D ¼ ðb~ bÞ

~ 1 b ¼ bD

ai ¼ a~ i D

Finally, we note that although the k-algorithm cannot be used to sample from the posterior when the restriction spðbi Þ spðHÞ is imposed for some i, the D-algorithm just described can be easily adapted to this case.

Calculating the probability that: {sp(bi) ¼ sp(b), i ¼ 1, . . . , N} ~ r be the model in which sp(bi) ¼ sp(b), for every i ¼ 1, . . . , N Let M ~ rÞ and rank(b) ¼ r. To compute the Bayes factor for this model ðM against the model M0 with r ¼ (0, . . . , 0), note that M0 arises when a~ ¼ ðvecð~a1 Þ0 ; . . . ; vecð~aN ÞÞ0 ¼ 0. Therefore, the SDDR can be estimated as:

B~ 0;r ¼

1=M

PM

m¼1 p

~ r ; SðmÞ ; CðmÞ ; . . . ; C ðmÞ ; b~ ðmÞ ; nðmÞ ; tðmÞ ; y

a~ jM N 1 a~ ¼0

~ p a~ jM r a~ ¼0

where the numerator and denominator are given, respectively, by the following two expressions: M

ð2pÞNnr=2 X 1 ðmÞ 0 ðmÞ

ðmÞ 1=2 V a~ ba~ exp ba~

V a~

2 M m¼1 and Gðnn =2Þ nn Nnr mn p Gððnv NnrÞ=2Þ

!Nnr=2

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where M is the number of (post burn-in) replications of the Gibbs sampler, (m) superscripts denote Gibbs sampler draws, and ðba~ ; V a~ Þ are the elements of ðbb~; V b~Þ that correspond to a~ . Calculating the probability that: sp(b)Dsp(H). Finally, we present the SDDR to obtain the posterior probability that sp(b)Dsp(H) in the case where sp(bi) ¼ sp(b). To evaluate the hypothesis ~ where j ~ ¼ H 0 b~ is a s r that sp(b)Dsp(H), we write b~ ¼ H j~ þ H ? l, 0 ~ ~ full rank matrix and l ¼ H ? b is a (ns) r full rank matrix. Note that ~ is normally distributed with mean l ¼ ðI r H 0 Þb ~ and covariance vecðlÞ ? b matrix V l~ ¼ ðI r H 0? ÞV b~ ðI r H ? Þ Therefore, the overidentifying restriction can be imposed by setting l~ ¼ 0. The SDDR can be estimated by:

M

X 1 ðmÞ0 ðmÞ

ðmÞ 1=2 l l V exp V

~ ~ l 2 Mðnt =ðmt pnÞÞðnsÞr=2 ðGððnt þ ðn sÞrÞÞ=2Þ=Gðnt =2Þ m¼1 l ð2pÞðnsÞr=2

INVESTIGATING NONLINEAR PURCHASING POWER PARITY DURING THE POST-BRETTON WOODS ERA – A BAYESIAN EXPONENTIAL SMOOTH TRANSITION VECM APPROACH Deborah Gefang ABSTRACT This paper proposes a Bayesian procedure to investigate the purchasing power parity (PPP) utilizing an exponential smooth transition vector error correction model (VECM). Employing a simple Gibbs sampler, we jointly estimate the cointegrating relationship along with the nonlinearities caused by the departures from the long-run equilibrium. By allowing for nonlinear regime changes, we provide strong evidence that PPP holds between the US and each of the remaining G7 countries. The model we employed implies that the dynamics of the PPP deviations can be rather complex, which is attested to by the impulse response analysis.

Bayesian Econometrics Advances in Econometrics, Volume 23, 471–500 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23014-8

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1. INTRODUCTION Given its importance in open economy macro modeling and policy advice, the validity of purchasing power parity (PPP) over the post-Bretton Woods era has been the subject of intensive study in the literature. Employing unit root tests or cointegration tests in a linear framework, earlier work generally fails to conﬁrm the presence of PPP over the modern ﬂoating exchange rate regime (e.g., Meese & Rogoff, 1988; Edison & Fisher, 1991; Mark, 1990). Inspired by the theoretical arguments that emphasize the role of the transaction cost as proposed by Dumas (1992) and Sercu, Uppal, and Van Hulle (1995), among others, recent studies turn to analyze whether PPP adjustment follows a nonlinear process. This research has led to evidence in favor of relative PPP (e.g., Michael, Nobay, & Peel, 1997; Baum, Barkoulas, & Caglayan, 2001; Sarno, Taylor, & Chowdhury, 2004; Peel & Venetis, 2005).1 The majority of the literature modeling PPP in a nonlinear framework uses univariate models. In these models, the variable of concern is the real exchange rate which is calculated by imposing a cointegrating vector on the nominal exchange rates and the foreign and domestic price levels.2 However, given the interrelationships among the three variables that constitute PPP, multivariate models, especially nonlinear vector error correction models (VECM), can be more effective in capturing both the long-run and shortrun dynamics of PPP adjustment. Perhaps the reason why researchers have not followed this route is due to the lack of a fully developed statistics tool that can directly test the cointegration (or no cointegration) null in a nonlinear VECM against its both linear and nonlinear alternatives (see Seo, 2004, 2006; Kapetanios, Shin, & Snell, 2006 for the latest developments in the nonlinear VECM tests). This paper proposes a Bayesian approach to investigate PPP within an exponential smooth transition VECM (ESVECM) framework. Speciﬁcally, we follow the Bayesian cointegration space approach introduced by Strachan and Inder (2004) and the Bayesian logistic smooth transition vector autoregressive (LSVAR) approach of Gefang and Strachan (2007).3 Our method jointly captures the equilibrium and the presence of nonlinearity in the ESVECM in a single step. Compared with the available classical estimation techniques, which often require multiple steps and Taylor expansions, our approach is less susceptible to the sequential testing and inaccurate approximations problems. Furthermore, the commonly used maximum likelihood estimation in classical works is subject to the multimode problem caused by the nuisance parameters in the transition function

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of ESVECM. Yet, jagged likelihood functions do not create any particular problems in our Gibbs sampling scheme. In our empirical investigation, we analyze the validity of PPP between the US and the remaining six G7 countries over the post-Bretton Woods era. We take account of model uncertainty through Bayesian model selection and Bayesian model averaging. We use Bayes factors to calculate the posterior model probabilities. Speciﬁcally, Bayes factors are derived from the Savage–Dickey density ratio (SDDR) approach of Koop and Potter (1999) and the Bayesian information criterion (BIC) (Schwarz, 1978) approximation approach of Kass and Raftery (1995). Here, different models are distinguished by the presence of the cointegration relationship, the order of the model, whether there exist nonlinear effects, and the transition variables which trigger the regime changes. Our estimation results strongly support that PPP holds, while the dynamics of the adjustment process to PPP is nonlinear. Furthermore, our results from the general impulse response functions show that the dynamics of the misalignment from PPP is determined by the sources and magnitudes of the original shocks. The rest of the paper is structured as follows. Section 2 introduces the ESVECM model and Bayesian inferences. Section 3 reports empirical results and Section 4 concludes.

2. THE EXPONENTIAL SMOOTH TRANSITION VECM Under the relative PPP, the nominal exchange rates and domestic and foreign prices should follow a cointegration relationship. However, as argued by Dumas (1992), among others, due to the presence of the transaction cost, the adjustment toward PPP should follow a nonlinear process, where small deviations from PPP are left uncorrected for the proﬁt is not large enough to cover the transaction costs. In the multivariate framework, this type of nonlinear adjustment can be captured by a threshold VECM (TVECM) or an ESVECM. In a VECM, the adjustment process induced by deviations from the long-run equilibrium is a linear function of the magnitude of the deviations from that long-run equilibrium. In contrast, in a TVECM or an ESVECM, the dynamics of the adjustment process change across regimes, and the driving force of the regime changes is governed by the observed deviations from the equilibrium through the transaction function. In a TVECM, the regime changes are assumed to be

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discrete, whereas in an ESVECM, the regimes change smoothly. Since the market force driving PPP adjustment is an aggregated process, following the suggestions of Tera¨svirta (1994), we use an ESVECM to model the nonlinear convergence toward PPP between two countries. ESVECM appears to have another attractive property for it allows the same dynamics of regime changes for deviations above and below the equilibrium level. Let yt ¼ ½st pt pt , where st, pt, and pt are the logarithms of the foreign price of the domestic currency and respective domestic and foreign price levels. Assuming the cointegration relationships are common across different regimes, we model PPP in the exponential smooth transition VECM for t ¼ 1, . . . , T as follows. Dyt ¼ yt1 ba þ d t n þ Sph¼1 Dyth Ch þ Fðzt Þðyt1 baz þ d t nz þ Rph¼1 Dyth Czh Þ þ t

ð1Þ

where Dyt ¼ ytyt1. The error term et is a Gaussian white noise process, with E(et) ¼ 0, Eð0s t Þ¼ S for s ¼ t, and Eð0s t Þ¼ 0 for s 6¼ t. The dimensions of Ch and Czh are 3 3, and the dimensions of b, au, and azu are 3 r, where r is the rank of the cointegration space, with r can be either 0 or 1. If PPP holds, the value of r should be equal to 1. In model (1), the regime changes are assumed to be caused by a past deviation from the equilibrium relationship, and the dynamics of the regime changes is captured by the symmetric U-shaped exponential smooth transition function proposed by Tera¨svirta (1994): Fðzt Þ ¼ 1 expðgðzt cÞ2 Þ

(2)

where the transition variable zt ¼ ytdb is the cointegrating combination among s, p, and p at period td;4 c, the equilibrium level of the cointegrating relationship, also the threshold around which the regime changes; g, the smooth parameter that governs the speed of the transition process between extreme regimes, with higher values of g implying faster transition. The transition function F(zt) is bounded by 0 and 1. It is seen that F(zt) ¼ 0 when ztc ¼ 0, and F(zt) ¼ 1 when ztc-7N. As convention, we deﬁne F(zt) ¼ 0 and F(zt) ¼ 1 corresponding to the middle and outer regimes, respectively. In the middle regime, model (1) becomes a linear VECM, with the adjustment process governed by (a, n, Ch); while in the outer regime, model (1) becomes a different linear VECM, where the dynamics of the model are determined by ða þ az ; n þ nz ; Ch þ Czh Þ. Between the two extreme regimes, the speed of PPP adjustment is determined by the

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deviations from the equilibrium. For small deviations from PPP, the model is more dependent on the parameters of the middle regime. Once the deviations get larger, the adjustment process will be more inﬂuenced by the parameters in the outer regime. Finally note that Eq. (1) allows a set of models which vary in the rank of the cointegration vector (0 or 1), the order of the autoregressive process, the lag length of the transition variable, and the presence of the nonlinearity.

2.1. The Likelihood Function For notational convenience, we can rewrite model (1) as Dyt ¼ x1;t1 ba þ x2;t U þ Fðzt Þðx1;t1 baz þ x2;t Uz Þ þ t

(3)

where x1,t1 ¼ yt1, x2,t ¼ (dt, Dyt1, . . . , Dytp), U ¼ ðn0 ; C0 1 ; . . . ; C0 p Þ0 , 0 0 0 Uz ¼ ðnz ; Cz1 ; . . . ; Czp Þ0 . To simplify the notation, we ﬁrst deﬁne the T n matrix X0 ¼ ðDy01 ; Dy02 ; . . . ; Dy0T Þ0 and T 2(rþ2þnp) matrix X ¼ (X1b X2 FzX1b FzX2), where X1 ¼ ðx01;1 ; x01;2 ; . . . ; x01;T Þ0 , X2 ¼ ðx02;1 ; x02;2 ; . . . ; x02;T Þ0 , and Fz ¼ 0 0 diag(F(z1), F(z2), . . . , F(zT)). Next, we set B ¼ ða0 U0 az Uz Þ0 . Finally, we stack the error terms et in the T n matrix E, where E ¼ ð01 ; 02 ; . . . ; 0T Þ0 . Hence, model (1) can be written as X0 ¼ X1 ba þ X2 U þ Fz X1 baz þ Fz X2 Uz þ E ¼ XB þ E The likelihood function of model (4) is as follows: 1 T=2 1 0 LðyjR; B; bÞ / jRj exp trR E E 2

(4)

(5)

Vectorizing model (4), we transform model (1) into x0 ¼ xb þ e

(6)

where x0 ¼ vec(X0), x ¼ In X, b ¼ vec(B), and e ¼ vec(E). Note that E(eeu) ¼ Ve ¼ R IT. Hence, trS1 E0 E ¼ e0 ðR1 IT Þe ^ 0 V1 ðb bÞ ^ ¼ s2 þ ðb bÞ

(7)

where s2 ¼ x00 Mv x0 ; Mv ¼ R1 ½IT XðX0 XÞ1 X0 , b^ ¼ ½In ðX0 XÞ1 X0 x0 and V ¼ R ðX0 XÞ1 . Thus, the likelihood function in Eq. (5) can be

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rewritten as 1 ^ 0 V1 ðb bÞ ^ LðyjR; B; bÞ / jRjT=2 exp ½s2 þ ðb bÞ 2

(8)

2.2. Priors Although the strict version of PPP states that the combination stþpt p should be stationary, there is no theoretical guidelines to specify the values of b in the cointegration relationship for the relative PPP. Furthermore, it is impossible to impose meaningful informative priors for the coefﬁcients of the long-run/short-run adjustment in the VECM or for the parameter that indicates the speed of regime changes in the transition function. Therefore, we use the uninformative or weakly informative priors to allow the data information to dominate any prior information. To start with, we assume that all possible models are to be, a priori, equally likely. Before eliciting our priors of the parameters, it is worthwhile to stress the identiﬁcation problems in our model setting. Note that both a linear VECM and a simple smooth transition VAR model suffer from the identiﬁcation problems. As well documented in the literature, a linear VECM suffers from both the global and local nonidentiﬁcation of the cointegration vectors and parameters corresponding to the long-run adjustment. In Bayesian literature, a great effort has been made to surmount this problem. In earlier research, to set uninformative prior for the cointegration vector b, researchers ﬁrst normalize b into b ¼ [Ir Vu]u, then impose uninformative prior on the sub-vector V. However, as argued by Strachan and van Dijk (2004), this approach has an undesirable side-effect that it favors the regions of cointegration space where the imposed linear normalization is actually invalid. In most recent work, researchers have worked on putting uninformative priors on the cointegration space (e.g., Strachan, 2003; Strachan & Inder, 2004; Villani, 2005). As pointed out by Koop, Strachan, van Dijk, and Villani (2006) in their survey on the Bayesian approaches to cointegration, since only the space of the cointegration vector can be derived from the data, it is better to elicit priors in terms of the cointegration space than in terms of cointegration vectors. With regards to the smooth transition part of the model, as explained by Lubrano (1999a), since Bayesians have to integrate over the whole domain of the smooth parameter, the identiﬁcation problem that arises from g ¼ 0 (the so-called

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Davies’ problem (Davies, 1977), see Koop and Potter (1999) for further explanation) becomes more serious in Bayesian context than in the classical framework. Bauwens et al. (1999) and Lubrano (1999a, 1999b) introduce a number of prior settings to solve the problem. Following Gefang and Strachan (2007), we tackle this problem by simply setting the prior distribution of g as Gamma. The nonidentiﬁcation problem faced by the ESVECM model is slightly different. Although the Davies’ problem remains relatively the same as in a smooth transition VAR, the problem in identifying the cointegration vector and its adjustment parameters is subject to the additional inﬂuence from the transition parameter. Here the cointegration vector comes forth in three combinations, namely ba, bg, and baz. However, this difference does not render the identiﬁcation problem more complicated than what we have to deal with in a single linear VECM and a LSTVAR. As long as we can rule out the possibility that g ¼ 0, we can identify b, a, az, and g sequentially once we choose a way to normalize b. Following the arguments of Koop et al. (2006), we elicit the prior of b indirectly from the prior expressed upon the cointegration space.5 While we adopt the general approach developed in Strachan and Inder (2004), we diverge in two aspects important for this application. First, as is standard in the cointegration analysis of PPP (as there is considerable empirical evidence and theoretical support for this restriction), we only consider a single equilibrium cointegrating relationship. Second, we restrict ourselves to the economically justiﬁable region of the cointegrating space where the signs of the elements in the cointegration vector are [þ þ ] or [ þ]. Therefore, our method does not explore if there are any other long-run equilibrium relationships different from PPP. Having restricted the support to this region, the prior is otherwise uninformative on this space. Speciﬁcally, we set the prior of b as following. First, we specify the space of the three by one vector b to be uniformly distributed over the twodimensional Grassman manifold G1,2. pðbÞ ¼

1 c31

(9)

R where c31 ¼ G1;2 dg31 is a constant, which is the volume of the compact space G1,2 (James, 1954; Muirhead, 1982). Next, we restrict bub ¼ 1 for the purpose of identiﬁcation as the normalization method does not distort the distribution of the cointegration space (see Strachan & Inder, 2004, for

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DEBORAH GEFANG

further explanation). Thus, we can use polar coordinates to denote the semiorthogonal b as follows: b ¼ ½sinðy1 Þ sinðy2 Þ

sinðy1 Þ cosðy2 Þ

cosðy1 Þ0

To describe the uniform distribution of the cointegration space in polar coordinates, we multiply the uninformative prior of the space of b in Eq. (9) by sin(y1), the Jacobian of the transformation from rectangular coordinates to polar coordinates (Muirhead, 1982, p. 55). As explained before, we want to restrict the signs of the elements in b instead of allowing the space of b to move freely. The objective can be easily achieved by restricting the range of y1 to be from p/2 to p, and the range of y2 to be from 0 to p/2. With regards to the variance covariance matrix of the error terms, following Zellner (1971), we set standard diffuse prior for R: pðRÞ / jRjðnþ1Þ=2 For the purpose of our research, we need to calculate posterior model probabilities to compare across different possible models. As the dimension of b changes across different model speciﬁcations, to have Bayes factors well deﬁned, we are not allowed to set ﬂat prior for b (see Bartlett, 1957; O’Hagan, 1995, for details). Therefore, following Strachan and van Dijk (2006), we set weakly informative conditional proper prior for b as: PðbjS; y1 ; y2 ; g; c; M o Þ / Nð0; VÞ where b ¼ vec(B), V ¼ RZ1Ik, k ¼ 2(rþlþnp), Z is the shrinkage prior as proposed by Ni and Sun (2003). As practiced in Koop, Leon-Gonzalez, and Strachan (2006), we draw Z from the Gibbs sampler. In our case, we set the relatively uninformative prior distribution of Z as Gamma with mean mZ , and degrees of freedom nZ , where mZ ¼ 10, nZ ¼ 0:0001. Note that in our prior setting, the conditional weakly informative priors for a and az are the same, which are normal with zero mean and covariance matrix R ðb0 ZI3 bÞ1 . To avoid the Davies’s problem in the nuisance parameter space, following Gefang and Strachan (2007), we set the prior distribution for g as Gamma, which excludes a priori the point g ¼ 0 from the integration range. Since the nonlinear part of b can still be a vector of 0s as gW0, the prior speciﬁcation of g does not render model (1) in favor of the nonlinear effect. In the empirical work, we set the prior distribution of g as Gamma(1, 0.0001), which is relatively uninformative.

Investigating Nonlinear Purchasing Power Parity

479

Finally, to interpret our results more sensibly, we elicit the conditional prior of c as uniformly distributed between the upper and lower limits of the middle 80% of the transition variables (which, in our case, is the product of ½std ptd ptd and the cointegrating vector b). Note that the bounds of the support for c are both data dependent and dependent upon b.

2.3. Posterior Computation We use full conditional Gibbs sampler for posterior computations. From the priors just elicited and likelihood function derived in Section 2.1, we ﬁnd that the posterior of R is inverted Wishart (IW) with scale matrix EuE, and the degrees of freedom T, while the conditional posterior of b is normal with mean b ¼ vec½ðX0 X þ ZIk Þ1 X0 X0 and covariance matrix V ¼ R ðX0 X þ ZIk Þ1 . Note that the posterior distributions of y1, y2, g, and c are not of any standard form. However, the ranges of y1 and y2 are restricted as explained in the previous section, and in each run of the Gibbs sampler, the range of c can be predetermined based on the current draws of y1 and y2. Thus, we can use Griddy Gibbs sampling introduced in Ritter and Tanner (1992) to draw y1, y2, and c within the main Gibbs sampler. With respect to g, we resort to Metropolis–Hastings algorithms (Chib & Greenberg, 1995) within Gibbs for the estimation. In order to carry out all the aforementioned posterior analysis, we need to know the posterior of Z as well. The conditional posterior of Z is calculated as 1 (10) pðZjB; R; g; c; Y; XÞ / pðZÞjR1 ZIk j1=2 exp b0 ðR1 ZIk Þb 2 which indicates that the conditional posterior of Z is distributed as Gamma with the mean mZ ¼ ðnZ mZ Þ=ðnZ þ mZ trðB0 BS1 ÞÞ, and the degrees of freedom nZ ¼ nk þ nZ . The Gibbs sampling scheme can be summarized as follows: 1. 2. 3. 4. 5. 6. 7. 8.

Initialize (b, S, y1, y2, g, c, Z); Draw S| b, y1, y2, g, c, Z from IW(EuE, T); Draw b|S, y1, y2, g, c, Z from N(b; V b ); Draw y1, y2|S, b, g, c, Z numerically by Griddy Gibbs; Draw g|S, b, y1, y2, c, Z through Metropolis–Hastings method; Draw c|b, S, y1, y2, g, Z numerically by Griddy Gibbs; Draw Z|b, S, y1, y2, g, c from GðmZ ; nZ Þ; Repeat steps 2–7 for a suitable number of replications.

480

DEBORAH GEFANG

In case the draws from Metropolis–Hastings simulator get stuck in a local mode, we try different starting values for the sampler. One of the main concerns of our study is to examine the posterior probabilities of different possible models and trace the effects of cointegration and nonlinearity. For this purpose, we resort to two approaches, namely the SDDR approach of Koop and Potter (1999) and the BIC approximation approach of Kass and Raftery (1995), to calculate Bayes factors. As explained in Koop and Potter (1999), by penalizing parameter-rich models, using Bayes factors to calculate posterior odds ratio can resolve the over ﬁtting problems that generally exist in nonlinear models. Following Koop and Potter (1999) and Koop, Strachan et al. (2006), Koop, LeonGonzalez et al. (2006), we use SDDR to compute Bayes factors comparing every restricted model nested within the general model (1) with the general model itself. Using this information, we back out the posterior model probabilities for each country pair through a base model (e.g., the model where all the parameters in b are restricted to be zero). Note that the restricted linear VECM model occurs when all the elements of az and Uz are equal to zero. Likewise, the restricted linear VAR model with neither the cointegration nor the nonlinear effect occurs when we impose all the elements of a, az, and Uz to be equal to zeros.6 Hence, we can use the conditional posterior distribution and the conditional priors of b to compute the Bayes factor for the restricted model M1 (nested in model M2) versus the unrestricted model M2 using the SDDR, which is given by the expression: B1;2 ¼

PrðM 1 jyÞ pðbjM 2 ; yÞjbi ¼0 ¼ pðbjM 2 Þjbi ¼0 PrðM 2 jyÞ

where the restrictions are bi ¼ 0. Note that this method penalizes parameterrich models as explained in Koop and Potter (1999). With regard to the second approach, as illustrated by Kass and Raftery (1995), using the BIC to approximate Bayes factors cannot produce correct values even for very large samples. However, the approach is appealing in many aspects. For example, it can produce reasonable results in large samples, and it is easy to implement (see Kass & Raftery, 1995, and Koop, Potter, & Strachan, 2005, for further illustrations). Similar to the procedure in the SDDR approach, we integrate BIC calculation into the Gibbs sampler, then back out the posterior model probabilities by comparing the BIC of each possible model with that of a common base model.

Investigating Nonlinear Purchasing Power Parity

481

3. EMPIRICAL RESULTS In this section, we investigate whether PPP holds between the US and the other six G7 countries – Canada (CAN), France (FRA), Germany (GER), Italy (ITA), Japan (JAP), and the UK. In all cases, the US is considered the foreign country. We extract monthly nominal exchange rates and consumer price index (CPI) series from the International Financial Statistics database. For Canada, Japan, and the UK, the data span the period 1973:1 to 2006:12. For France and Italy, the sample period covers from 1973:1 until the ﬁxing of the Euro conversion rate 1998:12. For Germany, we use the former West Germany data running from 1973:1 to 1991:12. The Gibbs sampler is run for 12,000 passes with the ﬁrst 2,000 discarded. The convergence of the sequence draws is checked by the Convergence Diagnostic measure introduced by Geweke (1992). We use the MATLAB program from LeSage’s Econometrics Toolbox (LeSage, 1999) for the diagnostic. The parameter estimates are presented in Tables 1–3. Given the large amount of parameters being estimated, we only report the estimation results for the cointegration relationship, the threshold, and the smooth variable which indicate the speed of the regime changes. To aid comprehension, both the angles in the polar coordinates and corresponding elements in the cointegration vector are reported.7

3.1. Model Comparison Results In this section, we report results relating to the posterior model probabilities of 85 different models (namely 1 model with only the error terms, 6 linear VARs, 6 linear VECMs, 36 exponential smooth transition VAR models (ESVAR), and 36 ESVECM models) for each country pair.8 Among these models, both the maximal order of the autoregressive process and the longest lag length of the transition indicator are allowed to be 6. We assume the 85 models are exhaustive and mutually independent. Table 4 summarizes the total posterior probabilities of the models based on the two model comparison methods. Based on SDDR, in all cases, ESVECM models receive over 90% of the posterior model probabilities, which provides strong evidence that PPP holds, and the adjustment process towards PPP is nonlinear. When employing the BIC approach, we observe that the ESVECM models are not as dominant as using the SDDR approach. However, since the ESVECM and ESVAR models jointly account for more than 80% of the posterior model probabilities across

482

DEBORAH GEFANG

Table 1. d¼l US–CAN g 12.6520 3.5105 2.3150 y1 0.0537 1.4372 y2 0.0805 c 0.2820 0.4604 0.7291 b1 0.0980 b2 0.6774 b3 US–FRA g y1 y2 c b1 b2 b3

30.4830 4.8129 3.0893 0.0758 1.0376 0.4789 4.9709 0.4376 0.0450 0.0266 0.9986

Parameters (a).

d¼2

d¼3

d¼4

d¼5

d¼6

13.7470 4.1453 2.0431 0.0865 1.5365 0.0246 1.3063 0.5750 0.8900 0.0305 0.4550

13.3550 4.0240 2.0648 0.0685 1.4857 0.0519 1.3917 0.4554 0.8773 0.0749 0.4742

15.5270 5.2090 1.9085 0.0790 1.5337 0.0316 2.2738 0.5565 0.9429 0.0350 0.3313

13.8080 4.0288 1.9747 0.0930 1.5171 0.0463 1.8870 0.5909 0.9182 0.0493 0.3930

13.9260 3.4595 1.9319 0.0373 1.5283 0.0309 2.1457 0.2492 0.9346 0.0398 0.3534

28.7710 4.3462 3.0240 0.0498 1.0875 0.3591 4.4999 0.3262 0.1039 0.0545 0.9931

0.1131 0.3682 2.2722 0.0480 0.4227 0.1585 1.7649 0.4055 0.3134 0.6967 0.6453

25.9230 6.5391 3.0322 0.0954 1.0655 0.4284 4.5704 0.5818 0.0956 0.0529 0.9940

0.0339 0.1182 2.2540 0.0846 0.5105 0.1852 1.9835 0.6248 0.3790 0.6767 0.6313

0.0382 0.1242 2.3632 0.0698 0.6029 0.1697 1.1213 0.6242 0.3981 0.5783 0.7121

Note: Standard deviations are in italics.

all samples, the existence of PPP and nonlinear effects triggered by the deviations from PPP are also evident in the BIC approach. Overall, both the SDDR and BIC approaches put most probability on nonlinear models, suggesting it is improper to model the interrelationship among the nominal interest rate and domestic and foreign price levels in a linear framework. It may also be illuminating to look into the support for the VECM and VAR models in the linear context. Using the SDDR approach, we ﬁnd the linear VECM models are more favored over linear VAR models in all countries, especially in the cases of US–GER and US–UK. However, the results from the BIC indicate that in all cases, the linear VECM and linear VAR models receive relatively the same amount of posterior model probabilities, which would cast doubt over the validity of PPP in our

483

Investigating Nonlinear Purchasing Power Parity

Table 2.

US–GER g y1 y2 c b1 b2 b3 US–ITA g y1 y2 c b1 b2 b3

Parameters (b).

d¼l

d¼2

d¼3

d¼4

d¼5

d¼6

0.0136 0.0653 1.8754 0.0499 0.0186 0.0109 3.2665 0.3401 0.0177 0.9538 0.2999

0.9400 1.8586 1.8200 0.0609 0.0180 0.0112 3.6215 0.3841 0.0174 0.9690 0.2466

0.6267 1.1395 1.9436 0.0114 0.0130 0.0057 2.7841 0.0835 0.0121 0.9312 0.3642

9.0952 1.6984 1.8318 0.1280 0.0365 0.0347 3.5507 0.8344 0.0353 0.9655 0.2580

0.0517 0.1453 1.9217 0.0355 0.0139 0.0074 2.9426 0.2554 0.0131 0.9390 0.3438

14.4090 2.9474 1.9702 0.0033 0.0384 0.0255 2.6839 0.1183 0.0353 0.9206 0.3889

4.3500 2.9948 2.4866 0.0076 0.0615 0.0590 0.3737 0.0732 0.0374 0.6080 0.7931

7.8474 3.8444 2.4649 0.0033 0.0487 0.0409 0.2265 0.1011 0.0305 0.6255 0.7797

42.2050 8.1170 3.0858 0.0451 0.8980 0.4882 4.8580 0.3030 0.0436 0.0347 0.9985

6.4511 2.2292 2.5294 0.0048 0.0248 0.0158 0.8063 0.0546 0.0143 0.5745 0.8184

30.5660 2.0644 3.0539 0.0820 1.1982 0.3517 4.6900 0.5221 0.0816 0.0319 0.9962

3.4249 1.7186 2.4769 0.0187 0.0339 0.0362 0.3258 0.1963 0.0209 0.6165 0.7871

Note: See notes in Table 1.

sample if we were to choose to neglect the nonlinear effects in the ﬁrst instance. To provide information on the degree of model uncertainty, in Table 4 we also report the total posterior mass of the top 20 models. Observe that there is a great deal of model uncertainty when we resort to the BIC to calculate Bayes factors. Table 5 contains results of the sum of the posterior probabilities of the ESVECM models distinguished by the transition variables for each country pair. Using the BIC approach, we ﬁnd that in all cases, the posterior probabilities received by each transition indicator are relatively the same. However, the model comparison results derived from the SDDR show that the transition indicators with longer lag lengths are generally preferred over the shorter ones in modeling the nonlinear effects. Given the time lags

484

DEBORAH GEFANG

Table 3.

US–JAP g y1 y2 c b1 b2 b3 US–UK g y1 y2 c b1 b2 b3

Parameters (c).

d¼l

d¼2

d¼3

d¼4

d¼5

d¼6

22.7700 32.2510 2.6538 0.0856 0.7510 0.2628 1.4580 0.6592 0.3198 0.3426 0.8834

34.7190 7.3199 1.9233 0.0206 1.5155 0.0460 3.0393 0.2016 0.9371 0.0519 0.3452

85.1300 59.0440 1.8639 0.0517 0.9572 0.6698 3.7788 0.5806 0.7827 0.5513 0.2889

15.2390 6.5107 2.1151 0.0019 0.0763 0.0147 2.1029 0.0610 0.0652 0.8530 0.5178

1.5974 2.9569 2.3533 0.1240 0.3485 0.2272 0.7404 0.7972 0.2422 0.6665 0.7051

48.3000 11.0160 1.8933 0.0497 1.5201 0.0568 3.3569 0.4702 0.9472 0.0481 0.3170

12.6260 5.0581 2.0099 0.2439 1.2821 0.2369 2.4621 0.8302 0.8677 0.2577 0.4251

16.8620 6.9726 2.3977 0.0288 0.8151 0.0489 0.7374 0.2304 0.4929 0.4644 0.7358

17.0540 5.2787 2.3327 0.0596 0.9159 0.0901 1.1670 0.3802 0.5739 0.4407 0.6903

15.2000 5.2428 2.3090 0.0645 0.9517 0.0895 1.3126 0.3856 0.6024 0.4293 0.6729

14.2380 4.5599 1.7875 0.0891 1.5146 0.0741 3.1515 0.2692 0.9751 0.0549 0.2150

14.5250 6.1151 2.0869 0.2538 1.2118 0.2544 2.1100 0.9630 0.8143 0.3055 0.4935

Note: See notes in Table 1.

between the contract and settlement in international trade, this result is not surprising. However, in the case of US–FRA, it turns out that the most preferred lag length of the transition indicator is 2, and it receives nearly 100% of the posterior mass. To shed more light on the properties of the posterior probabilities, we report the individual top 20 models for each country pairs in Tables 6–8. In all cases, observe that except for a few exceptions, all the top 20 models chosen by both the SDDR and BIC are nonlinear. Observe that there is substantial model uncertainty when the BIC is employed. As a matter of fact, there is little evidence in favor of any single model. Turning to the results calculated from the SDDR, we also ﬁnd model uncertainty, nonetheless less obvious than what we ﬁnd when using the BIC approach.

Model

ESVECM ESVAR Top 20 models VECM/VARa a

US–CAN

Summarized Posterior Model Probabilities.

US–FRA

US–GER

US–ITA

US–JAP

US–UK

SDDR

BIC

SDDR

BIC

SDDR

BIC

SDDR

BIC

SDDR

BIC

SDDR

BIC

0.9451 0.0549 0.9911 8.0570

0.4286 0.4250 0.2413 1.0063

0.9997 0.0003 1.0000 4.4462

0.4363 0.4171 0.2478 1.0007

0.9442 0.0548 0.9898 10108.5000

0.4209 0.4337 0.2438 0.9714

0.9179 0.0821 1.0000 0.7355

0.4334 0.4221 0.2446 0.9774

0.9989 0.0011 1.0000 3.1522

0.4324 0.4208 0.2433 0.9998

0.9917 0.0083 0.9934 118.7717

0.4279 0.4265 0.2414 1.0030

Investigating Nonlinear Purchasing Power Parity

Table 4.

Reports the ratio of the total posterior probabilities of the linear VECM models to that of the linear VAR models.

485

486

Posterior Model Probabilities of the Transition Variables.

Table 5. Model

d¼1 d¼2 d¼3 d¼4 d¼5 d¼6

US–CAN

US–FRA

US–GER

US–ITA

US–JAP

US–UK

SDDR

BIC

SDDR

BIC

SDDR

BIC

SDDR

BIC

SDDR

BIC

SDDR

BIC

0.0000 0.0000 0.0000 0.0117 0.1058 0.8825

0.1667 0.1667 0.1667 0.1666 0.1667 0.1666

0.0040 0.9960 0.0000 0.0000 0.0000 0.0000

0.1643 0.1647 0.1687 0.1649 0.1687 0.1687

0.0020 0.0021 0.0020 0.1228 0.0020 0.8691

0.1667 0.1666 0.1669 0.1664 0.1668 0.1665

0.0000 0.0000 0.2501 0.0000 0.7499 0.0000

0.1677 0.1681 0.1641 0.1681 0.1641 0.1680

0.0000 0.0000 0.0000 0.0000 0.0000 1.0000

0.1674 0.1651 0.1659 0.1677 0.1682 0.1658

0.0002 0.0237 0.3377 0.6385 0.0000 0.0000

0.1666 0.1666 0.1666 0.1667 0.1667 0.1668

Note: d is the lag length of the transition variable.

DEBORAH GEFANG

Number

US–CAN SDDR

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Top 20 Most Preferred Models (a).

ESVECM(6)d ¼ 6 ESVECM(5)d ¼ 6 ESVECM(4)d ¼ 6 ESVECM(2)d ¼ 6 ESVECM(3)d ¼ 6 ESVECM(1)d ¼ 6 ESVAR(6)d ¼ 6 ESVECM(5)d ¼ 5 ESVECMb(5)d ¼ 5 ESVECM(4)d ¼ 5 ESVECM(2)d ¼ 5 ESVECM(3)d ¼ 5 ESVECM(1)d ¼ 5 ESVAR(1)d ¼ 6 ESVAR(1)d ¼ 5 ESVAR(6)d ¼ 5 ESVAR(5)d ¼ 6 ESVECM(6)d ¼ 4 ESVECM(5)d ¼ 4 ESVECM(4)d ¼ 4

US–FRA BIC

0.1415 0.1413 0.1402 0.1395 0.1390 0.1356 0.0377 0.0164 0.0164 0.0163 0.0162 0.0162 0.0158 0.0042 0.0039 0.0029 0.0026 0.0018 0.0018 0.0018

ESVECM(1)d ¼ 1 ESVECM(1)d ¼ 2 ESVECM(1)d ¼ 3 ESVECM(1)d ¼ 6 ESVECM(1)d ¼ 5 ESVECM(1)d ¼ 4 VECM(1) ESVECM(2)d ¼ 1 ESVECM(2)d ¼ 2 ESVECM(2)d ¼ 3 ESVECM(2)d ¼ 6 ESVECM(2)d ¼ 5 ESVECM(2)d ¼ 4 VECM(2) ESVECM(3)d ¼ 1 ESVECM(3)d ¼ 2 ESVECM(3)d ¼ 3 ESVECM(3)d ¼ 6 ESVECM(3)d ¼ 5 ESVECM(3)d ¼ 4

SDDR 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0119

ESVECM(6)d ¼ 2 ESVECM(3)d ¼ 2 ESVECM(2)d ¼ 2 ESVECM(4)d ¼ 2 ESVECM(1)d ¼ 2 ESVECM(5)d ¼ 2 ESVECM(6)d ¼ 1 ESVECM(3)d ¼ 1 ESVECM(4)d ¼ 1 ESVECM(5)d ¼ 1 ESVAR(5)d ¼ 2 ESVAR(4)d ¼ 2 ESVECM(2)d ¼ l ESVAR(3)d ¼ 2 ESVECM(6)d ¼ 4 ESVECM(1)d ¼ 1 ESVAR(2)d ¼ 2 ESVAR(6)d ¼ 2 ESVECM(3)d ¼ 4 ESVECM(4)d ¼ 4

BIC 0.8478 0.0417 0.0350 0.0266 0.0226 0.0220 0.0035 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

ESVECM(1)d ¼ 1 ESVECM(1)d ¼ 2 ESVECM(1)d ¼ 4 ESVAR(6)d ¼ 5 ESVAR(6)d ¼ 6 ESVECM(2)d ¼ l ESVAR(6)d ¼ 3 ESVECM(2)d ¼ 2 ESVECM(2)d ¼ 4 ESVAR(3)d ¼ 3 ESVAR(3)d ¼ 6 ESVAR(3)d ¼ 5 ESVECM(3)d ¼ 1 ESVECM(3)d ¼ 2 ESVECM(3)d ¼ 4 ESVAR(4)d ¼ 3 ESVAR(4)d ¼ 5 ESVAR(4)d ¼ 6 ESVAR(2)d ¼ 3 ESVAR(1)d ¼ 6

0.0126 0.0126 0.0126 0.0125 0.0125 0.0125 0.0125 0.0124 0.0124 0.0123 0.0123 0.0123 0.0123 0.0123 0.0123 0.0123 0.0123 0.0123 0.0123 0.0122

Investigating Nonlinear Purchasing Power Parity

Table 6.

