Unit Root Tests in Time Series Volume 1: Key Concepts and Problems, Volume 1

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Unit Root Tests in Time Series Volume 1

10.1057/9780230299306 - Unit Root Tests in Time Series Volume 1, Kerry Patterson

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Palgrave Texts in Econometrics General Editor: Kerry Patterson Titles include: Simon P. Burke and John Hunter MODELLING NON-STATIONARY TIME SERIES Michael P. Clements EVALUATING ECONOMETRIC FORECASTS OF ECONOMIC AND FINANCIAL VARIABLES

Terence C. Mills MODELLING TRENDS AND CYCLES IN ECONOMIC TIME SERIES Kerry Patterson A PRIMER FOR UNIT ROOT TESTING Kerry Patterson UNIT ROOT TESTS IN TIME SERIES VOLUME 1 Key Concepts and Problems

Palgrave Texts in Econometrics Series Standing Order ISBN 978–1–4039–0172–9 (hardback) 978–1–4039-0173–6 (paperback) (outside North America only) You can receive future titles in this series as they are published by placing a standing order. Please contact your bookseller or, in case of difﬁculty, write to us at the address below with your name and address, the title of the series and the ISBNs quoted above. Customer Services Department, Macmillan Distribution Ltd, Houndmills, Basingstoke, Hampshire RG21 6XS, England

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Lesley Godfrey BOOTSTRAP TESTS FOR REGRESSION MODELS

Unit Root Tests in Time Series Volume 1 Key Concepts and Problems

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Kerry Patterson

10.1057/9780230299306 - Unit Root Tests in Time Series Volume 1, Kerry Patterson

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c Kerry Patterson 2011 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6-10 Kirby Street, London EC1N 8TS.

The author has asserted his right to be identiﬁed as the author of this work in accordance with the Copyright, Designs and Patents Act 1988. First published 2011 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, of Houndmills, Basingstoke, Hampshire RG21 6XS. Palgrave Macmillan in the US is a division of St Martin’s Press LLC, 175 Fifth Avenue, New York, NY 10010. Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world. R R and Macmillan are registered trademarks in the United States, Palgrave the United Kingdom, Europe and other countries ISBN: 978–0–230–25024–6 hardback This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin. A catalogue record for this book is available from the British Library. Library of Congress Cataloging-in-Publication Data Patterson, K. D. Unit root tests in time series volume 1 : key concepts and problems / Kerry Patterson. p. cm. Includes bibliographical references and index. ISBN 978–0–230–25024–6 1. Econometrics. 2. Time-series analysis. I. Title. HB139.P374 2010 519.5 5—dc22

2010027477

10 9 8 7 6 5 4 3 2 1 20 19 18 17 16 15 14 13 12 11 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne

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Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages.

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To Bella and a continuing treasure at Auton Farm

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Detailed Contents List of Tables List of Figures Symbols and Abbreviations Preface 1 Introduction to Random Walks and Brownian Motion 2 Why Distinguish Between Trend Stationary and Difference Stationary Processes? 3 An Introduction to ARMA models 4 Bias and Bias Reduction in AR Models 5 Conﬁdence Intervals in AR Models 6 Dickey-Fuller and Related Tests 7 Improving the Power of Unit Root Tests 8 Bootstrap Unit Root Tests 9 Lag Selection and Multiple Tests 10 Testing for Two (or More) Unit Roots 11 Tests with Stationarity as the Null Hypothesis 12 Combining Tests and Constructing Conﬁdence Intervals 13 Unit Root Tests for Seasonal Data Appendix 1: Random Variables; Order Notation Appendix 2: The Lag Operator and Lag Polynomials References Author Index Subject Index

ix xxii xxvii xxxii xxxiii 1 38 68 123 158 189 260 319 348 385 434 497 519 597 603 619 633 637

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Contents

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Detailed Contents xxii xxvii xxxii xxxiii

1 Introduction to Random Walks and Brownian Motion 1.1 Random walks 1.1.1 The random walk as a partial sum process (psp) 1.1.2 Random walks: visits to the origin (sign changes and reﬂections) 1.1.3 Random walk: an example of a stochastic process 1.1.4 Random walk: an example of a nonstationary process 1.1.4.i A strictly stationary process 1.1.4.ii Weak or second-order stationarity (covariance stationarity) 1.1.4.iii The variance of a random walk increases over time 1.1.4.iv The autocovariances of a random walk are not constant 1.1.5 A simple random walk with Gaussian inputs 1.1.6 Variations on the simple random walk 1.1.7 An empirical illustration 1.1.8 Finer divisions within a ﬁxed interval: towards Brownian motion 1.2 Deﬁnition of Brownian motion 1.3 Functional central limit theorem (FCLT) 1.4 Continuous mapping theorem (CMT) 1.5 Background to unit root and related tests 1.5.1 What is a unit root? 1.5.1.i Generalising the random walk 1.5.1.ii Integrated of order d: the I(d) notation 1.5.2 The development of unit root tests 1.6 Are unit root processes of interest? 1.6.1 Are there constant ‘great ratios’? 1.6.2 Purchasing power parity 1.6.3 Asset returns 1.7 Concluding remarks Questions ix

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List of Tables List of Figures Symbols and Abbreviations Preface

Detailed Contents

2 Why Distinguish Between Trend Stationary and Difference Stationary Processes? 2.1 What accounts for the trend? 2.2 Inappropriate detrending and inappropriate ﬁrst differencing 2.2.1 Spurious detrending 2.2.1.i Limit distributions and test statistics 2.2.1.ii Spurious periodicity 2.2.2 Spurious ﬁrst differencing 2.2.2.i The over-differencing effect 2.2.2.ii Implications of over-differencing for the spectral density function 2.3 Persistence and the impact of shocks 2.3.1 The role of shocks 2.3.1.i A simple model 2.3.1.ii More complex models 2.4 Measures of the importance of the unit root 2.4.1 The variance ratio for a simple random walk 2.4.2 Interpretation in terms of autocorrelations and spectral density 2.4.3 Interpretation in terms of a permanent-temporary (P-T) decomposition 2.4.4 Interpretation in terms of unconditional and conditional variances 2.4.5 The relationship between the persistence measure and the variance ratio 2.5 Illustration: US GNP 2.6 Unit root tests as part of a pre-test routine 2.6.1 Spurious regression 2.7 Concluding Remarks Questions 3 An Introduction to ARMA Models 3.1 ARMA(p, q) models 3.1.1 Deterministic terms 3.1.2 The long run and persistence 3.2 Roots and stability 3.2.1 The modulus of the roots of the AR polynomial must lie outside the unit circle 3.2.2 The special case of unit roots 3.2.2.i Unit roots in the AR polynomial 3.2.2.ii A notational convention 3.2.2.iii Unit roots in the MA polynomial

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Inﬁnite moving average representation of the ARMA model 3.3.1 Examples 3.3.2 The general MA representation 3.3.3 Inversion of the MA polynomial 3.4 Approximation of MA and ARMA models by a ‘pure’ AR model 3.5 Near-cancellation of roots 3.6 The Beveridge-Nelson (BN) decomposition of a lag polynomial 3.6.1 The BN decomposition of a lag polynomial 3.6.2 The BN decomposition for integrated processes (optional on ﬁrst reading) 3.6.2.i The BN decomposition in terms of the ARMA lag polynomials 3.6.2.ii Precise and efﬁcient computation of the BN decomposition 3.6.2.iii Illustrations of the BN decomposition 3.7 The Dickey-Fuller (DF) decomposition of the lag polynomial and the ADF model 3.7.1 The DF decomposition 3.8 Different ways of writing the ARMA model 3.8.1 Alternative ways of representing the dynamics 3.8.1.i An error dynamics model 3.8.1.ii Common factor model 3.8.1.iii Direct speciﬁcation 3.8.2 A pseudo t test of the null hypothesis, γ = 0: τˆ 3.8.3 DF n-bias tests: δˆ 3.9 An outline of maximum likelihood (ML) estimation and testing for a unit root in an ARMA model 3.9.1 ML estimation 3.9.1.i The ARMA model (Shin and Fuller, 1998) 3.9.1.ii The log-likelihood functions 3.9.1.iii AR(1), stationary case with niid errors 3.9.2 Likelihood-based unit root test statistics 3.9.2.i ML: conditional and unconditional approaches 3.9.3 Estimation, critical values and ﬁnite sample performance 3.9.3.i Estimation 3.9.3.ii Asymptotic distributions 3.9.3.iii Critical values and estimation 3.9.3.iv Finite sample performance 3.10 Illustration: UK car production 3.10.1 Estimated model and test statistics 3.10.2 The BN decomposition of the car production data

76 76 77 78 79 80 81 81 83 83 86 87 89 90 91 92 92 94 96 98 98 100 100 100 101 102 104 105 107 107 108 108 109 110 111 112

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3.3

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3.11 Concluding remarks Questions

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4 Bias and Bias Reduction in AR Models 4.1 Finite sample bias of the LS estimator 4.1.1 The bias to order T 4.1.2 First-order bias and the long-run multiplier 4.2 Bias reduction 4.2.1 Total bias and ﬁrst-order bias 4.2.1.i Total bias 4.2.1.ii First-order bias 4.2.2 First-order unbiased estimators in the AR model 4.2.3 Simulating the bias 4.2.3.i Illustration of linear bias correction using the AR(1) model 4.2.4 Constant bias correction 4.2.5 Obtaining a linear bias corrected estimator 4.2.6 Linear bias correction in AR(p) models, p ≥ 2 4.2.7 The connection 4.2.8 The variance and mean squared error of the FOBC estimators 4.2.8.i Mean squared error comparison 4.3 Recursive mean adjustment 4.3.1 AR(1) model 4.3.1.i Constant mean 4.3.1.ii Trend in the mean 4.3.2 Extension to AR(p) models 4.3.2.i Constant mean, AR(p) models 4.3.2.ii Trend in the mean, AR(p) models 4.4 Bootstrapping 4.4.1 Bootstrapping to reduce bias 4.5 Results of a simulation: estimation by least squares, ﬁrst-order bias correction and recursive mean adjustment 4.6 Illustrations 4.6.1 An AR(1) model for US ﬁve-year T-bond rate 4.6.2 An AR(2) model for US GNP 4.7 Concluding remarks Questions 4.8 Appendix: First-order bias in the constant and linear trend cases

123 124 124 127 128 129 129 129 129 131

5 Conﬁdence Intervals in AR models 5.1 Conﬁdence intervals and hypothesis testing 5.1.1 Conﬁdence intervals

158 160 160

132 133 133 134 135 137 138 138 139 139 140 141 141 142 142 142 145 147 147 149 150 151 155