Note: The order of the model is in parenthesis, and the subscript d denotes the lag length of the transition variable.

487

488

Table 7. No.

US–GER SDDR ESVECM(6)d ¼ 6 ESVECM(3)d ¼ 6 ESVECM(4)d ¼ 6 ESVECM(2)d ¼ 6 ESVECM(1)d ¼ 6 ESVECM(5)d ¼ 6 ESVAR(1)d ¼ 6 ESVECM(6)d ¼ 4 ESVECM(4)d ¼ 4 ESVECM(3)d ¼ 4 ESVECM(5)d ¼ 4 ESVECM(1)d ¼ 4 ESVECM(2)d ¼ 4 ESVAR(1)d ¼ 4 ESVAR(6)d ¼ 6 ESVAR(3)d ¼ 6 ESVAR(4)d ¼ 6 ESVAR(4)d ¼ 4 ESVAR(2)d ¼ 6 ESVAR(3)d ¼ 4

Note: See notes in Table 6.

US–ITA BIC

0.2128 0.1824 0.1468 0.1074 0.1071 0.0817 0.0211 0.0201 0.0178 0.0176 0.0159 0.0153 0.0152 0.0127 0.0037 0.0033 0.0030 0.0020 0.0019 0.0018

ESVAR(1)d ¼ 3 ESVAR(1)d ¼ 5 ESVAR(1)d ¼ 1 ESVAR(1)d ¼ 2 ESVAR(2)d ¼ 3 ESVAR(2)d ¼ 5 ESVAR(1)d ¼ 6 ESVAR(2)d ¼ 1 ESVAR(1)d ¼ 4 ESVAR(2)d ¼ 2 ESVECM(1)d ¼ 3 ESVAR(3)d ¼ 3 VAR(1) ESVECM(1)d ¼ 6 ESVECM(1)d ¼ 5 ESVAR(3)d ¼ 5 ESVECM(1)d ¼ 1 ESVAR(2)d ¼ 6 ESVECM(1)d ¼ 4 ESVAR(3)d ¼ 1

SDDR 0.0123 0.0123 0.0123 0.0123 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121

ESVECM(2)d ¼ 5 ESVECM(6)d ¼ 5 ESVECM(3)d ¼ 5 ESVECM(4)d ¼ 5 ESVECM (5)d ¼ 5 ESVECM(1)d ¼ 5 ESVECM(6)d ¼ 3 ESVECM(4)d ¼ 3 ESVECM(3)d ¼ 3 ESVECM(2)d ¼ 3 ESVECM(5)d ¼ 3 ESVAR(2)d ¼ 5 ESVAR(3)d ¼ 5 ESVAR(4)d ¼ 5 ESVECM(1)d ¼ 3 ESVAR(5)d ¼ 5 ESVAR(4)d ¼ 3 ESVAR(3)d ¼ 3 ESVAR(4)d ¼ 3 ESVAR(2)d ¼ 3

BIC 0.1600 0.1470 0.1287 0.1134 0.0734 0.0708 0.0583 0.0529 0.0395 0.0331 0.0313 0.0185 0.0152 0.0136 0.0096 0.0093 0.0086 0.0062 0.0056 0.0049

ESVECM(1)d ¼ 3 ESVECM(1)d ¼ 5 ESVECM(2)d ¼ 3 ESVECM(2)d ¼ 5 ESVAR(1)d ¼ 4 ESVAR(1)d ¼ 6 ESVECM(1)d ¼ 2 ESVECM(1)d ¼ 6 ESVAR(1)d ¼ 2 ESVECM(1)d ¼ 1 ESVAR(2)d ¼ 4 ESVECM(1)d ¼ 4 ESVAR(2)d ¼ 2 ESVAR(3)d ¼ 4 ESVAR(2)d ¼ 6 ESVAR(3)d ¼ 2 ESVECM(2)d ¼ 2 ESVECM(2)d ¼ 6 ESVAR(3)d ¼ 6 ESVECM(2)d ¼ 1

0.0124 0.0124 0.0123 0.0123 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122

DEBORAH GEFANG

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Top 20 Most Preferred Models (b).

No.

US–JAP SDDR

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

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ESVECM(3)d ¼ 6 ESVECM(2)d ¼ 6 ESVECM(6)d ¼ 6 ESVECM(4)d ¼ 6 ESVECM(5)d ¼ 6 ESVECM(1)d ¼ 6 ESVAR(3)d ¼ 6 ESVAR(2)d ¼ 6 ESVAR(6)d ¼ 6 ESVAR(4)d ¼ 6 ESVAR(5)d ¼ 6 ESVECM(6)d ¼ 2 ESVECM(5)d ¼ 2 ESVECM(6)d ¼ 5 ESVECM(4)d ¼ 5 ESVECM(5)d ¼ 5 ESVECM(3)d ¼ 5 ESVAR(6)d ¼ 2 ESVAR(5)d ¼ 5 ESVECM(2)d ¼ 5

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0.2838 0.2083 0.2045 0.1473 0.0989 0.0562 0.0004 0.0003 0.0002 0.0002 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

ESVECM(1)d ¼ 1 ESVECM(1)d ¼ 6 ESVECM(1)d ¼ 3 ESVECM(1)d ¼ 2 ESVAR(1)d ¼ 5 ESVECM(1)d ¼ 5 ESVECM(1)d ¼ 4 ESVAR(2)d ¼ 5 ESVECM(2)d ¼ l ESVECM(2)d ¼ 3 ESVECM(2)d ¼ 6 ESVAR(3)d ¼ 56 ESVECM(2)d ¼ 2 ESVAR(5)d ¼ 5 ESVAR(4)d ¼ 5 ESVAR(6)d ¼ 5 ESVECM(2)d ¼ 5 ESVECM(2)d ¼ 4 VECM(1) ESVECM(3)d ¼ 2

SDDR 0.0123 0.0123 0.0123 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121

ESVECM(1)d ¼ 4 ESVECM(2)d ¼ 4 ESVECM(6)d ¼ 4 ESVECM(3)d ¼ 4 ESVECM(4)d ¼ 4 ESVECM(5)d ¼ 4 ESVECM(1)d ¼ 3 ESVECM(6)d ¼ 3 ESVECM(2)d ¼ 3 ESVECM(3)d ¼ 3 ESVECM(4)d ¼ 3 ESVECM(5)d ¼ 3 ESVECM(1)d ¼ 2 ESVECM(6)d ¼ 2 ESVECM(2)d ¼ 2 ESVECM(3)d ¼ 2 ESVECM(4)d ¼ 2 ESVECM(5)d ¼ 2 ESVAR(2)d ¼ 4 ESVAR(3)d ¼ 4

BIC 0.1293 0.1235 0.1049 0.1044 0.0897 0.0828 0.0617 0.0613 0.0575 0.0552 0.0500 0.0486 0.0046 0.0040 0.0039 0.0038 0.0033 0.0030 0.0010 0.0009

ESVECM(1)d ¼ 2 ESVECM(1)d ¼ 4 ESVECM(1)d ¼ 3 ESVECM(1)d ¼ 6 ESVECM(1)d ¼ 1 ESVECM(1)d ¼ 5 VECM(2) ESVECM(2)d ¼ 2 ESVECM(2)d ¼ 4 ESVECM(2)d ¼ 3 ESVECM(2)d ¼ 6 ESVECM(2)d ¼ l ESVECM(2)d ¼ 5 ESVAR(1)d ¼ 6 ESVAR(1)d ¼ 5 ESVAR(1)d ¼ 4 ESVAR(1)d ¼ 1 ESVAR(1)d ¼ 3 ESVECM(2)d ¼ 5 VECM(3)

0.0122 0.0122 0.0122 0.0122 0.0122 0.0122 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121 0.0121 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120 0.0120

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Table 8.

Note: See notes in Table 6.

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When we employ the SDDR, for all country pairs, the top 20 models account for more than 99% of the total posterior mass. However, the degree of model uncertainties is rather different across country pairs. In the case of US–FRA, the single most preferred model obtains 84.78% of the posterior model probabilities, more than 99% of the posterior model probabilities are taken by the top six models. While in other cases, although a great majority of the posterior mass is also taken by the top six models, the posterior model probabilities tend to spread across the six models more evenly. For example, in the case of US–CAN, each of the top six models accounts for relatively 14% of the posterior model probabilities. In the case of US–DEU, the posterior model probabilities of the top six models range from 8.17% to 21.27%. The most obvious case of model uncertainty can be found in the US–UK pair. For US–UK, 96.89% of the posterior mass is scattered across 12 models, with their posterior models probabilities ranging from 4.85% to 12.93%. Given the deferent degrees of model uncertainty found via the SDDR and the BIC, we report the time proﬁles of the smooth transition functions of the most probable models based on the SDDR in Fig. 1, and the time paths of the smooth transition functions derived by Bayesian model averaging (BMA) based on the BIC in Fig. 2. The graphs show that the regime switching processes are rather smooth for all cases, thus it is improper to adopt an abrupt function to model the nonlinear effects. In Fig. 1, observe that throughout the years, in the cases of US–CAN and US–UK, the dynamics of regime changes is gradually switching from the outer regimes toward the middle regimes. In the cases of US–FRA and US–ITA, we observe U-shaped time proﬁles, with the former hit the middle regime in November, 1980, and the latter hit the middle regime in August, 1980. In the case of US–GER, the dynamics of the PPP adjustment remains very close to the middle regime. In contrast, for US–JAP, the dynamics of the regime changes is in the outer regime during most of the time. For comparison, we plot the BMA-based smooth transition functions in Fig. 2. Observe that the corresponding graphs in Figs. 1–2 display a relatively similar pattern. However, since BMA based on the BIC gives roughly equal weights across different models, the range of the transition functions becomes narrower in Fig. 2.

3.2. Impulse Response Analysis It is acknowledged that the impulse response functions of the nonlinear models are history- and shock-dependent (e.g., Potter, 1994; Koop,

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Pesaran, & Potter, 1996 ). We use the generalized impulse response function proposed in Koop et al. (1996) to examine the effect of a shock on the PPP relationship. In particular, we examine the generalized impulse response functions of GIP for a shock, ut, and a history, ot1 as follows GI P ðn; ut ; ot1 Þ ¼ E½ptþnjut ;ot1 E½ptþn jot1

(11)

where n is the time horizon. By averaging out the future shocks, in Eq. (11), we treat the impulse responses as an average of what might happen given what has happened. Using Bayesian approach, we calculate the generalized impulse responses by averaging out the history uncertainties, the future uncertainties, the parameter uncertainties, and model uncertainties. To examine the impulse response functions of the cointegrating PPP combination, we allow a shock amounting to 70.01 and 70.02, respectively, to hit each of the three variables (namely st, pt, and pt ). The time horizon of the impulse responses is set to 60 months. Note that for each country pair, we have 85 models, 12 different shocks, and 2 methods for model comparison (based on the SDDR and BIC). For brevity, we only present the impulse response functions of the PPP combinations for the most preferred models in Figs. 3–5.9 Inspecting the impulse response functions, we have two main ﬁndings. i. The dynamics of PPP deviations are determined by the sources and magnitudes of the initial shocks that hit st, pt, and pt . ii. Deviations from PPP are mean-reverting in the next 5 years in all cases except for when shocks are originated from Canada’s price levels. However, the types of convergence processes are rather different across different cases. Our ﬁndings in the impulse response functions of PPP might shed some light on the discussions regarding the half life of PPP adjustment.10 As shown in our study, the impacts on PPP relationship varies with the sources and magnitudes of the initial shocks hitting st, pt, and pt . In the cointegrating context, an amount of deviation from PPP can be traced to a myriad combinations of initial shocks that hit st, pt, and pt . Hence, we suggest that any assertions on the speed of PPP convergence which neglect the causes of the deviation can be misleading.11

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Investigating Nonlinear Purchasing Power Parity

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4. CONCLUSION In this paper, we introduce a Bayesian approach to estimating an ESVECM model to investigate whether PPP holds between the US and the other six G7 countries. The model comparison results are in accord with the theoretical assertion that in the long run PPP holds, and the adjustment to PPP is a nonlinear process with the regime changes governed by the magnitude of deviations from the long-run PPP equilibrium. Furthermore, our research casts doubt over the practice of estimating the half life of PPP deviations. The analysis of the impulse response functions shows that the mean-reverting process of the PPP misalignment can be rather complex.

NOTES 1. Note that the research adopting a panel data framework (e.g., Lothian, 1997; Lopez & Papell, 2006) usually ﬁnds support for PPP in the real exchange rates under the recent ﬂoating exchange rate regime. However, the panel data approach is not free from controversies (e.g., O’Connell, 1998; Sarno & Taylor, 1998). In Bayesian framework, Kai Li (1999) proposes a system of equations model with hierarchical priors to surmount the problems associated with panel data unit root tests. 2. Generally, the imposed cointegrating vector is either in accord with the strict version of PPP or is pre-estimated through a linear VECM. 3. Their approach is based on the univariate smooth transition model estimation technique introduced by Bauwens, Lubrano, and Richard (1999). 4. Note that the driving force of the regime changes can be any exogenous or endogenous variables of concern. In this study, we only examine the nonlinear effects caused by the misalignments from PPP. 5. To our knowledge, in literature, only Sugita (2006) applies the Strachan and Inder (2004) methodology in deﬁning the prior density for cointegrating vector in a nonlinear VECM. In his model, the regime changes are assumed to follow a Markov switching process. 6. It is important to stress that as explained by Koop, Strachan et al. (2006), Koop, Leon-Gonzalez et al. (2006), in the linear VECM model, the rank of the cointegration relationship is equal to 0 if and only if a ¼ 0. 7. We identify the parameters by normalizing bub ¼ 1; linear identiﬁcation can be achieved by ﬁrst dividing the reported b by one of its element that of concern, then transform the reported g and c accordingly. 8. Note that by imposing restrictions on the long-run adjustment parameters a and az in the unrestricted model, the linear and nonlinear VAR models we considered are in differences. 9. The full set of graphs are available on request. 10. The half-life estimates has been extensively used in the literature to indicate the speed of PPP adjustment on real exchange rates (e.g., Cheung & Lai, 1994; Lothian & Taylor, 1996; Lopez, Murray, & Papell, 2005).

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11. Chortareas and Kapetanios (2005) also claim that using the half-life measure to analyze PPP adjustment might be problematic. However, their reasoning is different from ours.

ACKNOWLEDGMENTS I thank the referee, Gary Koop, Roberto Leon-Gonzalez, Rodney Strachan, and the participants at 6th Annual Advances in Econometrics Conference at LSU for their helpful comments and suggestions. This paper also has beneﬁted from the productive discussions with Roderick McCrorie and Emi Mise. Any remaining errors are the author’s responsibility. Financial support from the Department of Economics, University of Leicester is gratefully acknowledged.

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James, A. T. (1954). Normal multivariate analysis and the orthogonal group. Analysis of Mathematical Statistics, 25, 40–75. Kapetanios, G., Shin, Y., & Snell, A. (2006). Testing for cointegration in nonlinear smooth transition error correction models. Econometric Theory, 22, 279–303. Kass, R., & Raftery, A. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795. Koop, G., Leon-Gonzalez, R., & Strachan, R. W. (2006). Bayesian inference in a cointegration panel data model. Discussion Papers in Economics 06/2, Department of Economics, University of Leicester. Koop, G., Pesaran, M. H., & Potter, S. M. (1996). Impulse response analysis in nonlinear multivariate models. Journal of Econometrics, 74, 491–499. Koop, G., & Potter, S. M. (1999). Bayes factors and nonlinearity: Evidence from economic time series. Journal of Econometrics, 88, 251–281. Koop, G., Potter, S. M., & Strachan, R. W. (2005). Re-examining the consumption-wealth relationship: the role of model uncertainty. Discussion Papers in Economics 05/3, Department of Economics, University of Leicester. Koop, G., Strachan, R. W., van Dijk, H., & Villani, M. (2006). Bayesian approaches to cointegration. In: T. Mills & K. Patterson (Eds), The Palgrave handbook of econometrics, Vol. 1: Theoretical econometrics. Basingstoke: Palgrave-Macmillan. LeSage, J. (1999). Applied econometrics using MATLAB. http://www.spatialeconometrics. com/ Li, K. (1999). Testing symmetry and proportionality in PPP: A panel-data approach. Journal of Business and Economic Statistics, 17, 409–418. Lopez, C., Murray, C. J., & Papell, D. H. (2005). State of the art unit root tests and purchasing power parity. Journal of Money, Credit and Banking, 37(2), 361–369. Lopez, C., & Papell, D. H. (2006). Convergence to purchasing power parity at the commencement of the Euro. Review of International Economics, 14, 1–16. Lothian, J. R. (1997). Multi-country evidence on the behaviour of purchasing power parity under the current ﬂoat. Journal of International Money and Finance, 16, 19–35. Lothian, J. R., & Taylor, M. P. (1996). Real exchange rate behavior: The recent ﬂoat from the perspective of the past two centuries. Journal of Political Economy, 104(3), 488–509. Lubrano, M. (1999a). Bayesian analysis of nonlinear time series models with a threshold. In: Nonlinear econometric modelling. Cambridge: Cambridge University Press. Lubrano, M. (1999b). Smooth transition GARCH models: A Bayesian perspective. Universite Aix-Marseille III G.R.E.Q.A.M. 99a49. Mark, N. C. (1990). Real and nominal exchange rates in the long run: An empirical investigation. Journal of International Economics, 28, 115–136. Meese, R. A., & Rogoff, K. S. (1988). Was it real? The exchange-rate interest differential relation over the modern ﬂoating rate period. Journal of Finance, 43, 933–948. Michael, P., Nobay, A. R., & Peel, D. A. (1997). Transaction costs and nonlinear adjustment in real exchange rates: An empirical investigation. Journal of Political Economy, 105(4), 862–879. Muirhead, R. J. (1982). Aspects of multivariate statistical theory. New York: Wiley. Ni, S. X., & Sun, D. (2003). Noninformative priors and frequentist risks of Bayesian estimators of vector-autoregressive models. Journal of Econometrics, 115, 159–197. O’Connell, P. G. (1998). The overvaluation of purchasing power parity. Journal International Economics, 44, 1–19.

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O’Hagan, A. (1995). Fractional Bayes factors for model comparison. Journal of the Royal Statistical Society, B57, 99–138. Peel, D. A., & Venetis, I. A. (2005). Smooth transition models and arbitrage consistency. Economica, 72, 413–430. Potter, S. (1994). Nonlinear impulse response functions. Working Paper, Department of Economics, University of California, Los Angeles, CA. Ritter, C., & Tanner, M. A. (1992). Facilitating the Gibbs sampler: The Gibbs stopper and the Griddy–Gibbs sampler. Journal of the American Statistical Association, 87, 861–868. Sarno, L., & Taylor, M. P. (1998). Real exchange rate under the recent ﬂoat: Unequivocal evidence of mean reversion. Economic Letters, 60, 131–137. Sarno, L., Taylor, M. P., & Chowdhury, I. (2004). Nonlinear dynamics in deviations from the law of one price: A broad-based empirical study. Journal of International Money and Finance, 23, 1–25. Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics, 6(2), 461–464. Seo, B. (2004). Testing for nonlinear adjustment in smooth transition vector error correction models. Econometric Society, Far Eastern Meetings. Seo, M. (2006). Bootstrap testing for the null of no cointegration in a threshold vector error correction model. Journal of Econometrics, 143, 129–150. Sercu, P., Uppal, R., & Van Hulle, C. (1995). The exchange rate in the presence of transaction costs: Implications for tests of purchasing power parity. Journal of Finance, 50, 1309–1319. Strachan, R. W. (2003). Valid Bayesian estimation of the cointegrating error correction model. Journal of Business and Economic Statistics, 21(1), 185–195. Strachan, R. W., & Inder, B. (2004). Bayesian analysis of the error correction model. Journal of Econometrics, 123, 307–325. Strachan, R. W., & van Dijk, H. K. (2004). Bayesian model selection with an uninformative prior. Keele Economics Research Papers KERP 2004/01, Centre for Economic Research, Keele University. Strachan, R. W., & van Dijk, H. K. (2006). Model uncertainty and Bayesian model averaging in vector autoregressive processes. Discussion Papers in Economics 06/5, Department of Economics, University of Leicester. Sugita, K. (2006). Bayesian analysis of Markov switching vector error correction model. Discussion Paper/2006/13, Graduate School of Economics, Hitotsubashi University. Tera¨svirta, T. (1994). Speciﬁcation, estimation and evaluation of smooth transition autoregressive models. Journal of the American Statistical Association, 89(425), 208–218. Villani, M. (2005). Bayesian reference analysis of cointegration. Econometric Theory, 21, 326–357. Zellner, A. (1971). An introduction to Bayesian inference in econometrics. New York: Wiley.

BAYESIAN FORECAST COMBINATION FOR VAR MODELS$ Michael K. Andersson and Sune Karlsson ABSTRACT We consider forecast combination and, indirectly, model selection for VAR models when there is uncertainty about which variables to include in the model in addition to the forecast variables. The key difference from traditional Bayesian variable selection is that we also allow for uncertainty regarding which endogenous variables to include in the model. That is, all models include the forecast variables, but may otherwise have differing sets of endogenous variables. This is a difﬁcult problem to tackle with a traditional Bayesian approach. Our solution is to focus on the forecasting performance for the variables of interest and we construct model weights from the predictive likelihood of the forecast variables. The procedure is evaluated in a small simulation study and found to perform competitively in applications to real world data.

$

The views expressed in this paper are solely the responsibility of the authors and should not be interpreted as reﬂecting the views of the Executive Board of Sveriges Riksbank. We have beneﬁted from discussions with Martin Sko¨ld and Mattias Villani and the comments from an anonymous referee.

Bayesian Econometrics Advances in Econometrics, Volume 23, 501–524 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23015-X

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MICHAEL K. ANDERSSON AND SUNE KARLSSON

1. INTRODUCTION The increasing availability of data has spurred the interest in forecasting procedures that can extract information from a large number of variables in an efﬁcient manner. Examples include the diffusion indexes of Stock and Watson (2002b) and procedures based on combining forecasts from many models as in Jacobson and Karlsson (2004), see Stock and Watson (2006) for a recent review and additional references. While this development has clear implications for policy makers such as central banks (see Bernanke & Boivin, 2003), procedures of this type are not particularly widespread in these banks.1 One possible reason for the apparent lack of interest in the possibilities offered by these procedures is that the literature has largely focused on univariate forecasting procedures. This paper attempts to bridge this gap by proposing a Bayesian procedure for combining forecasts from multivariate forecasting models, e.g., VAR models. Standard applications of Bayesian model averaging suffer from a basic difﬁculty in this context, when additional variables are included and modeled the connection between the overall measure of ﬁt for the model, the marginal likelihood, and the expected forecasting performance for the variables of interest is lost. It is easy to see that the (multivariate) marginal likelihood can change when a model is modiﬁed by adding, removing, or exchanging variables without this having the corresponding effect on the predictive ability for the variable of interest. We circumvent this problem by focusing on the predictive performance for the variables of interest and base the forecast combination on the predictive likelihood as proposed by Eklund and Karlsson (2007) in the context of univariate forecasting models. While the basic predictive likelihood is also multivariate it is meaningful to marginalize the predictive distribution with respect to the auxiliary variables yielding a univariate predictive distribution and corresponding predictive likelihood. Forecasts from different models can then be combined using weights based on the univariate predictive likelihood. Speciﬁcally we consider forecast combination and, indirectly, model selection for VAR models when there is uncertainty about which additional variables to include in the model. Given a set of auxiliary variables that are expected to be useful for modeling and forecasting the variable of interest we consider the set of models that arise when taking all possible combinations of the auxiliary variables. The forecasts from these models are then combined using weights based on the predictive likelihood at the relevant forecast horizon.

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In most cases the predictive likelihood will not be available in closed form. Instead we use MCMC methods to simulate the predictive distribution and estimate the density function from the MCMC output. In addition the MCMC output is used to obtain forecast intervals both for forecasts based on a single model and the combined forecast. The procedure is evaluated in a simulation study and found to perform competitively in an application to forecasting the growth rate of US GDP. The remainder of the paper is organized as follows. Section 2 introduces the predictive likelihood as a basis for forecast combination and model averaging and shows how it can be used in the current context of multivariate, dynamic, forecasting models. Section 3 outlines our prior speciﬁcation. Sections 4 and 5 evaluate the performance of forecast combinations based on the predictive likelihood in a Monte Carlo study and an application to forecasting US GDP growth. Section 6 concludes.

2. BAYESIAN FORECAST COMBINATION Bayesian forecast combination is a straightforward application of Bayesian model averaging.2 Suppose that the forecaster has a set, M ¼ fM1 ; . . . ; MM g, of M possible forecasting models available, each speciﬁed in terms of a likelihood function Lðyjyi ; Mi Þ and prior distribution for the parameters in the model, pðyi jMi Þ. In addition the forecaster assigns prior probabilities, pðMi Þ, to each model, reﬂecting the forecaster’s prior conﬁdence in the models. The posterior model probabilities can then be obtained by routine application of Bayes theorem pðMi jyÞ ¼

mð yjMi ÞpðMi Þ M P

(1)

mðyjMj ÞpðMj Þ

j¼1

where

Z mðyjMi Þ ¼

Lðyjyi ; Mi Þpðyi jMi Þdyi

(2)

is the marginal likelihood of model Mi . The combined forecast is obtained as Eð yTþh jyÞ ¼

M X j¼1

Eð yTþh jy; Mj ÞpðMj jyÞ

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MICHAEL K. ANDERSSON AND SUNE KARLSSON

by weighting the forecasts from each model by the posterior model probabilities. It is easily seen that the Bayesian forecast combination is a special case of the general result that the marginal (over all models) posterior distribution for some function f of the parameters is pðfjyÞ ¼

M X

pðfjy; Mj ÞpðMj jyÞ

(3)

j¼1

The crucial feature of the marginal distribution (3) is that it takes account of both parameter and model uncertainty. It is thus relatively easy to produce prediction intervals that incorporate model uncertainty. The marginal likelihood (2) is the basic Bayesian measure of ﬁt of a model and is a joint assessment of how well the likelihood and parameter prior agrees with the data. It is the key quantity for determining the posterior model probabilities and hence the weights assigned to the forecasts from the different models.

2.1. Predictive Likelihood The marginal likelihood is well suited for combination of univariate forecasting models but, unfortunately, problematic when it comes to the combination of forecasts from multivariate forecasting models. Multivariate forecasting models, e.g., VAR-models, are typically built with the express purpose of forecasting a single variable and the remaining dependent variables in the model are only included if they are deemed to improve the forecasting performance for the variable of interest. As the marginal likelihood measures the ﬁt of the whole model, it is easy to see that the forecast performance can remain unaffected by a change in the model that either increases or decreases the marginal likelihood. This can happen when a dependent variable is exchanged for another variable or the dimension of the model changes as variables are added or dropped from the model. To overcome these problems with the marginal likelihood we propose to base the forecast combination on the predictive likelihood as suggested by Eklund and Karlsson (2007) in the context of univariate forecasting models. Our primary motivation for using the predictive likelihood is that it is meaningful to marginalize this over the nonforecasted variables to obtain a measure that is focused on the variable of interest. An added beneﬁt of the predictive likelihood is that it is a true out-of-sample measure of ﬁt whereas the marginal likelihood depends on the predictive content of the parameter

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prior. When combining the forecasts from a large set of models it is often too time consuming to provide well thought out parameter priors for all the models. Instead uninformative default priors such as the ones suggested by Ferna´ndez, Ley, and Steel (2001) are used and with this type of prior the marginal likelihood essentially reduces to an in-sample measure of ﬁt. Our use of the predictive likelihood is based on a split of the data, Y ¼ ðy0 1 ; y0 2 ; . . . ; y0 T Þ0 , into two parts, the training sample, Ynn ¼ ~n ¼ ðy0 1 ; y0 2 ; . . . ; y0 n Þ0 of size n, and an evaluation or hold out sample, Y ðy0 nþ1 ; y0 nþ2 ; . . . ; y0 T Þ0 of size m ¼ Tn, where yt ¼ (y1t, . . . ,yqt) is the vector of modeled variables. The training sample is used to convert the prior into a posterior and the predictive likelihood is obtained by marginalizing out the parameters from the joint distribution of data and parameters, Z ~ n jyi ; Yn ; Mi Þpðyi jYn ; Mi Þdyi ~ n jYn ; Mi ¼ LðY (4) p Y n n n ~n Technically this is the predictive distribution of an unknown Y conditional on the training sample, Ynn . When evaluated at the observed ~ n , Eq. (4) provides a measure of the out-of-sample predictive performance Y and we refer to this as the predictive likelihood. Since our primary interest is to forecast a subset of the q modeled variables the multivariate predictive likelihood (4) suffers from the same drawback as the marginal likelihood in that it is not directly informative about the forecasting performance for the variable of interest. To overcome this we marginalize the predictive ~ n with respect to the auxiliary variables, with y1 the distribution of Y variable of interest we have Z ~ n jYn ; Mi d y~ 2;n . . . d y~ q;n (5) p y~ 1;n jYnn ; Mi ¼ p Y n the marginal predictive likelihood for the hold of sample of y1 as a measure of the average predictive performance for the variable of interest. Replacing the marginal likelihood with the marginal predictive likelihood in Eq. (1) yields the predictive weights pð~y1;n jYnn ; Mi ÞpðMi Þ w Mi j~y1;n ; Ynn ¼ M P pð~y1;n jYnn ; Mj ÞpðMj Þ j¼1

and the combined forecast y^Tþh ¼

M X E yTþh jY; Mj ÞwðMi j~y1;n ; Ynn j¼1

(6)

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MICHAEL K. ANDERSSON AND SUNE KARLSSON

Note that the forecasts from each model is conditional on all available data up to time T. That is, the model-speciﬁc posterior is based on the full sample and the forecast is the expected value of yTþh with respect to this posterior. The sample split is only used for the purpose of calculating the predictive weights. Comparing Eq. (6) with the posterior model probabilities in Eq. (1) it is clear that there are two distinct differences between using predictive weights for forecast combination and standard Bayesian model averaging. The ﬁrst difference is the use of prior model probabilities in Eq. (6) instead of posterior model probabilities based on the training sample as suggested by Eq. (1). This is the sample split idea; the training sample is used to learn about the parameters of each model and the hold out sample is used to assess the forecasting performance and update the model weights. The second difference is that we marginalize out the auxiliary variables from the predictive likelihood to produce a measure of forecast performance that focuses on the variable of interest. While the predictive weights given by Eq. (6) strictly speaking can not be interpreted as posterior probabilities, they have several appealing properties in addition to providing a basis for meaningful marginalization with respect to the auxiliary variables in the model. Proper prior distributions are not required for the parameters. The predictive likelihood is, in contrast to the marginal likelihood, well deﬁned as long as the posterior distribution of the parameters conditioned on the training sample is proper. The predictive likelihood is not an absolute measure of forecasting performance. Instead it is relative to the precision of forecasts implied by the model and models with a good in-sample ﬁt are penalized when a ‘‘good’’ and ‘‘bad’’ model both forecasts poorly. This is illustrated in Fig. 1. If the forecast error is small (1, solid lines) as can be expected from a model with good in-sample ﬁt, the predictive likelihood prefers the ‘‘good’’ model but the ‘‘bad’’ model is favored if the forecast error (2, dashed lines) is larger than what can be expected from the ‘‘good’’ model. The predictive weights will thus be small for models that overﬁt the data or models that suffer from structural breaks. 2.2. Dynamic Models The predictive densities given by Eqs. (4) and (5) are joint predictive distributions for lead times h ¼ 1 through h ¼ m ¼ Tn. For dynamic

507

Bayesian Forecast Combination for VAR Models Predictive likelihood

-5

-4

-3

-2

-1

0

1

2

3

4 5 Forecast error

Fig. 1. Predictive Likelihood for a ‘‘Good’’ and a ‘‘Bad’’ Model. Predictive Likelihood for a ‘‘Good’’ Model (Circles) and a ‘‘Bad’’ Model (Dots) Evaluated at Small (Solid Lines) and Large (Dashed Lines) Forecast Errors.

models where the forecast precision typically deteriorates as the lead time increases these will not be appropriate measures of forecast performance if the focus is on producing forecasts for a few select lead times. One solution is to set m to the largest lead time, H, considered but this will typically be small (say eight quarters) and the Monte Carlo experiments in Eklund and Karlsson (2007) indicate that the hold out sample should be large, on the order of 70% of the data. To combine these two requirements we suggest using a series of short horizon predictive likelihoods, gðY; njMi Þ ¼

Th Yk

p y1;tþh1 ; . . . ; y1;tþhk jYnt ; Mi

(7)

t¼n

where h1, . . . , hk represents the lead times at which we wish to evaluate the forecast performance. The use of the predictive likelihood in dynamic models is complicated by the fact that the predictive likelihood is not available in closed form for lead times hW1. Instead the predictive distribution must be simulated and the predictive likelihood estimated from the simulation output. Standard density estimation techniques can be used for this purpose and works quite well if the predictive likelihood is evaluated at a single lead time. Evaluating the predictive likelihood at multiple horizons leads to more complex multivariate density estimation.

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MICHAEL K. ANDERSSON AND SUNE KARLSSON

To facilitate the use of multiple horizon predictive likelihoods we take advantage of the model structure and use the idea of Rao–Blackwellization to estimate the predictive likelihood. Consider the task of evaluating the unknown density fu at u ¼ x when we have draws from the joint distribution of (u, v) or only the marginal Rdistribution of v and R the conditional density fu|v is known. We want fu(x) ¼ fu,v (x, v) dv ¼ fu|v (x, v) fv (v) dv ¼ Ev [ fu|v (x, v)] where we make the dependence of fu|v on v explicit by including it as an argument toP the function. A simple Monte Carlo estimate is then given n n by f^u ðxÞ ¼ 1=R R i¼1 f ujv ðx; vi Þ where vi are the draws from the marginal distribution of v. The Rao–Blackwellized estimate will in general be quite precise even for moderate sample sizes and preserves any smoothness properties of the underlying density. In our case the conditioning variables are the parameters of the VARmodel, C, and W and we wish to estimate the predictive likelihood for the subset of variables and lead times, y1;tþh1 ; . . . ; y1;tþhk . For the VAR-model yt ¼

p X

yti Ai þ xt C þ ut

i¼1

¼ zt C þ ut

ð8Þ

with zt ¼ (yt1, . . . , ytp, xt) and normally distributed errors, utBN(0, W), the joint lead time 1 through H ¼ max (h1, . . . , hk) predictive distribution conditional on the parameters, pðytþ1 ; :::; ytþH jYnt ; Mi ; C; WÞ, is multivariate normal (see Lu¨tkepohl, 1993 for details). Consequently, the conditional predictive distribution for the subset of interest, pðy1;tþh1 ; . . . ; y1;tþhk jYnt ; Mi ; C; WÞ is also multivariate normal. The Rao–Blackwellized estimate of pð y1;tþh1 ; . . . ; y1;tþhk jYnt ; Mi Þ is then obtained as ^ y1;tþh1 ; . . . ; y1;tþhk jYnt ; Mi Þ ¼ pð

R 1X pð y1;tþh1 ; . . . ; y1;tþhk jYnt ; Mi ; CðiÞ ; WðiÞ Þ R i¼1

by averaging over draws C(i) and W(i) from the posterior distribution based on Ynt . The draws from the posterior distribution of the parameters are, in our case, obtained from a standard Gibbs sampler. The estimates of the predictive weights, ﬁnally, are formed as ^ njMi ÞpðMi Þ gðY; w^ Mi j~y1;n ; Ynn ¼ M P ^ njMj ÞpðMj Þ gðY; j¼1

(9)

Bayesian Forecast Combination for VAR Models

509

with ^ njMi Þ ¼ gðY;

Th Yk

p^ y1;tþh1 ; . . . ; y1;tþhk jYnt ; Mi

(10)

t¼n

3. PRIOR SPECIFICATION We use a normal-diffuse prior on the parameters in the VAR-model (8), i.e., vec (C) B N(c0, R0) and p(W)p|W|(qþ1)/2, see Kadiyala and Karlsson (1997) for details and the Gibbs sampler for simulating from the posterior distribution of C and W. The prior for C is a Litterman-type prior. That is, c0 is zero except for elements corresponding to the ﬁrst own lag of variables. These are set to unity for variables believed to be nonstationary and to 0.9 for stationary variables. R0 is a diagonal matrix and the prior standard deviations are given by p1 , own lags, k ¼ 1, . . . , p k p3 si p1 p2 , lags of variable j in equation i, k ¼ 1, . . . , p s j k p3 p4, deterministic variables where si is the residual standard deviation for equation i from the OLS ﬁt of the VAR-model. The model prior is given by pðMj Þ /

K Y

ddkk ð1 dk Þ1d k

k¼1

where dk ¼ 1 if variable k is included in the model and dk is the prior inclusion probability of variable k.

4. MONTE CARLO EXPERIMENT We use three small Monte Carlo experiments to evaluate the forecasting performance of forecast combinations based on the predictive weights (9). The data generating processes are a bivariate VAR(l), 0:5 0:2 (11) DGP 1 : yt ¼ yt21 þut 0:5 0:5

510

MICHAEL K. ANDERSSON AND SUNE KARLSSON

a bivariate VAR(2), DGP 2 : yt ¼ yt21

0:5

0:2

0:5

0:5

þ yt2

0:1

0:1

0:2 0:3

þ ut

(12)

a trivariate VAR(1), 0

0:5 B DGP 3 : yt ¼ yt21 @ 0:5 0:5

1 0:2 0:1 C 0:5 0:1 A þ ut 0:3 0:2

(13)

and, ﬁnally, a univariate AR(2) DGP 4 : yt ¼ 0:5yt1 þ 0:3yt2 þ ut

(14)

In addition we generate a set of ﬁve extraneous variables as z1;t ¼ 0:5y1;t1 þ 0:5z1;t1 þ e1;t z2;t ¼ 0:5y2;t1 þ 0:5z2;t1 þ e2;t z3;t ¼ 0:7z3;t1 þ e3;t z4;t ¼ 0:2z4;t1 þ e4;t z5;t ¼ e5;t with ui,t and ei,t are i.i.d. standard normal random variables. The last, white noise, extraneous variable is dropped with the trivariate VAR-model and a second white noise series is added for the univariate AR-model so that the generated datasets in each Monte Carlo experiment consist of seven variables. For each experiment we generate 100 datasets of length 112 with the last 12 observations set aside for forecast evaluation. The variable to be forecasted is y1,t. For the bivariate DGPs we consider the 42 models arising from modeling y1,t alone or together with combinations of y2,t and z1,t, . . . , z5,t with a maximum of four variables in the model, for the trivariate DGP we consider the 57 possible models when allowing a maximum of ﬁve variables in the model. For the univariate AR(2) z2 simpliﬁes to z2,t ¼ 0.5 z2,t1þe2,t and we consider models with a maximum of four variables. We use two settings for the lag length of the VAR-models, p ¼ 2 and p ¼ 4. We are particularly concerned about the number of observations needed for the hold out sample, for this we consider three cases, m ¼ 30, m ¼ 50, and m ¼ 70 (m ¼ 70 is not used in combination with lag length 4 in the estimated models since this would reduce the number of available

Bayesian Forecast Combination for VAR Models

511

observations too much) and the effect of the lead time used for the calculation of the predictive weights, here we consider eight alternatives, the single lead times h ¼ 1, 2, 3, 4, and 8 and the multiple lead times h ¼ (1, 2, 3, 4), h ¼ (1, 2, 3, 4, 5, 6, 7, 8), and h ¼ (1, 4, 8). We also experiment with two speciﬁcations of the model prior, setting dk ¼ 0.2 implying a prior expected model size of 2.15 when we allow for four variables in the model and 2.19 when we allow ﬁve variables. The other settings dk ¼ 0.5, with all models equally likely and prior expected model sizes 3.29 and 3.74. The prior for C is speciﬁed with p1 ¼ 0.5, p2 ¼ 0.5, p3 ¼ 1, and p4 ¼ 5.0. When conducting the Monte Carlo exercise we simplify the estimation of the predictive likelihoods by not updating the posterior distribution of the parameters as t increases in the product (10), this allow us to perform all the calculations for the predictive weights within a single Gibbs sampler run instead of running one Gibbs sampler for each value of t.3 The predictive likelihoods are estimated based on 5,000 draws from the Markov chain and the ﬁnal forecast, Eð yTþh jY; Mj Þ, is estimated from 5,000 draws from the Markov chain based on the full sample. To increase the precision of the estimate we use antithetic variates where an antithetic draw of C, conditional on W, is obtained in each step of the Markov chain.

4.1. Results We will focus on DGP1, a bivariate VAR(1), when the models are estimated with lag length p ¼ 2 when reporting the results. The qualitative results are similar for the other DGPs as well as models estimated with p ¼ 4.4 Table 1 reports on the posterior variable inclusion ‘‘probabilities,’’ or more precisely the sum of the predictive weights for the set of models containing the variable. It is clear that the procedure is able to discriminate between the variable y2 which is in the true model and the extraneous variables. The strongest discrimination is achieved when the predictive likelihood is evaluated at h ¼ 1. This is not too surprising given that prediction intervals rapidly become very wide as the forecast horizon increases with a correspondingly diminishing discriminatory power. Longer lead times might, however, be important for seasonal or cyclical data. This is to some extent indicated by the results for DGP2 which contains a cycle. Evaluating the predictive likelihood at multiple horizons discriminates almost as well as the single h ¼ 1 and can be a useful alternative. Increasing the size of the hold out sample is beneﬁcial for discriminating between the variables although the estimation sample can obviously not be made too

512

Table 1.

MICHAEL K. ANDERSSON AND SUNE KARLSSON

Predictive Weights for Variables, DGP1, Models Estimated with Lag Length p ¼ 2. Hold Out Sample, m ¼ 30

h p( y2)

Hold Out Sample, m ¼ 70

max[ p(zi)]

pð y2 Þ max½ pðzi Þ

p( y2)

max[ p(zi)]

pð y2 Þ max½ pðzi Þ

Model prior, 1 4 8 1–4 1–8 1, 4, 8

dk ¼ 0.2 0.79 0.42 0.31 0.76 0.70 0.76

0.17 0.19 0.19 0.17 0.18 0.17

4.71 2.26 1.57 4.38 3.81 4.49

0.92 0.49 0.28 0.79 0.66 0.76

0.15 0.20 0.20 0.19 0.18 0.16

6.11 2.47 1.40 4.10 3.76 4.68

Model prior, 1 4 8 1–4 1–8 1, 4, 8

dk ¼ 0.5 0.88 0.60 0.49 0.85 0.78 0.85

0.31 0.36 0.37 0.30 0.28 0.29

2.79 1.67 1.32 2.89 2.80 2.88

0.96 0.63 0.40 0.84 0.71 0.82

0.28 0.32 0.32 0.26 0.22 0.23

3.48 1.96 1.27 3.25 3.27 3.64

Note: p( ) denotes the predictive weight for the variable.

small (in particular when the posterior is not updated with new observations and always based on the ﬁrst Tm observations). As can be expected we also achieve better discrimination with the dk ¼ 0.2 model prior which favors small models. Table 2 summarizes the model selection properties of the predictive likelihood. The predictive weights for the true model are not particularly large but the performance is reasonable in terms of model selection. With the dk ¼ 0.2 model prior the correct model is selected in between 70 and 87% of the Monte Carlo replicates when the predictive likelihood is evaluated at h ¼ 1. Performance is, on the other hand, quite poor with the uninformative model prior which favors large models. Fig. 2 summarizes the forecast performance for DGP1 and models estimated with lag length p ¼ 2. The ﬁgure compares the root mean square forecast error (RMSE) for the forecast combination to that of the forecasts from the model with only y1,t, i.e., an AR(2). There is clearly a substantial gain for shorter forecast lead times. The larger hold out sample, m ¼ 70, provides the best forecasts together with predictive criteria that puts weight on lead time 1. The difference between the dk ¼ 0.2 and dk ¼ 0.5 model

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Model Selection, DGP1, Models Estimated with Lag Length p ¼ 2.

Table 2.

Model Prior, dk ¼ 0.2

H

Model Prior, dk ¼ 0.5

Hold out, m ¼ 30

Hold out, m ¼ 70

Hold out, m ¼ 30

Hold out, m ¼ 70

Weight

Selected

Weight

Selected

Weight

Selected

Weight

Selected

0.31 0.16 0.12 0.33 0.33 0.34

0.87 0.29 0.19 0.61 0.50 0.66

0.44 0.23 0.12 0.42 0.31 0.41

0.70 0.34 0.13 0.46 0.34 0.45

0.08 0.05 0.05 0.13 0.19 0.14

0.20 0.18 0.25 0.28 0.30 0.31

0.18 0.15 0.10 0.33 0.28 0.34

0.39 0.26 0.15 0.37 0.28 0.40

1 4 8 1–4 1–8 1, 4, 8

Note: Average predictive weight and proportion selected for true model.

1,05

RMSE relative to AR(2)

1,00

0,95

0,90

h=1, m=30 h=8, m=30 h=1-8, m=30 h=1, m=70 h=8, m=70 h=1-8, m=70

0,85

h=4, m=30 h=1-4, m=30 h=1,4,8, m=30 h=4, m=70 h=1-4, m=70 h=1,4,8, m=70

0,80 1

Fig. 2.

2

3

4

5

6 7 8 Forecast lead time

9

10

11

12

RMSE for Forecast Combination Relative to AR(2), DGP1, dk ¼ 0.2.

priors is small for this DGP and models estimated with lag length p ¼ 4 gives slightly worse forecasts. The results for DGP2 shown in Fig. 3 show a larger improvement from the forecast combination at lead time 1 than for DGP1 but the results are slightly worse than an AR(2) at the longer lead times. Again, the forecasts

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MICHAEL K. ANDERSSON AND SUNE KARLSSON 1,05

RMSE relative to AR(2)

1,00

0,95

0,90

h=1, m=30 h=8, m=30 h=1-8, m=30 h=1, m=50 h=8, m=50 h=1-8, m=50

0,85

h=4, m=30 h=1-4, m=30 h=1,4,8, m=30 h=4, m=50 h=1-4, m=50 h=1,4,8, m=50

0,80 1

Fig. 3.

2

3

4

5

6 7 8 Forecast lead time

9

10

11

12

RMSE for Forecast Combination Relative to AR(2), DGP2, dk ¼ 0.2.

combinations based on the predictive likelihood evaluated at h ¼ 4 and 8 provides the least improvement on an AR(2). Performance is slightly better for the dk ¼ 0.5 model prior with smaller differences between combinations based on predictive likelihoods evaluated at different horizons. With DGP3 (Fig. 4) the forecast combination improves on an AR(2) at all but the longest lead times. The difference between the different forecast combinations is small except for when the predictive likelihood is evaluated at h ¼ 8 which performs worse than the other combinations. The difference between model priors is very small, the dk ¼ 0.5 prior does slightly better at longer lead times and the dk ¼ 0.2 prior does slightly better at short lead times. DGP4, ﬁnally, provides a check on the forecast performance when the true model is a univariate AR(2). The performance of the predictive likelihood forecast combination, depicted in Fig. 5, is very close to the AR(2) and the RMSE never exceeds that of the AR(2) by more than 1.6% in the experiments we performed. The forecast performance is very stable across the prior settings, the horizons at which the predictive likelihood is evaluated, and the size of the hold out sample. There are, on the other hand, substantial differences in terms of variable and model selection. With the

515

Bayesian Forecast Combination for VAR Models 1,05

RMSE relative to AR(2)

1,00

0,95

0,90

h=1, m=50 h=8, m=50 h=1-8, m=50 h=1, m=70 h=8, m=70 h=1-8, m=70

0,85

h=4, m=50 h=1-4, m=50 h=1,4,8, m=50 h=4, m=70 h=1-4, m=70 h=1,4,8, m=70

0,80 1

Fig. 4.

2

3

4

5

6 7 8 Forecast lead time

9

10

11

12

RMSE for Forecast Combination Relative to AR(2), DGP3, dk ¼ 0.2.

1,05

RMSE relative to AR(2)

1,00

0,95

0,90

h=1, m=30 h=8, m=30 h=1-8, m=30 h=1, m=70 h=8, m=70 h=1-8, m=70

0,85

h=4, m=30 h=1-4, m=30 h=1,4,8, m=30 h=4, m=70 h=1-4, m=70 h=1,4,8, m=70

0,80 1

Fig. 5.

2

3

4

5

6 7 8 Forecast lead time

9

10

11

12

RMSE for Forecast Combination Relative to AR(2), DGP4, dk ¼ 0.2.

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uniform dk ¼ 0.5 prior, which favors large models, the correct univariate model is never selected in more than 38% of the replicates. This indicates the importance of the shrinkage prior on the parameters as a protection against overﬁtting. The model selection performs better with the dk ¼ 0.2 prior where the correct model is selected in 39–89% of the replicates. Overall it is clear that forecast combination based on the predictive likelihood can improve substantially on the common benchmark of a univariate AR-model. The improvement is larger for short lead times and is also larger for more complex DGPs. The performance is in general better when the predictive likelihood is evaluated at a single short horizon although the use of multiple horizons may be more robust. With a single horizon the use of standard density estimation techniques is uncomplicated and the procedure generalizes readily to situations where the model structure does not allow the use of the Rao–Blackwellization device.

5. FORECASTING US GDP This section illustrates the predictive likelihood forecast combination procedure at work. The forecast variable is US gross domestic product (GDP). The VAR models are of dimensions one to four and we use a dataset of 19 series (GDP included) ranging from second quarter of 1971 to the second quarter of 2007. The full list of variables can be found in the appendix. This implies estimation of 988 (unique) model combinations. The series are modeled in their ﬁrst differences or in the levels, but in the presentation the forecasts, as well as the data, are in the fourth logdifferences (as an approximation to yearly growth rates). The prior variable probabilities, dk, are all set to 0.2, but we have also tried a value of 0.5 (which is equivalent to a uniform prior over the model space). The ﬁnal results do not change much when the prior distribution is changed. However, the procedure puts larger predictive weights on larger systems when the prior 0.5 is used. The predictive likelihood is evaluated at lead time 1 and computed through 5,000 Gibbs samples and 50 evaluation points in time. The ﬁnal forecasts arises as the mean forecast from 1,000 Gibbs samples. The prior speciﬁcation for the parameters is of the same Litterman type as in the Monte Carlo experiment; we set the ﬁrst (own) lag mean to zero for difference stationary variables and the ﬁrst lag mean to 0.9 for stationary series. The overall tightness (p1) is 0.2, the cross-equation tightness (p2) is 0.5, the lag decay (p3) is 1, and the tightness on the constant term (p4) is 5.

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In order to compare the general forecasting performance of our procedure, we compute (pseudo out-of-sample) root mean squared errors (RMSE) for the combination estimator and compare it to the model with the highest model predictive weight (which may be a different model for different forecast occasions). Furthermore, the performance is also compared to a Bayesian second-order autoregressive model, a random walk forecasts, and a recent mean construct (based on the last eight quarters of data). The RMSEs, for horizons 1–8, are calculated for forecasts ranging from ﬁrst quarter of 2000 to second quarter of 2007. The reported results concern average performance of the procedure (in terms of RMSEs) but we also present a current situation analysis (in terms of forecasts and predictive weights).

5.1. Average Forecasting Performance Fig. 6 presents a cascade plot of forecasts (one to eight steps ahead) from different points in time. This picture reveals how well the forecasts track the development of GDP growth (if we neglect data revisions which may be sizeable). For example, the ﬁrst BMA forecast is constructed with data up to the last quarter of 1999. From the forecast cascade it is demonstrated that US GDP, yearly growth rates

6

Forecasts 5 4 3 2 1 0 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

1998

Fig. 6.

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

Sequential Forecasts from 2000:1 to 2008:3. Note: The ﬁgure presents the median of the predictive distribution.

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Table 3. Lead

1 2 3 4 5 6 7 8 SD (GDP)

Forecast Accuracy (RMSE) for US GDP Growth-Rate.

Forecast Combination

Top Model

AR(2)

Random Walk

Recent Mean

No Forecasts

0.41 0.58 0.93 1.23 1.36 1.41 1.33 1.22

1.11 1.10 1.01 0.96 0.94 0.94 0.95 0.95 1.15

1.23 1.23 1.11 1.04 0.97 0.95 0.95 0.96

1.61 1.73 1.44 1.33 1.31 1.36 1.47 1.64

3.32 2.71 1.88 1.56 1.49 1.48 1.54 1.61

30 29 28 27 26 25 24 23

Note: RMSE for forecast combination. Ratio of RMSE to RMSE for forecast combination for other procedures.

the BMA procedure underestimated the weakness of the economy during 2001, but predicts the period 2002–2005 as reasonably well. The forecasts did not quite catch the down turn in the recent past and GDP growth is somewhat overpredicted, but not to the same degree as in 2001. Turning to a more formal evaluation of the forecasts, Table 3 shows that the forecast combination improves on the top model and especially the AR(2) for shorter lead times but does slightly worse than the top model for lead times 4 and higher and worse than the AR(2) for lead times 5 and higher.5 Due to the small evaluation sample no formal testing is performed. This improvement is somewhat more articulated when we use the uniform prior for the models, dk ¼ 0.5. The two simplest alternative forecasts, namely the random walk and the recent mean forecasts, perform notably worse than the other forecasts. The size of the RMSE of the forecast combination for lead times h ¼ 4 and higher is approximately the same as the standard deviation of the GDP series. Our procedure can thus be regarded as a complement to traditional forecasts for short horizons. This is in line with previous studies, see for instance Galbraith and Tkacz (2006). 5.2. Contemporaneous Forecasts from the Procedure Fig. 7 presents the posterior median of the combination forecasts given data up to second quarter of 2007. The forecast cover the period 2007:3–2009:4. The ﬁgure also presents the associated 50 and 68% probability intervals for

519

Bayesian Forecast Combination for VAR Models 5 Prob 68% Prob 50% AR BMA GDP yearly diff

4,5 4 3,5 3 2,5 2 1,5 1 0,5 0

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

2000

Fig. 7.

2001

2002

2003

2004

2005

2006

2007

2008

2009

Forecast from 2007:2. Posterior Mean and Probability Intervals for Forecast Combination and Mean Forecast from a Bayesian AR(2).

the forecast combination and the forecasts from a Bayesian autoregression. The intervals demonstrate that there is considerable forecast uncertainty. The combination forecast suggests that the US economy will slowly approach the potential growth rate. The autoregressive forecast only considers the dynamics contained in GDP itself, whereas, the combination procedure also takes the other 18 variables into account. Fig. 7 demonstrates that the information contained in the indicator variables leads to a lower forecast for the whole forecast period compared to not using the indicator information. Thus, the indicators contain a signal of a weaker growth than the GDP series by itself. Fig. 8 presents the predictive weights (i.e., the sum of the predictive weights for the models containing the variable) for each variable, based on the present full dataset. This information may be useful by itself, e.g., this information may be incorporated in judgmentally based forecasting schemes. The highest variable inclusion probability is found for the jobless claims (JOBLESS). The other real variables exhibit notably lower predictive weights probabilities, and the nominal variables even lower weights. Some interesting patterns do, however, emerge if we consider groups of variables measuring the same underlying feature of the economy. Employment (EMP_payr, EMP, chic_EMP), stock prices (NASDAQ, NYSE),

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MICHAEL K. ANDERSSON AND SUNE KARLSSON

1

0,8

0,6

0,4

0,2

Fig. 8.

1 2 3 4 5 6 7 8 9 10

CarSales

FFR

Chic_emp

Chic_prod

ISM_pmi

NYSE

NASDAQ

M2

CPI_core

Variable Inclusion, Predictive Weights.

Table 4. Rank

PCE_core

Profits

SAVErate

COMPEMP

EMP

EMP_payr

JOBLESS

CONS

INDPROD

0

Top 10 Models.

Variables GDP JOBLESS GDP CONS JOBLESS GDP JOBLESS NASDAQ GDP JOBLESS NYSE GDP JOBLESS M2 GDP JOBLESS Chic_prod GDP CONS JOBLESS GDP CONS JOBLESS GDP JOBLESS EMP GDP JOBLESS M2 Equal weights for all speciﬁcations/models

Pred. Weight

NASDAQ NYSE Chic_prod

0.126 0.093 0.059 0.044 0.037 0.034 0.033 0.030 0.022 0.021 0.001

Note: The table presents the top ten models based on data from 1971:2 to 2007:2. The column Pred. Weight reports the predictive weight of each model.

consumption (CONS), money supply (M2), and production (INDPROD, ISM_pmi, chic_PROD) are all important factors. Table 4 presents posterior analysis for the top 10 models, using the current dataset. As a point of reference the table also gives the ‘‘posterior

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probability,’’ 1/988, for an equal weighting scheme. Given the variable predictive weights it is not a surprise that the top ranked model consists of GDP and jobless claims.

6. CONCLUSIONS This paper proposes to use weights based on the predictive likelihood for combining forecasts from dynamic multivariate forecasting models such as VAR-models. Our approach overcomes a basic difﬁculty with standard Bayesian forecast combination based on the marginal with multivariate forecasting models, that the marginal likelihood can change with the dimension of the model in ways that are unrelated to the forecasting performance for the variable of interest. This is achieved by considering the marginal predictive likelihood for the variable of interest rather than the joint predictive likelihood which suffers from the same problem. The predictive likelihood is not available in closed form for forecasts at lead times greater than 1 and we propose simulation strategies for estimating the predictive likelihood. Our approach is completely general and does not rely on natural conjugate priors or the availability of closed form solutions for the posterior quantities. All that is required is the ability to simulate from the posterior distribution of the parameters and to simulate one-step ahead forecasts. The approach is thus also well suited for nonlinear forecasting models. We evaluate the performance of the forecast combination procedure in a small Monte Carlo study and in an application to forecasting US GDP growth. Overall the forecast combinations perform very well. In the Monte Carlo study the forecast combination outperforms our benchmark autoregression by as much as 23% but does slightly worse for forecasts more than four quarters ahead.

NOTES 1. Notable practitioners are Sveriges Riksbank, the Bank of England, and the Bank of Canada. These central banks employ a wide variety of model approaches, ranging from simple univariate time series models to highly sophisticated multivariate nonlinear models. While a great many models are used, the procedures are easy to manage and highly automated (see Andersson and Lo¨f, 2007 and Kapetanios, Labhard, & Price, 2007).

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2. See Hoeting, Madigan, Raftery, and Volinsky (1999) for an introduction to Bayesian model averaging and Min and Zellner (1993), Jacobson and Karlsson (2004), and Koop and Potter (2004) for applications of Bayesian model averaging to forecasting and Timmermann (2006) for a review of forecast combination. 3. We do a limited check on the effect of not updating the prior by rerunning a few experiments for the ﬁrst DGP with the posterior updated as new observations are added. The results are slightly better when the posterior is updated, particularly for m ¼ 70, but overall the differences are small. 4. A comprehensive set of results are available in the working paper version of the paper, Andersson and Karlsson (2007). 5. The ability of autoregressions to compare well with more sophisticated approaches is a familiar phenomenon. See Stock and Watson (2002a) and Stock and Watson (2004).

REFERENCES Andersson, M., & Karlsson, S. (2007). Bayesian forecast combination for VAR models. Working Paper 2007:13. Department of Economics, Statistics and Informatics, O¨rebro University. Andersson, M. K., & Lo¨f, M. (2007). The Riksbank’s new indicator procedures. Sveriges Riksbank Economic Review, 1, 76–95. Bernanke, B. S., & Boivin, J. (2003). Monetary policy in a data-rich environment. Journal of Monetary Economics, 50, 525–546. Eklund, J., & Karlsson, S. (2007). Forecast combination and model averaging using predictive measures. Econometric Reviews, 26, 329–363. Ferna´ndez, C., Ley, E., & Steel, M. F. J. (2001). Benchmark priors for Bayesian model averaging. Journal of Econometrics, 100, 381–427. Galbraith, J., & Tkacz, G. (2006). How far can we forecast? Forecast content horizons for some important macroeconomic time series. Working Paper 2006-13. Department of Economics, McGill University. Hoeting, J. A., Madigan, D., Raftery, A. E., & Volinsky, C. T. (1999). Bayesian model averaging: A tutorial (with discussion). Statistical Science, 14, 382–417. Corrected version available at http://www.stat.washington.edu/www/research/online/hoeting1999.pdf Jacobson, T., & Karlsson, S. (2004). Finding good predictors for inﬂation: A Bayesian model averaging approach. Journal of Forecasting, 23, 479–496. Kadiyala, K. R., & Karlsson, S. (1997). Numerical methods for estimation and inference in Bayesian VAR-models. Journal of Applied Econometrics, 12, 99–132. Kapetanios, G., Labhard, V., & Price, S. (2007). Forecast combinations and the Bank of England’s suite of statistical forecasting models. Technical Report No. 323. Bank of England. Koop, G., & Potter, S. (2004). Forecasting in dynamic factor models using Bayesian model averaging. Econometrics Journal, 7(2), 550–565. Lu¨tkepohl, H. (1993). Introduction to multiple time series analysis (2nd ed.). Berlin: SpringerVerlag. Min, C.-K., & Zellner, A. (1993). Bayesian and non-Bayesian methods for combining models and forecasts with applications to forecasting and international growth rates. Journal of Econometrics, 56, 89–118.

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Stock, J. H., & Watson, M. W. (2002a). Forecasting using principal components from a large number of predictors. Journal of the American Statistical Association, 97(460), 1167–1179. Stock, J. H., & Watson, M. W. (2002b). Macroeconomic forecasting using diffusion indexes. Journal of Business and Economic Statistics, 20, 147–162. Stock, J. H., & Watson, M. W. (2004). Combination forecasts of output growth in a sevencountry data set. Journal of Forecasting, 23(6), 405–430. Stock, J. H., & Watson, M. W. (2006). Forecasting with many predictors. In: G. Elliott, C. W. J. Granger & A. Timmermann (Eds), Handbook of economic forecasting. Amsterdam: North-Holland. Timmermann, A. (2006). Forecast combinations. In: G. Elliott, C. W. J. Granger & A. Timmermann (Eds), Handbook of economic forecasting. Amsterdam: North-Holland.

APPENDIX. DATA USED FOR THE US GDP FORECASTS The dataset consists of real, nominal, and indicator type variables: GDP: National Income Account, Overall, Total, Constant Prices, SA (US Department of Commerce) INDPROD: Production, Overall, Total, SA (Federal Reserve) CONS: Personal Outlays, Overall, Total, Constant Prices, SA (US Department of Commerce) JOBLESS: Jobless claims, SA (US Department of Labor) EMP–payr: Employment, Overall, Nonfarm Payroll, Total, SA (Bureau of Labor Statistics) EMP: Civilian Employment, Business Cycles Indicators, SA (The Conference Board) COMPEMP: National Income Account, Compensation of Employees, Total, SA (US Department of Commerce) SAVErate: Personal Savings, Rate, SA (Federal Reserve) Proﬁts: National Income Account, Corporate Proﬁts, with IVA and CCAdj, Total, SA (US Department of Commerce) PCE–core: Price Index, PCE, Overall, Personal Consumption Expenditures less Food and Energy, SA (Bureau of Economic Analysis) CPI–core: Consumer Prices, All Items less Food and Energy, SA (Bureau of Labor Statistics) M2: Money Supply M2, SA (Federal Board of Governors) NASDAQ: Composite Index, Close (NASDAQ) NYSE: Composite Index, Close (NYSE)

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ISM–pmi: Business Surveys, ISM Manufacturing, PMI Total, SA (Institute for Supply Management) Chic–prod: Business Surveys, Chicago PMI, Production, SA (PMAC) Chic–emp: Business Surveys, Chicago PMI, Employment, SA (PMAC) FFR: Policy Rates, Fed Funds Effective Rate (Federal Reserve) CarSales: – Car Sales, Domestic, SA (US Department of Commerce) – Car Sales, Imported, SA (US Department of Commerce) – Truck Sales, Domestic Light, SA (US Department of Commerce) – Truck Sales, Imported Light, SA (US Department of Commerce)

BAYESIAN INFERENCE ON TIME-VARYING PROPORTIONS William J. McCausland and Brahim Lgui ABSTRACT Time-varying proportions arise frequently in economics. Market shares show the relative importance of ﬁrms in a market. Labor economists divide populations into different labor market segments. Expenditure shares describe how consumers and ﬁrms allocate total expenditure to various categories. We introduce a state space model where unobserved states are Gaussian and observations are conditionally Dirichlet. Markov chain Monte Carlo techniques allow inference for unknown parameters and states. We draw states as a block using a multivariate Gaussian proposal distribution based on a quadratic approximation of the log conditional density of states given parameters and data. Repeated draws from the proposal distribution are particularly efﬁcient. We illustrate using automobile production data.

INTRODUCTION A vector pt (pt1, . . . , ptp) of observed at time t satisﬁes ptiZ0 Pproportions p for all iA{1, . . . , p} and p ¼ 1. Time-varying proportions arise ti i¼1 frequently in economics. Market shares show the relative importance of

Bayesian Econometrics Advances in Econometrics, Volume 23, 525–544 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23016-1

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ﬁrms in a market. Labor economists divide the working age population into different labor market segments: people are either in the labor force or not, those in the labor force are either employed or unemployed, and the employed may be working full-time or part-time. Also relevant to labor economists is a person’s allocation of time to work, child care, home production, leisure, and other pursuits. Expenditure shares describe how consumers and ﬁrms allocate total expenditure to various categories of spending. Probabilities are also proportions, and we can study the probabilities of events implied by market prices of contracts in prediction markets such as the Iowa Electronic Markets (http://www.biz.uiowa.edu/ iem/) or Intrade (http://www.intrade.com/). Varying proportions also arise outside of economics. Biologists track the time-varying proportions of species in a habitat, such as a lake. Geologists study the variation of the composition of rocks in a spatial dimension. Modeling time-varying proportions allows us to learn about the joint dynamics of proportions and other observed data, to measure the response of proportions to changes in exogenous variables, and to make predictions. State space models have proved useful for exploring many kinds of dynamic relationships. With proportions data, difﬁculties arise because of the positivity and adding-up constraints. The most common resolution of this difﬁculty involves modeling suitably transformed data. Aitchison (1986) introduces two different transformations, both of which have been used extensively. The additive log ratio (alr) transformation of a proportion vector p is ðlog p1 log pp ; . . . ; log pp1 log pp Þ Note that the alr transformation of (p, 1p) is the same as the logit transformation of p. The alr transformation has the advantage of being a bijective map from the simplex to Rp1 , but it is not symmetric: the transformation depends on the choice of a reference proportion, here pp. The centered log ratio (clr) transformation is ! p p X X 1 1 log pi ; . . . ; log pp p log pi log p1 p i¼1

i¼1

This transformation is symmetric, but the image of the transformation is P fx 2 Rp : pi¼1 xi ¼ 0g, which is more difﬁcult to work with. Whatever the transformation used, models of transformed data are more difﬁcult to interpret than models for untransformed proportions. Grunwald, Raftery, and Guttorp (1993) introduce a state space model where the

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conditional distribution of proportions given latent states is explicit. They use the Dirichlet distribution, a natural distribution for proportions since its support is the simplex: if p has a Dirichlet distribution with parameter vector c (g1, . . . , gp)W0, then (p1, . . . , pp1) has density 8 Qp gi 1 < QGðGÞ pi 0 for all i 2 f1; . . . ; pg p i¼1 pi Gðg Þ i i¼1 f ðp1 ; . . . ; pp1 ; cÞ ¼ :0 otherwise P Pp where pp ¼ 1 p1 i¼1 pi and G i¼1 gi . Another advantage of the Dirichlet distribution is that the mean is a transparent function of the parameters: E½p ¼

1 c G

However, the Dirichlet distribution is fairly inﬂexible: once we choose the ratios gi/gj to set the mean of the distribution, only the sum G remains as a degree of freedom. We can interpret G as an overall precision parameter. We cannot freely choose covariances or even individual variances. The elements of p are neutral, meaning that for all i and j, pi and pj/(1pi) are independent. In state space models, state dynamics can compensate for the inﬂexibility of the Dirichlet distribution: while the conditional distribution of proportions given unobserved states is Dirichlet, the unconditional distribution of proportions may be much richer. In Grunwald et al. (1993), state dynamics follow the Dirichlet conjugate distribution, deﬁned in that paper. Here we introduce a state space model where observations are conditionally Dirichlet and latent state transitions are Gaussian. Thus, our method has the same advantage as that of Grunwald et al. (1993), the transparency of the conditional distribution of proportions given states. This facilitates the interpretation of state dynamics. Gaussian state transitions have several advantages over the Dirichlet conjugate distribution. They are easier to interpret, more ﬂexible, and more familiar. Finally, we do not have to settle for approximations of unknown normalization constants. Durbin and Koopman (1997) describe an approach to inference in state space models with Gaussian states and non-Gaussian observations. Given such a semi-Gaussian state space model, they construct, as a device, a fully Gaussian state space model where the posterior distribution of the latent states approximates that of the semi-Gaussian model. While they use this distribution as an importance distribution to compute the likelihood

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function using importance sampling, it could just as well be used as a proposal distribution in a Markov chain Monte Carlo (MCMC) method for posterior simulation. Although framed in different terms, the proposal distribution we describe below is the same as the approximation Durbin and Koopman (1997) propose as an importance distribution for the kind of state space model we consider here. We use an entirely different method for drawing states, however. Our procedure is not based on the Kalman ﬁlter, but rather the algorithm described in McCausland, Miller, and Pelletier (2007). McCausland et al. (2007) show that for many important cases, their method is more computationally efﬁcient than methods, such as that of Durbin and Koopman (1997), that use the Kalman ﬁlter. The McCausland et al. (2007) algorithm is particularly efﬁcient for repeated draws. In The Model section, we describe in detail a state space model for timevarying proportions. Then we describe MCMC methods for simulating states and parameters from their joint posterior distribution. We draw states and parameters in separate Gibbs blocks. In Drawing States section we show how to draw states from a proposal distribution that approximates their conditional posterior distribution. We show how to draw parameters from their conditional posterior distributions, using fairly standard methods, in Conditional Inference for Parameters section. We illustrate our methods in an empirical example in An Empirical Example section. We conclude in the last section.

THE MODEL We use state space models to describe the dynamics of time-varying proportions. The state vector at has length p and is a ﬁrst-order Gaussian vector autoregressive process. The observed vectors pt of proportions are conditionally independent and Dirichlet given the state process. The state vector process fat g1 t¼1 is deﬁned by 1 a1 Nðl1 ; H21 1 Þ at ja1 ; . . . ; at1 NðBxt þ Uat1 ; Ha Þ

where xt is a k-vector of covariates. We deﬁne the parameter vector h (l1, H1, B, U, Ha).

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529

Given the state sequence, the pt are conditionally independent and each pt has a Dirichlet distribution with parameter vector given by ct ¼ exp(at), where the exponential function is applied element by element to vector arguments. Thus, we have the following conditional density of p given states: Pm m Y g 1 G i¼1 gti Q ptiti (1) f ðpt ja1 ; . . .Þ ¼ m i¼1 Gðgti Þ i¼1

DRAWING STATES For posterior inference, we use a Metropolis–Hastings chain (Metropolis, Rosenbluth, Rosenbluth, Teller, & Teller, 1953; Hastings, 1970) whose stationary distribution is the joint posterior distribution of h and a. In this section, we describe an update block for a, so the target distribution is the conditional distribution of a given h and p. We describe a proposal distribution approximating this target distribution. The proposal has a multivariate Gaussian distribution based on a quadratic approximation of the log target density log f(a|h, p) at the mode a1 of the target distribution. For modest amounts of data and observation densities f(pt|at) that are reasonably nearly log-quadratic in at, the multivariate Gaussian approximation is close enough to the target distribution that it may be used as an importance distribution or a proposal distribution. Examples include those considered in Durbin and Koopman (1997) and the current paper. As the amount of data increases, however, numerical efﬁciency degrades. We can write log f ðajh; pÞ ¼ log f ðajhÞ þ log f ðpjaÞ þ k where k is a term not depending on a. The ﬁrst term is already quadratic in a. It is convenient to write it in terms and covector c of the conditional distribution of a given h, of the precision H rather than the more familiar mean and variance. This is because precisions and covectors combine additively when quadratic forms are summed. Therefore, we write 1 np log 2p a> Ha þ 2c> a þ c> H 1 c log f ðajhÞ ¼ ½log jHj 2

(2)

530

WILLIAM J. MCCAUSLAND AND BRAHIM LGUI

We compute 2 11 H 6 6 H21 6 6 0 6 ¼6 . H 6 . 6 . 6 6 0 4 0

12 H 22 H 32 H .. .

0 H23 33 H .. .

0

0

.. .

0 .. .

0 0

0 0

n1;n1 H Hn;n1

0

3

7 0 7 7 0 7 7 .. 7 7; . 7 7 n1;n 7 H 5 n;n H

2

3 c 1 6 c 2 7 6 7 6 7 6 c 3 7 6 7 c ¼ 6 . 7 6 .. 7 6 7 6 7 4 c n1 5

(3)

c n

where 11 ¼ H1 þ U> Ha U; H

nn ¼ Ha H

tt ¼ Ha þ U> Ha U for t ¼ 2; . . . ; n 1 H t;t1 ¼ Ha U for t ¼ 1; . . . ; n 1 t1;t ¼ U> Ha and H H c 1 ¼ H1 l1 U> Ha Bx2 ;

c n ¼ Ha Bxn

c t ¼ Ha Bxt U> Ha Bxtþ1 for t ¼ 2; . . . ; n 1 The second term, log f(p|a), is not quadratic but we can approximate it as such. A second-order Taylor expansion of log f(p|a) around some reference value a gives log f ðpjaÞ

1 > a Hða Þa 2cða Þ> a þ k0 2

(4)

where 2 6 6 6 Hða Þ 6 6 4

h1 ða Þ

0

0

0 .. . 0

h2 ða Þ .. . 0

.. .

0 .. .

hn ða Þ

3 7 7 7 7; 7 5

2

3 c1 ða Þ 6 . 7 . 7 cða Þ 6 4 . 5 cn ða Þ

and for t ¼ 1, . . . , n, ht ða Þ ¼

@2 log f ðpt jat Þ @at @a> t

(5)

531

Bayesian Inference on Time-Varying Proportions

ct ða Þ ¼

@ log f ðpt jat Þ @2 log f ðpt jat Þ at @at @at @a> t

(6)

and ku is a term not depending on a. We derive expressions for ht(a ) and ct(a ) in appendix. Combining the quadratic expression for log f(a|h) in Eq. (2) with the quadratic approximation of log f(p|a) in Eq. (4), we obtain the following quadratic approximation of the log conditional posterior density of a: i 1 h Þa 2c ða Þ> a þ k00 log f ðajh; pÞ a> Hða 2 þ Hða Þ; c ða Þ c þ cða Þ, and kv is a term not depending ÞH where Hða on a. Since the mode a1 of the conditional posterior density maximizes Þ must be positive deﬁnite and therefore the approximation logf ðajh; pÞ; Hða of log f (a|h,p) at a1 is the log of a multivariate Gaussian density.

Implementation Þ; and c ða Þ Drawing from the proposal distribution involves ﬁnding a ; Hða Þ1 c ða Þ; Hða Þ1 Þ. The quantities a ; Hða Þ, and then drawing an NðHða 1 1

Þ c ða Þ, since Hða Þ c ða Þ is the mean, and and c ða Þ satisfy a ¼ Hða Þ and therefore mode, of the Gaussian approximation with precision Hða

covector c ða Þ. We ﬁnd these values by iterating the following computation until numerical convergence: old Þ1 c ðaold Þ anew :¼ Hða old Þ is not positive deﬁnite, in which case we It sometimes happens that Hða replace the step with anew :¼ H

safe

ðaold Þ1 c

safe

ðaold Þ

safe safe old Þ is positive semi ðaold Þ is positive deﬁnite, H ðaold Þ Hða where H safe ðaold Þa deﬁnite, and the gradient of the quadratic approximation a> H t safe

2a> c ðaold Þ is still equal to the gradient of log f(p|a) at aold. We construct safe ðaold Þ c safe ðaold Þ by replacing h ðaold Þ and c ðaold Þ with hsafe ðaold Þ and H t t t t t t old ða Þ, whose precise deﬁnitions are given in appendix. csafe t t

532

WILLIAM J. MCCAUSLAND AND BRAHIM LGUI

In general, solving a large (here mn mn) system is very costly. However, it is band diagonal with one can take advantage of the special structure of H: bandwidth 1. We use the algorithm described in McCausland et al. (2007) ¼ c . The steps are the following, where to solve for a in the equation Ha and c analogous to those of t;tþ1 , and c t are deﬁned by partitions of H tt ; H H and c in Eq. (3): H Þ1 ; m ¼ R c . (1) Compute R ¼ ðH 1

11

1

1 1

(2) For t ¼ 2, . . . , n, compute H 1 Rt ¼ ðH tt t;t1 Rt1 Ht1;t Þ ;

mt ¼ Rt ðc t H t;t1 mt1 Þ

(3) Compute an ¼ mn. (4) For t ¼ n1, . . . ,1, compute at ¼ mt Rt H t;tþ1 atþ1 In practice, we never actually construct the Rt, only the R1 and their t Cholesky factors. This is more efﬁcient and delivers everything we need. Þ1 c ða Þ; Hða Þ1 Þ we again take advantage of To draw an NðHða Once we have a1, lt, and the Cholesky the band diagonal structure of H. , we use the algorithm in McCausland et al. decompositions of the R1 t (2007) to draw the proposal a as follows: (1) Draw ann Nðmn ; Rn Þ (2) For t ¼ n1, . . . , 1, draw n ant Nðmt Rt H t;tþ1 atþ1 ; Rt Þ We accept the proposal with probability f ðan jhÞf ðpjan Þ gðaÞ : min 1; f ðajhÞf ðpjaÞ gðan Þ where g is the proposal density. We evaluate g(a) as gðan Þ ¼ g ann g ann1 jann g an1 jan2 ; . . . ; ann using the conditional means and variances computed using the algorithm above. To evaluate g(a), we use the same algorithm to compute the appropriate conditional means and variances, but without drawing any values.

Bayesian Inference on Time-Varying Proportions

533

We note that multiple draws for a given value of h are particularly efﬁcient. For each t, we only need to compute Rt H t;tþ1 and the Cholesky decomposition of Rt once. Drawing at only requires two operations of order O(m2): premultiplying atþ1 by ðRt H t;tþ1 Þ and premultiplying a vector of independent Gaussian random variables by the Cholesky decomposition of Rt. The methods of de Jong and Shephard (1995) and Durbin and Koopman (2002) are both more computationally demanding (see McCausland et al., 2007, for a detailed comparison of computational costs for repeated draws of a).