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5.3

5.4

5.5

5.6

5.1.2 The link with hypothesis testing The quantile function 5.2.1 Constant quantiles 5.2.2 Nonconstancy of the quantiles 5.2.3 Complications in the AR case 5.2.3.i Simulation of the LS quantiles 5.2.3.ii Empirical example: an AR(2) model for US GNP Constructing conﬁdence intervals from median unbiased estimation 5.3.1 Inverse mapping and median unbiased estimation 5.3.2 Conﬁdence intervals based on median unbiased estimation 5.3.3 Extension of the median unbiased method to general AR(p) models Quantiles using bias adjusted estimators 5.4.1 Quantiles 5.4.2 Coverage probabilities 5.4.3 Section summary Bootstrap conﬁdence intervals 5.5.1 Bootstrap conﬁdence intervals 5.5.1.i The bootstrap percentile-t conﬁdence interval 5.5.1.ii The grid-bootstrap percentile-t conﬁdence interval Concluding remarks Questions

6 Dickey-Fuller and Related Tests 6.1 The basic set-up 6.1.1 An AR(1) process 6.1.1.i Back-substitution 6.1.1.ii Variances 6.1.1.iii Weakly dependent errors 6.1.2 The LS estimator 6.1.3 Higher-order AR models 6.1.4 Properties of the LS estimator, stationary AR(p) case 6.2 Near-unit root case; simulated distribution of test statistics 6.3 DF tests for a unit root 6.3.1 First steps: formulation and interpretation of the maintained regression 6.3.2 Three models 6.3.3 Limiting distributions of δˆ and τˆ test statistics

162 164 164 165 165 166 167 169 169 173 175 175 175 177 178 179 179 179 182 184 186 189 190 190 190 192 193 195 197 199 200 203 203 205 207

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xiv Detailed Contents

Obtaining similar tests 6.3.4.i Similarity and invariance in error dynamics/common factor approach 6.3.4.ii Similarity and invariance in the direct (DF) speciﬁcation 6.3.5 Continuous record asymptotics 6.3.6 The long-run variance as a nuisance parameter 6.3.7 Joint (F-type) tests in the DF representation 6.3.8 Critical values: response surface approach 6.4 Size and power 6.5 Nonlinear trends 6.6 Exploiting the forward and reverse realisations 6.6.1 DF-max tests (Leybourne, 1995) 6.6.2 Weighted symmetric tests 6.7 Non-iid errors and the distribution of DF test statistics 6.7.1 Departures from iid errors 6.7.1.i Distinguishing variances 6.7.1.ii Effects on the limit distributions 6.7.1.iii Expectations of the limit distributions 6.7.1.iv Effect of non-iid errors on the size of the tests 6.7.2 Illustrations 6.7.2.i MA(1) errors 6.7.2.ii AR(1) errors 6.7.2.iii Effect of non-iid errors on the size of DF tests 6.8 Phillips-Perron (PP) semi-parametric unit root test statistics 6.8.1 Adjusting the standard test statistics to obtain the same limit distribution 6.8.2 Estimation of the variances 6.8.2.i The unconditional variance, σz2 6.8.2.ii Estimating the long-run variance 6.8.3 Modiﬁed Z tests 6.8.4 Simulation results 6.8.4.i PP tests, Zˆρµ and Zˆτµ , and modiﬁed PP tests, MZˆρµ and MZˆτµ 6.9 Power: a comparison of the ADF, WS and MZ tests 6.10 Concluding remarks Questions 6.11 Appendix: Response surface function 7 Improving the Power of Unit Root Tests 7.1 A GLS approach 7.1.1 GLS estimation

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6.3.4

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7.2

7.3

7.4

Feasible GLS 7.1.2.i Simple examples 7.1.3 Approximating an MA process by an AR model 7.1.3.i Using the AR version of FGLS 7.1.3.ii Using the MA version of FGLS via an AR approximation 7.1.3.iii ARMA errors FGLS unit root tests 7.2.1 DF-type tests based on FGLS 7.2.2 GZW unit root test (ARMA variation) Illustration 7.3.1 US Unemployment rate 7.3.1.i Standard ADF estimation 7.3.1.ii PP tests 7.3.1.iii FGLS (GZW version) Tests that are optimal for particular alternatives (point-optimal tests) 7.4.1 Likelihood ratio test based on nearly-integrated alternatives 7.4.1.i Known deterministic components 7.4.1.ii Unknown deterministic components 7.4.1.iii A likelihood-based test statistic, with estimated trend coefﬁcients 7.4.1.iv The power envelope 7.4.2 A family of simple-to-compute test statistics 7.4.2.i The PT tests 2 7.4.2.ii Estimating σz,lr 7.4.3 GLS detrending and conditional and unconditional tests 7.4.4 Unconditional case: the QT family of test statistics 7.4.5 Exploiting the quasi-differencing interpretation 7.4.6 ADF tests using QD data 7.4.7 Critical values 7.4.8 Illustration of a power envelope and tangential power functions 7.4.8.i Graphical illustrations 7.4.8.ii Power gains? 7.4.9 Sensitivity to the initial condition 7.4.10 A test statistic almost invariant to the initial condition 7.4.11 Weighted test statistics to achieve robustness to the initial condition 7.4.11.i Combining test statistics 7.4.11.ii Power and the initial condition

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7.1.2

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7.6

7.4.12 Illustration: US industrial production Detrending (or demeaning) procedure by recursive mean adjustment Concluding remarks Questions

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8 Bootstrap Unit Root Tests 8.1 Bootstrap schemes with an exact unit root 8.1.1 The random inputs to the bootstrap replications 8.1.2 A bootstrap approach to unit root tests 8.1.3 The AR(1) case 8.1.3.i The AR(1) case: no constant 8.1.3.ii Variations due to different speciﬁcations of the deterministic terms 8.1.4 Extension to higher-order ADF/AR models 8.1.5 Sieve bootstrap unit root test 8.1.5.i Chang and Park sieve bootstrap 8.1.5.ii Psaradakis bootstrap scheme 8.2 Simulation studies of the bootstrap unit root tests 8.2.1 Chang and Park 8.2.2 Psaradakis 8.2.2.i Summary of results 8.2.2.ii Lag truncation criteria 8.2.3 A brief comparison of the methods of Chang and Park and Psaradikis 8.2.3.i AIC versus marginal-t as lag selection criteria 8.2.3.ii Summary 8.2.4 ‘Asymptotic’ simulations 8.3 Illustration: US unemployment rate 8.4 Concluding remarks Questions

319 321 322 323 323 324

9 Lag Selection and Multiple Tests 9.1 Selection criteria 9.1.1 ADF set-up: reminder 9.1.2 Illustration of the sensitivity of size and power to lag length selection 9.1.3 Consistency and simulation lag selection criteria 9.1.4 Lag selection criteria 9.1.4.i Fixed lag 9.1.4.ii Choose k∗ as a function of T 9.1.4.iii Data-dependent rules 9.1.5 Simulation results

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7.5

9.2

9.3 9.4

9.1.5.i MA(1) errors 9.1.5.ii AR(1) errors 9.1.5.iii Lag lengths Multiple tests 9.2.1 Test characteristics 9.2.2 Deﬁning the indicator sets 9.2.3 Overall type I error 9.2.4 Test power and test dominance 9.2.4.i Test dependency function 9.2.4.ii Test conﬂict function 9.2.5 Illustrations: different test statistics and different lag selection criteria 9.2.5.i Illustration 1: a comparison of τˆ µ and τˆ ws µ 9.2.5.ii Illustration 2: using different lag selection methods Empirical illustration Concluding remarks Questions

10 Testing for Two (or More) Unit Roots 10.1 Preliminaries 10.1.1 I(2) characteristics 10.1.2 Illustrative series 10.1.3 Alternatives to I(2) 10.1.4 Some factorisations and restrictions 10.2 The Hasza-Fuller (HF) decomposition of a lag polynomial 10.2.1 Derivation of the HF decomposition 10.2.1.i Example 1 10.2.1.ii Numerical example (continuation) 10.2.1.iii Example 2 10.3 DF test as a second unit root is approached 10.4 Testing the null hypothesis of two unit roots 10.4.1 The HF test 10.4.1.i Direct demeaning 10.4.1.ii Initial conditions 10.4.2 Critical values for HF F-type tests 10.4.3 Sen-Dickey version of the HF test 10.4.3.i Double-length regression for the SD symmetric test statistic 10.4.3.ii SD F-type test, p = 2 10.4.3.iii SD F-type test, p > 2 10.4.3.iv Deterministic components

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xviii Detailed Contents

11 Tests with Stationarity as the Null Hypothesis 11.1 A structural time series model interpretation 11.1.1 The local level model 11.1.2 The importance of the limits 11.1.3 The ‘ﬂuctuations’ testing framework 11.2 Tests for stationarity 11.2.1 The KPSS test 11.2.2 Modiﬁed rescaled range test statistic (MRS) (Lo, 1991) 11.2.3 Komogoroff-Smirnoff-type test (KS) (Xiao, 2001) 11.2.4 A Durbin-Watson-type test (SBDH) (Choi and Ahn, 1999) 11.3 A test with parametric adjustment 11.3.1 The Leybourne and McCabe (1994) test 11.3.2 The modiﬁed Leybourne and McCabe test 11.3.3 Data-dependent lag selection for the LBM test(s) 11.4 The long-run variance 11.4.1 Estimation of the long-run variance 11.4.2 Illustrative simulation results 11.5 An evaluation of stationarity tests 11.5.1 Selection of the bandwidth parameter 11.5.2 Summary tables of limiting distributions and critical values 11.5.3 Empirical size 11.5.3.i Illustrative simulations 11.5.3.ii Looking at the quantiles: comparing empirical distribution functions

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10.5 The Dickey-Pantula (DP) approach 10.5.1 An extension of the DF procedure 10.5.2 Numerical illustration 10.5.3 Start from the highest order of integration 10.6 Power 10.6.1 Asymptotic local power 10.6.2 Small sample simulation results 10.6.2.i SD comparison of the HF and SD versions of the F test 10.6.2.ii Illustrative simulations: F test and DP tests 10.7 Illustrations of testing for two unit roots 10.7.1 Illustration 1: the stock of US consumer credit 10.7.2 Illustration 2: CPI Denmark 10.8 Concluding remarks Questions

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486 486 487 491 494

12 Combining Tests and Constructing Conﬁdence Intervals 12.1 The importance of the initial condition 12.2 Problems with stationarity tests for highly correlated series 12.3 Conﬁdence intervals revisited 12.3.1 Inverting the PT unit root test statistic to obtain a conﬁdence interval 12.3.2 Constructing a conﬁdence interval using DF tests with GLS detrended data 12.3.2.i Test inversion 12.3.2.ii Illustration using time series on US GNP 12.4 Concluding remarks Questions