CONDITIONAL INFERENCE FOR PARAMETERS We use standard methods exploiting conditionally conjugate priors to draw the parameters from their exact conditional posterior distributions. All conditional distributions below are well known and we can draw from them using standard methods. We suppose that l1, H1, (B, U), and Ha are a priori independent. We let " # B> aþ vec > F The initial mean parameter l1 is multivariate Gaussian: 1 l1 Nðr1 ; P 1 Þ

and its conditional conjugacy leads to the following conditional posterior distribution: 1 Þ l1 ja; H1 ; B; U; Ha Nðr1 ; P 1 where P 1 þ H1 P 1

and

1 ðP 1 r1 þ H1 a1 Þ r1 P 1

The initial precision parameter H1 is Wishart: 1Þ H1 Wiðm 1 ; A giving the following conditional posterior distribution: Þ H1 a; l1 ; B; U; Ha Wiðm 1 ; A 1 where m 1 ¼ m þ 1 and

1 ¼ A 1 þ ða1 l Þða1 l Þ> A 1 1 1 1

534

WILLIAM J. MCCAUSLAND AND BRAHIM LGUI

With priors for l1 and H1, there is more than one way to express a large degree of prior uncertainty about a1. One way is to choose fairly precise priors for l1 and H1, favoring values of H1 with small eigenvalues. Such values of H1 imply a high degree of uncertainty about a1. In the limit, one can choose ﬁxed values for l1 and H1. Another way is to choose an imprecise prior for l1, and a precise prior for H1 favoring values of H1 with large eigenvalues. In the choice between the two, there is a computational do and thus H, trade-off. Values of H1 with large eigenvalues ensure that H, not have any small eigenvalues. This leads to faster convergence to a1 and a conditional distribution of a given p and h that is more nearly Gaussian. On the other hand, values of H1 with large eigenvalues lead to higher posterior correlation between a1 and l1, which implies slower mixing. The coefﬁcient vector aþ is multivariate Gaussian: 21 Þ aþ Nðr; P leading to the following conditional posterior distribution: 1

Þ aþ ja; l1 ; H1 ; Ha Nðr; P where 2

n P

6 t¼2 P þ Ha 6 P 6P 4 n t¼2

0

n P

xt x> t at1 x> t

02

n P

B B6 t¼2 B6 1 BP r P B r þ vecB6 P @ @4 n t¼2

t¼2 n P t¼2

xt a> t

x t a> t1 at1 a> t1 3

3 7 7 7 5

11

7 CC 7 CC 7 H a CC AA >5

at1 at

The state conditional precision parameter Ha has the following Wishart prior: aÞ Ha Wiðm a; A so that its conditional posterior distribution is Þ Ha ja; l1 ; H1 ; B; U Wiðm ; A a where X 1 21 þ ¼ A m a ¼ m a þ n 1 and A ðat Bxt Uat1 Þðat Bxt Uat1 Þ> a a n

t¼2

535

Bayesian Inference on Time-Varying Proportions

AN EMPIRICAL EXAMPLE As an empirical example, we analyze automobile production data. We use data tabulated in Grunwald et al. (1993), whose original sources were the Motor Vehicle Manufacturers Association of the U.S. and the 1989 Economic Report of the President to Congress. Fig. 1 illustrates the data. The top panel shows raw automobile production data for Japan, for the United States, and for all other countries during the period from 1947 to 1987. The middle panel shows world market share for the same three regions. The bottom panel shows annual percentage growth of the U.S. Gross National Product over the same period. We use our model to analyse these data. The proportions pt1, pt2, and pt3 represent the world market share of Japan, the USA, and all other countries, respectively. We use k ¼ 2 covariates. We let xt1 1 and xt2 be the U.S. GNP growth. We take m ¼ p ¼ 3 and Zt ¼ I, so that there is one state variable for each region. There are no covariates in the observation equation. For this special case we obtain, using the derivation in appendix, the following expressions for ht and ct:

2 ht ¼ c1 ðGt Þct c> t diag ½cðGt Þ cðcti Þ þ log pti cti c1 ðcti Þcti 2 cti ¼ ½cðGt Þ cðcti Þ þ log pti cti ð1 ati Þ c1 ðGt Þðc> t at Þcti þ c1 ðcti Þcti ati

We set prior hyperparameters to the following distribution for l1, B, U, H1, and Ha 2 0:25 6 1 ¼ 40 r 1 ¼ 2:0 6:0 6:0 ; P 0 02

0:5

B6 0:0 B6 B6 6 r ¼ vecB B6 0:9 B6 @4 0:0 0:0

0:5

0:5

31

C 0:0 7 7C 7C C 0:0 7 7 C; 7C 0:9 0:0 5A 0:0 0:9

0:0 0:0

2

02

values to specify a prior 0

0:25 0

0 7 5 0:25

4:0

6 B6 104 6 B6 6 B6 P ¼ diag6vecB6 100:0 6 B6 6 B6 4 4 @4 10 104

m 1 ¼ 100:0; m a ¼ 20:0; where I is the 3 3 identity matrix.

1 ¼ I A

a ¼ 0:002I A

3

0

4:0 104 104

4:0 104 104

100:0 104 104

100:0

313 7C7 7C7 7C7 7C7 7C7 7C7 5A5

WILLIAM J. MCCAUSLAND AND BRAHIM LGUI

536

5000 500 50

Japan USA Other

10

1000s of automobiles

Automobile Production

1950

1960

1970

1980

0.050

0.500

Production Share

0.001 0.005

Production share as a fraction of total

Year

Japan USA Other 1950

1960

1970

1980

2

4

6

8 10

USA GNP Growth

−2 0

Percentage change in USA GNP

Year

1950

1960

1970

1980

Year

Fig. 1.

Data for Automobile Production Example.

Bayesian Inference on Time-Varying Proportions

537

We noted in Contional Inference for Parameters section that uncertainty about the initial state a1 is expressed through the priors for l1 and H1. For computational reasons described there, we choose a quite informative prior for H1, centered around a value that assigns high probability to a1 being close to l1, and a relatively vague prior on l1. The combined result is a fairly vague prior on the initial state a1. The precision H1 has mean 100I with 100 degrees of freedom. At this mean value, the conditional standard deviation of the elements of a1, given l1 and H1, is 0.1. This is quite small compared to the prior standard deviation of the elements of l1, equal to 2.0. We now consider how the conditional distribution of p1 given a1 varies with a1. The prior mean of l1 is (2, 6, 6). We compute E[p1|a1 ¼ (2, 6, 6)] ¼ (0.0090, 0.4955, 0.4955). The corresponding standard deviations are 0.0033, 0.0175, and 0.0175, respectively. The asymmetry with respect to Japan reﬂects prior knowledge about the state of immediate postwar Japan. Changing one element a1i by 2.0, the standard deviation of l1i, changes the market share ratio E[p1i|a1]XE[p1j|a1] by a factor of e2 ¼ 7.4. Changing all elements of a1 by 2.0 in the same direction does not change E[p1|a1], but it does alter the standard deviations. The conditional standard deviations of p1i given a1 ¼ (4, 8, 8) are 0.0012, 0.0064, and 0.0064; given a1 ¼ (0, 4, 4), they are 0.0091, 0.0478, and 0.0478. We see that the prior for l1 covers much reasonable variation in a1, whatever the value of H1. The elements of B and U are mutually independent, with B1iBN(0.5, (0.5)2), B2iBN(0.0, (0.01)2), UiiBN(0.9, (0.1)2), and UijBN(0, (0.01)2) for j 6¼ i. The mean value of U implies a half life of ln 0.5/ln 0.9E6.6 years for shocks to at and for initial conditions. Changing one diagonal value of U to 0.8, one standard deviation below the mean, leads to a half life of ln 0.5/ ln 0.8E3.1 years for the region in question, which we consider low but plausible. A diagonal value of 0.98 corresponds to a half life of about 34.3 years, high but also plausible. There is high prior probability that offdiagonal elements of U are close to zero. With U and B equal to their mean values, the steady state mean of at is (5, 5, 5). Given this value of at, the conditional mean of p is (1/3, 1/3, 1/3) and the conditional standard deviations are 0.022. Change Bi1 by 0.1, and the steady state mean of ati becomes 6, corresponding to a change in prior market share ratio of e1E2.72. A prior standard deviation of 0.5 allows considerable variation in the steady state mean of ati for Uii as low as 0.8 or as high as 0.98. The slope coefﬁcients of the U.S. GNP growth, the Bi1, have a prior mean of 0.0 and a prior standard deviation of 0.01. Given that GNP growth is expressed as a percentage, the absolute value |Bi1| ¼ 0.01 implies that a difference in GNP

538

WILLIAM J. MCCAUSLAND AND BRAHIM LGUI

growth of one percentage point corresponds to a change of about 1% in market share. We choose a fairly informative prior for Ha, the precision of the at innovation. This is advisable in state space models to avoid overﬁtting. With Ha set to its prior mean of 104I, the standard deviation of a shock to any ati is 0.01, roughly corresponding to a 1% change in market share pti. The prior degrees of freedom of 20.0, together with n ¼ 41, implies that the weight of prior information is about half that of the data information. We illustrate the posterior distribution in Tables 1–4, showing the quantiles 0.25, 0.5, and 0.75 of various marginal posterior distributions. Table 1 shows the posterior median and interquartile range for elements of B. We see that there is considerable sensitivity of ati to U.S. GNP for i ¼ 2 (USA) and little evidence of any sensitivity for i ¼ 1 (Japan) or i ¼ 3 (other). Table 2 shows the same quantiles for the elements of U. The data are quite informative about the diagonal elements of U, just as the prior is quite informative about the off-diagonal elements. State sequences for all regions are fairly persistent, but this persistence is relatively high for Japan and relatively low for the USA. Since the conditional precision Ha of the at is difﬁcult to interpret, we report features of the conditional variance Ra H1 a . Table 3 shows the Table 1.

Posterior Median and Interquartile Range for Elements of B.

Japan USA Other

Table 2.

Japan USA Other

Constant

USA GNP

0.4077 (0.3498, 0.4650) 0.9139 (0.7139, 1.1258) 0.7245 (0.5758, 0.8692)

0.0005 (0.0051, 0.0053) 0.0178 (0.0132, 0.0226) 0.0043 (0.0089, 0.0000)

Posterior Median and Interquartile Range for Elements of U. Japan

USA

Other

0.924 (0.918, 0.929) 0.009 (0.015, 0.003) 0.004 (0.002, 0.009)

0.002 (0.005, 0.009) 0.819 (0.778, 0.856) 0.000 (0.007, 0.007)

0.001 (0.006, 0.008) 0.001 (0.008, 0.006) 0.871 (0.842, 0.900)

539

Bayesian Inference on Time-Varying Proportions

Posterior Median and Interquartile Range for the Square Roots of the Diagonal Elements of Ra.

Table 3. Japan 0.0103 (0.0093, 0.0115)

Table 4.

Japan USA Other

USA

Other

0.0105 (0.0095, 0.0119)

0.0104 (0.0093, 0.0116)

Posterior Median and Interquartile Range for Elements of Ca. Japan

USA

Other

1.000 (1.000, 1.000) 0.018 (0.140, 0.105) 0.001 (0.121, 0.122)

0.018 (0.140, 0.105) 1.000 (1.000, 1.000) 0.001 (0.114, 0.121)

0.001 (0.121, 0.122) 0.001 (0.114, 0.121) 1.000 (1.000, 1.000)

posterior median and interquartile range for the square roots of the diagonal elements of Ra, which are the standard deviations of the shocks. Table 4 shows the posterior median and interquartile range for the conditional correlations implied by the conditional variance Ra. There is little evidence that shocks to at are strongly correlated, either positively or negatively, across region. The posterior means of the standard deviation of shocks are similar to their prior means, reﬂecting the fairly informative prior for Ha. The estimated acceptance probability for a proposals is 0.414. We draw a three times for each new draw of h and estimate the probability of at least one acceptance in three as 0.714. The comparison of this value to 1(10.414)3 ¼ 0.799 suggests a moderately decreasing hazard rate.

CONCLUSIONS This work builds on the paper by McCausland et al. (2007), which describes new methods for drawing states in Gaussian state space models. Here we draw states in a semi-Gaussian state space model where the state is Gaussian and observations are Dirichlet. The general approach here can be applied to other semi-Gaussian state space models.

540

WILLIAM J. MCCAUSLAND AND BRAHIM LGUI

Variations and extensions are possible. In the case where k ¼ 1 and xt 1, it is reasonably straightforward to consider a stationary model. For such a model, the parameter vector is h (B, U, Ha), and l1 and H1 are the following functions of B, U, and Ha: l1 ¼ ðI UÞ1 B

1 21 vec H21 1 ¼ ðIp2 U UÞ vec Ha

See Hamilton (1994, p. 265) for details on the computation of H1 1 . In the stationary case, (B, U) and Ha are no longer conditionally conjugate, but we can use the conditional posterior distributions described in Conditional Inference for Parameters section as proposal distributions. We can generalize the observation equation to allow for covariates and factor models. We can specify c exp (DwtþZat), where D is a p q matrix of coefﬁcients, wt, a q-vector of covariates, and Z, a p m matrix of factor loadings. It is not much more difﬁcult to draw a from its conditional posterior distribution. The derivation of the ht and ct in appendix is already general enough to accommodate such a speciﬁcation. However, drawing D and Z from their conditional posterior distributions is more difﬁcult than drawing l1, B, U, H1, and Ha from theirs, since there is no obvious conditionally conjugate distribution. For the example we consider in this paper, the multivariate Gaussian approximation of the conditional distribution of states given parameters and data is close enough for posterior simulation of reasonable numerical efﬁciency. However, efﬁciency deteriorates as the amount of data increases. For the special case of univariate states, McCausland (2008) introduces reﬁnements to the proposal distribution of states so that it more closely approximates the conditional distribution of states given parameters and observed data. This leads to dramatic improvements in numerical efﬁciency. In future work, we plan to generalize these reﬁnements to multivariate state space models like the one in the current paper.

ACKNOWLEDGMENTS We thank Paolo Giordani, Gary Koop, and other participants of the 6th Annual Advances in Econometrics Conference for helpful suggestions.

REFERENCES Aitchison, J. (1986). The statistical analysis of compositional data: Monographs on statistics and applied probability. London, UK: Chapman & Hall Ltd.

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de Jong, P., & Shephard, N. (1995). The simulation smoother for time series models. Biometrika, 82(1), 339–350. Dhrymes, P. J. (2000). Mathematics for econometrics (3rd ed.). New York: Springer–Verlag. Durbin, J., & Koopman, S. J. (1997). Monte Carlo maximum likelihood estimation for non– Gaussian state space models. Biometrika, 84(3), 669–684. Durbin, J., & Koopman, S. J. (2002). A simple and efﬁcient simulation smoother for state space time series analysis. Biometrika, 89(3), 603–615. Grunwald, G. K., Raftery, A. E., & Guttorp, P. (1993). Time series of continuous proportions. Journal of the Royal Statistical Society Series B, 55, 103–116. Hamilton, J. D. (1994). Time series analysis. Princeton, NJ: Princeton University Press. Hastings, W. K. (1970). Monte Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109. McCausland, W. J. (2008). The HESSIAN method (Highly Efﬁcient State Smoothing, in a Nutshell). Cahiers de recherche du De´partement de sciences e´conomiques. Universite´ de Montre´al, No. 2008-03. McCausland, W. J., Miller, S., & Pelletier, D. (2007). A new approach to drawing states in state space models. Cahiers de recherche du De´partement de sciences e´conomiques, Universite´ de Montre´al, No. 2007-06. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., & Teller, E. (1953). Equation of state calculations by fast computing machines. Journal of Chemical Physics, 21, 1087–1092.

APPENDIX. DERIVATIONS OF ht AND ct In this appendix we derive expressions for ht and ct used in Drawing States section to draw states. We use the following notational conventions. The vector i has all elements equal to 1. Vector ei has elements equal to zero except the ith, which is 1. For vector arguments, the functions c, c1, log, exp, min, and max are applied element by element and return vectors of the same length. The Hadamard (element-by-element) product of matrices A and B is denoted AB and the Kronecker product is denoted A B. The digamma function c and the trigamma function c1 are deﬁned by

cðxÞ

@ log GðxÞ @x

and

c1 ðxÞ

@2 log GðxÞ @x2

where G is the gamma function. For matrix and vector calculus, we use the notational conventions in Dhrymes (2000). We begin with the most general results. Suppose f(yt|ct) is a density or probability mass function for observed data given a vector of parameters ct, and that ct is a deterministic twice differentiate function of at. Then the

542

WILLIAM J. MCCAUSLAND AND BRAHIM LGUI

gradient and Hessian of log f with respect to at are given by @ log f @ log f @ct ¼ @at @ct @at and " # @2 log f @ @ct > @ log f > ¼ @at @at @a> @at @ct t 2 > 2 @ log f @ ct @ct @ log f @ct ¼ Im þ @at @a> @at @at @ct @ct @c> t t We now specialize to the case that ct ¼ exp(BwtþZtat). We deﬁne gt BwtþZtat. We compute the following derivatives of ct: @ct ¼ diagðct ÞZt @at @ 2 ct @ ¼ ½Z> t diagðct Þ @at @a> @a t t @ ¼ ðIm Z> diagðct Þ t Þ @at @ > diagðct Þ Zt ¼ ðIm Zt Þ @g 2 t >3 ct1 e1 e1 6 7 6 7 .. 6 7Zt ¼ ðIm Z> Þ . t 6 7 4 5 > ctp ep ep In the case where f is the Dirichlet density in Eq. (1), we have @ log f ¼ ½cðGt Þi cðct Þ þ log pt > @ct

(A.1)

@2 log f ¼ c1 ðGt Þii> diagðc1 ðct ÞÞ @ct @c> t

(A.2)

543

Bayesian Inference on Time-Varying Proportions

Putting it all together, we have @ log f ¼ ½cðGt Þi cðct Þ þ log pt > diagðct ÞZt @at

@ diagðct Þ @2 log f ¼ ½cðGt Þi cðct Þ þ log pt > Im I Z> Zt t > @at @at @gt > þ Z> t diagðct Þ c1 ðGt Þii diagðc1 ðct ÞÞ diagðct ÞZt n o @ diagðc Þ > t ¼ cðGt Þi cðct Þ þ log pt Z> Zt t @gt > þ Z> t c1 ðGt Þct ct diagðct c1 ðct Þ ct Þ Zt > ¼ Z> t diagððcðGt Þi cðct Þ þ log pt Þ ct Þ þ c1 ðGt Þct ct Zt > þ Z> t c1 ðGt Þct ct diagðct c1 ðct Þ ct Þ Zt Finally, ht and ct are given by Eqs. (5) and (6). and csafe where We can also compute values hsafe t t (1) hsafe is guaranteed to be positive semi-deﬁnite; t (2) the gradient of the log density of the Gaussian approximation with and covector csafe equals the gradient of log f with respect precision hsafe t t to at; and ht is positive semi-deﬁnite. (3) hsafe t We compute hsafe and csafe as ht and ct, but we replace the Hessian of log f t t with respect to ct, given by Eq. (A.2), with the negative semi-deﬁnite matrix A deﬁned by ( A¼

ln c1ðGt Þii> diagðc1 ðct ÞÞ >

minð0; c1 ðGt ÞÞii diagðmaxð0; c1 ðct ÞÞÞ

where Pp n

l ¼

i¼1 c1 ðcti Þ

c1 ðGt Þ

ct 40 otherwise

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WILLIAM J. MCCAUSLAND AND BRAHIM LGUI

If ctW0, l solves the following equation in l: jAj ¼ jlc1 ðGt Þii> diagðc1 ðct ÞÞj ¼ jdiagðc1 ðct ÞÞj j1 lc1 ðGt Þi> ðdiagðc1 ðct ÞÞÞ1 ij ¼ 0 We use Dhrymes (2000, Corollary 2.6) to pass from the second to the third line. Given this result, it is easy to see that A is negative semi-deﬁnite and that safe the matrix A ð@2 log f Þ=ð@ct @c> t Þ is negative semi-deﬁnite. Therefore ht safe and ht ht are positive semi-deﬁnite.

IMPOSING STATIONARITY CONSTRAINTS ON THE PARAMETERS OF ARCH AND GARCH MODELS Christopher J. O’Donnell and Vanessa Rayner ABSTRACT In their seminal papers on ARCH and GARCH models, Engle (1982) and Bollerslev (1986) speciﬁed parametric inequality constraints that were sufﬁcient for non-negativity and weak stationarity of the estimated conditional variance function. This paper uses Bayesian methodology to impose these constraints on the parameters of an ARCH(3) and a GARCH(1,1) model. The two models are used to explain volatility in the London Metals Exchange Index. Model uncertainty is resolved using Bayesian model averaging. Results include estimated posterior pdfs for one-step-ahead conditional variance forecasts.

1. INTRODUCTION Many economic and ﬁnancial time series exhibit periods of high volatility followed by periods of relative tranquility. There is considerable interest in

Bayesian Econometrics Advances in Econometrics, Volume 23, 545–566 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23017-3

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CHRISTOPHER J. O’DONNELL AND VANESSA RAYNER

forecasting this special type of intertemporal variability. Forecasts of exchange rate volatility, for example, are useful for international portfolio management and macroeconomic analysis, while forecasts of volatility in asset returns are used to price derivative securities and design dynamic hedging strategies. Other examples in the ﬁeld of ﬁnancial economics are discussed in Bollerslev, Chou, and Kroner (1992). Engle (1982) was the ﬁrst to design an econometric model for estimating and forecasting time-varying levels of volatility. In his autoregressive conditional heteroskedasticity (ARCH) model, the unconditional error variance is constant but the conditional variance evolves as a moving average process. The model is consistent with the general notion that the conditional distribution of a series is paramount when it comes to forecasting. Indeed, the ability of the ARCH model to explain variations in the conditional moments of a series ensures it plays a key role in the analysis of economic behavior in the presence of uncertainty. The basic ARCH model has been extended in several ways. The most inﬂuential and widely used extension is Bollerslev’s (1986) generalized autoregressive conditional heteroskedasticity (GARCH) model, which assumes the conditional variance process is an autoregressive moving average (ARMA) process. The asymmetric GARCH models of Nelson (1991) and Glosten, Jagannathan and Runkle (1993) are also widely used, particularly in equity markets where bad news tends to have a more pronounced effect on volatility than good news (Black, 1976). ARCH and GARCH models are commonly estimated using maximum likelihood (ML) methods, taking no account of the fact that the conditional variance process must be non-negative and stationary. Consequently, point and interval estimates of parameters and variances may be statistically or economically implausible. This paper illustrates Bayesian methodology for imposing non-negativity and stationarity restrictions on ARCH and GARCH models. As it happens, in an application to data from the London Metals Exchange (LME), we ﬁnd that estimates of the parameters of two models are largely unaffected by the imposition of these restrictions. In our empirical application, and in most other practical settings, there is considerable uncertainty concerning the exact form of the conditional variance process. Sampling theorists usually resolve this uncertainty using common model selection criteria and/or hypothesis tests. One undesirable consequence of this strategy is that it can lead to a false picture of the reliability of estimates of economic quantities of interest (see, for example, Danilov & Magnus, 2004). The Bayesian solution involves computing posterior model probabilities and estimating economic quantities of interest

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using model averaging. In the case of inequality-constrained ARCH and GARCH models, a method proposed by Gelfand and Dey (1994) for computing posterior model probabilities is not immediately available due to the fact that inequality-restricted prior densities are only known up to a multiplicative constant. This paper illustrates the use of importance sampling to estimate these normalizing constants and, subsequently, posterior model probabilities. The paper is organized as follows. In Section 2, we present the standard linear regression model with GARCH errors and identify constraints that ensure the conditional variance process is both non-negative and stationary. In Section 3, we outline common sampling theory methods for selecting plausible ARCH and GARCH models for estimation. In Section 4, we discuss conventional ML estimation of GARCH models under the assumption that the errors are normally distributed. In Section 5, we describe Bayesian methods for incorporating non-negativity and stationarity constraints into the estimation process. In Section 6, we use both sampling theory and Bayesian methods to estimate an ARCH(3) and a GARCH(1,1) model of an LME time series. Model uncertainty is resolved through computation of posterior model probabilities and model averaging. In Section 7 we offer some concluding remarks.

2. ARCH AND GARCH REGRESSION MODELS Consider the standard linear regression model, yt ¼ x0t g þ t

(1)

for t ¼ 1, . . . , T, where xt is a K 1 vector containing an intercept and K1 explanatory variables, and et is a random error with Eðt Þ ¼ 0

(2)

Covðt ; s Þ ¼ 0 for tas; and

(3)

Varðt Þ ¼ s2

(4)

Volatility regression models are underpinned by an additional assumption concerning the conditional error variance, namely Varðt jt1 ; t2 ; . . .Þ ¼ ht 40

(5)

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CHRISTOPHER J. O’DONNELL AND VANESSA RAYNER

Error processes with this last property are said to be conditionally heteroskesdastic. Different assumptions concerning the evolution of ht give rise to different volatility models, including the ARCH (Engle, 1982), GARCH (Bollerslev, 1986), EGARCH (Nelson, 1991), and IGARCH (Engle & Bollerslev, 1986) models. The GARCH(p, q) model of Bollerslev (1986) is obtained under the speciﬁc assumption that ht evolves as an ARMA process: ht ¼ a 0 þ

p X i¼1

bi hti þ

q X

ai 2ti ¼ a0 þ bðLÞht þ aðLÞ2t

(6)

i¼1

where pZ0 and qW0. If p ¼ 0, the model collapses to the ARCH(q) model of Engle (1982). The strict inequality qW0 ensures the process is capable of exhibiting time-varying volatility, while inequality constraints on functions of the unknown parameters ensure the process is non-negative and covariance1 stationary. Speciﬁcally: C.1 ht is locally non-negative if a0 þ bðLÞht þ aðLÞ2t 0 for t ¼ 1, . . . , T; C.2 ht is globally non-negative if a0W0, biZ0 for i ¼ 1, . . . , p, and aiZ0 for i ¼ 1, . . . , q; and C.3 ht is covariance stationary if C.2 is satisﬁed and b(1)þa(l)o1 (Bollerslev, 1986, Thm 1; Engle, 1982, Thm 2). Conditions C.1 and C.2 follow immediately from Eq. (6). The local (i.e., within-sample) non-negativity condition C.1 is empirically relevant because it ensures the log-density of the observations is deﬁned. The global nonnegativity condition C.2 is equally important because of the role GARCH models play in computations that depend on out-of-sample forecasts (e.g., value-at-risk). Observe that condition C.2 is a sufﬁcient condition. Alternative sufﬁcient conditions for global non-negativity have been established by Nelson and Cao (1992). Condition C.3 is a sufﬁcient condition for covariance stationarity. That said, if C.2 holds then b(1) þ a(1)o1 is both necessary and sufﬁcient for covariance stationarity. Bougerol and Picard (1992) establish (stronger) necessary and sufﬁcient conditions for strict stationarity of the conditional variance process. The Bougerol–Picard conditions involve negativity of a Lyapunov exponent and could be imposed using the methodology described below. However, applied time series analysts are usually only concerned with covariance stationarity so the Bougerol–Picard conditions are not considered here. Rather, we focus on conditions C.2 and C.3. If these two

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549

conditions hold then the unconditional error variance is s2 ¼ a0[1b(1)a(1)]1. In empirical applications, it is common to estimate ARCH and GARCH models using unconstrained maximum likelihood methods and then check the estimated parameters for conformity with C.1 and C.2 (e.g., Engle, 1982). Negative variance estimates are sometimes avoided by imposing ﬁxed lag structures on the coefﬁcients (e.g., Engle, 1983; Engle, Lilien, & Robins, 1987), but this clearly and unnecessarily over-constrains the parameter space. This paper illustrates how Bayesian methodology can be used to impose C.1, C.2, and C.3 directly. Sampling theory methods are also available for imposing these constraints directly, but are generally unsatisfactory. For example, Iglesias and Linton (2007) have shown that common quasi-maximum likelihood (QML) estimators of GARCH models are biased when inequality constraints are imposed within the maximization algorithm (and the bias can be quite large).

3. PRELIMINARY MODEL SELECTION Estimation of volatility models is usually preceded by formal tests for the presence of ARCH or GARCH effects. With this in mind, it is useful to consider the deviation of 2t from its conditional mean: vt ¼ 2t Eð2t jt1 ; t2 ; . . .Þ ¼ 2t ht . Substituting ht ¼ 2t vt into the conditional variance function (6) reveals that the squares of GARCH ( p, q) errors evolve as an ARMA (m, p) process: 2t ¼ a0 þ fðLÞ2t þ vt bðLÞvt

(7)

where f (L) ¼ a(L)þb(L) is a polynomial of order m ¼ max(p, q). To test for GARCH effects, it is common to estimate the mean function (1) using OLS2 and then replace 2t in Eq. (7) with the squared OLS residuals, e2t , to obtain e2t ¼ a0 þ fðLÞe2t þ wt

(8)

where wtEvtb(L)vt is (approximately) a moving average process. Engle’s (1982) test for ARCH effects involves estimating Eq. (8) by OLS and testing H0: f1 ¼ f2 ¼ . . . fm ¼ 0. Other tests are also available, including the Ljung and Box (1978) test for serial correlation and a nonparametric test suggested by Gregory (1989). The small sample properties of various tests

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CHRISTOPHER J. O’DONNELL AND VANESSA RAYNER

have been investigated by, for example, Engle, Hendry, and Trumble (1985) and Gregory (1989). After conﬁrming the presence of GARCH effects, the problem of choosing between different ARCH and GARCH speciﬁcations is a matter of choosing the values of p and q. One method for choosing p is to examine the estimated autocorrelation function (ACF) and partial autocorrelation function (PACF) of the moving average process wt in Eq. (8). The value of m ¼ max (p, q) can sometimes be determined by testing the signiﬁcance of the estimated coefﬁcients from Eq. (8). Finally, m and q sometimes reveal themselves in the ACF and PACF of the e2t : if the ACF and PACF point to an ARMA(m, p) process with mWp then we conclude q ¼ m; if they suggest an ARMA (m, p) process with m ¼ p then we can only conclude 1rqrp. In most practical settings, this preliminary model selection strategy yields more than one plausible candidate model for estimation. Methods for resolving this model uncertainty are discussed in Section 6 below.

4. MAXIMUM LIKELIHOOD ESTIMATION Best linear unbiased estimates of the parameters of the mean function (1) can be obtained using OLS. However, the fact that OLS does not account for conditional heteroskedasticity implies that more efﬁcient nonlinear estimators of g are available. In any event, estimates of g alone do not allow us to forecast volatility. For ML estimation of both the mean function and the conditional variance function, it is common to assume et is normally distributed.3 The conditional likelihood function for the so-called normal-GARCH (or N-GARCH) model is (e.g., Bauwens, Lubrano, & Richard, 2003, pp. 208, 219): ! ! T T Y 1X ð1=2Þ T=2 1 0 2 pð yjy; F 0 Þ ¼ ð2pÞ ht h ðyt xt gÞ exp (9) 2 t¼1 t t¼1 where y ¼ (y1, . . . , yT)u; y is a P 1 vector of unknown parameters; and F 0 is a set of initial conditions. In this paper, we assume the following initial conditions: F.1 xt ¼ 0 for tr0; F.2 yt ¼ 0 for to0; and F.3 ht ¼ s2 for tr0, where s2 denotes the sample variance of yt.

The Parameters of ARCH and GARCH Models

551

Conditions F.1 and F.2 have been used by Nakatsuma (2000). Condition F.3 has been suggested by Bauwens et al. (2003, p. 219) and simply equates the presample conditional variances to an estimate of the unconditional variance, s2. Other sets of initial conditions are available and involve treating different sets of presample quantities as unknown parameters. Conditions F.1 to F.3 are computationally convenient because the only presample quantity that remains unknown is y0 ¼ e0. Thus, y ¼ (gu, bu, au, e0, a0)u where b ¼ (b1, . . . , bp)u and a ¼ (a1, . . . , aq)u. The number of unknown parameters in this formulation of the model is P ¼ K þ p þ q þ 2. Econometrics software packages such as EViews use standard optimization algorithms to maximize of the likelihood function (9). Unfortunately, except in special cases, the ﬁnite sample properties of ML estimators of GARCH models are not well known. It is partly for this reason we consider estimation within a Bayesian framework.

5. BAYESIAN ESTIMATION In the Bayesian approach to inference, postsample beliefs about the unknown parameters are summarized in the form of a posterior probability density function (pdf). This posterior pdf is proportional to a prior pdf multiplied by the likelihood function. The non-negativity and stationarity constraints C.1 and C.2 can be incorporated into the Bayesian estimation process using, for example, the following informative prior (e.g., Bauwens et al., 2003, p. 219): pðyÞ / Iðy 2 YÞ (10) where Y is the region of the parameter space where the constraints hold, and I( ) an indicator function that takes the value 1 if the argument is true and 0 otherwise. If all the parameters in the model are theoretically constrained, then this prior has compact support and is proper. However, if some of the parameters are theoretically unconstrained then Eq. (10) is improper. This may complicate hypothesis testing and Bayesian model comparison because posterior odds ratios may not be well-deﬁned (e.g., Bartlett, 1957). When comparing models using posterior odds ratios, a useful rule-ofthumb is ‘‘to use noninformative priors over parameters which are common to all models. However, informative, proper priors should be used over all other parameters’’ (Koop, 2003, p. 269). In this paper, where the parameters e0, a0, and g are common to all models, we use the following prior:4 pðyÞ / f N ðbjmb ; Sb Þ f N ðajma ; Sa Þ Iðy 2 YÞ

(11)

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CHRISTOPHER J. O’DONNELL AND VANESSA RAYNER

where mb, Sb, ma, and Sa are hyperparameters chosen by the researcher.5 In our empirical work, relative noninformativeness is achieved by setting Sb ¼ 1000Ip and Sa ¼ 1000Iq. The prior (11) combines with the likelihood function (9) to yield a posterior pdf of nonstandard form. Thus, simulation methods must generally be used to estimate the characteristics of marginal posterior pdfs. Unfortunately, Markov Chain Monte Carlo (MCMC) simulation of the posterior is complicated by the fact that ht ¼ a0 þ bðLÞht þ aðLÞðyt x0t gÞ2 , making it impossible to draw inferences on g conditionally on the ht. Some of the MCMC algorithms that have been used to deal with this problem include importance sampling (e.g., Geweke, 1988; Kleibergen & van Dijk, 1993), the Metropolis–Hastings (MH) algorithm (e.g., Geweke, 1994) and the Griddy–Gibbs sampler (e.g., Bauwens & Lubrano, 1998). In this paper, we use a random walk MH algorithm to draw iteratively from conditional posterior pdfs that take the form: pð0 jg; a0 ; b; a; yÞ / pð yjy; F 0 Þ Iðy 2 YÞ pðgj0 ; a0 ; b; a; yÞ / pð yjy; F 0 Þ Iðy 2 YÞ pða0 j0 ; g; b; a; yÞ / pð yjy; F 0 Þ Iðy 2 YÞ pðbj0 ; g; a0 ; a; yÞ / f N ðbjmb; Sb Þ pð yjy; F 0 Þ Iðy 2 YÞ pðaj0 ; g; b; a0 yÞ / f N ðajma ; Sa Þ pð yjy; F 0 Þ Iðy 2 YÞ Four iterations of the MH algorithm were used to draw one value from each conditional density, even though one iteration would have been enough to yield a Markov Chain with the correct invariant distribution (Chib & Greenberg, 1996). We used four iterations only to improve the mixing properties of the chain. In this paper, the algorithm was programmed in GAUSS.

6. FINAL MODEL SELECTION Sampling theorists usually select a ﬁnal GARCH speciﬁcation using conventional hypothesis tests, residual analysis, and standard model selection criteria. Conventional hypothesis testing procedures can be applied to the coefﬁcients of ARCH (q) and GARCH (1,1) models on the grounds that, under fairly weak conditions, ML and QML estimators of these models are consistent and asymptotically normal (Weiss, 1986; Li, Ling, & McAleer, 2002; Lumsdaine, 1996). However, little is known about the

The Parameters of ARCH and GARCH Models

553

asymptotic properties of ML and QML estimators for the parameters of other classes of GARCH model. Residual qﬃﬃﬃﬃ analysis typically involves formation of standardized residuals st ¼ et = h^t where h^t denotes an estimate of ht. If the model has been correctly speciﬁed, then st should not contain any ARCH or GARCH effects. Finally, standard model selection criteria, including the Akaike and Schwarz information criteria, can be used to select an appropriate GARCH speciﬁcation, although Enders (2004, p. 136) recommends using modiﬁed criteria when the conditional variance function is of more interest than the mean function. The recommended criteria are AIC n ¼ ln L þ 2P SBC n ¼ ln L þ P ln T where ln L is the maximized value of the log-likelihood function and P the number of parameters in the mean function and the conditional variance function combined. Irrespective of the (validity of ) methods used to select a ﬁnal GARCH model, sampling theorists typically report their chosen model in a way that ignores the non-zero probability that an alternative model may be correct. We regard this as unsatisfactory. In principle, the problem of model uncertainty can be satisfactorily resolved in a Bayesian framework using posterior model probabilities.

6.1. Posterior Model Probabilities Let Mr denote the rth model under consideration (r ¼ 1, . . . , R). We now modify our earlier notation so that each model is characterized by a vector of parameters, yr, a prior pdf, p(yr|Mr), a likelihood function, p( y|yr, Mr), and a posterior pdf, p(yr|y, Mr). If p(Mr) is the prior probability for model Mr, the posterior probability is pðM r jyÞ ¼

pð yjM r ÞpðM r Þ pðyÞ

where pðyÞ ¼

R X r¼1

pð yjM r ÞpðM r Þ

(12)

554

CHRISTOPHER J. O’DONNELL AND VANESSA RAYNER

and Z pð yjM r Þ ¼

pð yjyr ; M r Þpðyr jM r Þdyr

(13)

is the marginal likelihood. If we can estimate the R marginal likelihoods then a ﬁnal model can be selected on the basis of highest estimated posterior probability. 6.2. Marginal Likelihoods Several numerical methods have been developed for estimating marginal likelihoods, including bridge sampling (Meng & Wong, 1996), a harmonic mean estimator (Newton & Raftery, 1994), and reversible jump MCMC (Green, 1995). In this paper, we use an estimator proposed by Gelfand and Dey (1994). Gelfand and Dey (1994, p. 511) observe that the inverse of the marginal likelihood can be written Z 1 f ðyr Þ ¼ pðyr jy; M r Þdyr (14) pð yjM r Þ pð yjyr ; M r Þpðyr jM r Þ where f (yr) is any pdf with support contained in Y. This leads to the following reciprocal importance estimator of the marginal likelihood " #1 S 1X f ðyðmÞ r Þ ^ yjM r Þ ¼ pð ðmÞ S m¼1 pð yjyðmÞ r ; M r Þpðyr jM r Þ where yðmÞ is the mth draw from p(yr|y, Mr) and S the length of the MCMC r chain. The efﬁciency of the reciprocal importance estimator is sensitive to the choice of f(yr). Most importantly, the support conditions must be met. Geweke (1999) recommends choosing f(yr) to be a truncated multivariate normal pdf: ð1=2Þ 1 0 ^ 1 ^ ^ ^y ðy exp y Þ ðy y Þ Iðyr 2 Gr Þ S f ðyr Þ ¼ t1 ð2pÞP=2 S r r r r r yr 2 (15) ^ y are estimates of the mean and variance of the posterior pdf, where y^ r and S r p(yr|y, Mr), and tA(0, l) is a probability chosen by the researcher. The indicator function in Eq. (15) takes the value one if yr satisﬁes 1 ðyr y^ r Þ0 S^ yr ðyr y^ r Þ w21t ðPÞ

The Parameters of ARCH and GARCH Models

555

where w21t ðPÞ is the value that leaves an area of t in the left-hand tail of a w2 distribution with P degrees of freedom. In our empirical work we use t ¼ 0.1. To avoid computational difﬁculties it is sometimes convenient to normalize the marginal likelihoods by multiplying both sides of Eq. (14) by an arbitrary scalar k. Since k is ﬁxed across models, the normalized marginal likelihoods can be used in place of the unnormalized marginal likelihoods in Eq. (12).

6.3. Normalizing Constants Observe from Eq. (14) that the reciprocal importance estimator can only be implemented if both the likelihood and the prior are known. Unfortunately, the prior (11) is only known up to a factor of proportionality. Speciﬁcally, it takes the form: pðyr jM r Þ ¼ kr qðyr Þ where kr is an unknown normalizing constant. Thus, to apply the reciprocal importance estimator, we must ﬁrst estimate kr. The approach used in this paper is based on the result that Z 1 qðyr Þ mðyr Þ dyr ¼ kr mðyr Þ where m(yr) is an importance sampling density with the property that m(yr)W0 for all values of yr where q(yr)W0. Then an estimate of kr is " #1 Sn ðmÞ X 1 qðy Þ r k^r ¼ n S m¼1 mðyrðmÞ Þ where yðmÞ is the mth draw from m(yr) and S the number of draws. In our r empirical work we use mðyr Þ ¼ fN ðyr jy^ r ; cr S^ yr Þ

(16)

where crW1 and large enough to ensure that m(yr) is non-negligible for all values of yr where q(yr)W0 (equivalently, large enough to ensure the importance weights are not too large). Some alternative algorithms for estimating (ratios of) normalizing constants are reviewed by Chen and Shao (1997).

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CHRISTOPHER J. O’DONNELL AND VANESSA RAYNER

7. APPLICATION TO LONDON METALS EXCHANGE INDEX For an empirical illustration, we collected 556 daily observations on the LME Index, covering the period 11 April 2000 to 1 July 2002. Series plots and unit root tests were used to establish that the series and its logarithm were both difference-stationary. Accordingly, we chose to model the daily rate of return: LMEX t yt ¼ ln LMEX t1 where LMEXt denotes the value of the index in period t. The correlogram for the series provided unambiguous evidence that the mean function was of the form yt ¼ g 0 þ t The OLS estimate of g0 is reported in column A in Table 1 (with the standard error in parentheses). The associated residuals exhibit some volatility clustering, as seen in Fig. 1. To formally conﬁrm the presence of GARCH effects, the squared residuals were used to conduct the Lagrange multiplier test described in Section 3. Using m ¼ 4, the prob-value of the test was zero when rounded to four decimal places, so the null hypothesis of no GARCH effects was easily rejected. In conducting this particular test, we observed that estimates of f1 and f4 in Eq. (8) were statistically insigniﬁcant at usual levels of signiﬁcance, leading us to conclude that mr3. The correlogram of the squared OLS residuals appeared consistent with both AR(3) and ARMA(1,1) processes. For this reason, and bearing in mind that our empirical example is intended only as an illustration, we chose to regard only the ARCH(3) and GARCH(1,1) processes as plausible models for the conditional variance.

7.1. Estimated Parameters ML estimates of the parameters of the two models are reported in columns B and E in Table 1. In these columns, numbers in parentheses are estimated asymptotic standard errors. All estimated parameters are plausible insofar as they satisfy conditions C.1 through C.3. Recall that condition C.1 ensures the log-likelihood is deﬁned, condition C.2 ensures global non-negativity,

OLS (A)

ARCH (3) ML(B)

g0 a0

1.70E4 (3.95E4) –

b1

–

a1

–

a2

–

a3

–

s2

8.65E5

AIC SBC p(y|Mr)

1,805.4 1,796.41 –

Parameter Estimates and Model Selection Criteria. GARCH (1,1)

Bayes C.l (C)

Bayes C.2 (D)

Bayes C.3 (E)

ML (F)

Bayes C.l (G)

Bayes C.2 (H)

Bayes C.3 (I)

2.63E4 (3.78E4) 5.97E5 (0.69E5) –

2.88E4 (3.83E4) 5.98E5 (0.74E5) –

3.02E4 (3.82E4) 5.88E5 (0.71E5) –

2.99E4 (3.83E4) 5.89E5 (0.71E5) –

0.0342 (0.0397) 0.121 (0.048) 0.160 (0.048) 8.71E5 – 1,810.11 1,788.51 –

0.0503 (0.0457) 0.130 (0.060) 0.168 (0.061) 9.27E5 (1.07E5) – – –

0.0593 (0.0411) 0.130 (0.059) 0.171 (0.062) 9.31E5 (1.14E5) – – –

0.0587 (0.0412) 0.130 (0.060) 0.170 (0.062) 9.30E5 (1.13E5) – – 0.748

2.84E4 (3.78E4) 2.88E6 (1.45E6) 0.917 (0.024) 0.049 (0.016) –

3.23E4 (3.81E4) 1.15E5 (1.04E5) 0.791 (0.137) 0.078 (0.037) –

3.23E4 (3.83E4) 1.28E5 (1.29E5) 0.776 (0.158) 0.078 (0.037) –

3.23E4 (3.83E4) 1.20E5 (1.03E5) 0.783 (0.134) 0.079 (0.036) –

–

–

–

–

8.50E5 – 1,811.76 1,794.48 –

8.98E5 (1.33E4) – – –

8.80E5 (3.69E4) – – –

9.08E5 (7.81E5) – – 0.252

The Parameters of ARCH and GARCH Models

Table 1.

557

558

CHRISTOPHER J. O’DONNELL AND VANESSA RAYNER .05 .04 .03 .02 .01 .00 -.01 -.02 -.03 -.04 7/00

Fig. 1.

1/01

7/01

1/02

7/02

OLS Residuals for Random Walk Model of LME Index Returns.

and condition C.3 ensures both global non-negativity and covariance stationarity (C.3 is not satisﬁed unless C.2 is also satisﬁed). Observe, however, that the ARCH(3) estimate for a1 lies within one estimated standard error of zero, so asymptotic 95% conﬁdence intervals will easily include values that violate condition C.2. This may be sufﬁcient reason to reject the ARCH(3) model in favor of the more parsimonious GARCH(1,1) model. The AIC values reported at the bottom of Table 1 could be used to support this decision. At the same time, however, the SBC criterion, which attaches a relatively high penalty to parameters in the conditional variance function, would lead us to reject both models in favor of a model with no ARCH or GARCH effects. We resolved some of these issues by estimating the models in a Bayesian framework. Columns C to E in Table 1 report the results of estimating the ARCH(3) model subject to constraints C.1, C.2, and C.3, respectively. The GARCH(1,1) results are reported in columns G to I. The numbers in these columns are means and standard deviations of MCMC samples of size 100,000. The convergence properties of the MCMC chains were assessed visually, using plots that resembled those in Fig. 2. This particular ﬁgure plots the sequences of draws used to obtain the ARCH(3) results reported in column C in Table 1. None of the chains in this ﬁgure exhibit any signs of nonstationarity.

The Parameters of ARCH and GARCH Models

Fig. 2.

559

ARCH(3) Chains Satisfying C.1.

The estimates reported in columns B and C of Table 1 are generally similar, reﬂecting the following facts: (i) both sets of estimates satisfy condition C.1, even though the ML estimates are unconstrained; (ii) the Bayesian prior density is relatively noninformative, so the posterior pdf should closely resemble the likelihood function; and (iii) the MCMC chains are long and stable, so sampling errors should be small. That said, the Bayesian estimated posterior standard deviations are generally larger than the ML estimated asymptotic standard errors, and there is a large difference between the Bayesian and ML point estimates of a1. The difference in the standard deviations/errors is due to the fact that the ML standard errors fail to account for the uncertainty involved in estimating s2. The difference between the point estimates of a1 (and, to a lesser extent, a2 and a3) is due to the fact that the marginal likelihoods/posteriors are asymmetric, and the ML point estimate is a mode while the Bayesian point estimate is a mean. The importance of asymmetry is evident in Fig. 3 where we plot the

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CHRISTOPHER J. O’DONNELL AND VANESSA RAYNER

Fig. 3.

Estimated Posterior Pdfs for ARCH(3) Parameters Satisfying C.1.

estimated marginal posterior pdfs for aj( j ¼ 0, . . . , 3), g0 and a(1). The estimated pdf in the top-right-hand panel is the (relatively asymmetric) estimated pdf for a1. Observe from this particular pdf that there is nonzero probability that a1o0 (approximately 0.12). Columns D and E in Table 1 report the ARCH(3) parameter estimates obtained after imposing conditions C.2 and C.3. A comparison of columns C and D reveals that imposing C.2 has only impacted on the point estimate of a1, while a comparison of columns D and E reveals that imposing C.3 has had no discernable impact on any coefﬁcients. This is because, for this model and data, only C.2 is binding. The estimated coefﬁcients of the GARCH(1,1) model are also insensitive to the imposition of the constraints. This is because, for the GARCH(1,1) model and the LMEX data, the probability that C.2 is violated is less than 0.01.

561

The Parameters of ARCH and GARCH Models

7.2. Conditional Variance Forecasts The unconstrained ML coefﬁcient estimates reported in columns B and F in Table 1 were used to generate two sets of within-sample conditional variance forecasts for the last ten periods in the time series. Two more sets of forecasts were then generated using the Bayesian estimates reported in columns E and I. All four sets of forecasts are reported in Table 2. No standard errors are reported for the ML forecasts, partly because they are not as theoretically or intuitively appealing as the Bayesian forecasts, and partly because convenient computational methods are unavailable. The measures of reliability reported in parentheses below the Bayesian ARCH and GARCH forecasts are standard deviations of MCMC samples. From a statistical viewpoint, there is nothing remarkable about the forecasts reported in Table 2, except perhaps that the Bayesian forecasts are generally higher than the ML forecasts. Even then, the differences

Table 2.

ARCH (3)

t

547 548 549 550 551 552 553 554 555 556

Conditional Variance Forecasts 1.0Eþ4. GARCH (1,1)

ML

Bayes C.3

ML

Bayes C.3

0.600 na 0.599 na 0.605 na 0.649 na 0.745 na 0.786 na 0.688 na 0.837 na 0.854 na 0.691 na

0.594 (0.071) 0.593 (0.071) 0.600 (0.069) 0.670 (0.068) 0.749 (0.071) 0.796 (0.071) 0.724 (0.073) 0.849 (0.083) 0.867 (0.091) 0.748 (0.104)

0.626 na 0.604 na 0.587 na 0.622 na 0.599 na 0.593 na 0.650 na 0.626 na 0.606 na 0.706 na

0.644 (0.069) 0.625 (0.075) 0.614 (0.079) 0.685 (0.080) 0.650 (0.076) 0.648 (0.078) 0.749 (0.091) 0.700 (0.074) 0.667 (0.073) 0.836 (0.109)

BMA

0.607 (0.074) 0.601 (0.073) 0.603 (0.072) 0.674 (0.071) 0.724 (0.084) 0.759 (0.097) 0.730 (0.079) 0.812 (0.104) 0.816 (0.123) 0.770 (0.113)

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between the two sets of forecasts are not so large that the ML forecasts would not be spanned by Bayesian 95% highest posterior density (HPD) intervals. If we are interested in using conditional variance forecasts to inform decision-making in nonferrous metals markets, our focus should not be so much on the differences between the ML and Bayesian forecasts, but on the differences between the Bayesian forecasts obtained using the ARCH(3) and GARCH(1,1) speciﬁcations. Whereas we have strong a priori grounds for choosing the Bayesian results over the ML results, we have no such grounds for choosing one ARCH/GARCH model over the other – both speciﬁcations are considered equally likely. To resolve this model uncertainty, we have used the methods discussed in Section 6 to estimate posterior model probabilities. The normalizing constants were estimated by drawing 500,000 observations from the importance sampling density (16). We obtained estimates of k^1 ¼ 21:92 and k^2 ¼ 20:89 for the ARCH(3) and GARCH(1,1) models respectively. Associated estimates of the posterior ^ 2 jyÞ ¼ 0:252. Thus, ^ 1 jyÞ ¼ 0:748 and pðM model probabilities were pðM whereas sampling theory methods would lead us to use conditional variance forecasts from the GARCH(1,1) model, or perhaps from a model that held the conditional variance constant (recall the discussion in the ﬁrst paragraph of Section 7.1), the estimated posterior odds-ratio favors the constrained ARCH(3) model over the constrained GARCH(1,1) model by a factor of 3:1. The estimated posterior model probabilities have been used to obtain probability-weighted averages of the two sets of forecasts. A rationale for so-called Bayesian model averaging (BMA) is provided by Leamer (1978). The BMA forecasts are reported in the last column of Table 2. In that column, the standard errors in parentheses reﬂect uncertainties associated with the unknown parameters as well as uncertainties surrounding model choice. A more complete picture of the relationship between the three sets of Bayesian forecasts can be gleaned from Figs. 4 and 5. Fig. 4 presents estimated posterior pdfs for the conditional variance in period t ¼ 555, the second last period in the time series. Observe from Fig. 1 that this was a period when the series was relatively tranquil, so all estimated pdfs for the conditional variance in this period are plausibly centered closer to zero and fairly similar. Fig. 5 presents estimated posterior pdfs for period t ¼ 536 when the series was slightly more volatile. In this ﬁgure, where the axes are on a different scale to Fig. 4, the differences between the three pdfs are much more pronounced.

The Parameters of ARCH and GARCH Models

Fig. 4.

Estimated Pdf for Conditional Variance in Period t ¼ 555.

Fig. 5.

Estimated Pdf for Conditional Variance in Period t ¼ 536.

563

8. CONCLUSION The ARCH and GARCH models of Engle (1982) and Bollerslev (1986) are widely used to analyze intertemporal changes in volatility. Both speciﬁcations are usually estimated using unconstrained ML and QML methods. Unfortunately, the ﬁnite sample properties of these estimators are unknown.

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Moreover, unless the conditional variance processes are non-negative and stationary, their asymptotic properties are also unknown. This paper illustrates Bayesian methodology for imposing non-negativity and stationarity constraints on ARCH and GARCH models, and for drawing valid ﬁnite-sample inferences concerning nonlinear functions of the unknown parameters, including conditional variances. The methodology was applied to a time series of returns for commodities listed on the LME. Preliminary analysis led us to focus on two equallyplausible models for this series – an ARCH(3) model and a GARCH(1,1) model. Following estimation, we were able to resolve this model uncertainty and assign probabilities of 0.748 and 0.252 to the ARCH and GARCH models respectively. These estimated probabilities were subsequently used to obtain probability-weighted averages of conditional variance forecasts. In our empirical example, parameter estimates were generally found to be insensitive to the imposition of global non-negativity and weak stationarity constraints. Our conclusion is that there are at least some models and empirical contexts where the quality of statistical inference has not been diminished by failure to impose the regularity constraints statistical theory prescribes.

NOTES 1. A stochastic process is weakly or wide-sense or covariance stationary if its ﬁrst two moments are ﬁnite and do not depend on time (e.g., Bauwens et al., 2003, p. 221). 2. The error term in (1) is white noise, so OLS is still the best linear unbiased estimator of g. 3. Alternative distributions that have been used in empirical work include the Student-t (e.g., Bollerslev, 1987), the power exponential (e.g., Baillie & Bollerslev, 1989), the generalized exponential (e.g., Nelson, 1991), a normal-lognormal mixture (e.g., Hsieh, 1989) and a normal-Poisson mixture (e.g., Jorion, 1988) distribution. 4. For a slightly more informative prior, see Nakatsuma (2000). 5. The notation fN(a|b,C) indicates that a follows a multivariate normal distribution with mean vector b and covariance matrix C.

REFERENCES Baillie, R. T., & Bollerslev, T. (1989). The message in daily exchange rates: A conditional variance tale. Journal of Business and Economic Statistics, 7, 297–305. Bartlett, M. S. (1957). A comment on D.V. Lindley’s statistical paradox. Biometrika, 44(3/4), 533–534.

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Hsieh, D. A. (1989). Modeling heteroskedasticity in daily foreign exchange rates. Journal of Business and Economic Statistics, 7, 307–317. Iglesias, E. M., & Linton, O. B. (2007). Higher-order asymptotic theory when a parameter is on a boundary with an application to GARCH models. Econometric Theory, 23(6), 1136–1161. Jorion, P. (1988). On jump processes in the foreign exchange and stock markets. Review of Financial Studies, 1, 427–445. Kleibergen, F., & Van Dijk, H. K. (1993). Non-stationarity in GARCH models – a Bayesian analysis. Journal of Applied Econometrics, 8, S41–S61. Koop, G. (2003). Bayesian econometrics. England: Wiley. Leamer, E. E. (1978). Speciﬁcation searches: Ad hoc inference with non-experimental data. New York: Wiley. Li, W. K., Ling, S., & McAleer, M. (2002). Recent theoretical results for time series models with GARCH errors. Journal of Economic Surveys, 16, 245–269. Ljung, G. M., & Box, G. E. P. (1978). On a measure of lag of ﬁt in time series models. Biometrika, 67, 297–303. Lumsdaine, R. L. (1996). Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica, 64(3), 575–596. Meng, X. L., & Wong, W. H. (1996). Simulating ratios of normalising constants via a simple identity: A theoretical exploration. Statistica Sinica, 6, 831–860. Nakatsuma, T. (2000). Bayesian analysis of ARMA-GARCH models: A Markov chain sampling approach. Journal of Econometrics, 95, 57–69. Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59(2), 347–370. Nelson, D. B., & Cao, C. Q. (1992). Inequality constraints in the univariate GARCH model. Journal of Business and Economic Statistics, 10(2), 229–235. Newton, M. A., & Raftery, A. E. (1994). Approximate Bayesian inference by the weighted likelihood bootstrap. Journal of the Royal Statistical Society Series B, 56, 3–48. Weiss, A. A. (1986). Asymptotic theory for ARCH models: Estimation and testing. Econometric Theory, 2, 107–131.

BAYESIAN MODEL SELECTION FOR HETEROSKEDASTIC MODELS Cathy W. S. Chen, Richard Gerlach and Mike K. P. So ABSTRACT It is well known that volatility asymmetry exists in ﬁnancial markets. This paper reviews and investigates recently developed techniques for Bayesian estimation and model selection applied to a large group of modern asymmetric heteroskedastic models. These include the GJR-GARCH, threshold autoregression with GARCH errors, TGARCH, and double threshold heteroskedastic model with auxiliary threshold variables. Further, we brieﬂy review recent methods for Bayesian model selection, such as, reversible-jump Markov chain Monte Carlo, Monte Carlo estimation via independent sampling from each model, and importance sampling methods. Seven heteroskedastic models are then compared, for three long series of daily Asian market returns, in a model selection study illustrating the preferred model selection method. Major evidence of nonlinearity in mean and volatility is found, with the preferred model having a weighted threshold variable of local and international market news.

Bayesian Econometrics Advances in Econometrics, Volume 23, 567–594 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23018-5

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1. INTRODUCTION The family of autoregressive conditional heteroskedastic (ARCH) models by Engle (1982) and generalized ARCH (GARCH) by Bollerslev (1986), have become the widely accepted volatility models. However, they only allow symmetric responses to past shocks. Poon and Granger (2003) described a list of prominent characteristics or stylized facts for ﬁnancial time series. This includes dynamic volatility and volatility persistence, fat tailed (compared to normality), mean stationary returns, and also asymmetric volatility; in response to past positive and negative returns. The ﬁrst asymmetric volatility model was the exponential GARCH (EGARCH) of Nelson (1991). Since then numerous models such as the GJR-GARCH of Glosten, Jagannathan, and Runke (1993), threshold GARCH (TGARCH) of Zakoian (1994), and quadratic ARCH (QARCH) of Sentana (1995) have been proposed. Many of these are based on the nonlinear threshold autoregressive (TAR) model of Tong (1978). Li and Li (1996) introduced a double threshold ARCH (DT-ARCH) model, to capture volatility and mean asymmetry, and this was extended to the DT-GARCH by Brooks (2001). Chen, Chiang, and So (2003) proposed an exogenous mean factor and an exogenous threshold variable in a double threshold DTX-GARCH model, highlighting mean and volatility asymmetries in response to the US market news. Chen and So (2006) proposed a nonlinear GARCH model that employed a weighted average of local and exogenous factors as the threshold variable, also including t-distributed errors to capture fat-tailed returns. An advantage of a weighted average threshold is that information from different markets or industries can be combined to inﬂuence the change in regimes. The relative importance of these sources is then assessed by the estimated weights. The aforementioned studies have conﬁrmed the asymmetric volatility phenomenon; volatility in international stock markets is higher following ‘‘bad’’ local or international market news, that is, negative returns. They have also extended the phenomenon of asymmetry, ﬁnding that volatility persistence, and the spillover effect of international market news on local mean returns, are higher, and that average returns are often lower, following bad local or international market news. Practitioners are increasingly turning to Bayesian methods for the analysis of complicated heteroskedastic models. This move seems due to the advent of inexpensive high-speed computers and the development of stochastic integration methodology, especially Markov chain Monte Carlo (MCMC) approaches. MCMC is a computationally intensive simulation

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569

method for numerical integration developed in the 1980s, making it possible to tackle more complex, realistic models and problems. Bayesian methods have been successfully applied in similar nonlinear GARCH models by Geweke (1995), Bauwens and Lubrano (1998), Vrontos, Dellaportas, and Politis (2000), and many others. These methods have the further advantage of being valid under the stationarity and positivity parameter constraints usually required for such models (see Silvapulle & Sen, 2004 for problems with large sample theory under such constraints) and the ability to do joint ﬁnite sample inference on all model parameters, including delay lag and threshold cut-off level, which are alternately often estimated by information criteria (Li & Li, 1996) or simply ascribed a value (Brooks, 2001). However, comparison across models may not proceed in a completely similar fashion, due to the fact that formal model comparison via Bayes factors remains difﬁcult (Berg, Meyer, & Yu, 2004). We brieﬂy review three Bayesian model comparison methods, focusing on a recent sample relevant to heteroskedastic models. These include: importance sampling to produce posterior model odds ratios, as in Geweke (1995) and Gerlach, Carter, and Kohn (1999), also applied to choose between four competing GARCH models in Chen and So (2006) and between GARCH and stochastic volatility models in Gerlach and Tuyl (2006); the reversible-jump (RJ) MCMC method of Green (1995), applied to choose between a symmetric and nonlinear GARCH model in So, Chen, and Chen (2005), between GARCH and DT-GARCH in Chen, So, and Gerlach (2005), and between a GARCH and EGARCH model in Vrontos et al. (2000); and the recent direct method for approximate posterior model probability estimation, by Congdon (2006, 2007). All of these methods involve MCMC methods or extensions, to allow estimation of marginal model likelihoods, Bayes factors, and/or posterior model probabilities. Since in most cases the computation of marginal likelihoods involves high-dimensional integration, the computational time required is usually quite long; see the discussion in Carlin and Chib (1995), Chib (1995), and Godsill (2001). Recently, however, Congdon (2006, 2007) introduced a more efﬁcient approach to approximate posterior model probabilities, following on from work in Carlin and Chib (1995), Godsill (2001), and Scott (2002). We discuss these three methods and make a recommendation on the preferred option in Section 3. The purpose of this paper is to review model selection methods and then make a recommendation for heteroskedastic models. This method will then be applied to choose between seven competing heteroskedastic models for some ﬁnancial return data. We thus intend to ﬁnd the optimal

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heteroskedastic model in these ﬁnancial markets; and in doing so fully investigate and analyze the apparent nonlinear behavior in these markets. For the rest of the paper, Section 2 describes several nonlinear GARCH models, and in particular, a general double threshold GARCH model speciﬁcation with auxiliary variables; Section 3 discusses the three model comparison methods; Section 4 contains an analysis of returns from three major Asian stock indices; while Section 5 concludes.

2. THRESHOLD NONLINEAR HETEROSKEDASTIC MODELS We consider seven modern heteroskedastic models, all with Student-t distributed errors, with details given in this section. We consider single and double threshold models, including models with no asymmetry, only mean asymmetry, only volatility asymmetry, and both mean and volatility asymmetry. The simplest is the ARX-GARCH-t model in So, Chen, and Liu (2006) which assumes an autoregressive (AR) mean process with exogenous (X ) variables. Other asymmetric generalizations in the mean or variance equations are also in our list. For example, the ARXGJR-GARCH-t model in Chen, Gerlach, and So (2006), a threshold ARX or TARX-GARCH-t model (Li & Lam, 1995s model is a particular case), and ARX-TGARCH-t model in So et al. (2005), where ‘‘GJR’’ stands for the asymmetric variance speciﬁcation proposed by Glosten et al. (1993). Finally, three double threshold models are considered: a DTX-GARCH with exogenous threshold (Chen et al., 2006), a DT-GARCH with local market or self-exciting threshold (Chen et al., 2005), and a DTX-GARCH whose threshold is a linear combination of the local and exogenous variables, as in Chen and So (2006). Building on the work of Tong (1978), Li and Li (1996), Brooks (2001), and Chen et al. (2003), Chen and So (2006) introduced the following general threshold heteroskedastic model: yt ¼ fðjÞ 0 þ at ¼

pj X

pﬃﬃﬃﬃ ht t ;

ht ¼ aðjÞ 0 þ

fðjÞ i yti þ

i¼1

cðjÞ ki xk;ti þ at ;

if rj1 ztd orj

k¼1 i¼1 i:i:d:

t Dð0; 1Þ

dj X i¼1

qj lj X X

2 aðjÞ i ati þ

cj X l¼1

bðjÞ l htl

ð1Þ

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Bayesian Model Selection for Heteroskedastic Models

P where zt ¼ w1 z1;t þ þ wm zm;t ; 0 wi 1; m i¼1 wi ¼ 1, for j ¼ 1, . . . , g and D(0, 1) is an i.i.d. error distribution with mean 0 and variance 1. Here yt is the observed data; ht the conditional volatility of yt7y1, . . . , yt1; zt the threshold variable; xk,t refers to an exogenous variable; d a positive integer, commonly referred to as the delay lag; the number of regimes is g; and the model orders pj, qj, dj, cj, are nonnegative integers. The threshold values rj satisfy N ¼ r0or1o . . . org ¼ N, so the intervals [rj1, rj), j ¼ 1, . . . , g form a partition of the space of the threshold variable ztd. Empirical evidence in the literature shows that et tends to be more fat-tailed than normal, as assumed in Li and Li (1996), Brooks (2001); for example, see Chen et al. (2003). Here a Student-t distribution, standardized to have unit variance, is ﬁtted to et to capture this empirical leptokurtosis. A feature of this model is that the threshold variable zt is a linear combination of auxiliary variables zi,t; i ¼ 1, . . . , m. These can be any function of exogenous xi,t or endogenous variables yt, . . . , y1. Some examples are: 1. z1,t ¼ yt and zi,t ¼ xi,t for iW1. w1 ¼ 1 and wi ¼ 0 for i W1 give the models of Li and Li (1996) and Brooks (2001). 2. z1t ¼ at, where ! pj qj lj X X X ð jÞ ð jÞ ð jÞ fi yti cki xk;ti Iðrj1 ztd orj Þ a t ¼ yt f 0 i¼1

k¼1 i¼1

is a function of exogenous variables and yt’s and again w1 ¼ 1 and wi ¼ 0 for i W 1. The GJR-GARCH model falls into this category. 3. zi,t ¼ xi,t, the threshold is a linear combination of exogenous variables only. For identiﬁability and stationarity, we follow Chen and So (2006) to assume that fðpjÞj a0, the g AR vectors ðfð0jÞ ; . . . ; fðpjjÞ Þ; j ¼ 1; . . . ; g, are distinct and p X i¼1

max jfði jÞ jo1 j

where p ¼ max {p1, . . . , pg} and fði jÞ ¼ 0 for i W pj. We also have the following standard restrictions on the variance parameters which guarantee positivity and stationary: að0jÞ 40; aði jÞ ; bðl jÞ 0 and

dj X i¼1

aði jÞ þ

cj X l¼1

bðl jÞ o1

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3. MODEL SELECTION We consider several competing parametric Bayesian models for the same observation vector y1,n, which refers to a data set consisting of observations at times 1–n. In Bayesian model hypothesis testing, the decision between two models Mi versus Mj is made by posterior odds ratio; that is, choose Mi if PORijW1, where PORij ¼

pðM i jy1;n Þ pðy1;n jM i Þ pðM i Þ ¼ pðM j jy1;n Þ pðy1;n jM j Þ pðM j Þ

p(Mi) is the prior probability of model Mi and pðy1;n jM i Þ the marginal (or integrated) likelihood, of the vector of observations y1,n=(y1, . . . , yn), deﬁned as: Z pðy1;n jM j Þ ¼ pðy1;n jhj ; M j Þpðhj jM j Þdhj (2) where hj is the parameter vector for Mj and pðy1;n jhj ; M j Þ; pðhj jM j Þ the sampling density function and the prior density function under Mj, respectively. This method can be applied equally well to nested or nonnested models, a clear advantage over frequentist model selection criteria, for example, information criteria. Without any prior information on model choice, the prior odds ratio ¼ 1 and the PORij then becomes the well-known Bayes factor (Kass & Raftery, 1995). When there are more than two models, the model with highest posterior probability is chosen. Marginal likelihoods have proven a challenge to estimate in the Bayesian and frequentist literature, see Kass and Raftery (1995) for a review of the main issues. This is mainly because they involve a multidimensional integration over the parameter space, as in Eq. (2), which often has high dimension (being the number of parameters in the model), especially when hj involves latent variables or many exogenous factors. For many modern statistical models it is not possible to analytically complete this integral. Often the likelihood is not of a known distributional form in the parameters and even if it was, nonconjugate priors would make a closed-form solution to the integral generally impossible. Kass and Raftery (1995) recommended using a multivariate Laplacian approximation to the integrand in Eq. (1), but this can be inaccurate for complex multiparameter models. Thus, usually the integral is done numerically, but it has proven a challenge to do so efﬁciently and accurately. Standard approaches like adaptive quadrature are usually not sufﬁcient. Monte Carlo approaches, which usually become

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MCMC because of the high dimension involved, are now the common approach. Even then, before Congdon (2006), the suggested approaches were very difﬁcult to implement and often involved the inverse of the likelihood function, which can become numerically unstable. We review three recently suggested methods below. 3.1. Importance Sampling To estimate the marginal likelihood for each model, we ﬁrst consider the importance sampling MCMC method proposed in Geweke (1995) and Gerlach et al. (1999). When kZt, PN 1;t1 ½i t;k 1;t1 ½i p y jy ; h ; M Þ=pðy jy ; h ; M j j 1;t1 t i¼1 k k ; jM j ¼ p^ yt y PN ½i t;k 1;t1 ; hk ; M j i¼1 1=p y jy 1;k where y½i k is the ith MCMC iterate from the posterior pðhj jy ; M j Þ. This estimator can be calculated by running separate MCMC sampling schemes while increasing the sample size k sequentially, say k ¼ 100, 200, . . . , n, then ^ t jy1;t1 ; M j Þ with kt not greater than the increments. An evaluating pðy estimate of the marginal likelihood is then evaluated as: n Y p^ yt jy1;t1 ; M j ; p^ y1;n jM j ¼

j ¼ 1; . . . ; k

t¼1

This method was employed successfully in Chen and So (2006); Gerlach and Tuyl (2006) used it to compare a range of symmetric and Markov switching GARCH and stochastic volatility models; and Gerlach, Chen, Lin, and Huang (2006) to choose between two competing asymmetric DT-GARCH models. However, the method is quite computationally intensive, in comparison to those presented next, since it requires the MCMC sampling scheme to be run more than once (actually n/100, rounded up, times). Further, since the method involves an inverse likelihood term in both the numerator and denominator, it can be prone to numerical difﬁculties and instability, especially for smaller sample sizes k. A similar approach was adopted by Osiewalski, Pajor, and Pipien (2007) who compared bivariate GARCH and stochastic volatility models. They computed the marginal likelihood by the harmonic mean method of Newton and Raftery (1994): (Z )1 1;n 1;n 1 1;n p y jhj ; M j dP hj jM j ; y p y jM j ¼ hj

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CATHY W. S. CHEN ET AL.

where P(hj|Mj, y1,n) denotes the posterior cumulative distribution function. The authors pointed out that this method can be computationally very demanding, and we note it can again be numerically unstable. There is also the possibility that the standard error does not exist for these estimators. As such, Bayesian comparison of models may not be ideal under this method.

3.2. Reversible-Jump (RJ) MCMC Method We next consider the RJ MCMC method of Green (1995) which was employed to choose between pairs of GARCH models by Vrontos et al. (2000), So et al. (2005), and Chen et al. (2005). This method adds a model indicator to the MCMC sampling scheme and then ‘‘jumps’’ between the (potentially non-nested) models while maintaining the ‘‘detailed balance’’ conditions ensuring convergence of the Markov chain. To apply the RJ MCMC method we must choose prior speciﬁcations, a jumping rule, and proposal distributions. We consider jumps between Models Mi and Mj, requiring a one-to-one bijective transformation between the two models. In general, this bijection can be quite complex, however, we deﬁne ui ¼ hj and uj ¼ hi, thus implying a simple transformation Jacobian of jð@ðhj ; uj ÞÞ=ð@ðhi ; ui ÞÞj ¼ 1; ensuring the necessary condition that the dimensions of (hi, ui) and (hj, uj) are the same; that is, d(hi)+d(ui) ¼ d(uj)+d(hj). Such a Jacobian was employed by Vrontos et al. (2000), though others are possible. The detailed process is as follows: Step 1: Simulate a proposal hj from a proposal density qi (ui); Step 2: The jump to Mj is accepted with the probability min{1, P} where

P¼

pðy1;n jM j ; hj ÞPðhj jM j ÞpðM j ÞJðM i ; M j Þqj ðuj jhj Þ @ðhj ; uj Þ pðy1;n jM i ; hi ÞPðhi jM i ÞpðM i ÞJðM j ; M i Þqi ðui jhi Þ @ðhi ; ui Þ

(3)

If the jump is not accepted, stay in Mi and update the parameters hi. The term p(y1,n7Mj, hj) is the likelihood for Model Mj, P(hj7Mj) is the prior distribution and p(Mj) is the prior probability for each model. We have generally set the prior probability of a jump from Model Mi to Mj to one, that is, J(Mi, Mj) ¼ 1 allowing jumps at each MCMC iteration and p(Mi) ¼ 0.5 to reﬂect prior model ignorance (again other choices are viable);

Bayesian Model Selection for Heteroskedastic Models

575

the RJ acceptance probability for such a move can then be reduced to ! pðy1;n jM j ; hj ÞPðhj jM j Þqj ðuj Þ min 1; pðy1;n jM i ; hi ÞPðhi jM i Þqi ðui Þ with the kernels qi and qj being independent of hi and hj, respectively. The posterior model probability estimate for Mj is the proportion of times the MCMC sample chose Model Mj. This method has enjoyed recent popularity in the literature, but often applied to choose between two models only. There are also many technical issues in its implementation. The most important of these concerns the choice of proposal densities. The method can be quite sensitive to this choice, especially in more complex models such as those considered here, and the complexity and difﬁculty in implementation can increase signiﬁcantly when considering more than two models at one time. As such we will recommend and employ in this paper the ﬁnal method considered below, by Congdon (2006).

3.3. Direct Posterior Model Probabilities The proposal by Congdon (2006, 2007) following Carlin and Chib (1995), Godsill (2001), and Scott (2002), again uses a Monte Carlo, but this time direct, approximation to the posterior probability for each model. There is no increase in complexity for this method as the number of models increases beyond two. We consider K competing models for the same observation matrix yl,n. Separate MCMCQsamples fhð jÞ ¼ ðhð1jÞ ; . . . ; hðKjÞ Þ; j ¼ M þ 1; Ng are taken 1;n from pðhjy1;n Þ ¼ K k¼1 pðhk jy Þ as detailed in the next section. An approximate Monte Carlo estimate of p(Mi|y1,n) is obtained as ^ i jy1;n Þ ¼ pðM

N X pðM i jy1;n ; hðjÞ i Þ NM j¼Mþ1

where pðM i jy1;n ; hði jÞ Þ / pðy1;n jhði jÞ ; M i Þpðhði jÞ jM i ÞpðM i Þ ¼ Gði jÞ

(4)

where hi are the parameters from Model i; yði jÞ the jth MCMC iterate from the posterior distribution of Model i; p(Mi) the prior probability in favor of Model i; p(Yi|Mi) the prior distribution for Model i, and pðy1;n jhðjÞ i ; M i Þ the likelihood function for Model Mi. The normalizing factor in Eq. (4) is the

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CATHY W. S. CHEN ET AL.

sum of the terms GðkjÞ ; k ¼ 1; . . . ; K. We set p(Mk) ¼ 1/K to reﬂect prior model ignorance. For numerical efﬁciency, the calculation in Eq. (4) in practice is based on jÞ Þ scaled versions of the likelihoods. The scaling employs the maximum ðLðmax of the model log-likelihoods at each MCMC iteration and subtracts it from jÞ . each model likelihood, that is, LðkjÞ ¼ pðy1;n jhðkjÞ ; M k Þ becomes LðkjÞ Lðmax Exponentiating the scaled log-likelihoods then gives the new scaled terms GðkjÞ , so that: Gð jÞ PðM i jy1;n ; hð jÞ Þ ¼ PK i ð jÞ k¼1 Gk As noted by Congdon (2006), ‘‘while parallel sampling of the models is not necessary, such a form of sampling facilitates model averaging and assessing the impact of individual observations on the overall estimated Bayes factor.’’ The ﬁrst M MCMC samples are discarded as the burn-in period. In summary, the estimator can be formulated as: Z Z 1;n 1;n pðM i jy Þ ¼ pðM i ; hjy Þdh ¼ PðM i jh; y1;n Þpðhjy1;n Þdh N X 1 PðM i jy1;n ; hði jÞ Þ N M j¼Mþ1

(5)

N X 1 pðy1;n jhði jÞ ; M i Þpðhði jÞ jM i ÞpðM i Þ PK N M j¼Mþ1 l¼1 pðy1;n jhðl jÞ ; M l Þpðhðl jÞ jM l ÞpðM l Þ

(6)

As mentioned, in an interesting manuscript, by Robert and Marin (2008), the major difference between the method of Congdon (2006) and that of Carlin and Chib (1995) is the use of MCMC samples for hi generated from the product pðhi jy1;n ; M i Þ in the former, rather than from the joint posterior pðhjy1;n Þ in the latter. This separate MCMC sampling from each model does not exactly achieve the required sample for the integral above, and makes this method an approximation only. However, by focusing on the posteriors of hi separately and independently, the MCMC iterative scheme of Congdon (2006) can be much simpler and less computationally intensive than either Carlin and Chib’s (1995) method or RJ MCMC. In our experience with univariate and multivariate GARCH model selection, the approximation seems quite good (see below) and Congdon’s method seems to be a powerful model selection tool; see Gerlach and Chen (2008) as an example.