497 498 500 503

13 Unit Root Tests for Seasonal Data 13.1 Seasonal effects illustrated 13.2 Seasonal ‘split’ growth rates 13.3 The spectral density function and the seasonal frequencies 13.3.1 Frequencies, periods and cycles 13.3.2 Power spectrum and periodogram 13.3.3 Illustrations (randomising the input) 13.3.4 Aliasing (artiﬁcial cycles) 13.3.5 Seasonal integration from a frequency domain perspective 13.4 Lag operator algebra and seasonal lag polynomials 13.4.1 The seasonal operator in terms of L 13.5 An introduction to seasonal unit root tests: the DHF (1984) test 13.5.1 The seasonal DGP 13.5.1.i No deterministic components

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11.5.4 The power of tests for stationarity against the unit root alternative 11.5.4.i Power against a ﬁxed alternative 11.5.4.ii Power against local alternatives 11.6 Illustrations: applications to some US consumer price indices 11.6.1 US CPI (aggregate series) 11.6.2 US regional consumer prices: using tests of nonstationarity and stationarity in testing for convergence 11.6.2.i Framework for tests 11.6.2.ii Consumer prices for two regions of the US 11.7 Concluding remarks Questions

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13.6

13.7 13.8

13.9

13.5.1.ii Seasonal intercepts 13.5.1.iii Seasonal deterministic trends 13.5.2 Limiting distributions HEGY tests 13.6.1 The HEGY regression 13.6.2 The constructed variables and the seasonal frequencies 13.6.3 The structure of hypothesis tests 13.6.3.i Tests for the roots +1, −1 13.6.3.ii Tests for unit roots at the harmonic seasonal frequencies, λs , s = 0, s = S/2 13.6.3.iii Overall tests 13.6.4 The HEGY tests: important special cases 13.6.4.i Quarterly data 13.6.4.ii Monthly data 13.6.5 Limiting distributions and critical values 13.6.5.i Monthly case 13.6.5.ii Quarterly case 13.6.5.iii Nuisance parameters 13.6.5.iv Empirical quantiles 13.6.6 Multiple testing 13.6.7 Lag augmentation Can the (A)DF test statistics still be used for seasonal data? Improving the power of DHF and HEGY tests 13.8.1 Recursive mean adjustment for seasonal unit root tests 13.8.2 Improving power with monthly data 13.8.2.i Systematic sampling 13.8.2.ii QM-HEGY 13.8.2.iii Choice of test statistic for quarterly data 13.8.2.iv Critical values 13.8.2.v Illustration: US industrial production Finite sample results, DHF and HEGY 13.9.1 Power, initial assessment 13.9.2 Not all roots present under the null 13.9.2.i AR(1) DGP 13.9.2.ii Other DGPs: one cycle and two cycles per year 13.9.3 Extension to include RMA versions of the HEGY and DHF tests 13.9.4 Size retention

540 542 543 543 545 546 547 547 547 548 548 548 550 552 553 555 555 556 558 561 561 562 563 564 564 564 566 567 567 568 569 570 572 574 577 578

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Detailed Contents xxi

582 582 582 586 587 590 590 592 593 594

Appendix 1: Random Variables; Order Notation A1.1 Discrete and continuous random variables A1.2 The (cumulative) distribution function, cdf A1.2.1 CDF for discrete random variables A1.2.2 CDF for continuous random variables A1.3 The order notation: ‘big’ O, ‘small’ o

597 597 598 598 599 602

Appendix 2: The Lag Operator and Lag Polynomials A2.1 The lag operator, L A2.2 The lag polynomial A2.3 Differencing operators A2.4 Roots A2.4.1 Solving for the zeros A2.4.2 Graphical representation of the roots A2.4.3 Roots associated with seasonal frequencies A2.5 The inverse of a lag polynomial and the summation operator A2.5.1 The inverse lag polynomial A2.5.2 The summation operator ∆−1 as an inverse polynomial A2.6 Stability and the roots of the lag polynomial A2.7 Some uses of the lag operator A2.7.1 Removing unit roots A2.7.2 Data ﬁlter A2.7.3 Summing the lag coefﬁcients A2.7.4 Application to a (deterministic) time trend Questions

603 603 603 604 605 605 605 608 609 609

References Author Index Subject Index

619 633 637

610 610 612 612 613 614 615 615

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13.10 Empirical illustrations 13.10.1 Illustration 1, quarterly data: employment in US agriculture 13.10.1.i A linear trend alternative 13.10.1.iiA nonlinear trend alternative 13.10.2 Illustration 2, monthly data: UK industrial production 13.11 Some other developments 13.11.1 Periodic models 13.11.2 Stationarity as the null hypothesis 13.12 Concluding remarks Questions

List of Tables

Number of citations of key articles on unit roots

1.1

Contrasting properties of I(0) and I(1) series

26

2.1 2.2 2.2a 2.2b 2.3 2.3a 2.3b

Errors of speciﬁcation Estimated ARMA models for US GNP (real) ARMA (3, 2) model for yt ARMA (2, 2) model for ∆yt Roots of the polynomials of the ARMA models ARMA (3, 2) model for yt ARMA (2, 2) model for ∆yt

41 58 58 58 58 58 58

3.1

Finite sample critical values for conditional and unconditional LR-type test statistics LS, CML and ML: illustrative simulation results, MA(1) errors, T = 100 ARMA models for UK car production, coefﬁcient estimates Test statistics for a unit root, car production data

3.2 3.3 3.4

xxxiv

109 110 112 112

4.6 4.7

First-order least squares bias: limT→∞ T[E(φˆ i − φi )] Constant included in regression Constant and trend included in regression Total bias of LS estimators and ﬁxed points Constant included in regression Constant and trend included in regression A summary of the abbreviations used in the following sections Bias correction in an AR(1) model LS and bias adjusted estimates of an AR(2) model for US ﬁve-year T-bond LS and bias adjusted estimates of an AR(2) model for US GNP Roots for the zero bias case: constant included in regression

148 150 153

5.1 5.2

Quantiles to obtain a median unbiased estimator Simulated coverage probabilities

173 178

4.1 4.1a 4.1b 4.2 4.2a 4.2b 4.3 4.4 4.5

126 126 126 126 126 126 128 146

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P.1

6.1 6.2 6.3 6.4 6.5 6.6 6.6a 6.6b 6.7 6.8 6.9 6.10a 6.10b 6.11 6.11a 6.11b 6.12 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11

8.1

Percentage of times that tρ < −1.645 Summary of simple maintained regressions: error dynamics/common factor approach Summary of maintained regressions: DF representations Null hypothesis for DF joint F-type tests Illustration of response surface coefﬁcients Illustrative size and power comparisons of DF-type tests MA(1) error structure: zt = (1 + θ1 L)εt AR(1) error structure: (1 − ϕ1 L)zt = εt ˆ max Critical values of the DF-max tests, τˆ max µ and τ β Effect of non-iid errors on means of limit distribution of test statistics Effect on (asymptotic) size of DF tests with non-iid errors Effect of MA(1) errors on the size of PP-type tests, 5% nominal size Effect on AR(1) errors on the size of PP-type tests, 5% nominal size Illustrative size and power comparisons of DF-type tests MA(1) error structure AR(1) error structure Response surface coefﬁcients Summary of GLS matrices, general and special cases Size and power comparisons: CGLS and ML test statistics, AR(1) error structure Estimation results for US unemployment, Nelson and Plosser data Summary of unit root test values for US unemployment data Critical values (ﬁxed initial condition) Critical values (initial condition: drawn from unconditional distribution) ˆ (i) (g, λ) Critical values for Q Critical values for τˆ DFG , τˆ DFQ i i rma Critical values for τˆ µ and τˆ rma β Estimation results for US unemployment Summary of unit root test values for US unemployment data Simulated size and power of the Chang and Park ADF tests

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203 206 207 215 216 218 218 219 223 230 234 245 246 248 248 248 257 264 275 277 278 295 295 303 305 311 314 314 337

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List of Tables

8.2

8.3 8.3a 8.3b 8.3c 8.4 8.5 8.6 9.1 9.2 9.3 9.4 9.5 9.6 9.7

10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8

List of Tables

Simulated size (ρ = 1) and power (ρ = 0.9) of Psaradakis bootstrap tests for 5% nominal size, T = 100; θ1 = –0.8 throughout Comparison of bootstrapping procedures Simulated size: θ1 = 0 and ρ = 1; simulated power: θ1 = 0 and ρ = 0.95, θ1 = 0 and ρ = 0.9 Simulated size: θ1 = –0.4 and ρ =1; simulated power: θ1 = –0.4 and ρ = 0.95, θ1 = –0.4 and ρ = 0.9 Simulated size: θ = –0.8 and ρ = 1; simulated power: θ1 = –0.8 and ρ = 0.95, θ1 = –0.8 and ρ = 0.9 ‘Aysmptotic’ size: θ1 = –0.8 and ρ = 1 ML estimation of ARIMA(2, 0, 2) for US unemployment rate Test statistics for US unemployment rate MA(1) errors: empirical size and power of τˆ µ using different lag selection criteria AR(1) errors: size and power of τˆ µ , different lag selection criteria Intersection of sets using different lag selection criteria, % same choice, MA(1) Intersection of sets using different lag selection criteria, % same choice, AR(1) Joint probabilities for two test statistics Probabilities under independence of tests given H0 Illustrative probabilities if tests are distributed as joint normal, α = 0.05, with dependency indicated by η A taxonomy of possibilities in the two root case Critical values for HF F-type tests Empirical power of tests for two unit roots; data directly demeaned assuming a constant mean Empirical power of tests for two unit roots; data directly demeaned by a constant and linear trend Empirical power of tests for two unit roots; data directly demeaned by a constant, linear trend and a quadratic trend Number of unit roots; estimation details for log consumer credit, US Number of unit roots; estimation details for log of CPI, Denmark Dependent and explanatory variables in the simple symmetric test of Sen and Dickey

338 340 340 340 341 342 343 344

363 364 366 366 369 370 372 392 406 418 419 420 423 427 432

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List of Tables xxv

11.3 11.4 11.5 11.6a 11.6b 11.7 11.8 11.9 11.10 12.1 12.2 12.3 12.4 12.5 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 13.10 13.11 13.12 13.13 13.14

Limiting distributions of test statistics for stationarity Quantiles of null distributions for stationary tests (critical values) Effect of lag selection on quantiles of LBM(µ) Stationarity tests: empirical size for 5% nominal size AR(1) errors Stationarity tests: empirical size for 5% nominal size, MA(1) errors LBM tests, US CPI (logs) Stationarity tests, US CPI (logs) Long-run variance estimates and bandwidth parameters Unit roots tests on US regional CPIs; (log) levels, N-E and W US regional inﬂation contrast: LBM(µ) tests for stationarity US regional inﬂation contrast: tests for stationarity

459

Sensitivity of empirical size to the initial condition Upper quantiles of unit root test statistics when used for staionarity The roles of different values of c and ρ H0 : ρ = ρ∗ against HA : ρ = ρ¯ or, equivalently, c = c∗ against c = c¯ Stationarity tests for US GNP

500

Calculation of seasonal ‘split’ growth rates: ∆1 yn,s = yn,s – yn,s−1 Aliased frequencies: monthly data sampled at quarterly intervals Seasonal test statistics distinguished by mean and trend components Quantiles for DFH tests Seasonal frequencies, periods and roots: the quarterly case Constructed variables and seasonal frequencies: the quarterly case Hypothesis tests: the quarterly case Seasonal frequencies, periods and roots: the monthly case Constructed variables and seasonal frequencies: the monthly case Hypothesis tests: the monthly case Limiting null distributions of HEGY test statistics HEGY critical values: quarterly data HEGY critical values: monthly data Matrix of estimated pairwise probabilities (monthly tests)

460 461 462 463 485 485 485 490 491 492

504 505 508 512

525 533 542 544 549 549 549 552 553 553 554 557 559 561

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11.1 11.2

List of Tables

13.15 13.16 13.17 13.18 13.19 13.20 13.21 13.22 13.23 13.24

Some critical values for QM-HEGY, 5% tests (4% quarterly, 1% monthly) Summary of results for US industrial production Size and power of tests Size of tests: AR(1) errors Size of tests: MA(4) errors Average lags for AR(1) errors and MA(4) errors Test statistics for unit roots for US agricultural employment Test statistics for unit roots for UK industrial production (monthly) Constructed variables and seasonal frequencies for the weekly case The structure of hypotheses: the weekly case

567 568 578 580 581 582 585 589 596 596

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1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

Sample paths of a binomial random walk Distribution function of visits to the origin Sample paths of a Gaussian random walk Sample paths of asymmetric binomial random walks Sample paths of drifted symmetric binomial random walks Exchange rate (daily), SWFR:£ Scatter graph of daily, SWFR:£ Gaussian random walk as time divisions approach zero

2 6 13 14 15 15 16 17

2.1 2.2a 2.2b 2.3 2.4 2.5 2.6 2.7 2.8 2.9

AC functions for residuals of detrended series Simulated spectral density function, T = 100 Simulated spectral density function, T = 200 Power spectrum of over-differenced series US GNP (quarterly, s.a.) Impulse responses based on ARMA (2, 2) model Estimates of G(1) and variance ratio from AR(k) models Simulated density function of R2 Simulated density function of tβ2 Measures of persistence for an MA(1) process

44 45 45 47 57 59 60 63 63 67

3.1 3.2

UK car production (logs, p.a., s.a.) Permanent and transitory car production (logs)

111 115

4.1 4.2 4.3 4.4

Bias functions of estimators RMSE functions of estimators US ﬁve-year T-bond rate US GNP (in logs, constant prices)

131 147 148 149

5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

5% and 10% quantiles and t-function Quantiles for LS estimates Quantiles for LS estimates: US GNP Quantile functions Inverse quantile functions 90% conﬁdence interval 5% quantiles for different estimators 95% quantiles for different estimators

166 167 168 172 172 174 176 177

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List of Figures

List of Figures

5.9a 5.9b 5.10 5.11

Simulated cumulative distribution function Simulated inverse cumulative distribution function Bootstrap quantiles: US GNP Obtaining a conﬁdence interval

181 181 185 187

6.1a 6.1b 6.1c 6.2 6.3 6.4 6.5a 6.5b 6.5c 6.6a 6.6b 6.6c

Distribution of tρ (no constant in estimated model) Distribution of tρ (constant in estimated model) Distribution of tρ (constant and trend in estimated model) Distribution of tρ with drift in DGP Ratio of variances: long-run/unconditional (MA(1) errors) Ratio of variances: long-run/unconditional (AR(1) errors) Distribution of τˆ with MA(1) errors Distribution of τˆ µ with MA(1) errors Distribution of τˆ β with MA(1) errors Distribution of τˆ with AR(1) errors Distribution of τˆ µ with AR(1) errors Distribution of τˆ β with AR(1) errors

201 202 202 211 232 233 235 235 236 236 237 237

7.1 7.2 7.3 7.4 7.5

277 296 296 297

7.7 7.8 7.9 7.10a 7.10b 7.11a 7.11b

US unemployment rate (Nelson and Plosser series) Variation in 5% critical values for test statistics Illustration of power envelope for ρT,µ (c) Comparison of power envelopes Power as scale of initial observation varies, π(κ| ρ), constant in estimated model Power as initial observation varies, π(κ| ρ), trend in estimated model Power comparison, ρ = 0.95 Power comparison: the weighted tests US Industrial production (logs, p.a., s.a.) Estimated scaled residuals ξˆc Estimated weights αˆ DF and GLSC unit root test Weighted unit root test

300 305 306 307 308 308 308 308

8.1

US unemployment rate (monthly data)

343

9.1 9.2 9.3 9.4 9.5

MA (1) errors: size and power, δˆ µ MA(1) errors: size and power, τˆ µ MA(1) errors: relative frequency of lags chosen by AIC AR(1) errors: size and power, δˆ µ

352 353 353 355 355

7.6

AR(1) errors: size and power, τˆ µ

299

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9.6 9.7a 9.7b 9.8 9.9 9.10 9.11 9.12 9.13 9.14 9.15 9.16 9.17 9.18

AR(1) errors: relative frequency of lag chosen by AIC Bivariate normal with η = 0.75 Probability of conﬂict, two tests, bivariate normal Power functions for τˆ µ and τˆ ws µ Conditional test dependency functions Unconditional test conﬂict Power of τˆ µ using G-t-S and MAIC Conditional test dependencies, G-t-S and MAIC Unconditional test conﬂict τˆ µ with G-t-S and MAIC Power of τˆ µ using G-t-S and MAIC Conditional test dependencies τˆ µ with G-t-S and MAIC Unconditional test conﬂict τˆ µ with G-t-S and MAIC US (log) wheat production The impact of different lag lengths

356 371 372 374 375 376 377 378 379 379 380 380 381 382

10.1 10.2a 10.2b 10.3a 10.3b 10.4a 10.4b 10.5a 10.5b 10.6a 10.6b 10.6c 10.6d 10.7a 10.7b 10.7c 10.7d

Simulated I(d) series, d = 0, 1, 2, 3 US average hourly earnings (logs) Denmark CPI (logs) US M1 (logs) US consumer credit (logs) Empirical size of DF τˆ µ lower tail Empirical size of DF τˆ µ upper tail Empirical size of DF τˆ β lower tail Empirical size of DF τˆ β upper tail US Consumer credit, actual and trends (logs) Residuals from trends, US credit US credit, monthly growth rate and trend Residuals for linear trend for growth rate Denmark CPI (logs), actual and trends Residuals from trends, Denmark CPI Denmark CPI, annual inﬂation rate and trend Residuals from linear trend for growth rate

389 390 390 391 391 399 400 400 401 421 421 422 422 425 426 426 427

11.1 11.2 11.3 11.4 11.5 11.6a 11.6b 11.7 11.8a

Bartlett and quadratic spectral kernals Estimators of long-run variance: rmse, AR(1) QS, BW and QS (CAB) estimators: rmse, AR(1) Estimators of long-run variance: rmse, MA(1) Comparison by bandwidth criteria: rmse, MA(1) KSµ , empirical size for AR(1) errors: 4 kernels LBMµ , empirical size for AR(1) errors Best of each test, empirical size for AR(1) errors KPSSµ , empirical size for MA(1) errors: 4 kernels

453 455 456 457 457 464 464 465 466

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List of Figures

List of Figures

11.8b 11.9 11.10 11.11 11.12 11.13 11.14 11.15a 11.15b 11.15c

LBMµ , empirical size for MA(1) errors Best of each test, empirical size for MA(1) errors EDFs for KPSS(µ) and LBM(µ) EDFs for KPSS(µ) , AR(1) errors EDFs for LBM(µ) , AR(1) errors EDFs for KPSS(µ) , MA(1) errors EDFs for LBM(µ) , MA(1) errors Power of test statistics in standard versions (T = 100) Size-adjusted power of test statistics (T = 100, semi-parametric tests use m(qs) (4)) Size-adjusted power of test statistics (T = 100, (qs)

11.15d 11.15e

semi-parametric tests use mcab ) Size-adjusted power of test statistics (T = 100, semi-parametric tests use m(bw) (4)) Size-adjusted power of test statistics (T = 100, (bw)

467 467 468 469 469 470 471 473 473 474 474

11.19b 11.19c

semi-parametric tests use mcab ) Size-adjusted power of LBM test statistics (T = 100) Best power combinations (T = 100) Power comparison with best-size combinations (T = 100) Size-adjusted power, AR(1) DGP (T = 500, semi-parametric tests use m(qs) (4)) Best power and size, AR(1) DGP (T = 100) Best power combinations (T = 500) Power comparison with best size combinations (T = 500) Size adjusted power, AR(1) DGP (T = 500, semi-parametric tests use m(qs) (4)) Best power and size, AR(1) DGP (T = 500) Best power and size, AR(1) DGP (T = 500, semi-parametric

11.20 11.21 11.22 11.23 11.24 11.25

tests use mcab (4)) US CPI and linear trend (logs) US CPI inﬂation rate US regional CPIs (logs) CPI contrast (log) Regional CPI inﬂation rates Contrast of inﬂation rates

481 483 483 488 488 489 489

12.1 12.2

90% conﬁdence interval using QD data: US GNP 90% conﬁdence interval, QD data, with c(L): US GNP

514 515

13.1a 13.1b 13.2a

Consumers’ expenditure (UK) Consumers’ expenditure: seasonal component Consumers’ expenditure (logs)

522 522 523

11.15f 11.16a 11.16b 11.17a 11.17b 11.18a 11.18b 11.19a

(bw)

475 475 476 476 477 478 479 479 480 480

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xxx

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13.2b 13.3 13.4 13.5 13.6 13.7a 13.7b 13.8a 13.8b 13.9a 13.9b 13.10 13.11a 13.11b 13.12a 13.12b 13.12c 13.13 13.14a 13.14b 13.15a 13.15b 13.16a 13.16b 13.17a 13.17b 13.18a 13.18b 13.19 13.20 13.21 13.22 13.23 13.24 13.25

Consumers’ expenditure (logs): seasonal component Consumers’ expenditure: DV seasonal component Consumers’ expenditure Expenditure on restaurants and hotels (UK) The periodogram for a single frequency Periodogram for periods of 12 and 120 Periodogram for periods of 4 and 120 Detrended expenditure on restaurants and hotels Periodogram for levels Quarterly growth rate Periodogram for growth rate Aliasing of monthly cycles when sampled quarterly Periodogram for ∆4 yt = εt Seasonal splits for ∆4 yt = εt Periodograms for (1 + L)yt = εt and (1 + L2 )yt = εt Seasonal splits for yt = −yt−1 + εt Seasonal splits for yt = −yt−2 + εt The monthly, seasonal frequencies λj Simulated data for ∆12 yt = εt Periodogram for ∆12 yt = εt Power of DFH τˆ S,Sµ and HEGY FAll tests, no lags Power of DFH τˆ S,Sµ and HEGY FAll tests, 4 lags Power of tests with RW(1) DGP EDFs for τˆ S,Sµ with RW(1) DGP Power of tests with Nyquist DGP EDFs for τˆ S,Sµ with Nyquist DGP Power of tests with seasonal harmonic DGP EDFs for τˆ S,Sµ with seasonal harmonic DGP Power of DFH τˆ S,Sµ and HEGY FAll,Sµ Power of DFH τˆ S,Sµ,Sβ and HEGY FAll,Sµ,Sβ US agricultural employment Seasonal split growth rates Periodogram for US agricultural employment UK Index of Industrial Production (logs) n.s.a. Periodogram for UK IIP (log) n.s.a.