Bayesian Model Selection for Heteroskedastic Models

577

Congdon’s method requires some further assumptions. These are: (i)

Given Model Mi, the parameters between the K models are a priori independent; see Eq. (2) in Congdon (2006) and Carlin and Chib (1995, p. 475). (ii) The posterior density p(hi7y, Mi) is independent of hj where i 6¼ j, that is given the data and Mi, the parameters of Model Mi are independent of the parameters of Model Mj. See also Scott (2002). (iii) The pseudo-prior density p(hi7Mj) is uniform and proportional to 1; see Eq. (3) in Congdon (2006). If the current model is Mj, a priori this tells us nothing about the parameters of Model Mi. Assumption (i) seems reasonable since we do not need to know anything about Model Mj’s parameters to set a prior for Model Mi, where i 6¼ j. Assumption (ii) simply means that when we estimate hi, these are independent of the estimates of hj where i 6¼ j. This is true since we run each chain separately and independently for each model. Assumption (iii) also seems reasonable; we can only set the prior for Model Mj when we know the exact form of that model, not another Model Mk, where k 6¼ j. However, Robert and Marin (2008) have pointed out that assumption (iii) can imply that the joint model posterior (h7y) is improper, unless the parameters are restricted to a compact space. We make some points here. First, our priors are mostly ﬂat over ﬁnite ranges and thus this comment may not apply to the models and priors we consider in this paper. Further, Congdon (2007) points out that these ‘‘cross-model’’ priors are arbitrary and that their diffuseness could be viewed as a limiting argument that could be valid asymptotically; that is, these priors could be made only just proper, while still quite diffuse, and assumption (iii) viewed as a limit which approximates the truth in large samples, where the effect of priors is closer to negligible. Further, numerical experiments in Condgon (2006) showed that this choice does not yield any numerical problems in the implementation for a few examples. In fact, Robert and Marin (2008) demonstrated the close approximation of Condgon’s (2006) estimate to the true posterior probability in a toy example. To further improve the estimation by Eq. (6), we may use the idea of Congdon (2007) to reﬁne the pseudo-prior. In simulation studies using two toy examples, including the one in Robert and Marin (2008), Condgon’s (2007) method gives reliable estimates of the required posterior probability. The Winbugs code of the studies is available at http://webspace.qmul.ac.uk/pcongdon/ in the list of references (under ToyExamples.odc).

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CATHY W. S. CHEN ET AL.

Finally, numerous simulation studies on the quality of this approximation in multivariate GARCH and stochastic volatility model selections, which are not shown here, have been done by the authors (e.g., Gerlach & Chen, 2008). These studies, together with those published previously, lead us to conclude that for large samples Congdon’s (2006) method can give an accurate approximation to the posterior probability p(Mi|y1,n) in Bayesian model selection. The method delivers a powerful and sharp model selection tool in practice, in our experience.

3.4. Markov Chain Monte Carlo Methods All the model selection methods discussed require a sample to be able to be obtained from the joint posterior distribution of each model. In particular Congdon’s method relies on sampling from each model’s posterior separately and independently of the other models. The joint posterior density for each of the models we consider is not of a standard form, and thus we turn to MCMC methods to obtain these samples. We brieﬂy outline the details of the method we used, but refer the reader to Chen and So (2006) for more detail. The posterior distribution is the likelihood function multiplied by the prior distribution. We specify this function for the general model (1). Deﬁne s to be the maximal lag on any lagged term in Eq. (1), or d0, if it is larger than the maximal lag; ys+1,n=(ys+1, . . . , yn)u and Y be the vector of all unknown model parameters. The conditional likelihood function for Eq. (1) is: (

ðnþ1Þ=2 ) g n X Y sþ1;n Gððn þ 1Þ=2Þ 1 ðyt mt Þ2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ 1 þ jY ¼ I jt p y ðn 2Þht t¼sþ1 j¼1 Gðn=2Þ ðn 2Þp ht

where mt ¼ fð0jÞ þ

pj X i¼1

fði jÞ yti þ

qj X lj X

cðkijÞ xk;ti

k¼1 i¼1

and where Ijt is the indicator variable I(rj1rztdorj). MCMC methods for computational inference require groups of parameters to be sampled iteratively, from their conditional posterior distributions. We chose the groups: /j, aj, r, n, d, where the ﬁrst two groups respectively represent the vector of mean equation parameters and the vector of volatility equation parameters. Such blocking of parameters, whose MCMC samples will likely be highly intercorrelated, can lead to better mixing and convergence properties.

Bayesian Model Selection for Heteroskedastic Models

579

Drawing d is made easy by noting that it is discrete and has the posterior probabilities: pðysþ1;n jd; Hd ÞPrðd ¼ jÞ ; p d ¼ jjysþ1;n ; Hd ¼ Pd 0 sþ1;n jd ¼ i; H ÞPrðd ¼ iÞ d i¼1 pðy

j ¼ 1; . . . ; d 0

Our choice of prior distributions are model-speciﬁc and are thus presented in detail in Section 4.2. The remaining groups have conditional posteriors that are nonstandard. We thus incorporate the Metropolis and Metropolis– Hastings (MH) (Metropolis, Rosenbluth, Rosenbluth, & Teller, 1953; Hastings, 1970) methods to draw the MCMC iterates for these groups. To speed convergence and allow optimal mixing, we employ an adaptive MHMCMC algorithm that combines a random walk Metropolis (RW-M) and an independent kernel (IK)–MH algorithm. We employ the adaptive sampling scheme from Chen and So (2006). For the burn-in period, we use a Gaussian proposal RW-M algorithm. The proposal variance–covariance matrix, which might initially be chosen as, for example, a diagonal matrix of sample variances from least squares estimates, is subsequently tuned to achieve optimal acceptance rates, as in Gelman, Roberts, and Gilks (1996). After the burn-in period, a Gaussian proposal IK–MH algorithm is used: the proposal mean and variance– covariance are respectively the sample mean and sample variance– covariance of the burn-in iterates; see Chen and So (2006). Such an adaptive proposal updating procedure requires that the burn-in period has ‘‘covered’’ the posterior distribution. We extensively examine trace plots and autocorrelation function (ACF) plots from multiple runs of the MCMC sampler to conﬁrm convergence and to infer adequate coverage. We set the MCMC sampling size sufﬁciently large, discarding the burn-in iterates.

4. EMPIRICAL STUDY We illustrate the MCMC and model selection methods using the daily Nikkei 225 index, Hang Seng index (HSI), and the Taiwan stock market index (TAIEX), all obtained from Datastream International, from January 4, 1996 to October 17, 2006. We employ the US S&P500 composite index return as an exogenous variable in the mean equation in all models, and as a potential auxiliary variable for the threshold models. All subsequent analyses are performed on the daily log returns, yt=(log ptlog pt1) 100, pt is the price index at time t. To understand the characteristics of the data,

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CATHY W. S. CHEN ET AL.

Table 1.

Summary Statistics: Stock Index Returns for the Period January 4, 1996, to October 17, 2006.

n

Mean

Std

Skewness

Excess Kurtosis

Min.

Max.

Normality Testa

Nikkei 225

2,733

0.0079

1.4170

0.0374

1.9827

7.2339

7.66048

HSI

2,594

0.0144

1.6540

0.1540

11.7140

14.7347

17.2471

TAIEX

2,568

0.0076

1.5968

0.1216

2.5147

9.9360

8.5198

SP500

2,733

0.02899

1.1163

0.1053

3.4621

7.1127

5.5732

448.29 (0.0000) 14841.16 (0.0000) 682.94 (0.0000) 1369.95 (0.0000)

a

Jarque-Bera (1987) normality test.

summary statistics and time series plots are provided in Table 1 and Fig. 1, respectively. All return series exhibit the standard property of asset return data: they have fat-tailed distributions as indicated by the excess kurtosis, with all return series failing the Jarque–Bera test for (the absence of) normality, at the 1% level. 4.1. Models The seven competing heteroskedastic models are described below. Readers are reminded of the following notation: f denotes the AR terms (usually known as mean persistence), c denotes the mean-spillover effects of the exogenous variable x, d denotes the delay lag, r denotes the threshold cut-off parameter, while a, b are the ARCH and GARCH parameters, respectively. Model 1 is the ARX-GARCH-t model. yt ¼ f0 þ f1 yt1 þ c1 xt1 þ at pﬃﬃﬃﬃ at ¼ ht t ; t t n ht ¼ a0 þ a1 a2t1 þ b1 ht1 Model 2 is the ARX-GJR-GARCH model. The mean equation is the same as Model 1. The volatility equation is speciﬁed as: 2 ht ¼ a0 þ ða1 þ gS t1 Þat1 þ b1 ht1

( where

S t1

¼

1 0

if if

at1 0 at1 40

581

Bayesian Model Selection for Heteroskedastic Models Returns of Nikkei 225 5 0 −5 0

500

1000

1500

2000

2500

Returns of Hang Seng Index 15 10 5 0 −5 −10 −15 0

500

1000

1500

2000

2500

2000

2500

Returns of TAIEX 5 0 −5 −10 0

Fig. 1.

500

1000

1500

Daily Returns for the Period January 4, 1996, to October 17, 2006.

Model 3 is the TARX-GARCH model. The volatility equation is the same as Model 1. The mean equation is given as: 8 < fð1Þ þ fð1Þ yt1 þ cð1Þ xt1 þ at ; xtd r 0 1 1 yt ¼ ð2Þ ð2Þ : fð2Þ xtd 4r 0 þ f1 yt1 þ c1 xt1 þ at ; Model 4 is the ARX-TGARCH model. The mean equation is the same as Model 1. The volatility equation follows: 8 < að1Þ þ að1Þ a2 þ bð1Þ ht1 ; ytd r 0 1 t1 1 ht ¼ ð2Þ 2 ð2Þ : að2Þ þ a a þ b ytd 4r 0 1 t1 1 ht1

582

CATHY W. S. CHEN ET AL.

Model 5 is the DT-GARCH-t with an exogenous threshold variable: 8 < fð1Þ þ fð1Þ yt1 þ cð1Þ xt1 þ at ; xtd r 0 1 1 yt ¼ ð2Þ ð2Þ : fð2Þ xtd 4r 0 þ f1 yt1 þ c1 xt1 þ at ; at ¼

ht ¼

pﬃﬃﬃﬃ ht t ;

8 < að1Þ þ að1Þ a2 0 : að2Þ 0

þ

1 t1 2 að2Þ 1 at1

t t v

þ bð1Þ 1 ht1 ;

xtd r

bð2Þ 1 ht1 ;

xtd r

þ

Model 6 is the DT-GARCH-t with domestic threshold variable. The model is the same as Model 5 except the threshold variable is ytd instead of xtd. Model 7 is a DT-GARCH-t with a weighted threshold variable:

yt ¼

8 < fð1Þ þ fð1Þ yt1 þ cð1Þ xt1 þ at ;

ztd r

þ at ;

ztd r

0 : fð2Þ 0

þ

1 fð2Þ 1 yt1

at ¼

ht ¼

pﬃﬃﬃﬃ ht t ;

8 < að1Þ þ að1Þ a2 0 : að2Þ 0

þ

where zt ¼ w1 yt þ w2 xt ;

þ

1 cð2Þ 1 xt1

1 t1 2 að2Þ 1 at1

t tn

þ bð1Þ 1 ht1 ;

ztd r

bð2Þ 1 ht1 ;

ztd 4r

þ

0 wi 1;

w1 þ w2 ¼ 1:

Models 3 and 4 are special cases of Models 5 and 6, respectively. Model 5 is similar to Model 6 expect that the local threshold variable ytd is replaced by the exogenous xtd. Model 7 is from Chen and So (2006) and formulates the threshold variable as a linear combination of auxiliary variables. These are all asymmetric models, with the ﬁnal two being double threshold models, one with local or domestic market return threshold and one whose threshold is a weighted combination of the local and US market returns. Standard restrictions on the variance parameters for Models 1, 4–7 apply: 0oað0jÞ ob; að1jÞ ; bð1jÞ 0; að1jÞ þ bð1jÞ o1

(7)

Bayesian Model Selection for Heteroskedastic Models

583

which ensure each volatility ht is positive and that covariance stationarity is achieved. Note that for Model 1, j ¼ 1 only. Parameter restrictions on the variance parameters for Model 2 (GJR-GARCH) are: 0oaðjÞ 0 ob; a1 ; b1 0; a1 þ g 0; a1 þ b1 þ 0:5go1

(8)

4.2. Prior Distributions Bayesian inference requires a prior distribution for the unknown parameters. Weakly informative priors are adopted since computational model search methods need proper priors that can be evaluated. The general prior setups for Models 1–7 are: [1] The prior for the mean equation parameters is / jBN(/j0, Vj); that is, a Gaussian distribution with mean 0 and variance–covariance Vj, chosen to be a diagonal matrix with diagonal elements 0.1. [2] The prior for aj is a uniform over either Eqs. (7) or (8). [3] For the degrees of freedom n, we re-parameterize by deﬁning t ¼ n1. The prior for t is U(0, 1/4), so that nW4 and the ﬁrst four moments of the distribution of et are ﬁnite. [4] The prior for r is uniform: P(r) ¼ (1/(ul ))I(lorou) where l and u are the 1st and 3rd quantiles of the threshold variable. [5] For the delay term, d follows a discrete prior distribution on {1, 2, . . . , d0}: Pr(d ¼ i) ¼ (4i)/6, i ¼ 1,2,3, where d0 is 3. [6] The prior for the weight, w1 of Model 7, is a uniform distribution p I(B), where B is the set satisfying 0ow1o1 and w1+w2 ¼ 1 and I is an indicator. We set the value b ¼ 0.5s2 in Eq. (7), where s2 is the sample variance, as in Chen et al. (2006). The choice Vj with diagonal element set to 0.1 means the standard deviation of each parameter in /j is approximately 0.32 which fully covers the stationarity range of (1, 1).

4.3. Estimation Results We employed MCMC trace plots (or history plots) and ACF plots of MCMC iterates to check convergence for each model, which seemed to be satisﬁed in each case after a few hundred iterations. All delay lags were

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CATHY W. S. CHEN ET AL.

chosen as 1 with high probability (close to 1). All other parameter estimates for Models 1–7 are given in Tables 2–4 for the Nikkei 225, HSI and TAIEX, respectively. These models and estimates combine to describe the features of the mean and volatility of these data sets. In particular, Models 1–7 all agree that the Table 2. Estimates of the Parameters for the Nikkei 225 for Each Model; Standard Errors are in Parentheses.

fð1Þ 0 fð1Þ 0 cð1Þ 1 að1Þ 0 að1Þ 1 bð1Þ 1

M1

M2

M3

M4

M5

M6

M7

GARCH

GJR

TAR-GARCH

TGARCH

DTX

DTD

DTW

0.0132 (0.0213) 0.0665 (0.0181) 0.4269 (0.0223) 0.0228 (0.0079) 0.0731 (0.0107) 0.9167 (0.0121)

0.0032 (0.0215) 0.0628 (0.0180) 0.4249 (0.0215) 0.0308 (0.0091) 0.0378 (0.0113) 0.9061 (0.0125) 0.0823 (0.0218)

0.1247 (0.0482) 0.0795 (0.0347) 0.4443 (0.0407) 0.0222 (0.0077) 0.0743 (0.0108) 0.9160 (0.0119)

0.0109 (0.0217) 0.0638 (0.0191) 0.4270 (0.0221) 0.1238 (0.0313) 0.0827 (0.0207) 0.9013 (0.0251)

0.1267 (0.0495) 0.0739 (0.0351) 0.4559 (0.0434) 0.1141 (0.0410) 0.0550 (0.0127) 0.9155 (0.0228)

0.0689 (0.1012) 0.1095 (0.0598) 0.4156 (0.0404) 0.1881 (0.0475) 0.0892 (0.0200) 0.8840 (0.0301)

0.3200 (0.1381) 0.2060 (0.0636) 0.3798 (0.0620) 0.2445 (0.0517) 0.0662 (0.0176) 0.9083 (0.0264)

0.0092 (0.0074) 0.0668 (0.0124) 0.9061 (0.0140) 0.4031 (0.0697) 10.6521 (2.2215)

0.0608 (0.0276) 0.1210 (0.0211) 0.4159 (0.0281) 0.0123 (0.0083) 0.0047 (0.0042) 0.9773 (0.0089) 0.4510 (0.0488) 6.9987 (0.9880)

0.0141 (0.0291) 0.0495 (0.0295) 0.4340 (0.0270) 0.0106 (0.0080) 0.0473 (0.0114) 0.9035 (0.0137) 0.5656 (0.1169) 10.6472 (2.2211)

0.0299 (0.0288) 0.0764 (0.0262) 0.3974 (0.0325) 0.0189 (0.0087) 0.0459 (0.0109) 0.9032 (0.0132) 0.5813 (0.0546) 11.6968 (2.7389) 0.5703 (0.0789) 1

g fð2Þ 0

0.0673 (0.0262) 0.0900 (0.0244) 0.4261 (0.0264)

fð2Þ 1 cð2Þ 1 að2Þ 0 að2Þ 1 bð2Þ 1 r n

9.6842 (1.7029)

10.0794 (1.9199)

0.4071 (0.1219) 9.5077 (1.6671)

w1 d

1

1

2

1

585

Bayesian Model Selection for Heteroskedastic Models

Table 3. Estimates of the Parameters for the Hang Seng Index for Each Model; Standard Errors are in Parentheses.

fð1Þ 0 fð1Þ 1 cð1Þ 1 að1Þ 0 að1Þ 1 bð1Þ 1

M1

M2

M3

M4

M5

M6

M7

GARCH

GJR

TAR-GARCH

TGARCH

DTX

DTD

DTW

0.0372 (0.0200) 0.0154 (0.0174) 0.5222 (0.0229) 0.0120 (0.0045) 0.0654 (0.0108) 0.9296 (0.0115)

0.0284 (0.0208) 0.0094 (0.0177) 0.5165 (0.0228) 0.0160 (0.0052) 0.0418 (0.0129) 0.9240 (0.0119) 0.0521 (0.0197)

0.0527 (0.0426) 0.0842 (0.0339) 0.5101 (0.0367) 0.0117 (0.0042) 0.0649 (0.0107) 0.9302 (0.0110)

0.0377 (0.0198) 0.0122 (0.0178) 0.5263 (0.0226) 0.0902 (0.0291) 0.1004 (0.0217) 0.8918 (0.0244)

0.0646 (0.0384) 0.0933 (0.0305) 0.5431 (0.0379) 0.0085 (0.0066) 0.0556 (0.0087) 0.9418 (0.0091)

0.0226 (0.0543) 0.0816 (0.0397) 0.5562 (0.0453) 0.0688 (0.0299) 0.1023 (0.0214) 0.8899 (0.0234)

0.0530 (0.1048) 0.1413 (0.0500) 0.5573 (0.0553) 0.1029 (0.0344) 0.1073 (0.0268) 0.8825 (0.0292)

0.0708 (0.0251) 0.0157 (0.0235) 0.5347 (0.0308)

0.0061 (0.0049) 0.0568 (0.0136) 0.9179 (0.0146)

0.3391 (0.1218) 7.8342 (1.2272)

0.4079 (0.1153) 8.6296 (1.4472)

0.0568 (0.0249) 0.0018 (0.0202) 0.5399 (0.0307) 0.0014 (0.0011) 0.0008 (0.0008) 0.9979 (0.0014) 0.3924 (0.0205) 4.4967 (0.3078)

0.0268 (0.0263) 0.0186 (0.0238) 0.5055 (0.0323) 0.0073 (0.0053) 0.0404 (0.0123) 0.9395 (0.0145) 0.4597 (0.2291) 8.6717 (1.4445)

0.0371 (0.0224) 0.0179 (0.0207) 0.5118 (0.0282) 0.0113 (0.0058) 0.0443 (0.0121) 0.9343 (0.0140) 0.7042 (0.1042) 8.5399 (1.4488) 0.7072 (0.1680) 2

g fð2Þ 0 fð2Þ 1 cð2Þ 1 að2Þ 0 að2Þ 1 bð2Þ 1 r n

8.1400 (1.2613)

8.4372 (1.4068)

w1 d

2

1

3

2

mean persistence (f1) in the Nikkei is small, but signiﬁcant, and negative. This parameter is again always negative, but only signiﬁcant for Models 6–7, for the HSI, while it is never signiﬁcant for the TAIEX, but it is largest in magnitude for Models 6–7 again. There does not seem to be a strong

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Table 4.

fð1Þ 0 fð1Þ 1 cð1Þ 1 að1Þ 0 að1Þ 1 bð1Þ 1

Estimates of the Parameters for the TAIEX for Each Model; Standard Errors are in Parentheses.

Ml

M2

M3

M4

M5

M6

M7

GARCH

GJR

TAR-GARCH

TGARCH

DTX

DTD

DTW

0.0306 (0.0243) 0.0138 (0.0191) 0.3391 (0.0257) 0.0197 (0.0078) 0.0621 (0.0109) 0.9318 (0.0122)

0.0187 (0.0242) 0.0180 (0.0185) 0.3324 (0.0251) 0.0281 (0.0115) 0.0368 (0.0113) 0.9236 (0.0159) 0.0591 (0.0173)

0.0786 (0.0582) 0.0181 (0.0329) 0.3492 (0.0429) 0.0203 (0.0081) 0.0638 (0.0110) 0.9299 (0.0121)

0.0343 (0.0235) 0.0144 (0.0193) 0.3390 (0.0258) 0.1488 (0.0497) 0.0592 (0.0249) 0.9277 (0.0274)

0.1207 (0.0605) 0.0022 (0.0382) 0.3580 (0.0547) 0.2745 (0.0685) 0.0692 (0.0274) 0.9138 (0.0340)

0.1677 (0.1256) 0.0843 (0.0579) 0.3113 (0.0506) 0.2848 (0.0674) 0.0578 (0.0184) 0.9314 (0.0220)

0.2064 (0.1633) 0.0992 (0.0683) 0.3073 (0.0619) 0.3171 (0.0790) 0.0555 (0.0190) 0.9323 (0.0227)

0.0112 (0.0109) 0.0678 (0.0159) 0.9079 (0.0197) 0.4504 (0.0875) 6.7883 (0.9133)

0.0714 (0.0313) 0.0232 (0.0228) 0.3007 (0.0327) 0.0096 (0.0088) 0.0357 (0.0159) 0.9478 (0.0112) 0.4688 (0.0790) 5.0355 (0.4863)

0.0258 (0.0305) 0.0234 (0.0291) 0.3494 (0.0300) 0.0271 (0.0137) 0.0480 (0.0131) 0.8990 (0.0197) 0.7802 (0.0377) 7.2607 (1.0596)

0.0274 (0.0283) 0.0265 (0.0267) 0.3453 (0.0307) 0.0345 (0.0167) 0.0497 (0.0132) 0.8959 (0.0217) 0.8178 (0.0706) 7.3805 (1.1341) 0.9161 (0.0624) 1

g fð2Þ 0

0.0933 (0.0360) 0.0094 (0.0277) 0.3401 (0.0404)

fð2Þ 1 cð2Þ 1 að2Þ 0 að2Þ 1 bð2Þ 1 r n

6.5559 (0.8281)

6.9290 (0.9639)

0.2201 (0.2727) 6.6342 (0.8411)

w1 d

1

2

2

1

asymmetric effect for this parameter in these markets for Models 3, 5–7. These mostly negative estimates and symmetric effects for f1 are in agreement with results in Li and Lam (1995) and Chen and So (2006). Further, the mean-spillover effect (c1) from the US market return is strong,

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positive, and highly signiﬁcant, in all regimes, models, and markets, but again seems not to be signiﬁcantly asymmetric across models. The volatility persistence is strong and positive (i.e., a1+b1E1) in all regimes across all models and markets, agreeing with the well-known empirical fact, but again quite symmetric in effect. Asymmetric effects are however, strong and clear in all nonlinear models in each market. For the Nikkei 225 and TAIEX indices, models with mean asymmetry (Models 3, 5–7) clearly indicate that the mean return is lower and negative following bad news from the US or the domestic market (or combined as in Model 7), with higher and mostly positive average ð2Þ ð1Þ ð2Þ ð1Þ ð2Þ returns following good news; that is, fð1Þ 0 of0 ; f1 f1 ; c1 c1 in Models 3 and 5–7, in these two markets. However, this effect is not as apparent in Hong Kong. The threshold cut-offs (r) for Models 3–7 were all estimated between 0.40 and 0.58 for Nikkei; 0.34 and 0.70 for Hang Seng; and 0.22 to 0.82 for Taiwan. These are almost all signiﬁcantly lower than the usual value of 0 employed in similar studies: that is, bad news is not ‘‘ﬁnancially signiﬁcant’’ until a threshold somewhat below zero and closer to 0.5% (on average). Asymmetric volatility was also very strong and clear across markets. In particular volatility levels are signiﬁcantly higher following bad news, as ð2Þ ð1Þ ð2Þ ð1Þ ð2Þ highlighted by g W 0 in Model 2 and að1Þ 0 4a1 ; a1 a1 ; b1 b1 in Models 4–7. It seems the asymmetric effects are clear and apparent in the mean and variance intercept terms, but perhaps not present at all in the other parameters, except possibly in the ARCH terms a1 in the Hang Seng market. This is highlighted in Figs. 2 and 3, showing boxplots of the MCMC iterates for the mean parameters (Fig. 2) and the volatility parameters (Fig. 3) from Model 7. From left to right, the ﬁrst three boxplots are the parameters in regime 1, while the next three are from regime 2. It is quite apparent that the only clearly signiﬁcant differences between regimes are in the intercept parameters f0 and a0, except for differences in a1 for the Hang Seng market. For Model 7, the weight chosen for the US return in the weighted threshold variable is 0.57 (Nikkei) and 0.70 (Hang Seng), but these are not signiﬁcantly different from 0.5. The weight for the TAIEX was estimated to be 0.92, signiﬁcantly higher than 0.5. We should not interpret these weights too precisely, since the two threshold variables have slightly different variances (although both on the same scale of percentage returns, it is debatable as to whether these returns should be standardized in the threshold variable: we chose not to), however it seems each market has ‘‘close’’ to an equal effect on this threshold variable that drives asymmetric

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Fig. 2.

ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ Boxplots for Parameter Estimates of ðfð1Þ 0 ; f1 ; c1 ; f0 ; f1 ; c1 Þ in Model 7 for the Nikkei Market.

Fig. 3.

ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ Boxplots for Parameter Estimates of ðað1Þ 0 ; a1 ; b0 ; a0 ; a1 ; b0 Þ in Model 7 for the Nikkei Market.

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nonlinear behavior in Japan and Hong Kong, but that the local TAIEX has a far stronger inﬂuence than the US in driving asymmetric behavior in Taiwan. Finally, all degrees of freedom parameters are estimated between 6 and 12 across markets and models, far below what would be expected from normality and justifying the use of t-distributed residuals.

4.4. Model Selection Results We apply the method of Congdon (2006) to estimate approximate posterior model probabilities for all seven models considered in each market. These are shown in Table 5 and Fig. 4. For exposition, two additional scenarios are shown for Japan, where only the ﬁrst ﬁve and then the ﬁrst six models are considered. The same MCMC iterates are used in all three scenarios. We ﬁrst summarize the extended analysis of the Nikkei market. For the ﬁrst scenario, shown in the ﬁrst row of Table 5, Models 1–5 are compared via their estimated posterior probabilities. In contrast to results in Chen et al. (2006), the GJR-GARCH (GJR) model, with local shock threshold, zero threshold value, and a linear mean equation including the exogenous US return effect, is preferred to the double threshold DTXGARCH (DTX) model with exogenous threshold and potentially nonzero threshold value. We note that in Chen et al. (2006) there was no exogenous mean equation term for the GJR model. Its inclusion in this study may Table 5.

Posterior Probabilities for Each Model Across the Three Markets.

Note: The Probabilities in bold show the best model when only 5 and 6 models, instead of the full set of 7, were ﬁt to the dataset.

590

Fig. 4.

CATHY W. S. CHEN ET AL.

The Estimated Posterior Model Probabilities for the Nikkei Market, Over Three Scenarios.

partly explain our result, but also time-extended data series have been used here. The symmetric GARCH is favored next, followed closely by the TARX-GARCH, with little weight given to the ARX-TGARCH (TGARCH) model. This suggests that there is a strong asymmetric ARCH effect, driven by local market shocks, and a strong symmetric exogenous US mean-spillover effect, as described in the GJR model. Next we add the DT-GARCH model with local threshold variable (DTD). While the GJR-GARCH is still favored, in 2nd place, the DTGARCH is clearly the superior model among these six candidates. Little weight is given to the other four models. Again we see a strong nonlinear local market effect, but this time the (only) model allowing for both asymmetry in mean and volatility, in response to local market returns, is clearly favored. Again, we note that a US exogenous mean effect is included in the DT-GARCH model with local threshold, potentially explaining our differing result from that in Chen et al. (2006). Finally, we add Model 7, allowing a weighted combination of US and local returns as the threshold. While some weight is still given to the GJR model, the battle is now clearly between the DT-GARCH with local threshold (DTD) and the DT-GARCH with weighted threshold (DTW). The DTW wins out with posterior probability 0.40, compared to 0.29 for Model 6. The data clearly favor nonlinearity in mean and volatility being driven by the weighted threshold variable. This makes sense from a ﬁnancial

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viewpoint, since this threshold uses all the news information available, that is, local and US returns, from the previous trading day. For the HSI and TAIEX, we ﬁnd supporting evidence that again DTD and DTW are the preferred models. The marginally best model for HSI is DTD, indicating that the domestic return threshold variable is slightly more appropriate than a weighted average threshold variable. DTW and GJR are equally supported in this market. Although we ﬁnd very marginal superior performance of DTW to DTD for TAIEX, the difference is small. This is not surprising as o1 is so close to one in this market that these models are almost the same, for practical application. An alternate way to compare between models is to use a criterion based on a tradeoff between the ﬁt of the data to the model and the corresponding complexity of the model. Spiegelhalter, Best, Carlin, and Van der Linde (2002) proposed a Bayesian model comparison criterion based on Deviance Information Criterion (DIC), such that DIC ¼ goodness of ﬁt+complexity. Fit is measured via the deviance D(y) ¼ 2logL(data7y), while complexity is measured by the estimate of the effective number of parameters: PD ¼ E yjy ½D DðE yjy ½yÞ ¼ D DðyÞ This is the posterior mean deviance minus the deviance evaluated at the posterior mean of the parameters. The DIC is then deﬁned analogosly to the AIC as þ 2p DIC ¼ DðyÞ D ¼ D þ pD Models with smaller DIC are better supported by the data. To support our results via Congdon’s (2006) method, we also present the DIC results for these datasets and models in Table 5. The results are nicely in agreement in terms of the model selected, with similar rankings across the models in each market.

5. CONCLUSION We brieﬂy review three computational methods for estimating marginal likelihoods in this paper and highlight the advantages of Congdon (2006)’s approximate approach, for heteroskedastic models. The proposed approach

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is a special case of composite space MCMC sampling where jumps between models are not involved and so only evidence on the model ratios need to be accumulated. The construction of proposal densities for RJM-CMC are also avoided in this approach. Seven asymmetric volatility models, which differ in the mean structure, in the volatility structure or in the construction of the threshold variable, are evaluated. Our results show that the weighted average threshold formulation introduced in Chen and So (2006) is preferred for the Nikkei 225 and the TAIEX market data with a local threshold and double threshold model also scoring highly. This latter model is preferred for the HSI. Results using DIC agree with our conclusions. These results can initiate new discussion on how to combine information from different sources in determining regimes for asymmetric heteroskedastic models. It is easy to apply the proposed method to compare complex models and to the comparison of non-nested models. All the methods discussed here can also be applied to other volatility models, like Markov switching GARCH models and stochastic volatility models, retaining the advantages discussed. Choice of auxiliary variables in Chen and So (2006)’s speciﬁcation and the determination of the number of regimes (beyond two) in threshold processes are important topics for further research.

ACKNOWLEDGMENTS The authors thank the Editors and referee for their helpful suggestions and comments. Cathy Chen is supported by the grant: 95-2118-M-035-001 from the National Science Council (NSC) of Taiwan and grant 06G27022 from Feng Chia University.

REFERENCES Bauwens, L., & Lubrano, M. (1998). Bayesian inference on GARCH models using Gibbs sampler. Econometrics Journal, 1, 23–46. Berg, A. k, Meyer, R., & Yu, J. (2004). Deviance information criterion for comparing stochastic volatility models. Journal of Business and Economic Statistics, 22, 107–120. Bollerslev, T. (1986). Generalized autoregressive heteroskedasticity. Journal of Econometrics, 31, 307–327. Brooks, C. (2001). A double-threshold GARCH model for the French Franc/Deutschmark exchange rate. Journal of Forecasting, 20, 135–143.

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Carlin, B. P., & Chib, S. (1995). Bayesian model choice via Markov chain Monte Carlo. Journal of the Royal Statistical Society, Series B, 57, 473–484. Chen, C. W. S., Chiang, T. C., & So, M. K. P. (2003). Asymmetrical reaction to US stockreturn news: Evidence from major stock markets based on a double-threshold model. The Journal of Economics and Business, 55, 487–502. Special issue on Globalization in the New Millennium: Evidence on Financial and Economic Integration. Chen, C. W. S., Gerlach, R., & So, M. K. P. (2006). Comparison of non-nested asymmetric heteroscedastic models. Computational Statistics and Data Analysis, 51, 2164–2178. Special issue on Nonlinear Modelling & Financial Econometrics. Chen, C. W. S., & So, M. K. P. (2006). On a threshold heteroscedastic model. International Journal of Forecasting, 22, 73–89. Chen, C. W. S., So, M. K. P., & Gerlach, R. H. (2005). Assessing and testing for threshold nonlinearity in stock returns. Australian New Zealand Journal of Statistics, 47, 473–488. Chib, S. (1995). Marginal likelihood from the Gibbs output. Journal of the American Statistical Association, 90, 1313–1321. Congdon, P. (2006). Bayesian model choice based on Monte Carlo estimates of posterior model probabilities. Computational Statistics and Data Analysis, 50, 346–357. Congdon, P. (2007). Model weights for model choice and averaging. Statistical Methodology, 4, 143–157. Engle, R. F. (1982). Autoregressive, conditional heteroskedasticity with estimates of the variance of United Kingdom inﬂation. Econometrica, 50, 987–1007. Gelman, A., Roberts, G., & Gilks, W. (1996). Efﬁcient metropolis jumping rules. In: J. M. Bernardo, J. O. Berger, A. P. Dawid & A. F. M. Smith (Eds), Bayesian statistics 5. London: Oxford University Press. Gerlach, R., Carter, C. K., & Kohn, R. (1999). Diagnostics for time series analysis. Journal of Time Series Analysis, 20, 309–330. Gerlach, R., & Chen, C. W. S. (2008). Bayesian inference and model comparison for asymmetric smooth transition heteroskedastic models, Statistics and Computing (to appear in the Special Issue on Adaptive Monte Carlo Methods). Forthcoming, DOI:10.1007/s11222088-9063-1. Gerlach, R., Chen, C. W. S., Lin, S. Y., & Huang, M. H. (2006). Asymmetric responses of international markets to trading volume. Physica A: Statistical Mechanics and its Applications, 360, 422–444. Gerlach, R., & Tuyl, F. (2006). MCMC methods for comparing stochastic volatility and GARCH models. International Journal of Forecasting, 22, 91–107. Geweke J. (1995). Bayesian comparison of econometric models. Working Paper 532. Research Department, Federal Reserve Bank of Minneapolis. Glosten, L., Jagannathan, R., & Runke, D. (1993). On relationship between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance, 487, 1779–1801. Godsill, S. J. (2001). On the relationship between Markov chain Monte Carlo methods for model uncertainty. Journal of Computational and Graphical Statistics, 10, 1–19. Green, P. J. (1995). Reversible jump MCMC computation and Bayesian model determination. Biometrika, 82, 711–732. Hastings, W. K. (1970). Monte-Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109.

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Kass, R. E., & Raftery, A. E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795. Li, W. K., & Lam, K. (1995). Modelling asymmetry in stock returns by a threshold ARCH model. The Statistician, 44, 333–341. Li, C. W., & Li, W. K. (1996). On a double-threshold autoregressive heteroscedastic time series model. Journal of Applied Econometrics, 11, 253–274. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., & Teller, E. (1953). Equations of state calculations by fast computing machines. Journal of Chemical Physics, 21, 1087–1091. Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 59, 347–370. Newton, M. A., & Raftery, A. E. (1994). Approximate Bayesian inference by the weighted likelihood bootstrap (with discussion). Journal of the Royal Statistical Society, Series B, 56, 3–48. Osiewalski, J., Pajor, A., & Pipien, M. (2007). Bayesian comparison of bivariate GARCH, SV and hybrid models, MACROMODELS 2006. Proceedings of the 33rd international conference (pp. 247–277). Poon, S. H., & Granger, C. W. J. (2003). Forecasting volatility in ﬁnancial markets: A review. Journal of Economic Literature, 41, 478–539. Robert, C., & Marin, J.-M. (2008). On some difﬁculties with a posterior probability approximation technique. Unpublished INRIA working paper; available at http:// hal.inria.fr/inria-00260507/en Scott, S. (2002). Bayesian methods for hidden Markov models: Recursive computing in the 21st Century. Journal of the American Statistical Association, 97, 337–351. Sentana, E. (1995). Quadratic ARCH models. Review of Economic Studies, 62, 639–661. Silvapulle, M. J., & Sen, P. K. (2004). Constrained statistical inference: Inequality, order, and shape restrictions. New York: Wiley. So, M. K. P., Chen, C. W. S., & Chen, M. T. (2005). A Bayesian threshold nonlinearity test in ﬁnancial time series. Journal of Forecasting, 24, 61–75. So, M. K. P., Chen, C. W. S., & Liu, F. C. (2006). Best subset selection of autoregressive models with exogenous variables and generalized autoregressive conditional heteroscedasticity errors. Journal of the Royal Statistical Society, Series C, 55, 201–224. Spiegelhalter, D., Best, N. G., Carlin, B. P., & Van der Linde, A. (2002). Bayesian measures of model complexity and ﬁt (with Discussion). Journal of the Royal Statistical Society B, 64, 583–616. Tong, H. (1978). On a threshold model. In: C. H. Chen (Ed.), Pattern recognition and signal processing. Amsterdam: Sijhoff & Noordhoff. Vrontos, I. D., Dellaportas, P., & Politis, D. N. (2000). Full Bayesian inference for GARCH and EGARCH models. Journal of Business Economic Statistics, 18, 187–198. Zakoian, J. M. (1994). Threshold heteroskedastic models. Journal of Economic Dynamics and Control, 18, 931–955.

BAYESIAN STUDENT-t STOCHASTIC VOLATILITY MODELS VIA SCALE MIXTURES S. T. Boris Choy, Wai-yin Wan and Chun-man Chan ABSTRACT The normal error distribution for the observations and log-volatilities in a stochastic volatility (SV) model is replaced by the Student-t distribution for robustness consideration. The model is then called the t-t SV model throughout this paper. The objectives of the paper are twofold. First, we introduce the scale mixtures of uniform (SMU) and the scale mixtures of normal (SMN) representations to the Student-t density and show that the setup of a Gibbs sampler for the t-t SV model can be simpliﬁed. For example, the full conditional distribution of the log-volatilities has a truncated normal distribution that enables an efﬁcient Gibbs sampling algorithm. These representations also provide a means for outlier diagnostics. Second, we consider the so-called t SV model with leverage where the observations and log-volatilities follow a bivariate t distribution. Returns on exchange rates of Australian dollar to 10 major currencies are ﬁtted by the t-t SV model and the t SV model with leverage, respectively.