523 524 525 526 529 530 530 531 531 532 532 533 537 537 538 538 539 551 551 552 571 571 573 573 575 575 576 576 579 579 583 584 584 587 588

A1.1 A1.2

Binomial distribution, pmf and cdf Standard normal distribution, pdf and cdf

600 602

A2.1a A2.1b

Finding the roots of φ(z) φ(z) with complex roots

606 607

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List of Figures

⇒D →p → → ⇒ ∼ ≡ = Φ(z) R ℜ+ N+ εt ∏n j=1 xj n x ∑j=1 j

convergence in distribution (weak convergence) convergence in probability tends to; for example, ε tends to zero, ε → 0 mapping implies is distributed as deﬁnitional equality not equals the cumulative distribution function of the standard normal distribution the set of real numbers; the real line (–∞ to ∞) the positive half of the real line the set of non-negative integers white noise, unless explicitly excepted the product of xj , j = 1, . . . , n

0+ −0 L ∆ ∆s ∆s ⊂ ⊆ ∩ ∪ ∈ |a| iid m.d.s N(0, 1) niid B(t) W(t)

approach zero from above approach zero from below the lag operator, Lj yt ≡ yt−j the ﬁrst-difference operator, ∆ ≡ (1 – L) the s-th difference operator, ∆s ≡ (1 − Lj ) the s-th multiple of the ﬁrst difference operator, ∆s ≡ (1 − L)s a proper subset of a subset of intersection of sets union of sets an element of the absolute value (modulus) of a independent and identically distributed martingale difference sequence the standard normal distribution, with zero mean and unit variance independent and identically normally distributed standard Brownian motion; that is, with unit variance nonstandard Brownian motion

the sum of xj , j = 1, . . . , n

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Symbols and Abbreviations

The purpose of this book is to provide a review and critical assessment of some key procedures related in one way or another to the problem of testing for a unit root in a stochastic process. As is now well known, the presence of a unit root implies a form of nonstationarity that is considered to be relevant for economic time series, so that a non-standard inferential and distributional framework is required. The research and literature on this topic has grown almost exponentially since Nelson and Plosser’s seminal article published in 1982. Therein they applied the framework due to Dickey (1976) and Fuller (1976) to testing for a unit root in a number of macroeconomic time series. Subsequent key articles by Dickey and Fuller (DF) (1979, 1981) developed some aspects of the initial testing framework. The basic set-up for a DF unit root test is now familiar enough, being taught in most intermediate, if not introductory, courses in econometrics; however, the underlying distribution theory is somewhat more advanced, and the many complications that have arisen in practice has meant the development of a voluminous literature that, because of its extent, is difﬁcult to comprehend, especially for the non-specialist. Indeed, it is probably the case that a simple survey of the ﬁeld of methods and applications is virtually infeasible; indeed, the topic is so extensive that even some 20 years ago Diebold and Nerlove (1990) noted the scale of the literature on this topic. The articles on unit root tests are amongst the most cited in economics and econometrics and have clearly inﬂuenced the direction of economic research at a much wider level than simply testing for a unit root. A citation summary for articles based on univariate processes is presented in Table P.1. The numbers shown here clearly indicate that there has been a sustained interest in the topic over the last 30 years or so and, looking at the wider inﬂuence, the unit root literature led to the concept of cointegration and to some of the most cited of econometric articles, including Engle and Granger (1987), Johansen (1988, 1991) and Johansen and Juselius (1990) (see the note to Table P.1). The appropriate prerequisites for this book include some knowledge of econometric theory at an intermediate or graduate level as, for example, in Davidson (2000), Davidson and Mackinnon (2004) or Mittlehammer et al. (2000), and, possibly, with some additional directed study, as in good introductory books such as Gujarati (2006), Dougherty (2007), Ramanathan (2002) and Stock and Watson (2007). It would also be helpful to have had an introduction to the

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Table P.1 Number of citations of key articles on unit roots.

Dickey and Fuller (1979) Phillips and Perron (1988) Dickey and Fuller (1981) Perron (1989) Kwiatkowski, Phillips, Schmidt, Shin (1992) Nelson and Plosser (1982) Phillips (1987a) Zivot and Andrews Elliott, Rothenberg and Stock (1996) Said and Dickey (1984)

Number of citations 1 2 3 4 5 6 7 8 9 10

7,601 4,785 4,676 3,371 3,280 3,035 1,881 1,694 1,556 1,342

Notes: Articles relate to univariate unit root tests. Prominent articles, on a citation basis, involving largely multivariate methods are: Engle and Granger (1987): 12,366; Johansen (1988): 8,236; Johansen and Juselius (1990): 4,886; Johansen (1991): 4,150; and in econometric methods more generally, White (1980), 12,359. (The last article is the most highly cited on the basis of several citation methods.) On the index of economics articles since 1970, compiled by Han Kim et al. (2006), Dickey and Fuller (1979) ranks 7, whilst Engle and Granger (1987) and Johansen (1988) rank 4 and 8, respectively. Source: Google Scholar, accessed 22 February 2010.

methods of maximum likelihood and generalised least squares (GLS), for example as provided in Greene (2006). An introduction to time series analysis and mathematical statistics would also be useful; for example, for the former at the level of Chatﬁeld (2004) and the latter along the lines covered by Mittelhammer (1996). A book designed especially as a primer for this one is Patterson (2010). Some familiarity with the application of unit root tests would also be helpful to set the context. I have taken the brief of this book to include issues that are related to but theoretically separate from the central concern of testing for a unit root. For example, one of these is the problem of the bias in estimating the coefﬁcients in an autoregressive model; whilst this is strictly a ﬁnite sample effect, it is of practical interest and serves as a ‘lead in’ to the problems associated with unit testing. This book is, therefore, not about listing tests for a unit root. Not only would there not be enough space for such an enterprise, it is not the best way to indicate which test statistics, and methods more generally, have been taken up by practitioners. The main tests are of course presented and their rationale explained, together with examples to illustrate how they are used. However, the problem suggested by the presence of a unit root or near-unit root is more than just the design of a test statistic. There are two other important practical issues that a researcher has to face. The ﬁrst is to consider what the appropriate alternative hypothesis is. In the Dickey-Fuller paradigm, followed by many in practice, the alternative to a stochastic trend is a deterministic trend, usually

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Author(s)

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characterised as a low order polynomial in time. However, the choice of the order of the polynomial, typically representing reversion to a constant or a linear trend, has a critical effect on the power of the test if over-speciﬁed; on the other hand, if under-speciﬁed, a test will have no power to detect a true alternative. The representation of the ‘attractor’ as being generated by a low order polynomial trend is likely to be a shorthand, or reduced form, for a far more complex process. In part, the pre-eminent role of a simple deterministic trend in providing the mean or trend reverting alternative to a non-reverting process (nonstationary by way of a unit root or roots), is historical and, of itself, deserves further study and evaluation. A second key, practical aspect arises from the usual need to choose some form of a truncation parameter. In the context of the familiar augmented Dickey-Fuller tests, this is a lag truncation parameter and in the context of the semi-parametric tests, which require an estimate of the long-run variance, a parameter limiting the bandwidth in the formation of a sum of autocovariances. Whilst familiar criteria, such as the AIC, BIC and general-to-speciﬁc (g-t-s), rules are in frequent use, the combination of each one of these with a test statistic deﬁnes test procedures with potentially differing characteristics; their use in combination then leads to the accumulation of type I error. Some of the developments covered in this book are as follows. •

• • • • • •

The distinction between difference stationary and trend stationary processes and the implications of this distinction for the permanence or otherwise of shocks. An outline of the autoregressive moving average (ARMA) modelling framework and its role in testing for a unit root. The ﬁnite sample bias in estimating models and its implications for inference even when seemingly well into the region of stationarity. Forming conﬁdence intervals that are robust to the problem of quantiles that are not constant. The DF unit root tests and developments of them to account for weak serially correlated processes. Bootstrapping conﬁdence intervals and unit root tests. Tests that: – are based on a direct maximum likelihood approach; – are based on a GLS, or quasi-GLS, approach, including the inﬂuential tests by Elliott, Rothenberg and Stock (1996); – combine the backward and forward recursions of a random walk; – are based on recursive estimation of the mean or trend; – are robust to the initial condition; – allow for more than one unit root;

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Preface

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Preface

The results of a number of Monte Carlo studies are reported in various chapters. Indeed, simulation is a key tool that is used throughout to provide some guidance on ﬁnite sample issues. Consider, for example, the problems caused by the presence of weakly dependent errors when testing for a unit root. Then under fairly weak assumptions, the asymptotic properties of several frequently used test statistics are unaffected by such errors, but typically, the ﬁnite sample properties do not reﬂect the asymptotic properties, an example being the difﬁculty caused by the near cancellation of a root, especially a near-unit root, in the AR and MA components of an ARMA model. To understand the ﬁnite sample nature of such problems, many more simulations were run than are reported in the various chapters; the results are then typically illustrated for one or two sample sizes where they are representative of a wider range of sample sizes. There are a number of developments and problems not covered in this volume, but which are included in Volume 2. These include the following. Nonparameteric tests: the tests that have been considered in this volume, such as the family of Dickey-Fuller tests, are parametric tests in the sense that they are concerned with direct estimation in the context of the parametric structure of an AR or ARMA model. Nonparametric tests use less structure in that no such explicit parametric framework is required and inference is based on other information in the data, such as ranks, signs and runs. Semi-parametric tests use some structure, but it falls short of a complete parametric setting; an example here is the rank score based test, which is based on ranks. Fractional integration: this considers the case of fractional values of the integration parameter. That is, suppose that a stochastic process generates a time series that is integrated of order d, where d is a fractional number. What meaning can we attribute to such an operation and how can the parameter d be estimated? There are two general approaches to the analysis and estimation of fractional I(d) process, as they may be either analysed in the time domain or the frequency domain. Bounded random walks: the application of random walk models to some economic time series can be inappropriate, as where there are natural bounds or limits to the values that the series can take, such as in the case of unemployment rates and nominal interest rates. One way of modelling this is to allow unit root behavior; for example, persistence and the absence of mean reversion over a range of possible values but reversion at other values. These models have

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– are based on stationarity as the null hypothesis; – allow for unit roots in seasonal data.

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Structural breaks: Perron’s (1989) seminal article began another thread of the unit root literature. What if, instead of a unit root process generating the data, there was a trend subject to a break due to ‘exceptional’ events? How would standard unit root tests perform? For example, what would be their power characteristics, if the break was ignored in the alternative hypothesis? The idea of regime change that could affect led to a fundamental re-evaluation of the simplicity of the simple ‘opposing’ mechanisms of a unit root process, on the one hand, and a trend stationary process, on the other. In practice, although there are likely to be some contemporaneous and, later, historical indications of regime changes, there is almost inevitably likely to be uncertainty not only about the dating of such changes but also the nature of the changes. This poses another set of problems for econometric applications. If a break is presumed, when did it occur? Which model captures the nature of the break? If multiple breaks occurred, when did they occur? My sincere thanks go to Lorna Eames, my secretary at the University of Reading, for her unfailing assistance in the many tasks needed to bring the manuscript into shape. The graphs in this book were prepared with MATLAB www.mathworks.co.uk, which was also used, together with TSP (www.tspintl.com) and RATS (www.estima.com), for the numerical examples. Martinez and Martinez (2002) provide an invaluable guide to statistics with many MATLAB examples; guides to MATLAB include Hanselman and Littleﬁeld (2004), Moler (2004) and Hahn and Valentine (2007). If you have comments on any aspects of the book, please contact me at my email address given below.

Author’s email address: [email protected] Palgrave Macmillan Online: http://www.palgrave.com/economics/ Palgrave Macmillan email address: [email protected]

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in common that they involve some form of nonlinearity. Perhaps the simplest from of nonlinearity actually arises from piecewise linearity; that is, an overall model comprises two or more linear models for subperiods where the component models differ not in their form, for example all are AR(p), but in their parameters. A popular class of such models is the smooth transition autoregressive – or STAR – class, of which the exponential and logistic members are the most frequent in application, giving rise to the acronyms STAR and LSTAR.

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1

Introduction The ﬁrst part of this chapter introduces the random walk initially in a form with stochastic shocks generated by a random variable with a binomial distribution. In the simplest version of this process the random variable has two equally likely outcomes resulting in a symmetric binomial random walk. The idea is simple enough and the terminology is due to a problem posed by Pearson (see Hughes, 1995, p.53), although the concept dates from much earlier, originating in games of chance. Starting from the origin, at regular intervals a walker takes equally spaced steps either to the left (north) or to the right (south), with equal probability. The walker’s progress can be plotted in two dimensions by recording the distance from the origin on the vertical axis and the elapsed time on the horizontal axis; for example, one step to the north followed by one step to the south returns the walker to the origin. Such a graph will look like a series of equally sized steps, see Figure 1.1; perhaps surprisingly, the resulting path does not generally oscillate around zero, the theoretical mean of the process. This lack of ‘mean reversion’ is one of the key characteristics of a random walk. In this form the process has its origins in gambling, where the gambler gains or loses an equal amount at each gamble, with equal probability, and intuition might suggest that the gambler is not systematically losing or winning. The ‘distance from the origin’ corresponds to the gambler’s win/lose tally, which is one-dimensional and can be represented on the vertical axis. There are several ways to generalise the random walk described above. Indeed, Pearson’s problem was originally posed in a more general form. In particular, whilst the walker takes equally spaced steps of length x, he is allowed to pivot through any angle before taking the next step, not simply going north or south, which is an angle of ±90◦ to his orientation. For example, taking an equally spaced step at 45◦ would place the walker in a north-east direction. The

1

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Introduction to Random Walks and Brownian Motion

2 Unit Root Tests in Time Series

40 30 20 10 0 –10 –20 –30 –40

0

50

100

150

200

250

300

350

400

450

500

t Figure 1.1 Sample paths of a binomial random walk.

walk could therefore be represented as a bird’s-eye view of his progress in twodimensional geographic space (with time an implicit third dimension). In this case starting at the origin, one step in a northward direction followed by one step in a southward direction will not necessarily return the walker to the origin, as the step angle may differ from ±90◦ . By focusing on, say, the north-south direction, this random walk can be represented in two dimensions with time as one of the dimensions. The step size in the single dimension will depend on the pivoting angle and so will no longer be a constant; moreover, the resulting random variable is continuous as the step size deﬁned in this dimension can vary continuously between 0 and x. The random walk is of interest in its own right in economics as it provides a statistical model that is paired with rational expectations and some versions of the efﬁcient markets hypothesis. For example, suitably deﬁned, a random walk has the martingale property that the expected value of the random variable yt at time t – 1, conditional on the information set Ψ, comprising lagged values of yt , is yt−1 ; that is, Et−1 (yt |Ψ) = yt−1 , where Et−1 is the expectation formed at time t – 1. An implication of this property is that Et [yt − Et−1 (yt |Ψ)] = 0, so that the difference between outturn and expectation, yt − Et−1 (yt |Ψ), is ‘news’ relative to the information contained in Ψ, a property associated with the rational expectations hypothesis. An important property of a random walk process arises in the limit as the time divisions, the time steps, are ‘shrunk’ toward zero. The limiting process is

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yt

3

Brownian motion, knowledge of which is essential to an understanding of the distribution of several important unit root test statistics. Hence an introduction to the random walk also serves as an introduction to Brownian motion. It is a matter of choice as to the order in which the random walk, and developments thereof, and the concept of a ‘unit root’ can be introduced. The preference expressed here is motivated by the fascination often expressed in the counter-intuitive properties of a simple random walk and how quickly one can link such processes to economic time series. Given this background, the more formal testing framework then follows quite naturally, informed by some useful features of random walk type behaviour that can be illustrated graphically. Section 1.1 outlines the basic random walk with a number of generalisations and key properties including, for example, its lack of mean reversion, sometimes referred to as mean aversion. Section 1.2 deﬁnes Brownian motion (BM) and sections 1.3 and 1.4 state two key theorems, based on BM, required in unit root statistics, that is the functional central limit theorem (FCLT) and the continuous mapping theorem (CMT); these are often used together to obtain the limiting distribution of a unit root test statistic. Section 1.5 provides a brief background to the development of unit root and related tests and section 1.6 provides some selective economic examples in which unit root processes are of interest. This chapter assumes a basic familiarity with probability and some time series concepts, which are the more detailed subjects of later chapters. It is, for example, almost impossible to talk of a unit root without the context of an autoregressive model, which is itself a special case of the ARMA (autoregressive moving average) class of models, considered in greater detail in Chapter 3. Similarly, the idea of a unit root is greatly aided by ﬁrst studying the lag operator and lag polynomial. The reader may ﬁnd it useful to review the material in Appendices 1 and 2: Appendix 1 is a brief introduction to random variables and Appendix 2 provides some background material on the lag operator. Where necessary, the ARMA model of Chapter 3 is anticipated in this chapter in just sufﬁcient detail to make the context self-sufﬁcient.

1.1 Random walks The concept of a random walk has two roles of interest. First, it is prototypical model for the representation of the time series of many economic variables, including real variables such as output and employment and nominal or ﬁnancial variables such as price levels, inﬂation rates and exchange rates. It is often taken as a ‘default’ or ‘baseline’ model from which other models are evaluated. Second, random walks, or their limiting forms, appear as partial sum processes, psp, in econometric estimators, especially in distribution theory for estimators and test statistics. In the limit, here interpreted as taking smaller and smaller

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Introduction to Random Walks and Brownian Motion

4 Unit Root Tests in Time Series

steps in a given time interval, the random walk leads to Brownian motion, which is considered in section 1.7. 1.1.1 The random walk as a partial sum process (psp) We start with the random variable yt , with a sequence of length T of such variables written as {yt }Tt=1 . Apart from the starting value y0 , values of yt are determined by the following one period recursion: t = 1, . . . , T

(1.1)

Thus, yt is determined as its previous value, yt−1 , plus the intervention of a random variable denoted εt , usually referred to as a ‘shock’ in the sense that yt would otherwise just be yt−1 . The sequence {εt }Tt=1 is assumed to be independent white noise. (White noise, WN, requires E(εt ) = 0, E(ε2t ) = σε2 for all t, and E(εt εs ) = 0 for t= s; if all members of the sequence {εt }Tt=1 are independently and identically distributed, then {εt }Tt=1 is referred to as independent or strong white noise sequence.) This random walk is a partial sum process (psp) as, by successive back substitution, yt can be expressed as y0 plus the cumulated sum of the sequence, to that point, of stochastic inputs: yt = y0 + ∑j=1 εj t

(1.2)

One of the insights in distinguishing the behaviour of a time series generated by a random walk yt = yt−1 + εt (and generalisations of this process) is the nature of its evolution over time. Viewing {εt }t1 as a sequence of ‘shocks’, a particular sample path of yt is determined by the starting position y0 and the cumulated shocks, each of which receives an equal weight – in particular, there is no sense in which the past is forgotten, a feature that is sometimes referred to as inﬁnite ‘memory’. Such processes occur quite readily in gambling and have been the subject of considerable study and extension (see, for example, the classic text by Feller, 1968). A simple case that illustrates much of interest is the symmetric, binomial random walk generated by a sequence of gambles on the outcome of the toss of a fair coin: a coin is tossed, with the gambler winning one ( + 1) ‘chip’ if it lands heads and losing one (–1) ‘chip’ if it lands tails; the game continues sequentially with further tosses of the coin, indexed by the counter t = 1, . . . , T. For simplicity, the games are assumed to be played at the rate of one per period t, so t increments in units of 1 from 1 to T. Each toss of the coin is referred to as a ‘trial’, a term that originates from Bernoulli trials, resulting in the binary outcomes ‘success’ or ‘failure’, with probabilities p and q, respectively. Successive trials are independent in the sense that the outcome on any one is unaffected by any of the others.