Bayesian Econometrics Advances in Econometrics, Volume 23, 595–618 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23019-7

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1. INTRODUCTION The volatility of an asset or the volatility of the return of an asset plays an important role in managing risks in ﬁnancial world. To measure the volatility, different time series models have been proposed and studied. Engle (1982) introduced the autoregressive conditional heteroskedasticity (ARCH) models. These models are superior to many conventional autoregressive-type and moving average-type time series models because they allow the conditional variances of the time series data to be a deterministic function of the past observations. Thereafter, a number of variations of ARCH models, such as the Generalized ARCH (GARCH), Exponential GARCH (EGARCH), Integrated GARCH (IGARCH), Fractionally Integrated GARCH (FIGARCH), Factor ARCH (FACTORARCH), Threshold GARCH (TGARCH), were proposed to deal with a variety of data, such as interest rates, exchange rates, equity returns, Treasury Bills, option pricing. See Engle and Bollerslev (1986), Bollerslev (1987), Tong (1990), Engle, Ng, and Rothschild (1990), Nelson (1991), and Baillie, Bollerslev, and Mikkelson (1996) for details. In addition, readers are referred to Eagle (1995) for a comprehensive review of the ARCH models. An alternative class of model for volatility is the stochastic volatility (SV) model. Unlike the ARCH-type models, the SV models formulate the volatility by an unobservable process that allows the volatilities to vary stochastically. See Shephard (2005) for a comprehensive review of the SV models. Harvey (1989) expressed the SV model in the form of a state-space model and a simple normal-normal (N-N) SV model is deﬁned as Observation equation: State equation:

yt jht ¼ eht =2 t

t ¼ 1; 2; . . . ; n

ht jht1 ; m; f; t2 ¼ m þ fðht1 mÞ þ tZt t ht jm; f; t2 ¼ m þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Zt t ¼ 1 1 f2

t ¼ 2; 3; . . . ; n

where yt is the asset value or the return of an asset at time t, ht the unobserved log-volatility of yt, f 2 ð1; 1Þ the persistence in the volatility, and t the standard deviation of the log-volatility. The modal instantaneous volatility is a scaling factor given by b ¼ expðm=2Þ. In this paper, yt denotes the mean-corrected log-return of an asset. Conditional independence is

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assumed in the above equations. The innovations, t Nð0; 1Þ and Zt Nð0; 1Þ, are also assumed to be independent. This simple N-N SV model can be modiﬁed in many ways. Meyer and Yu (2000), Chib, Nardari, and Shephard (2002), and Omori, Chib, Shephard, and Nakajima (2007) studied the SV model with Student-t observation errors. Chib et al. (2002) also extended the SV model to include covariates. Meyer and Yu (2000) and Yu (2005) allowed et and Zt to be correlated and included a leverage effect in the SV model. Jacquier, Polson, and Rossi (2004) and Omori et al. (2007) considered a bivariate Student-t distribution in the SV model with leverage. Harvey, Ruiz, and Shephard (1994) and Shephard (1996) introduced two independent AR(1) processes in the observation equation. Chib et al. (2002) and Berg, Meyer, and Yu (2004) added a jump component and/or lagged observations to the observation equation. The extension from the univariate to multivariate observations was considered in Harvey et al. (1994), Asai, McAleer, and Yu (2006), Yu and Meyer (2006), and Chib, Nardari, and Shephard (2006). Parameter estimation and forecasting for the SV-type models are very difﬁcult, and the Bayesian approach via simulation-based Markov chain Monte Carlo (MCMC) algorithms has been widely used in SV modeling since early 1990s. For example, Jacquier, Polson, and Rossi (1994) adopted the Gibbs sampling scheme and Shephard and Pitt (1997) employed the Metropolis-Hastings scheme. Chib and his coauthors reformulated the SV models and approximated the log -w21 distribution with ﬁnite mixtures of normal distributions (Kim, Shephard, & Chib, 1998; Chib et al., 2002; Omori et al., 2007), while Yu and his coauthor implemented the SV models using the WinBUGS package (Meyer & Yu, 2000; Yu, 2005). This paper provides a full Bayesian analysis for SV models with and without leverage where the innovations follow Student-t distributions. The structure of the paper is as follows. In Section 2, we introduce the scale mixtures of normal (SMN) and the scale mixtures of uniform (SMU) representations for the Student-t density function. Section 3 presents a t-t SV model and the full conditional distributions of the model parameters for the Gibbs sampler are derived. The t SV model with leverage is considered in Section 4. The bivariate Student-t distribution for the observations and log-volatilities is expressed into an SMN distribution. Section 5 presents the results of ﬁtting the returns on the exchange rates of Australian dollar (AUD) to 10 major currencies using the t-t SV model and t SV model with leverage. Finally, concluding remarks are given in Section 6.

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2. SCALE MIXTURES DENSITY REPRESENTATIONS FOR THE STUDENT-t DISTRIBUTION The class of univariate SMN distribution was characterized by Andrews and Mallows (1974) using the Laplace transformation approach. Suppose that X is an SMN random variable with zero mean and unit variance. It can be shown that X ¼ Z l, where Z is a standard normal random variable and l is a positive random variable having a density function p. l is referred to as an SMN mixing parameter and p is the mixing distribution. The Student-t, symmetric stable and exponential-power distributions are well-known members of the SMN family. West (1987) and Choy and Smith (1997a) have explored robust inference using these distributions. Let m and s be the location and scale parameters of the SMN random variable X. Then the probability density function (p.d.f.) of X has the following mixture form: Z Nðxjm; ls2 ÞpðlÞdl (1) f ðxjm; sÞ ¼ <þ

where NðjÞ denotes the normal density and pðÞ a p.d.f. deﬁned on the <þ ¼ ð0; 1Þ. In Bayesian framework, Eq. (1) is equivalent to a two-stage hierarchy of the form Xjm; s; l Nðm; ls2 Þ l pðlÞ For a suitably chosen p, a wide class of symmetric and unimodal SMN distribution can be obtained. In particular, the Student-t distribution with n degrees of freedom corresponds to an inverse gamma mixing distribution, that is, n n l IG ; 2 2 where IG(a,b) is the inverse gamma distribution with density ba 1 aþ1 b=l e l40; a; b40 f ðlÞ ¼ GðaÞ l Meyer and Yu (2000) and Chib et al. (2002) considered the Student-t error distribution for the SV models via this SMN representation and proposed MCMC algorithms for implementation. However, in these papers, most of

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the full conditional distributions for the Gibbs sampling do not have a standard form. To facilitate a simpler Gibbs sampler setup and an efﬁcient computation for the SV models, we make use of the SMU representation for the normal density in this paper. Suppose that X is a normal random variable with mean m and variance s2, it can be easily shown that the normal p.d.f. can be written as Z 1 3 1 1 pﬃﬃﬃ Ga u ; du Nðxjm; s2 Þ ¼ 2 2 z2 2s u where z ¼ ðx mÞ=s and Ga(u|a,b) is the gamma density function with density f ðuÞ ¼

ba a1 bu u e GðaÞ

u40; a; b40

See Walker and Gutie´rrez-Pen˜a (1999) and Choy and Walker (2003) for details about the class of SMU distribution. Theoretically, an SMN distribution can be expressed in terms of an SMU distribution. Therefore, we can rewrite the Student-t distribution with a degrees of freedom into the following hierarchy: pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ Xjm; s; l; u U m s lu; m þ s lu (2) l IG

a a ; 2 2

u Ga

3 1 ; 2 2

where U(a,b) is a uniform distribution deﬁned on the interval (a,b) and aob.

3. BAYESIAN t-t SV MODELS Let rt be the value of an asset at time t ¼ 0,1, . . . , n. Deﬁne the meancorrected log-return as n P rt ri 1n ln ri1 yt ¼ ln ; t ¼ 1; 2; :::; n rt1 i¼1 In this section, we assume that the mean-corrected log-return has a Student-t distribution with n1 degrees of freedom. In addition, the log-volatility, ht, is

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assumed to have another Student-t distribution with n2 degrees of freedom. To facilitate a simple and efﬁcient Gibbs sampler, the Student-t distribution of yt is expressed as an SMU distribution while the Student-t distribution of ht is expressed as an SMN distribution. Therefore, the setup of the proposed t-t SV model is pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ Observation equations: yt jhu ; lt ; ut U lt ut eht =2 ; lt ut eht =2 n n 1 1 ; lt IG 2 2 3 1 ut Ga ; 2 2 8 > Nðm þ fðht1 mÞ; ot t2 Þ t ¼ 2; 3; :::; n > < State equations: ht jht1 ; m; f; t2 ; ot o 1 t2 > t¼1 N m; > : 1 f2 n n 2 2 ot IG ; 2 2 where ot ; t ¼ 1; 2; :::; n, are the mixing parameters of the SMN form of the Student-t distribution in the state equations. To complete a Bayesian framework, we assign the following independent priors to m, f, t2,n1 and n2: m Nðm0 ; s2 Þ fþ1 Beðaf ; bf Þ c¼ 2 t2 IGðat ; bt Þ n1 Gaða1 ; b1 Þ n2 Gaða2 ; b2 Þ where Be(a, b) is the beta distribution. Expressing the Student-t distribution into the SMN and SMU forms, the proposed Bayesian t-t SV model can be rewritten hierarchically as pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ yt jht ; lt ; ut U lt ut eht =2 ; lt ut eht =2 n n 1 1 ; lt IG 2 2 3 1 ut Ga ; 2 2

Bayesian Student-t Stochastic Volatility Models via Scale Mixtures

601

8 > Nðm þ fðht1 mÞ; ot t2 Þ t ¼ 2; 3; :::; n > < ht jht1 ; m; f; t2 ; ot o1 t2 > t¼1 N m; > : 1 f2 n n 2 2 ot IG ; 2 2 m Nðm0 ; s2 Þ c¼

fþ1 Beðaf ; bf Þ 2 t2 IGðat ; bt Þ n1 Gaða1 ; b1 Þ n2 Gaða2 ; b2 Þ

This hierarchical form of the t-t SV model simpliﬁes the setup of a Gibbs sampler scheme. The use of Student-t distributions for the observation error and log-volatility provides a robust analysis against extreme meancorrected log-returns and extreme log-volatilities, which can be easily identiﬁed using the appropriate mixing parameters of the SMN and SMU forms. 3.1. Gibbs Sampler for the Bayesian t-t SV Model Deﬁne y ¼ ðy1 ; y2 ; . . . ; yt Þ; h ¼ ðh1 ; h2 ; . . . ; hn Þ; k ¼ ðl1 ; l2 ; . . . ; ln Þ; u ¼ ðu1 ; u2 ; . . . ; un Þ; x ¼ ðo1 ; o2 ; . . . ; on Þ. For t ¼ 1; 2; . . . ; n, let ht ¼ ðh1 ; . . . ; ht1 ; htþ1 ; . . . ; hn Þ; kt ¼ ðl1 ; . . . ; lt1 ; ltþ1 ; . . . ; ln Þ; ut ¼ ðu1 ; . . . ; ut1 ; utþ1 ; . . . ; un Þ, and xt ¼ ðo1 ; . . . ; ot1 ; otþ1 ; . . . ; on Þ. With arbitrarily chosen starting values for h, k, u, O, f, m, t2, n1, and n2, a Gibbs sampler iteratively samples random variates from a system of full conditional distributions to mimic a random sample from the intractable joint posterior distribution. Let IðÞ be an indicator variable that is equal to 1 if the argument is correct and equal to 0 otherwise. In the context, the system of the full conditionals is given below. 1. Full conditional distribution for ht ; t ¼ 1; 2; . . . ; n: ht jht ; k; u; x; f; m; t2 ; n1 ; n2 ; yÞ Nðmt ; gt t2 ÞIðht 4 ln y2t ln lt ln ut Þ

602

S. T. BORIS CHOY ET AL.

where 8 1 f2 h1 m > t2 > > m þ f þ g 1 > 2 > o1 o2 o1 > > > < 1 f2 ht m htþ1 m 2 mt ¼ g t mþ f t2 þ þ > ot otþ1 ot otþ1 > > > > 2 > o t > n > : m þ fðhn1 mÞ 2

8 1 > 1 f2 f2 > > þ > > > o1 o2 < 2 1 gt ¼ 1 f > > þ > > o o > t tþ1 > : on

t¼1 t ¼ 2; . . . ; n 1 t¼n

t¼1 t ¼ 2; . . . ; n 1 t¼n

2. Full conditional distribution for lt ; t ¼ 1; 2; . . . ; n: n1 þ 1 n1 y2t ht lt jh; kt ; u; x; f; m; t ; n1 ; n2 ; yÞ IG I lt 4 e ; 2 ut 2 2

3. Full conditional distribution for ut ; t ¼ 1; 2; . . . ; n: 1 y2t ht 2 I ut 4 e ut jh; k; ut ; x; f; m; t ; n1 ; n2 ; yÞ exp lt 2 4. Full conditional distribution for ot: n2 þ 1 n2 1 f 2 2 ðh1 mÞ ; þ o1 jh; k; u; x1 ; f; m; t ; n1 ; n2 ; yÞ IG 2 2t2 2 2

ot jh; k; u; xt ; f; m; t2 ; n1 ; n2 ; yÞ n2 þ 1 n2 1 IG ; þ 2 ðht m fðht1 mÞÞ2 2 2t 2

t ¼ 2; . . . ; n

5. Full conditional distribution for m: 2

mjh; k; u; x; f; t ; n1 ; n2 ; yÞ N gm

! ! n X h1 ht fht1 1 þ ð1 fÞ ; gm o1 ot t¼2

Bayesian Student-t Stochastic Volatility Models via Scale Mixtures

603

where

gm ¼

n X 1 1 þ ð1 f2 Þ o1 o t¼2 t

!1

6. Full conditional distribution for t2: t2 jh; k; u; x; f; m; n1 ; n2 ; yÞ n n 1 ð1 f2 Þh21 X 1 IG at þ ; bt þ þ ððht mÞ fðht1 mÞÞ2 o1 2 2 o t t¼2

!!

7. Full conditional distribution for f:

o1 t2 fjh; k; u; x; m; t2 ; n1 ; n2 ; yÞ / N h1 jm; 1 f2 n Y Nðht jm þ fðht1 mÞ; ot t2 Þ pðfÞ t¼2

where p(f) is the prior density of f that can be derived from the prior distribution of c. 8. Full conditional distribution for n1: n1 jh; k; u; x; m; t2 ; f; n2 ; yÞ /

n n 1 1 IG lt ; Gaðn1 ja1 ; b1 Þ 2 2 t¼1

n Y

9. Full conditional distribution for n2: n2 jh; k; u; x; m; t2 ; f; n1 ; yÞ /

n n 2 2 IG ot ; Gaðn2 ja2 ; b2 Þ 2 2 t¼1

n Y

We note that the full conditional distributions for ht ; lt ; ut ; ot ; t ¼ 1; 2; . . . ; n, m, and t2 are the truncated normal, truncated inverse gamma, truncated exponential, inverse gamma, normal and inverse gamma distributions,

604

S. T. BORIS CHOY ET AL.

respectively. Generating random variates from the truncated exponential distribution will be straightforward using the inversion method while the algorithms proposed by Robert (1995) and Philippe (1997) can be used to simulate from the truncated normal and gamma distributions, respectively. For f, n1, and n2, the full conditional distributions are of a nonstandard form and various techniques, such as the ratio-of-uniforms (Wakeﬁeld, Gelfand, & Smith, 1992), adaptive rejection sampling (see Gilks & Wild, 1992), and Metropolis-Hastings methods, can be used to simulate random variates from these distributions. In WinBUGS, the slice sampling method (Damien, Wakeﬁeld, & Walker, 1999) is used instead.

4. BAYESIAN t SV MODELS WITH LEVERAGE Black (1976) and Nelson (1991) revealed that an increase in volatility associates with a drop in equality return. This ﬁnancial leverage effect is measured by the correlation between the error distribution of the log-return at time t and the error distribution of the log-volatility at time tþ1 (Harvey & Shephard, 1996). The SV model with leverage is speciﬁed by Observation equation: State equation:

yt jht ; r ¼ eht =2 t

t ¼ 1; 2; . . . ; n

htþ1 jht ; m; f; t2 ; r ¼ m þ fðht mÞ þ tZt

t ¼ 1; 2; . . . ; n 1

where t Zt

!

0 N ; 0

1

r

r

1

!! (3)

r is the correlation coefﬁcient of et and Zt, and h1 Nðm; ðt2 =ð1 f2 ÞÞÞ. Meyer and Yu (2000) implemented this model using Bayesian software WinBUGS, while Omori et al. (2007) extended the Kim, Shephard, and Chib (KSC) approach (Kim et al., 1998) of approximating the log-w2 distribution by a mixture of 7 normal distributions to a mixture of 10 normal distributions for the SV model with leverage. To make the SV model with leverage robust to outliers, we replace the bivariate normal distribution in Eq. (3) with a bivariate Student-t distribution with n degrees of freedom, that is, ! !! t 1 r 0 t2 ; (4) tn and h1 tn m; Zt r 1 0 1 f2

Bayesian Student-t Stochastic Volatility Models via Scale Mixtures

605

and the model is now called a t SV model with leverage. Since the bivariate Student-t distribution is also an SMN distribution, Eq. (4) can be written hierarchically as ! !! ! t 1 r 0 ; lt N Zt r 1 0 ! !! ! 0 yt eh t rteht =2 N ; lt m þ fðht mÞ htþ1 rteht =2 t2 l0 t 2 h1 N m; 1 f2 n n lt IG ; 2n 2n l0 IG ; 2 2 Jacquier et al. (2004) also considered this t SV model with leverage via the SMN, but they adopted the KSC approach. In this paper, we extend the approach of Meyer and Yu (2000, p. 208) from the bivariate normal distribution to bivariate t distribution via the SMN representation. Now, the t SV model with leverage can alternatively be speciﬁed by r yt jhtþ1 ; ht ; m; f; t2 ; lt ; r N eht =2 ðhtþ1 m fðht mÞÞ; lt ð1 r2 Þeht =2 t htþ1 jht ; m; f; t2 ; lt ; r Nðm þ fðht mÞ; lt t2 Þ l0 t 2 2 h1 jm; f; t ; l0 ; r N m; 1 f2 To complete a Bayesian framework, prior distributions for m, f, and t2 are chosen as in the t-t SV model. The prior for r is U(1,1) and the prior for n is Gaðan ; bn Þ. All prior distributions are assumed to be independent. Bayesian analysis of this t SV model with leverage can be easily implemented using WinBUGS as in Meyer and Yu (2000).

5. AN EMPIRICAL STUDY 5.1. The Data and the Prior Speciﬁcations To demonstrate the proposed SMU presentation for the Student-t density function in the application of the SV model, we analyze the daily closing

606

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exchange rates of AUD against 10 major currencies during a 2-year period from January 2006 to December 2007. The data are 510 mean-corrected logreturns of AUD to each of the 10 currencies: American dollar (USD), European Euro (EUR), Great British Pound (GBP), Switzerland Francs (CHF), Canadian dollar (CAD), New Zealand dollar (NZD), Japanese Yen (JPY), Chinese Yuan (CNY), Hong Kong dollar (HKD), and Singapore dollar (SGD). They are downloaded from the ofﬁcial Web site of the Reserve Bank of Australia (www.rba.gov.au). Time series plots of the data are given in Fig. 1. Obviously, some trading days produce extreme meancorrected log-returns and probably high volatilities. The data are ﬁtted by a t-t SV model. The prior distributions adopted are as follows. m Nð10; 1Þ t2 IGð0; 0Þ c Beð20; 1:5Þ n1 Gað1; 0:2ÞIð1; 30Þ n2 Gað1; 0:2ÞIð1; 30Þ Here, the vague and noninformative priors are assigned to m and t2 to express ignorance about these parameters. In Section 5.5, the data are ﬁtted by a t SV model with leverage. The prior distribution for the leverage effect parameter r is reUð1; 1Þ and the prior distribution for the degrees of freedom n of the bivariate t distribution is neGað1; 0:2ÞIð1; 30Þ. We run a Gibbs sampler for a single chain of 25,000 iterations. The ﬁrst 5,000 iterations are discarded as the ‘‘burn-in’’ period. To avoid high correlation between successive drawings of the model parameters, we picked up simulated values at every 20th iteration to mimic a random sample of 1,000 drawings from the joint posterior distribution. As a matter of fact, posterior means of the model parameters obtained from this systematic sample are very close to those obtained from the ergodic averages of all simulated values collected after the burn-in period. To show how well the Gibbs sampler mixes, Kim et al. (1998) and Chib et al. (2002) reported the simulation inefﬁciency factor (SIF). Given a set of simulated values of a parameter of interest from the Gibbs sampler, the SIF of the estimation of the posterior mean is deﬁned as (Chib et al., 2002, p. 296) SIF ¼ 1 þ 2

1 X k¼1

rðkÞ

Bayesian Student-t Stochastic Volatility Models via Scale Mixtures Time Series Plot of AUD/CHF

0.03

0.03

0.02

0.02

0.01

0.01

0.00

y(t)

y(t)

Time Series Plot of AUD/USD

-0.01

607

0.00 -0.01

-0.02

-0.02

-0.03

-0.03

-0.04

-0.04 1

1

50 100 150 200 250 300 350 400 450 500

50 100 150 200 250 300 350 400 450 500 Time t

Time t Time Series Plot of AUD/CAD

Time Series Plot of AUD/NZD

0.02

0.02

0.01 0.01

y(t)

y(t)

0.00 -0.01 -0.02

0.00 -0.01

-0.03 -0.02

-0.04 1

50 100 150 200 250 300 350 400 450 500

1

Time t Time Series Plot of AUD/EUR

50 100 150 200 250 300 350 400 450 500 Time t Time Series Plot of AUD/JPY

0.02

0.050

0.01

0.025 0.000 y(t)

y(t)

0.00 -0.01

-0.025

-0.02

-0.050

-0.03 -0.04 1

50 100 150 200 250 300 350 400 450 500 Time t

Fig. 1.

-0.075 1

50 100 150 200 250 300 350 400 450 500 Index

Time Series Plot of the Mean-Corrected Log-Returns of the Exchange Rates of AUD to each of the 10 Currencies.

where r(k) is autocorrelation at lag k. In practice, the inﬁnite sum in the calculation of SIF is replaced by a ﬁnite sum of 500 lags in this paper. The SIF can also be interpreted as the ratio of the variance of the posterior mean evaluated from the correlated MCMC samples to the variance of the posterior mean evaluated from the same number of uncorrelated posterior samples. If the Gibbs sampler mixes well, then the SIF will be close to one.

608

S. T. BORIS CHOY ET AL. Time Series Plot of AUD/GBP

Time Series Plot of AUD/CNY

0.02

0.03 0.02

0.01

0.01 y(t)

y(t)

0.00 -0.01

0.00 -0.01 -0.02

-0.02

-0.03 -0.04

-0.03 1

1

50 100 150 200 250 300 350 400 450 500 Time t

50 100 150 200 250 300 350 400 450 500 Time t Time Series Plot of AUD/SGD

Time Series Plot of AUD/HKD 0.03

0.02

0.02

0.01 0.00

0.00

y(t)

y(t)

0.01

-0.01

-0.01

-0.02

-0.02

-0.03

-0.03

-0.04

-0.04 1

50 100 150 200 250 300 350 400 450 500 Time t

Fig. 1.

1

50 100 150 200 250 300 350 400 450 500 Time t

(Continued )

Table 1. Simulation Inefﬁciency Factors of Selected Parameters in the t-t SV Model. Parameters

B¼1

B ¼ 10

B ¼ 20

B ¼ 50

B ¼ 100

n1 n2 m f t h250 l250 u250 o250

113.4 68.8 95.0 20.5 80.7 96.9 3.7 5.15 2.4

30.9 34.3 12.7 16.5 27.8 13.9 1.1 1.1 2.4

32.8 28.3 8.1 8.6 22.6 7.1 1.4 1.0 1.0

14.3 16.5 3.0 6.0 11.3 3.2 1.0 1.0 1.0

7.7 8.4 2.8 3.3 6.9 1.8 1.0 1.0 1.0

These values are evaluated based on 1,000 Gibbs output taken at every Bth iteration.

For the AUD/JPY exchange rates data, 1,000 posterior samples, collected at every Bth iteration where B ¼ 1, 10, 20, 50, and 100, are used to evaluate the SIF of model parameters and the results are displayed in Table 1. Generally speaking, the Gibbs sampler mixes very well for the mixing

Bayesian Student-t Stochastic Volatility Models via Scale Mixtures

609

parameters lt, ut, and ot at all B values. However, it does not mix well for n1, n2, m, t, and ht for small B values because the SIFs are large. In this paper, B ¼ 20 is chosen and the SIFs of all parameters, except n1, n2, and t, are less than 10 (see Table 1). The Gibbs sampler is quite effective for this choice of B value.

5.2. Parameter Estimation For comparison, an N-N SV model is also ﬁtted to the data. Table 2 exhibits the posterior means (posterior standard deviations in parentheses) of m, f, t, n1, n2, and the deviance information criterion (DIC) values for the 10 currencies. The degrees of freedom of the Student-t distributions for the mean-corrected log-return and for the log-volatility range from 7.8 to 18.5 and from 3.0 to 9.3, respectively. Obviously, the log-volatility distribution is heavier-tailed than the mean-corrected log-return distribution. The posterior mean of the standard deviation of the log-volatility t is smaller in the t-t SV model than in the N-N SV model. Fig. 2 displays the boxplots of the posterior samples of f, t, n1, and n2 in the t-t SV model.

5.3. Model Selection To choose between the N-N and t-t SV models, we use the DIC. It is widely used in Bayesian model selection for complicated hierarchical models in which the number of parameters is not clearly deﬁned. Spiegelhalter, Best, Carlin, and van der Linde (2002) deﬁned the DIC as DIC ¼ E½DðyÞ þ pD where E½DðyÞ is the posterior expectation of the deviance and is a Bayesian measure of goodness of ﬁt, and pD is the effective number of parameters in the model and is a measure of model complexity. The DIC can be easily calculated using the MCMC output. See Berg et al. (2004) for the application in SV models. The model with a smaller DIC is a better model. From Table 2, one can see that the t-t SV model is superior to the N-N SV model in the exchange rates of each of the 10 currencies. Using the DIC as a model comparison tool may be somewhat controversial and some Bayesian statisticians prefer using the Bayes factor. However, the Bayes factor involves the calculation of marginal likelihood

610

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Table 2. Posterior Means and Standard Errors (in Parentheses) of Parameters in the N-N SV and t-t SV Models for Exchange Rates Data. Currency USD

CAD

EUR

GBP

CHF

NZD

JPY

CNY

HKD

SGD

Model

DIC

n1

n2

m

f

t

N-N

3685.5

–

–

t-t

4215.4

N-N

3790.5

17.8 (5.8) –

3.2 (5.4) –

t-t

4312.4

N-N

3888.9

12.0 (5.7) –

8.7 (6.6) –

t-t

4422.9

N-N

3867.1

15.3 (5.9) –

4.4 (6.2) –

t-t

4381.9

N-N

3725.4

18.5 (5.6) –

6.5 (6.5) –

t-t

4255.8

N-N

3915.3

14.4 (5.8) –

3.0 (4.9) –

t-t

4437.4

N-N

3563.6

17.7 (5.9) –

9.3 (6.7) –

t-t

4111.8

N-N

3714.0

7.8 (3.4) –

3.9 (6.0) –

t-t

4245.2

N-N

3691.2

16.6 (5.6) –

3.1 (3.6) –

t-t

4215.0

N-N

3880.5

17.0 (5.5) –

3.7 (6.2) –

t-t

4408.4

13.0 (5.2)

4.2 (5.0)

10.30 (0.2989) 10.49 (0.3037) 10.47 (0.2156) 10.61 (0.3045) 10.71 (0.2045) 10.89 (0.3155) 10.73 (0.3186) 10.85 (0.3068) 10.42 (0.3031) 10.61 (0.3097) 10.75 (0.1973) 10.82 (0.1900) 10.09 (0.3890) 10.39 (0.4151) 10.34 (0.2989) 10.53 (0.2757) 10.29 (0.3055) 10.48 (0.2808) 10.87 (0.3555) 11.05 (0.3525)

0.9803 (0.0114) 0.9788 (0.0110) 0.9658 (0.0274) 0.9500 (0.0496) 0.9758 (0.0145) 0.9766 (0.0136) 0.9792 (0.0130) 0.9809 (0.0108) 0.9774 (0.0128) 0.9733 (0.0136) 0.9690 (0.0182) 0.9625 (0.0233) 0.9768 (0.0123) 0.9798 (0.0111) 0.9794 (0.0125) 0.9784 (0.0113) 0.9790 (0.0119) 0.9763 (0.0120) 0.9741 (0.0170) 0.9774 (0.0131)

0.1492 (0.0326) 0.0970 (0.0309) 0.1250 (0.0420) 0.1009 (0.0263) 0.1678 (0.0388) 0.1159 (0.0304) 0.1488 (0.0354) 0.1105 (0.0286) 0.1758 (0.0418) 0.1116 (0.0375) 0.1194 (0.0265) 0.1069 (0.0285) 0.2342 (0.0474) 0.1378 (0.0463) 0.1540 (0.0358) 0.0902 (0.0264) 0.1565 (0.0354) 0.1058 (0.0336) 0.1818 (0.0454) 0.1124 (0.0356)

The DIC values are for Bayesian model selection.

Bayesian Student-t Stochastic Volatility Models via Scale Mixtures Boxplot of phi

611

Boxplot of tau

1.00

0.25

0.98 0.20 tau

psi

0.96 0.94 0.92

0.15 0.10

0.90 0.05

0.88 USD CAD EUR GBP CHF NZD JPY CNY HKD SGD

USD CAD EUR GBP CHF NZD JPY CNY HKD SGD Boxplot of v_2 30

25

25

20

20 v_2

v_1

Boxplot of v_1 30

15

15

10

10

5

5

0

0 USD CAD EUR GBP CHF NZD JPY CNY HKD SGD

Fig. 2.

USD CAD EUR GBP CHF NZD JPY CNY HKD SGD

Boxplots of the Posterior Samples of f, t, n1, and n2 in the t-t SV Model for the 10 Currencies.

functions or the normalizing constants of the posterior distributions of two different models. Various methods have been proposed to stabilize the estimators of the marginal likelihoods. For example, Chib (1995) proposed a direct MCMC approach to approximate the marginal likelihoods and hence the Bayes factor. This method requires data augmentation in the Gibbs sampling algorithm and the Gibbs outputs. In the present context, the two mixing parameters lt and ut in the observation equation can be considered to be the latent variables and Chib’s approach can then be adopted to approximate the marginal likelihoods and hence the Bayes factor for model comparison. Note that this approach requires a proper joint prior distribution. If vague and noninformative prior distributions are used in part of the model building, Congdon’s (2006) method can be used.

5.4. Robust Analysis and Outlier Diagnostics It is well known that heavy-tailed distributions can accommodate outliers and hence provide a robust inference. The advantage of using heavy-tailed

612

S. T. BORIS CHOY ET AL.

Student-t distribution is obvious when we compare the results of the N-N and t-t SV models in Table 1. The Student-t error distribution automatically downweighs the inﬂuence of extreme mean-corrected log-returns in parameter estimation and hence provides more accurate estimates. Similarly, the Student-t distribution in the state equation also discounts the inﬂuence of extreme log-volatilities in statistical inference. For the AUD/JPY exchange rates data, Fig. 3 displays the time series plots of the posterior means of the log-volatility, ht, in the N-N and t-t SV models. The general trend of the unobserved log-volatility is that it decreases in the ﬁrst 7 months in 2006 and increases since then. Obviously, the log-volatilities are smaller in the t-t SV model than in the N-N SV model. The posterior mean of the standard deviation of the log-volatility, t is 0.1378 in the t-t SV model compared with 0.2342 in the N-N SV model. Choy and Smith (1997b) and Choy and Chan (2008) exhibited the detection of possible outliers using the mixing parameters of the SMN and SMU density representations, respectively. In the context of the SV models, we propose to use the SMU form of the Student-t distribution in the observation equation and the SMN form in the state equation to facilitate a simple setup of the Gibbs sampler. Under the SMU representation, an outlier is accommodated by increasing the range, and hence the variance of

Log-volatilities h(t) - AUD/JPY -7.0 1

51

101

151

201

251

301

351

401

451

-8.0

h(t)

-9.0 -10.0 -11.0 -12.0 time t N-N

t-t

Bivariate-t

Fig. 3. Comparison of Posterior Means of Log-Volatilities in the N-N SV Model (Broken Line), t-t SV Model (Solid Line), and t SV Model with Leverage (Gray Line).

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613

pﬃﬃﬃﬃﬃﬃﬃﬃ the uniform distribution in Eq. (2).pSince ﬃﬃﬃﬃﬃﬃﬃﬃ the range is determined by lt ut , we can compare the magnitude of lt ut for each mean-corrected log-return and large values correspond to possible outlying log-returns. Similarly, we use the mixing parameter, ot, in the SMN representation to identify outlying log-volatilities. For the AUD/JPY exchange rates data in Fig. 1, there may have some possible outliers but we do not know whether they are outlying log-returns or outlying log-volatilities. However, from Fig. 4(a), which displays the pﬃﬃﬃﬃﬃﬃﬃﬃ posterior means of the uniform mixing parameter l t ut , we can identify pﬃﬃﬃﬃﬃﬃﬃﬃ large lt ut values that are associated with outlying mean-corrected logreturns. The ﬁve most outlying log-returns come from Day 467 (Nov 12, 2007), Day 173 (Sept 8, 2006), Day 254 (Jan 5, 2007), Day 104 (Jun 1, 2006), and Day 294 (Mar 5, 2007). Similarly, Fig. 4(b) shows the posterior means of ot, and Day 394 (July 27, 2007) and Day 393 (July 26, 2007) are identiﬁed to contribute the largest two log-volatilities.

5.5. t SV Model with Leverage Table 3 presents the posterior means and standard deviations of m, f, t, r, n, and DIC of the exchange rates data when a t SV model with leverage is ﬁtted to the data. The posterior mean of r is positive for USD, NZD, CNY, and HKD and no leverage effect is observed in these exchange rates. Although CAD, GBP, JPY, and SGD have negative r values, they are not signiﬁcant at the 5% level in the hypothesis test that r is negative. EUR and CHF are the only two currencies that show the existence of the leverage effect. Compared with the results from the t-t SV model in Table 2, the posterior mean of the degrees of freedom n of the bivariate Student-t distribution in this leverage model is somewhere between the posterior means of n1 and n2 in the t-t SV model for each of the 10 currencies. Since the t-t SV model shows a difference in the n1 and n2 values but the bivariate Student-t distribution forces the marginal distributions of the log-return and the logvolatility to have the same degrees of freedom, the leverage model has to compromise by adjusting the degrees of freedom of the bivariate Student-t distribution. Moreover, the posterior mean of the standard deviation of the log-volatility t in the leverage model is greater than t in the t-t SV model but is less than t in the N-N model for each of the currencies. This means that the leverage model is superior to the N-N SV model but may not be better than the t-t SV model. This is conﬁrmed by the DIC values of the SV models and leverage model (Tables 2 and 3). The estimates of the

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S. T. BORIS CHOY ET AL. sqrt(lambda(t)*u(t)) for y(t) - AUD/JPY

sqrt(lambda(t)*u(t))

4.5

3.5

2.5

1.5

0.5 1

51

101

151

201

(a)

251

301

351

401

451

time t

omega(t) for log-volatilities h(t) - AUD/JPY 2.00

omega(t)

1.75

1.50

1.25

1.00 1 (b)

51

101

151

201

251

301

351

401

451

time t

Fig. 4. pThe ﬃﬃﬃﬃﬃﬃﬃﬃ Mixing Parameters in the t-t SV Model for Outlier Diagnostics: (a) lt ut for Identifying Outlying yt and (b) ot for Identifying Outlying ht.

unobserved log-volatilities are given in Fig. 3 as a comparison to those from the N-N and t-t SV models. Fig. 5 displays the posterior means of the normal mixing parameters lt of the leverage model. Those outliers identiﬁed in the t-t SV model are also detected by lt.

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Table 3. Posterior Means and Standard Errors (in Parentheses) of Parameters in the t SV Model with Leverage for Exchange Rates Data. Currency

DIC

n

r

m

f

t

USD

3732.5

CAD

3925.8

EUR

3969.5

GBP

3933.0

CHF

3798.2

NZD

3964.6

JPY

3639.3

CNY

3762.7

HKD

3742.7

SGD

3960.9

16.1 (5.7) 10.3 (4.9) 11.0 (5.5) 17.0 (6.0) 11.6 (5.5) 17.5 (5.8) 6.0 (2.7) 14.8 (5.8) 15.0 (6.1) 10.4 (4.6)

0.15 (0.21) 0.28 (0.27) 0.38 (0.19) 0.32 (0.21) 0.34 (0.19) 0.25 (0.20) 0.23 (0.18) 0.19 (0.20) 0.17 (0.20) 0.08 (0.24)

10.44 (0.32) 10.7 (0.19) 10.86 (0.27) 10.85 (0.33) 10.57 (0.30) 10.85 (0.20) 10.39 (0.38) 10.51 (0.32) 10.45 (0.33) 11.09 (0.37)

0.9822 (0.0108) 0.9551 (0.0330) 0.9765 (0.0121) 0.9843 (0.0100) 0.9771 (0.0121) 0.9682 (0.0191) 0.9830 (0.0091) 0.9846 (0.0093) 0.9828 (0.0102) 0.9806 (0.0125)

0.1295 (0.0273) 0.0988 (0.0306) 0.1309 (0.0305) 0.1241 (0.0270) 0.1443 (0.0347) 0.1087 (0.0266) 0.1408 (0.0354) 0.1243 (0.0281) 0.1226 (0.0286) 0.1242 (0.0317)

Denotes that r is signiﬁcantly less than zero at the 5% level of signiﬁcance.

lambda(t) for log-volatilities h(t) - AUD/JPY (with leverage)

lambda(t)

4.5

3.5

2.5

1.5

0.5 1

51

101

151

201

251

301

351

401

451

time t

Fig. 5.

The Mixing Parameters in the t SV Model with Leverage for Outlier Diagnostics: lt for Identifying Outlying Pairs (yt, htþ1).

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6. CONCLUDING REMARKS The Student-t distribution has been widely used for modeling heavy-tailed events and for robustness consideration. In Bayesian computational inference, the Student-t distribution is always expressed into an SMN form so as to simplify the Bayesian posterior calculations, in particular when Gibbs sampling scheme is used. See Wakeﬁeld, Smith, Racine-Poon, and Gelfand (1994) and Choy and Smith (1997b) for details. For some complicated models where the data are dependent such as the GARCH and SV models, this SMN representation may not be able to provide a substantial improvement in computational efﬁciency. The signiﬁcance of this paper is on the introduction of a two-stage scale mixtures representation for the Student-t density. This new SMU representation enables us to simplify the Bayesian computation for the SV models with Student-t innovations. For outlier diagnostics, relatively pﬃﬃﬃﬃﬃﬃﬃﬃ large posterior mean values of lt ut are associated with outlying meancorrected log-returns and relatively large posterior mean values of ot are associated with outlying log-volatilities. Multivariate SV models have become more popular in ﬁnancial time series over the last few years. However, the proposed SMU representation is only valid to the univariate Student-t distribution. The SMU representation of the multivariate t distribution is still not available and work on the development of a multivariate SMU representation is in progress.

ACKNOWLEDGMENTS The authors thank Siddhartha Chib for constructive comments on an earlier version of the paper and Jun Yu for providing two WinBUGS programs for our reference.