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yt = yt−1 + εt

5

To put the process more formally, let xt be the random variable representing the coin toss on the t-th toss of the coin; the sample space associated with each single toss of the coin is Ωj = (H, T), and the associated probability measure is Pxj = (px1 , px2 ); note that an italicised T, T, refers to the outcome that the coin toss is a tail. Next let εj be the random variable that is derived from xj , such that the original sample space Ωj is mapped to Ωεj = ( + 1, –1). The probability measures of xj and εj are the same; for example, for a symmetric random walk p = q = 1/2, so that Pxj = Pεj = (1/2, 1/2); if p = q then the resulting process is an asymmetric random walk. The random variable yt is the partial sum of the derived random variables εj , j = 1, . . . , t, and is usually referred to as the ‘tally’. We assume that there is no ‘charge’ to enter the game, so that y0 = 0. The sample space of yt is y the t-dimensional product space of Ω1 , that is Ωt = (Ωε1 )t = Ωε1 × Ωε1 × . . . × Ωε1 and, by independence, the probability measure associated with yt is the prody uct measure Pt = (Pε1 )t = Pε1 × Pε1 × . . . × Pε1 . Note that E(εt ) = 0, t = 1, . . . , T and E(yt ) = E(y0 ) + ∑ti=1 E(εi ) = 0, so that the theoretical mean of the tally is zero. The counterpart of the random walk sequence generated by the gambler is the random walk of the ‘banker’: yB,t = yB,t−1 − εt

(1.3)

= yB,0 − ∑j=1 εj t

Note that this is an example of a ‘zero sum’ game since yB,t + yt = yB,0 − ∑tj=1 εj + y0 + ∑tj=1 εj = yB,0 + y0 , where the latter equals zero if both parties start with zero capital. It will occasionally be useful to look at the random walk from the banker’s perspective. The random walk of Equation (1.1) was simulated with T = 500 and binomial inputs; ten sample paths were shown in Figure 1.1. These paths tend to confound intuition. A line of reasoning that seems attractive is that as the expected value of each component random variable εj is zero, the expectation of the tally, E(yt ), is zero, hence the sample paths can be expected to ﬂuctuate reasonably evenly about zero. The ﬁgure shows that this line of reasoning is false. There is very little reversion to the expected value of zero; indeed, once started in a particular direction, whether that is in the positive or negative half of the diagram, a sample path only rarely crosses (traverses), or is reﬂected from, the zero axis; these are collectively referred to as ‘visits to the origin’ or mean reversions, the number of which is a key characteristic that is considered further in section 1.1.3. 1.1.2 Random walks: visits to the origin (sign changes and reﬂections) As noted in the previous section, one of the characteristics of a random walk is that there is very little mean reversion; that is, although E(yt ) = 0, which is the

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Introduction to Random Walks and Brownian Motion

6 Unit Root Tests in Time Series

‘origin’ for this process, this is not an ‘attractor’ for the sample path. To formalise this idea, let VT be the number of visits to the origin (including reﬂections) of a √ symmetric binomial random walk of length T, then ζT = VT / T is the normalised number of such visits with distribution function F(ζT ) for ﬁnite T. It turns out that F(ζT ) has a simple limiting distribution as follows: F(ζT ) ⇒D F(ζ) = 2Φ(ζ) − 1

where Φ(ζ) is the (cumulative) standard normal distribution function. Equally, one can write in terms of the random variables that ζT ⇒D ζ, where ζ is a random variable with the half-normal distribution. Thus, the distribution function F(ζ) is that of the absolute value of a normally distributed random variable with mean µ = 0 and variance σ2 (see Feller, 1968; Burridge and Guerre, 1996; Garci´a ´ 2006). If σ2 = 1, as in the symmetric binomial random walk, then and Sanso, √ E(ζ) = 1/2π = 0.7979, so that E(VT ) = 0.7979 T, which is to the right of the median; if T = 500, then the integer part of E(V500 ) is 17. The median of F(ζ) is 0.674, so that the median number of mean reversions for (large) T trials is √ about 0.674 T (about because of the need to be an integer); for example, if T = 500, then median is about 15. The distribution function, F(ζ), and its mean and median are illustrated in Figure 1.2.

1 0.9 0.8 0.7 VT /√T

0.6

normalised number of visits to the origin of a symmetric, binomial random walk

0.5 0.4 0.3 0.2 median = 0.674

0.1 0

0

0.5

mean = 0.798 1

1.5

2

2.5

3

VT /√T Figure 1.2 Distribution function of visits to the origin.

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(1.4)

Introduction to Random Walks and Brownian Motion

7

Note that a random walk is an example of a stochastic process and, in the language of such processes, the realisations in Figure 1.1 are trajectories or sample paths. Our interest generally lies not in the outcome of a random variable at a single point in time, but in a sample path, or the distribution of sample paths, of a sequence of random variables over an interval of time. To conceptualise how such sample paths arise the idea of a stochastic process involves a sample space Ω, a probability space and time. In the case of the stochastic process deﬁned by the symmetric, binomial random walk then y y Ω = Ωt , with probability measure Pt and t = 1, . . . , T; note that if T → ∞, then the sample space and the probability space associated with the product measure become of inﬁnite dimension. This can also occur if T is ﬁxed and then partitioned into a grid with ‘mesh’ size ∆t and ∆t→ 0. Let Θ be the set of possible values taken by the time index, then in the random walk of Equation (1.1), time is discrete, so that Θ has a ﬁnite or countably inﬁnite number of elements. Indeed, in this case Θ is the set of positive integers or, if the process is viewed as starting at t = 0, the set of nonnegative integers, t ∈ Θ = N + = (0, 1, 2, . . . ); equally the process might be viewed as starting in the inﬁnite past and, hence, t ∈ Θ = N = (0, ±1, ±2, . . .). In the continuous time case, Θ is an interval, for example Θ = R, or the positive half line Θ = ℜ + or an interval on R, for example Θ = [0, 1] or Θ = [0, T]. Stochastic processes may be viewed as taking place in discrete time or in continuous time, which are represented, respectively, as follows: Y = (yt (ω) : t ∈ Θ, ω ∈ Ω)

discrete time

(1.5a)

Y = (y(t, ω) : t ∈ Θ ⊆ ℜ, ω ∈ Ω)

continuous time

(1.5b)

The continuous-time stochastic process represented at a discrete or countably inﬁnite number of points is then written either as y(s) or y(t) if only two or three points in time are being referenced or, more generally, as y(t1 ), y(t2 ), . . . . , y(tn ). Note that reference to ω may be suppressed if it is not material to the presentation. For given t ∈ Θ, yt (ω), or y(t, ω), is a function of ω ∈ Ω and is, therefore, a random variable. A realisation is a single number – the point on the sample path relating to, say, t = s. By varying the element of Ω, whilst keeping t = s, there is a distribution of outcomes at that point. For given ω ∈ Ω, yt (ω) is a function of time, t ∈ Θ. In this case an ‘outcome’ is a complete sample path; that is, a function of t ∈ Θ, rather than a single number. A description of the sample path would require a functional relationship rather than a single number. By varying ω a different complete sample path is obtained; that is, (potentially) different realisations for all t ∈ Θ.

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1.1.3 Random walk: an example of a stochastic process

The component random variables in the binomial random walk are discrete, with a simple countable number of outcomes; indeed, just two in this case. Hence this speciﬁcation of the random walk is an example of a discrete time, discrete variable stochastic process. Later we will consider a random walk where the stochastic inputs, εt , are distributed as N(0, σε2 ), in which case the stochastic process so generated is an example of a discrete time, continuous variable stochastic process. A case of particular interest that arises in the context of Brownian motion is a continuous time stochastic process. One can view this as the limit of a discrete time process in which a given interval of time is divided into a ﬁner and ﬁner grid, so that ∆t → 0 and Θ = R, or the positive half-line Θ = ℜ + or an interval on R, for example, Θ = [0, 1]. As noted, often the reference to ω ∈ Ω is suppressed and a single random variable in the stochastic process is written yt , but the underlying dependence on the sample space should be recognised and means that different ω ∈ Ω give rise to potentially different sample paths. 1.1.4 Random walk: an example of a nonstationary process A key distinction in econometrics and statistics is between processes that are stationary and those that are nonstationary. In a time series context, these are said to generate time series that are, respectively, stationary or nonstationary, it being understood that it is the underlying generating process that is stationary or nonstationary. Intuitively, stochastic processes that are stationary are unchanging in some key aspects, which give rise to several deﬁnitions of stationarity, differences between them depending on what is assumed to be unchanging. A strong form of stationarity requires that the joint probability distribution of the random variables that comprise the stochastic process is unchanging; however, the most often used deﬁnition in econometrics relates to a weakly (or second-order) stationary process. These deﬁnitions are considered in the next two subsections. 1.1.4.i A strictly stationary process Let τ = s and T be arbitrary, if Y is a strictly stationary, discrete time process for a discrete random variable, yt , then: P(yτ + 1 , yτ + 2 , . . . , yτ + T ) = P(ys + 1 , ys + 2 , . . . , ys + T )

(1.6)

That is, the joint probability mass function (pmf) for the sequence of length T starting at time τ + 1 is the same for any shift in the time index from τ to s and for any choice of T. This means that it does not matter which T-length portion of the sequence we observe. Since a special case of this result in the discrete case is for T = 1; that is, P(yτ ) = P(ys ), the marginal pmfs must also be the same for τ = s, implying that E(yτ ) = E(ys ). These results imply that other moments, including joint moments, such as the covariances, are invariant to arbitrary time shifts.