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BAYESIAN ANALYSIS OF THE CONSUMPTION CAPM Veni Arakelian and Efthymios G. Tsionas ABSTRACT In this paper we take up Bayesian inference for the consumption capital asset pricing model. The model has several econometric complications. First, it implies exact relationships between asset returns and the endowment growth rate that will be rejected by all possible realizations. Second, it was thought before that it is not possible to express asset returns in closed form. We show that Labadie’s (1989) solution procedure can be applied to obtain asset returns in closed form and, therefore, it is possible to give an econometric interpretation in terms of traditional measurement error models. We apply the Bayesian inference procedures to the Mehra and Prescott (1985) dataset, we provide posterior distributions of structural parameters and posterior predictive asset return distributions, and we use these distributions to assess the existence of asset returns puzzles. The approach developed here, can be used in sampling theory and Bayesian frameworks alike. In fact, in a samplingtheory context, maximum likelihood can be used in a straightforward manner.

Bayesian Econometrics Advances in Econometrics, Volume 23, 619–643 Copyright r 2008 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 0731-9053/doi:10.1016/S0731-9053(08)23020-3

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1. INTRODUCTION The consumption capital asset pricing model (C-CAPM) has played a central role in the modern theory of ﬁnance and macroeconomics. It has been used by Mehra and Prescott (1985) to investigate the so-called ‘‘equity premium puzzle’’: Over the past hundred years, the average real stock return in the United States has been about 7% and the average real return on Treasury bills has been about 1%. In their seminal paper, Mehra and Prescott (1985) showed that the C-CAPM is unable to replicate these facts which we refer to as ‘‘asset return puzzles.’’ Kocherlakota’s (1996) thorough literature review concludes that only market frictions, incompleteness, or non-standard preferences could solve the equity premium and risk-free rate puzzles. Moreover, there is an abundance of ﬁndings in the generalized method of moments (GMM) literature that the C-CAPM is rejected under fully non-parametric conditions. Indeed, Hansen, Heaton, and Yaron (1996) reject even after elaborate corrections for the ﬁnite-sample properties of GMM tests. It is also known, however, that the ﬁnite-sample properties of GMM could be very different compared to their asymptotic properties. GMM for asset pricing and other dynamic stochastic models is based on Euler equations derived from ﬁrst-order conditions. Typically, these are conditional expectations of non-linear functions that are set equal to zero at the optimal solution. Using certain instrumental variables, which are part of the ‘‘information set,’’ these conditional expectations can be converted to unconditional expectations, and then an appropriate distance function is minimized to obtain consistent estimates of the parameters. Leading presentations include Hansen (1982), Hansen and Singleton (1982), and Newey and West (1987). Hansen and Singleton (1983) is an excellent presentation of GMM in the context of asset pricing. A textbook presentation is included in Altug and Labadie (1994, pp. 87–108). For a very useful summary, see Ogaki (1993). Of course, GMM depends on the choice of instruments. Moreover, in view of the results in Hansen et al. (1996) GMM results may behave quite erratically in ﬁnite samples compared to their asymptotic properties in Hansen (1982). In a sense, GMM cannot allow for the incorporation of all available information since it uses instruments. There are many choices for instruments and some of them can even be weak, to make a difﬁcult problem even harder. In principle, likelihood-based inference is free from these criticisms (up to distributional assumptions) and makes full use of information. In fact, the score provides the answer to Gallant and Tauchen’s

Bayesian Analysis of the Consumption CAPM

621

(1996) question in their seminal paper. Here, we do not provide explicit expressions for the exact score – although this is not difﬁcult to compute. We do provide the exact likelihood and approximate expressions for the information matrix (in Appendix B). Both should be useful in a samplingtheory context. For reasons clearly explained in the excellent paper by Geweke (1999) an exact likelihood-based analysis of the C-CAPM or similar general equilibrium models, is very complicated. The model implies exact relationships between the endogenous variables as there are many endogenous variables but only a small number of shocks. Therefore, the model will be rejected by every possible dataset. In that sense, the GMM-based frequent rejections of the C-CAPM are not surprising at all. Several attempts have been made to provide parameter estimates for the model, including calibration, variants of indirect inference, GMM methods etc. All of them solve only part of the problem since they do not confront the data with a model-based exact likelihood function to provide inferences. The fundamental reason behind the widespread application of GMM methods in asset pricing models is the presumption that the implications of C-CAPM are exhausted in the provision of ﬁrst-order Euler conditions that cannot be solved to obtain closed-form expressions for asset returns. In fact, this is not true, since Labadie (1989) has shown that Euler equations can be solved to provide closed-form expressions. Since closed-form expressions are available, inferences based on the likelihood function, appropriately constructed, should be possible. The purpose of this paper is to explore the implications of this idea, and apply the results to the Mehra and Prescott (1985) dataset to assess the existence of an ‘‘equity premium puzzle’’ using formal means.1 Artiﬁcial datasets are also considered to validate the new methods presented here, and examine their sensitivity. The approach developed here, can be used by frequentists and Bayesians alike, and this is an important point that we want to emphasize. Since we construct a regular likelihood function, maximum likelihood estimation can be used for sampling-theory inference, and Monte Carlo simulation can be used to provide draws from the kernel posterior of the ‘‘deep,’’ structural parameters of the model, to conduct Bayesian inference. The likelihood function is highly complicated in terms of parameters that are severely constrained, but still maximum likelihood can be used without the need to resort to simulations. Moreover, it can be used without resort to approximations (Christiano, 1990), use of approximating linear state–space techniques (Hansen & Sargent, 1980) or even maximum simulated likelihood techniques (see Pakes & Pollard, 1989 for a general discussion).

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Ruge-Murcia (2007) presents several methods and, generally, favors the methods developed here in a wider class of models. In fact, in a model which is exactly like ours, Tauchen and Hussey (1991) have used numerical integration to solve the Euler equations. Gallant and Tauchen (1989) have proposed procedures based on computationally intensive simulated GMM. They all have ignored the fact that asset returns are, in fact, available in closed form. These closed form expressions are, however, of tremendous importance in a likelihood-based approach, be it frequentist or Bayesian. Other interesting approaches in the context of general equilibrium models, are explored in Ireland (2004), DeJong, Ingram, and Whiteman (2000), Lee and Ingram (1991), Schorfheide (2000), and Smith (1993). See also Uhlig (1999) for treatment from the practical point of view. An important class of asset pricing models includes models with habit persistence (Boldrin, Christiano, & Fisher, 1991; Constantinides, 1990; Ferson & Constantinides, 1991). For an excellent and lucid discussion, see Campbell and Cochrane (1999). Of course, many models can be used to explain the ‘‘equity premium puzzle.’’ The important thing, however, is whether such models are compatible with the real-business cycle studies. This sine qua non question is taken up in Lettau and Uhlig (2000). Our purpose here is not to compare directly with such models but to focus on two things: First, the simple, standard Mehra and Prescott (1985) model, is indeed compatible with large real equity returns and low real risk-free rates. Second, the standard model can be estimated easily in a likelihood framework.

2. THE ASSET PRICING MODEL Consider an economy with a representative agent endowed with constant relative risk aversion preferences. The preferences of the representaP (RRA) t tive agent are E0 1 t¼0 b uðct Þ, where bW0 is the discount factor, and the spot utility function is given by: 8 1g > < c 1 ; g40; ga1 1g (1) uðcÞ ¼ > : log c; ga1 where g is the coefﬁcient of RRA and ct the consumption of date t. There is a single productive unit that produces without cost the consumption good

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each period. Let Yt be the date t endowment, and deﬁne: Y tþ1 ytþ1 ¼ Yt

(2)

The stochastic process driving endowment is given by log ytþ1 ¼ k þ d log yt þ u1;tþ1 ;

t ¼ 0; 1; 2; . . .

(3)

where u1;t INð0; s2u Þ is an error term, for t ¼ 1,2, . . . and k, d the parameters. The reason for indexing the shock with a subscript ‘‘1’’ will become clear later on. The AR(1) speciﬁcation is supported by evidence in Labadie (1989) and Cecchetti, Lam, and Mark (1993). Let qt denote the price of equity and Pt the discount price of risk-free real bonds in terms of the consumption good. The real risk-free interest rate is denoted by Rft and is given in closed form as: 1 2 2 gd f 1 (4) 1 þ Rt ¼ Pt ¼ b exp gk g su ytþ1 ; t ¼ 0; 1; 2; . . . 2 The real return on equity is denoted by Rqt , and is given by: 1 þ Rqt ¼

qtþ1 þ Y tþ1 ; qt

t ¼ 0; 1; 2; . . .

(5)

Asset returns can be obtained using a modiﬁcation of Labadie’s (1989) procedure to obtain them in a series expansion. If r ¼ 1 g, deﬁne the recursive coefﬁcients: 1 Ajþ1 ¼ Aj b exp ðajþ1 þ rÞ k þ ðajþ1 þ rÞs2u 2 ajþ1 ¼ dðaj þ rÞ;

j ¼ 2; 3; . . .

(6)

with A1 ¼ b exp½rðk þ 12rs2u Þ and a1 ¼ rd. Then, stock prices are given by: qt ¼ Y t

1 X

a

A j yt j

(7)

j¼1

Gross real stock returns are given by the expression: P aj 1þ 1 j¼1 Aj ytþ1 q 1 þ Rtþ1 ¼ ytþ1 P1 aj j¼1 Aj yt and the gross real risk-free rate is provided in Eq. (4).

(8)

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The ‘‘deep’’ parameters (say h) have to satisfy certain constraints. First, the resulting matrix R must be positive deﬁnite. This can be ensured by reparameterizing using R ¼ CuC, where C is the Cholesky decomposition, and its elements are unrestricted. Second, we must have: jdjo1 Third, the resulting y must be consistent with existence of equilibrium with ﬁnite stock prices (Tsionas, 2003). Because of Eqs. (6) and (7) the relevant conditions are: 1 2 b exp z k þ zsu o1; and jzjo1 2 where z ¼ r=ð1 dÞ. Fourth, we have the constraints: 0obo1;

and

g40

3. THE ECONOMETRIC MODEL It is clear that the model has implications for asset returns that should be confronted with the data, not only to assess the model’s ability to replicate the behavior of observed returns but also in order to provide parameter estimates, and allow formal statistical inferences for parameters and functions of interest. Since we are able to express asset returns in closed form, via Eqs. (4) and (8), it is obvious that the likelihood function should be derived based on the system consisting of Eqs. (3), (4), and (8). The problem is, of course, that there is only one shock, u1t, so the model is not complete, and the singularity problem mentioned in Geweke (1999) exists, thus preventing the formulation of a likelihood function and the conduct of formal statistical inferences. We propose to solve this problem by following a familiar path in econometrics, namely the view that the endogenous variables are measured with error, leading us to introduce error terms in equations (4) and (8). However, ﬁrst we have to truncate the inﬁnite a summations. We do this by computing the index j so that Aj yt j , where e is a small positive constant. Deﬁne M t ðkÞ ¼ j, where k ¼ ½b; g; k; d; su 0 is the parameter vector. Therefore, we obtain: PM t ðlÞ aj 1 þ j¼1 Aj ytþ1 q þ u2;tþ1 (9) 1 þ Rtþ1 ¼ ytþ1 PM t ðlÞ aj j¼1 Aj yt

Bayesian Analysis of the Consumption CAPM

and the gross real risk-free rate is: 1 2 2 gd f 1 þ Rt ¼ b exp gk g su ytþ1 þ u3t 2

625

(10)

We let ut ¼ ½u1t ; u2t ; u3t 0 INð0; RÞ where R is a 3 3 covariance matrix. Denote the parameter vector by h ¼ ½b; g; k; d; vecðRÞ0 0 ½b; g; k; d; s11 ; s12 ; s13 ; s22 ; s23 ; s33 0 . The peculiarity of this system is that it is highly non-linear with respect to y, the structural equations depend explicitly on s11 ¼ s2u , and also the number of terms in each summation, Mt(k), is parameter-dependent. In that sense, s2u s11 is not a nuisance parameter as it is often the case in normal linear models. It is a ‘‘deep’’ structural parameter of the model and, of course, an element of R. Moreover, we adopt the general case and we do not assume s12 ¼ s13 ¼ 0. This would make R ‘‘block’’ diagonal and would facilitate the analysis considerably. However, we wish to proceed using the more general case. We wish to emphasize that our methodological contribution is the recognition that there are closed form expressions for asset returns given by Labadie’s methodology, a point also emphasized in a somewhat different context by Tsionas (2003, 2005). Previous research in asset pricing models neglected to recognize this fact and resorted to GMM-based procedures. Since, however, we have closed-form expressions for asset returns we can proceed to derive a closed-form likelihood function by introducing measurement error (in the familiar econometric way) in the closed-form expressions deﬁning asset returns. This is important because it will allow us to develop likelihood-based methods not only for parameter inferences (and, indeed, important parameters like the coefﬁcient of RRA) but also for a likelihood-based examination of the celebrated equity premium puzzle. Suppose we have a random sample X ¼ ½X01 ; X02 ; . . . ; X0T 0 , where Xt ¼ ½ln yt ; Rtf ; Rqt 0 , t ¼ 1; . . . ; T. All components of Xt are assumed to be observed. Then, the log-likelihood function of the model is: Lðh; XÞ ¼

3T T 1 log 2p log jRj tr ðXt Xt ðhÞÞ0 ðXt Xt ðhÞÞR1 (11) 2 2 2

where Xt ðhÞ represents the theoretical predictions of the model derived from Eqs. (9) and (10), h ¼ ½b; g; k; d; vechðRÞ0 0 is the parameter vector. The asset pricing model raises several issues in connection with the conduct of formal statistical inference. First, how should we formulate the priors? Second, how sensitive are the posterior results to different priors? Third, how should we

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evaluate the model’s ability to reproduce properties of observed asset returns, and assess the existence of an ‘‘equity premium puzzle’’? Fourth, what can we say about the performance of the Bayesian methods proposed in this paper when applied to artiﬁcial samples? These are the questions that we set out to answer in the following discussion. By application of Bayes’ theorem, we obtain the posterior distribution, pðhjXÞ / Lðh; XÞpðhÞ, where p(h) represents the prior. Our objective is to make inferences on parameters and functions of interest. Important functions of interest in our case are asset returns that can be used to assess formally the asset return puzzles. We use Markov Chain Monte Carlo (MCMC) techniques to generate a (non-random) sample fhðrÞ ; r ¼ 1; . . . ; Rg that converges to the distribution whose kernel density is p(h|X). We use an adaptive Metropolis algorithm that updates each component of h sequentially (so that it resembles a Gibbs sampling with complete blocking). In our context, we have p ¼ 10 parameters in total. For each i ¼ 1; :::; p, we generate a candidate yi from a proposal distribution that is uniform in the interval [ai, bi]. With probability ðrþ1Þ ¼ yi , else we set AðyiðrÞ ; yi Þ ¼ minf1; ðpðyi jXÞÞ=ðpðyðrÞ i jXÞÞg we set yi ðrþ1Þ ðrÞ ¼ yi . We determine the ai’s and bi’s during a preliminary run to yi ensure that the acceptance rate is not too high or too low. In the application reported in Section 5, we set the desired acceptance rate to 50%. The ﬁnal acceptance rate in the after burn-in samples was close to 30%, which is considered satisfactory in most Bayesian studies. Our MCMC scheme resembles the Gibbs sampler (Gelfand & Smith, 1990) in the sense that draws are being constructed from the posterior conditional distribution of each parameter. These draws, however, are implemented using the Metropolis algorithm (Tierney, 1992). Moreover, the Metropolis algorithm is used in an adaptive manner. Of course, the candidate parameter vector, y, has to satisfy the constraints that we introduced in Section 2. This is implemented using acceptance sampling within the Metropolis algorithm. Denote the admissible parameter space by Y Rp .

4. PRIOR ELICITATION The parameters of interest are b; g; k; d; and vechðRÞ0 . The discount factor, b, should be in the interval (0, 1) and values in excess of about 0.8 are highly likely. Therefore, as its prior distribution, we assume a b(20, 1). According to the real-business cycle studies, the constant RRA parameter, g, should be between 1 and 10, and values between 1 and 3 seem to be quite successful in

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627

reproducing the business cycle facts so such values should be considered more plausible. We choose2 a log-normal prior, ln g Nðm0 ; s20 Þ. Setting s20 ¼ 1, we examine the prior distribution of g, for two different values of m0, 0.5 and 1.445, which give EðgÞ ¼ 1 and EðgÞ ¼ 7, respectively. For the parameters k and d, we use independent normal priors. We assume that k Nð0; 0:025Þ and d Nð0; 0:0025Þ. The speciﬁc priors3, reﬂect our prior notion that these parameters are likely to be small, that for k we do not wish to be particularly informative, and that with probability 95%, parameter d lies in the interval from 0.05 to 0.05. These values, are quite reasonable in view of the fact that these parameters refer to an AR(1) process for output growth. For su, we adopt an inverted gamma distribution4, eb=su ; a; b; su 40. su IGða; bÞ with density pðsu Þ ¼ ðba =GðaÞÞsðaþ1Þ u We examine two cases, (a, b) ¼ (2.11, 0.011) and (a, b) ¼ (4.77, 0.19). In this way, the variance of su is 0.032 whereas the means are 0.01 and 0.05, respectively. For the remaining elements of R, we are completely uninformative, in the sense that we adopt ﬂat priors on the non-zero elements of its Cholesky factor. To assess whether the priors are reasonable, it is important to examine their implications in some detail. The natural way to examine the implications is to investigate what the priors imply about asset returns. So far, we have assumed a prior distribution pðhjnÞ for the parameters of interest, y, conditional on certain values for the hyper-parameters, x. It is possible to draw a random sample of values, say fhðmÞ ; m ¼ 1; . . . ; Mg from the prior pðhjnÞ, for some ﬁxed choice of x. For each h(m), we can use Eqs. (4) and (5) to draw samples ; t ¼ 1; . . . ; P Tg corresponding to each parameter draw. fRft ;ðmÞ ; Rq;ðmÞ t P f ;ðmÞ q;ðmÞ ¼ T 1 Tt¼1 R ft ;ðmÞ and R ¼ T 1 Tt¼1 Rq;ðmÞ , the sample Denote R t means of risk-free rate and equity return for each parameter draw. The joint q f f ;ðmÞ and distribution of R and R can be estimated using the draws R q;ðmÞ R ; m ¼ 1; . . . ; M. Clearly, this is an approximation to the prior predictive distribution of asset returns, and the approximation becomes better as the number of simulations, M, increases. Using our distributional assumptions on priors, and ﬁxing the discount factor to 0.98, we draw 5,000 random samples from the priors, to use them as parameters. We can use Eqs. (4) and (5) to evaluate the returns. From each evaluation, we save the mean of the equity and bond returns. The joint distributions of average returns are presented in Fig. 1. The message from these implied priors is that they are concentrated around large values of the equity premium and large values of the risk-free rate, a fact that is well known from the theoretical literature on asset pricing. We now proceed with presenting the results of posterior analysis.

628 VENI ARAKELIAN AND EFTHYMIOS G. TSIONAS

Fig. 1.

Contour Plots for Risk-Free Rate and Equity Returns.

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629

5. POSTERIOR ANALYSIS USING ARTIFICIAL DATA In order to investigate the performance of the new techniques, we will perform Bayesian inference for h using seven sets of artiﬁcial data (we refer to each dataset as a ‘‘case’’). The values of the parameters are chosen close to the ones reported in Labadie (1989), that is the risk aversion is set to 1, the discount factor to 0.95, the standard deviation to 0.035, and the constant (k) and the coefﬁcient (d) of the AR(1) endowment process, are set to 0.01 and 0.02, respectively. Generating artiﬁcial data is easy. We ﬁrst use Eq. (3) to generate yt. Asset returns are generated using the expressions (9) and (10). Our main concern has been the tolerance e required so that asset returns in Eq. (10) have indeed converged. In other words, the concern is about the number of terms, Mt(k) in Eq. (10). After extensive experimentation, values close to 105 were adequate. Additional evidence is provided in Appendix A. However, it must be noted that the series involved in Eq. (10) converge quite slowly. In principle, this is not a problem, but in practice it means that more computing time is required. All priors are assumed to be uniform distributions, that is we pretend, we are ‘‘completely uninformative’’ so to speak. We adopt a uniform prior for su as well. For the elements of R, we adopt a ﬂat prior on the different elements of its Cholesky factorization, R ¼ CuC. Adopting a Wishart prior for R would be difﬁcult, because our MCMC should be further restricted to ensure that R is positive deﬁnite, which is a horrendous task. We should, perhaps, point out that any prior on C – or the entire vector h for that matter – can be easily accommodated by our MCMC scheme. We have used 100,000 draws in total. The ﬁrst 50,000 are discarded to mitigate the impact of start up effects. Since our MCMC scheme produces correlated draws, for ﬁnal posterior computations we have retained only every other tenth draw. This mitigates the effect of autocorrelation. Finally, we have 5,000 draws in total. Standard convergence diagnostics (Geweke, 1993) show that our MCMC chains have converged, so they can be used with some degree of conﬁdence. The posterior distributions of the parameters are presented in Figs. 2–4. The posterior distributions, although not the same, they are not excessively different in terms of their fundamental characteristics. It should be mentioned that we do not expect robustness of the posterior in ﬁnite samples. Indeed, this is one major point in Bayesian inference.

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Fig. 2.

Fig. 3.

Posterior Distributions of Discount Factor, b.

Posterior Distributions of Relative Risk Aversion (RRA), g.

Bayesian Analysis of the Consumption CAPM

Fig. 4.

631

Posterior Distributions of Standard Deviation, su.

6. POSTERIOR ANALYSIS IN THE MEHRA AND PRESCOTT (1985) DATA Endowment, in the present model, can be measured using aggregate variables like consumption, GNP or dividends (see Labadie, 1989). This could be avoided in a model with production, but as noted in Mehra and Prescott (1985) production does not, by itself, solve the equity premium puzzle. In this paper, we use the annual data for per capita real consumption for the time period 1889–1977 of Grossman and Shiller (1981). The speciﬁc dataset was used since comparisons with previous studies are desirable (see Mehra & Prescott, 1985; Rietz, 1988; Labadie, 1989). Kandel and Stambaugh (1990) have used quarterly data but Mehra and Prescott (1985) report that their conclusions were not sensitive to the choice of data frequency. Deriving posterior distributions of asset returns is important in judging the model’s ability to reproduce key statistics. What is needed is to incorporate all available information in the posterior distributions of parameters to perform asset return inference. Dufﬁe and Singleton (1994) formalize the moment matching criterion implicit in Mehra and Prescott’s (1985) approach by using a simulated moments estimator. While extremely

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useful, this technique does not deliver distributions of asset returns in ﬁnite samples. The method adopted here (due to Meng, 1994) is the following: Given a sample fhðrÞ ; r ¼ 1; . . . ; Rg from the posterior distribution, p(h|X), asset returns Rq(h) and R f(h) in Eqs. (4) and (5), are functions of interest whose evaluation is possible for each draw. Speciﬁcally, given h(r), a set of S time series fyðsÞ t ; t ¼ 1; . . . ; ng is generated (for s ¼ 1; . . . ; S), asset returns RqðsÞ;t ðhðrÞ Þ and RfðsÞ;t ðhðrÞ Þ are computed for each date t, and each draw r, and their time series averages: q R ðhðrÞ Þ ¼ ðSTÞ1

S P T P s¼1 t¼1

f

R ðhðrÞ Þ ¼ ðSTÞ1

S P T P s¼1 t¼1

RqðsÞ;t ðhðrÞ Þ R0ðsÞ;t ðhðrÞ Þ

are computed for each r ¼ 1; . . . ; R. Standard kernel density procedures can then be used to estimate the posterior distribution of average asset returns. In this case, T ¼ 91 and S ¼ 500. To compute stock prices, qt, the inﬁnite PL sum ain ðLÞ l denote the series q ¼ Y Eq. (9) needs to be truncated. Let qðLÞ t t t l¼1 Al yt , that is stock prices computed with the inﬁnite sum truncated at L iterations. 5 qðLÞ We determine L so that jqðLþ1Þ t t j for all t ¼ 1; . . . ; T, and e ¼ 10 . This criterion is quite stringent since we are interested in absolute, not relative convergence of the series qðLÞ t . See Appendix A on how the value of e affects the joint posteriors of asset returns. We will use four different prior distributions:

ln ln ln ln

gBN(0.5, 1) and suBIG (4.77, 0.19) gBN(0.5, 1) and suBIG (2.11, 0.011) gBN(1.445, 1) and suBIG (2.11, 0.011) gBN(1.445, 1) and suBIG (4.77, 0.19).

We will, additionally, examine a ﬁfth case, where we assume that we are completely uninformative, adopting ﬂat priors for all the parameters. Marginal posterior distributions for the structural parameters of the model, are presented in Figs. 5–7. Since Case 4 turned out to be quite similar to Cases 1–3, we omit the posterior corresponding to Case 4 for clarity in presentation. Fig. 5 presents the posterior distributions of discount factor, b. The distribution clearly assigns almost no probability mass to values less than about 0.7, the mean is about 0.88 with values closer to 1 being more probable.

Bayesian Analysis of the Consumption CAPM

Fig. 5.

Marginal Posterior Distributions of Discount Factor, b.

Fig. 6.

Marginal Posterior Distributions of RRA Parameter, g.

633

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VENI ARAKELIAN AND EFTHYMIOS G. TSIONAS

Fig. 7.

Marginal Posterior Distributions of su.

Fig. 6 shows the posterior distributions of RRA parameter, g. These posteriors, assign almost no mass to values W5 and the means are close to 1.5. These posteriors provide considerable support for business cycle studies as well as for the calibration experiments of Mehra and Prescott (1985). The marginal posteriors of su, presented in Fig. 7, show that the posterior mean is about 0.015, a value close to what is commonly used in the literature.5 The results are not excessively sensitive to the prior distributions, a fact that may put at rest the view that prior information could dominate the data to produce artiﬁcial conclusions about parameters or functions of interest (like asset returns). The plots for the posterior distributions of b, g, and d, are extremely asymmetric. Consequently, it would be dangerous to rely on methods that depend on asymptotic approximations. One such method is GMM. The frequent phenomenon of rejecting the model when confronted with data in the context of GMM-based procedures, could be explained by the asymmetries evident in marginal posteriors of the ‘‘deep’’ structural parameters. Of course, in addition, we have to provide evidence that the joint posterior predictive distribution of asset returns is consistent with what we know from the data. In Figs. 8–11, we present the joint posterior predictive

Bayesian Analysis of the Consumption CAPM

Fig. 8. Marginal Posteriors of Asset Returns, Case 1.

Fig. 9. Marginal Posteriors of Asset Returns, Case 2.

635

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VENI ARAKELIAN AND EFTHYMIOS G. TSIONAS

Fig. 10.

Marginal Posteriors of Asset Returns, Case 3.

Fig. 11.

Marginal Posteriors of Asset Returns, Case 5.

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637

distributions of average real risk-free rate and equity return, computed using the simulation methodology that we described above. These posterior predictive distributions can be used to assess whether, and to what extent, there is an asset returns puzzle. Since the sample means 1 and 7% are not far out in the tails of the joint posterior, it turns out that the model is consistent with reality.

7. CONCLUSIONS In this paper, we have shown that Bayesian inferences for the structural parameters of asset pricing models can be routinely performed using available closed form solutions for asset returns. We have proposed priors for the structural parameters and examined their implications for asset returns. The methodological contribution is twofold. First, we use the closed form solutions for asset returns to propose an econometric model in the traditional sense. Second, we use Bayesian posterior predictive distributions of asset returns to draw conclusions about what the model really implies for the observed quantities of interest. We applied the new methods to artiﬁcial data as well as in the Mehra and Prescott (1985) celebrated dataset. It has been found that large equity premia and low risk-free rates are not incompatible with the posterior predictive distributions of asset returns resulting from the econometric model. The techniques should also be useful to sampling-theory researchers, since we provide an exact likelihood function in closed form, without the need to resort to questionable approximations or simulation.

NOTES 1. Mehra and Prescott (1985) advocate the use of calibration techniques to assess the model in terms of behavior of asset returns. Calibration is widespread practice in the real-business cycle and asset pricing literature. We do not wish to take a view on whether calibration is or is not useful. Our purpose is solely to explore the implications and capabilities of formal, exact, likelihood-based analysis of the asset pricing model. 2. What constitutes a ‘‘reasonable prior’’ is a matter of debate that still goes on and, most likely, it will go on in the foreseeable future. Here, our particular measure of ‘‘reasonableness’’ is based on implications of the given priors of ‘‘deep’’ parameters for important objects like asset returns. See Geweke (1989) for a presentation in a different context.

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3. In an absolute sense, the priors of k and d are obviously quite tight. However, given estimates obtained by Labadie (1989) and Mehra and Prescott (1985) these priors are ﬂat relative to the likelihood. The purpose of such priors is simply not to allow the prior dominate the likelihood. On this point, see also Note 2. 4. Clearly, most Bayesians would use an inverted gamma distribution for the variance rather than for the standard deviation su, in view of conjugacy. Given the non-linearity of the model there is no conjugacy here. We prefer to work directly with su since this parameter has attracted considerable discussion in the equity premium puzzle. See, however, Note 2 on this point. 5. The value 0.035 is commonly used. The value 0.05 could be interpreted as an upper bound when taking account of the fact that the consumption data is measured with error (Labadie, 1989).

ACKNOWLEDGMENTS We would like to thank Dek Terrell and an anonymous reviewer for their numerous constructive comments. Efthymios G. Tsionas wishes to acknowledge his intellectual debt to John F. Geweke for many of the ideas and implementations included here.

REFERENCES Altug, S., & Labadie, P. (1994). Dynamic choice and asset markets. New York: Academic Press. Boldrin, M., Christiano, L. L. J., & Fisher, J. D. M. (1991). Habit persistence, asset returns, and the business cycle. American Economic Review, 91, 149–166. Campbell, J. Y., & Cochrane, J. H. (1999). By force of habit: A consumption-based explanation of aggregate stock market behavior. Journal of Political Economy, 107, 205–251. Cecchetti, S., Lam, P., & Mark, N. (1993). The equity premium and the risk-free rate. Journal of Monetary Economics, 31, 21–45. Christiano, L. J. (1990). Linear-quadratic approximation and value function iteration. Journal of Business and Economic Statistics, 8, 99–113. Constantinides, G. (1990). Habit formation: A resolution of the equity premium puzzle. Journal of Political Economy, 98, 519–543. DeJong, D. N., Ingram, B. F., & Whiteman, C. H. (2000). A Bayesian approach to dynamic macroeconomics. Journal of Econometrics, 98, 203–223. Dufﬁe, D., & Singleton, K. J. (1994). Simulated moments estimation of Markov models of asset prices. Econometrica, 62, 929–952. Ferson, W., & Constantinides, G. (1991). Habit persistence and durability in aggregate consumption. Journal of Financial Economics, 29, 199–240. Gallant, A. R., & Tauchen, G. (1989). Seminonparametric estimation of conditionally constrained heterogeneous processes: Asset pricing applications. Econometrica, 57, 1091–1120.

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Gallant, A. R., & Tauchen, G. (1996). Which moments to match? Econometric Theory, 12, 657–681. Gelfand, A. E., & Smith, A. F. M. (1990). Sampling based approaches to calculating marginal densities. Journal of the American Statistical Association, 87, 523–532. Geweke, J. (1989). Exact predictive densities for linear models with ARCH disturbances. Journal of Econometrics, 40, 63–86. Geweke, J. (1993). Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments. In: J. O. Berger, M. Bernardo, A. P. Dawid & A. F. M. Smith (Eds.), Proceedings of the Fourth Valencia International Meeting on Bayesian Statistics (pp. 169–194). New York: Oxford University Press. Geweke, J. (1999). Computational experiments and reality. Manuscript, Department of Economics, University of Iowa, Iowa, IA. Grossman, S., & Shiller, R. J. (1981). The determinants of the variability of stock market prices. American Economic Review, 71, 222–227. Hansen, L., & Singleton, K. J. (1982). Generalized instrumental variables estimation of nonlinear rational expectations models. Econometrica, 50, 1269–1286. Hansen, L. P. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50, 1029–1054. Hansen, L. P., Heaton, J., & Yaron, A. (1996). Finite-sample properties of some alternative GMM estimators. Journal of the American Statistical Association, 14, 262–280. Hansen, L. P., & Sargent, T. (1980). Formulating and estimating dynamic linear rational expectations models. Journal of Economic Dynamics and Control, 2, 7–46. Hansen, L. P., & Singleton, K. J. (1983). Stochastic consumption, risk aversion, and the temporal behavior of asset returns. Journal of Political Economy, 91, 249–265. Ireland, P. (2004). A method of taking models to the data. Journal of Economic Dynamics and Control, 28, 1205–1226. Kandel, S., & Stambaugh, R. F. (1990). Expectations and volatility of consumption and asset returns. Review of Financial Studies, 3, 207–232. Kocherlakota, N. (1996). The equity premium puzzle: It’s still a puzzle. Journal of Economic Literature, 34, 42–71. Labadie, P. (1989). Stochastic inﬂation and the equity premium. Journal of Monetary Economics, 24, 277–298. Lee, B.-S., & Ingram, B. F. (1991). Simulation estimation of time-series models. Journal of Econometrics, 47, 195–205. Lettau, M., & Uhlig, H. (2000). Can habit formation be reconciled with business cycle facts? Review of Economic Dynamics, 3, 79–99. Mehra, R., & Prescott, E. C. (1985). The equity premium: A puzzle. Journal of Monetary Economics, 15, 145–162. Meng, X. L. (1994). Posterior predictive p-values. Annals of Statistics, 22, 1142–1160. Newey, W., & West, K. (1987). Hypothesis testing with efﬁcient method of moments estimation. International Economic Review, 28, 777–787. Ogaki, M. (1993). Generalized method of moments: Econometric applications. In: G. Maddala, C. Rao & H. Vinod (Eds), Handbook of Statistics: Econometrics (Vol. 11). Amsterdam: North-Holland. Pakes, A., & Pollard, D. (1989). Simulation and the asymptotics of optimization estimators. Econometrica, 57, 1027–1057. Rietz, T. A. (1988). The equity premium: A solution. Journal of Monetary Economics, 22, 117–131.

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Ruge-Murcia, F. J. (2007). Methods to estimate dynamic stochastic general equilibrium models. Journal of Economic Dynamics and Control, 31, 2599–2636. Schorfheide, F. (2000). Loss function-based evaluation of DSGE models. Journal of Applied Econometrics, 15, 645–670. Smith, A. A. (1993). Estimating nonlinear time-series models using simulated vector autoregressions. Journal of Applied Econometrics, 8, 6–84. Tauchen, G., & Hussey, R. (1991). Quadrature-based methods for obtaining approximate solutions to nonlinear asset pricing models. Econometrica, 59, 371–396. Tierney, L. (1992). Exploring posterior distributions using Markov chains. Computer Science and Statistics: Proceedings of the 23rd Symposium on the Interface, Fairfax Station, Interface Foundation (pp. 563–570). Another version is available in The Annals of Statistics, 22, 1701–1728 in 1994, with discussion. Tsionas, E. G. (2003). Exact solution of asset pricing models with arbitrary shock distributions. Journal of Economic Dynamics and Control, 27, 843–851. Tsionas, E. G. (2005). Likelihood evidence on the asset return puzzles. Review of Economic Studies, 72, 917–946. Uhlig, H. (1999). A toolkit for analyzing nonlinear dynamic stochastic models easily. In: R. Marimon & A. Scott (Eds), Computational methods for the study of dynamic economies (pp. 30–61). Oxford: Oxford University Press.

APPENDIX A. SENSITIVITY OF RESULTS TO VALUE OF e Figure A1 shows the sensitivity of the results with respect to the tolerance e used in order to truncate the inﬁnite sum that gives the stock prices. The results remain almost the same for the three last values of the tolerance, so setting e ¼ 105 should be sufﬁcient in practice.

APPENDIX B. THE JEFFREYS PRIOR An interesting question is the formulation of Jeffreys prior for this model. Given that the Jeffreys prior is analogue to the square of the determinant of Fisher’s information matrix and the complexity of the model due to its high non-linearity with respect to its parameters, we will adopt some simpliﬁcations. Assume that the variance–covariance matrix is known and diagonal and that the convergence of Labadie’s algorithm is succeeded within a single iteration. Therefore, the Jeffreys prior is given by the following: pðyÞ / jInfðyÞj1=2

Bayesian Analysis of the Consumption CAPM 641

Figure A1.

Sensitivity of Joint Priors of Asset Returns to the Value of Tolerance, e.

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VENI ARAKELIAN AND EFTHYMIOS G. TSIONAS

where

@2 log pðyjyÞ InfðyÞ ¼ E @yi @yj

and 2 1 InfðyÞ½1; 1 ¼ Ts1 11 ð1 gÞ s22 2 1 Inf y ½1; 1 ¼ Tmðs1 11 ð1 gÞ s22 Þ 1 2 1 Inf y ½1; 2 ¼ Tmðs1 11 þ ð1 gÞs22 þ g s33 Þ T 1 Inf y ½1; 3 ¼ ðs1 22 þ gs33 Þ b Inf y ½1; 4 ¼ Tððk þ ð1 gÞs22 þ dmÞð1 gÞs1 22 þ ðk gs11 þ dmÞgs33 Þ

Inf y ½2; 1 ¼ Tvðð1 gÞ2 s22 þ g2 s33 Þ Inf y ½2; 2 ¼ Tmðð1 gÞ2 s22 þ s11 þ g2 s33 Þ Tv Inf y ½2; 3 ¼ ðð1 gÞs22 þ gs33 Þ b Inf y ½2; 4 ¼ Tððk þ ð1 gÞs11 þ dmÞð1 gÞs22 þ ðk gs11 þ dmÞgs33 Inf y ½3; 1 ¼ 0:5Tððk þ ð1 gÞs11 dmÞðs11 s22 þ 2ðk gs11 Þs22 þ dmÞÞ þ ðk þ ð1 gÞs11 dmÞðs11 s33 þ 2s33 ðk gs11 þ dmÞÞÞ Inf y ½3; 2 ¼ 0:5Tbðs11 s33 þ 2ðk gs11 þ dmÞðs22 þ s33 ÞÞ Inf y ½3; 3 ¼ 0:5Tðð1 gÞðs11 s22 þ 2s22 ðk gs11 þ dmÞÞ þ gðs11 s33 þ 2s33 ðk gs11 þ dmÞÞÞ Inf y ½3; 4 ¼ 0:5Tðð1 gÞðs11 s22 þ 2s22 ðk gs11 þ dmÞÞðs22 þ s33 ÞÞ þ gmðs11 s33 þ 2s33 ðk gs11 þ dmÞÞÞ Inf y ½4; 1 ¼

T s22 b2

T ðð1 gÞs22 gs33 Þ b Tm Inf y ½4; 3 ¼ ðð1 gÞs22 gs33 Þ b T Inf y ½4; 4 ¼ ððk þ ð1 gÞ2 s11 þ dmÞs22 þ ðk gs11 þ dmÞs33 Þ b Inf y ½4; 2 ¼

Bayesian Analysis of the Consumption CAPM

643

where m Eðln lt Þ ¼ k=ð1 dÞ; v Eðln l2t Þ ¼ s11 =ð1 dÞ2 þ k2 =ð1 d2 Þ The ﬁgure below shows the Jefreys prior evaluated for ﬁxed parameters of the AR(1) that drives the endowment process ðk ¼ 0:017165; d ¼ 0:018145Þ.

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