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8 Unit Root Tests in Time Series

Introduction to Random Walks and Brownian Motion

9

If the random variables are continuous and also deﬁned in continuous time, a strictly stationary random process must satisfy the following: F[y(τ + t1 ), y(τ + t2 ), . . . , y(τ + tN )] = F[y(s + t1 ), y(s + t2 ), . . . , y(s + tN )]

(1.7)

where t1 < t2 . . . < tN , τ = s and F(.) is the joint distribution function, (see Appendix 1). If the probability density functions (pdfs) exist, then an analogous condition holds where F(.) is replaced by the joint pdf, denoted f(.): (1.8)

1.1.4.ii Weak or second-order stationarity (covariance stationarity) A less demanding form of stationarity is weak or second-order stationarity, which requires that the following three conditions are satisﬁed for arbitrary τ and s, τ = s: SS1: E(yτ ) = E(ys ) = µ

(1.9a)

SS2: var(yτ ) = var(ys ) = σy2

(1.9b)

SS3: cov(yτ , yτ + k ) = cov(ys , ys + k )

(1.9c)

The moments in SS1–SS3 are assumed to exist. The ﬁrst condition states that the mean is constant, the second that the variance is constant and the third that the k-th order autocovariance is invariant to an arbitrary shift in the time origin. The extension to continuous time is straightforward, replacing yτ by y(τ), and so on. From these three conditions, it is evident that a stochastic process could fail to be weakly stationary because its mean is changing; and/or its variance is changing; and/or the k-th order autocovariances depend on time for some k. A stochastic process that is not stationary is said to be nonstationary. A nonstationary process could be: nonstationary in the mean; nonstationary in the variance; and/or nonstatonary in the autocovariances. Usually it is apparent from the context whether the stationarity being referred to is strict or weak. When the word ‘stationary’ is used without qualiﬁcation, it is taken to refer to weak stationarity, shortened to WS, but, perhaps, most frequently referred to as covariance stationarity. (Weak or covariance stationarity is also referred to as wide-sense stationary, leading to the initials WSS.) Two particular cases of interest relate to difference stationarity and trend stationarity, generally referred to as DS and TS, respectively. A process that is DS is nonstationary in the levels of its component random variables, but stationary in their ﬁrst differences. Thus, the stochastic process Y is DS if: Y = (yt (ω) : t ∈ Θ, ω ∈ Ω) DY = (∆yt (ω) : t ∈ Θ∗ ⊆ Θ, ω ∈ Ω)

nonstationary process stationary process

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f[y(τ + t1 ), y(τ + t2 ), . . . , y(τ + tN )] = f[y(s + t1 ), y(s + t2 ), . . . , y(s + tN )]

Unit Root Tests in Time Series

The random walk yt = yt−1 + εt is an example of a DS process for, as elaborated below, neither its variance nor its autocovariances satisfy conditions SS2 and SS3; however, ∆yt = εt satisﬁes these conditions for t ∈ Θ∗ ⊆ Θ. The nonstationarity in Y is due to the implied accumulation of shocks, which is evident from the representation in Equation (1.2). A TS process is one that is stationary after the removal of a deterministic trend. Typically, a linear trend is assumed for the generating process, thus observations that are generated from such a process will tend to have the direction given by the sign of the trend implying E(yt ) = constant, and will not, therefore, satisfy SS1, even though they may satisfy SS2 and SS3. However, removal of the trend gives the detrended series, which will be stationary. Y = (yt (ω) : t ∈ Θ, ω ∈ Ω)

nonstationary process

Y˜ = (y˜ t (ω) : t ∈ Θ, ω ∈ Ω)

stationary process

y˜ t (ω)≡ yt (ω) − µt

detrended observation

The deterministic components are captured by the term µt ; for example, µt = β0 + β1 t, so that y˜ t (ω) is the detrended data for period t. For practical applications, an estimate of µt will be required. The next two subsections show that the random walk is not a stationary process by virtue of an increasing variance and autocovariances that are not invariant to a translation of the time index. 1.1.4.iii The variance of a random walk increases over time One of the problems for an intuitive understanding of the behaviour of a random walk sample path, is that the variance of yt is not constant; indeed, it increases linearly with t – this means that the range of yt increases with t. This characteristic reﬂects the lack of stationarity of the distribution of yt as t varies. The variance of yt is as follows: var(yt ) = ∑j=1 var(εj ) + 2 ∑i=1 ∑j>i cov(εi , εj ) t

t

t

(1.10a)

= ∑j=1 var(εj )

as cov(εi , εj ) = 0, i = j

(1.10b)

= tσε2

as var(εj ) = σε2 for all j

(1.10c)

t

In the case of the symmetric, binomial random walk σε2 = 1 and E(yt ) = 0 for all t; but the variance increases with t, such that the variance of y1 is 1, the variance of y100 is 100 and the variance of y500 is 500. Note that provided cov(εi , εj ) = 0, i = j and var(εj ) = σε2 for all j, then, holds, so the result would also hold for εt ∼ niid(0, σε2 ) or, weaker still, that εt is white noise.

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10

Introduction to Random Walks and Brownian Motion

11

1.1.4.iv The autocovariances of a random walk are not constant The k-th order autocovariance, γ(k), is a measure of the (linear) dependence between yt and its k-th lag, yt−k , equivalently, the k-th lead, if the process generating the data is covariance stationary. γ(k) is deﬁned as follows: γ(k) = E[yt − E(yt )][(yt−k − E(yt−k )] k = ± 1, 2, 3, . . .

(1.11)

Clearly, the expectations in (1.11) must exist for γ(k) to be deﬁned. The variance γ(0) is given by setting k = 0: γ(0) = E[yt − E(yt )]2

(1.12)

For a stationary process, the k-th order autocorrelation coefﬁcient ρ(k) is γ(k) scaled by the variance, γ(0) (which is constant on this assumption), so that: ρ(k) =

γ(k) γ(0)

(1.13)

The scaling ensures that 0 ≤ |ρk | ≤ 1. Considered as a function of k, γ(k) and ρ(k) give rise to the autocovariance and autocorrelation functions; the latter portrayed graphically, with k on the horizontal axis and ρk on the vertical axis, is referred to as the correlogram. (See equation (1.16) for an adjustment to ρ(k) for a nonstationary process.) Covariance (or second-order) stationarity requires that the γ(k) should be invariant to a translation of the time index, provided that a distance of k periods is maintained between the random variables. This is not the case for a random walk. To illustrate, assume for simplicity that y0 is a ﬁxed constant, so that var(y0 ) = 0, then γ(1) for t = 2 and t = 3, are, respectively, as follows: cov(y1 , y2 ) = cov(ε1 , ε1 + ε2 ) = var(ε1 ) + cov(ε1 , ε2 ) = σε2 cov(y2 , y3 ) = cov(ε1 + ε2 , ε1 + ε2 + ε3 ) = var(ε1 ) + var(ε2 ) + 2cov(ε1 , ε2 ) + cov(ε1 , ε3 ) + cov(ε2 , ε3 ) = 2σε2

Hence cov(y1 , y2 ) = cov(y2 , y3 ) although both relate to an index value k = 1. These derivations exploit the properties of white noise E(ε2t ) = σε2 for all t, and

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= cov(yt , yt−k )

12

Unit Root Tests in Time Series

E(εt εs ) = 0 for t = s. In general, γ(1) for arbitrary t is given by: cov(yt−1 , yt ) = cov(∑j=1 εj , ∑j=1 εj ) t−1

t

(1.14)

= (t − 1)σε2 Hence, γ(1) varies as the time index varies, increasing linearly with t. This result generalises to γ(k), so that t−k

t

(1.15)

= (t − k)σε2 As noted above, the k-th order autocorrelation coefﬁcient, ρ(k), is the standardised, or scaled, k-th order autocovariance. If the sequence {yt }Tt=1 is stationary, such that (inter alia) var(yt−k ) = var(yt ) = γ(0), for all t given k, then the appropriate scaling is γ(0). However, in the nonstationary case, such as the random walk of Equation (1.1), var(yt−k ) = var(yt ), leading to the following variation: cov(yt−k , yt ) ρ(k) = var(yt−k )var(yt )

(1.16)

(t − k)σε2 = ((t − k)σε2 )(tσε2 ) = (1 − k/t) For ﬁnite t, ρ(k) depends on t for a given k and is not, therefore, invariant to the time index t. Note that ρ(k) → 1 as t → ∞. 1.1.5 A simple random walk with Gaussian inputs An obvious extension of the symmetric random walk is to generate the stochastic inputs as draws from a normal distribution or some other symmetric continuous distribution. This gives a smother pattern to the sample paths, but otherwise replicates the pattern of long sojourns of the paths in one half or the other. This is illustrated in Figure 1.3 where εt ∼ niid(0, 1), but otherwise the details are √ as for Figure 1.1. In the case of Gaussian inputs, E(VT ) = 0.6363 T compared √ to E(VT ) = 0.7979 T for the binomial inputs. For example, if T = 500 then the integer part of E(VT ) is 14, compared to 17 for binomial inputs. 1.1.6 Variations on the simple random walk There are several interesting variations on the basic or ‘pure’ random walk of yt = yt−1 + εt . One of the most useful imparts a direction to the random walk, which can be done in one of two ways. First, the random walk can be made asymmetric. This is very simple to do in the case of binomial inputs, and corresponds to p = q; for example, to continue the gambling example, suppose

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cov(yt−k , yt ) = cov(∑j=1 εj , ∑j=1 εj )

Introduction to Random Walks and Brownian Motion

13

40 30 20 10 0 yt –20 –30 –40 –50

0

50

100

150

200

250

300

350

400

450

500

t Figure 1.3 Sample paths of a Gaussian random walk.

that p > q, then this will impart a positive direction to the walk. To illustrate, the simulations underlying Figure 1.1 were repeated but with p = 0.55, 0.6, 0.65 and 0.7, with the results shown in Figure 1.4. Even in the case of p = 0.55, this change is sufﬁcient to make the walk almost entirely positive, and as p increases further the walk has a clear positive direction. The second and perhaps more familiar way to impart a direction to the random walk is to introduce ‘drift’, so that the random walk becomes: yt = µ + yt−1 + εt

(1.17)

Thus, ceteris paribus, the increment/decrement to the random walk each period is µ, and the sign of µ will determine the direction of the drift subject to the realisation of εt . As in (1.2), by repeated back substitution yt can be expressed as y0 plus the cumulated stochastic inputs, but in this case there is an additional deterministic time trend due to the accumulation of drift: yt = y0 + µt + ∑j=1 εj t

(1.18)

The direction to the sample path of yt is imparted by the term µt, and the random walk generated by the cumulated sum of shocks will, depending on the sign and magnitude of µ, tend to be observed mostly on either the positive or the negative side of the zero axis. In the context of a gambling game µ > 0 could be the cost per play if the random walk is viewed from the banker’s perspective, whereas it is the negative of the cost per play if the random walk is from the gambler’s perspective.

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–10

14

Unit Root Tests in Time Series

100

200

80

150

p = 0.55

p = 0.60

60 100 yt

yt 40

50

20

0

0 0

100

200

300

400

500

–50

0

100

200

400

500

300

400

500

300

250 200

300 t

t

p = 0.65

p = 0.70

200

150 yt 100

yt 100

50 0 0 –50 0

100

200

300

400

t

500

–100

0

100

200 t

Figure 1.4 Sample paths of asymmetric binomial random walks.

The drifted random walk is important because it is a possible characterisation of economic time series that inherently have a direction, as is usually the case with macroeconomic aggregates such as the expenditure components of GNP, employment and price indices. It offers an explanation that is alternative to serially correlated deviations about a deterministic time trend. To illustrate, ten sample paths are shown in Figure 1.5 for a symmetric binomial random walk with µ = 0.05, 0.1, 0.15, 0.2. As the standard deviation of εt is unity, the drift coefﬁcient is in units of σε . The positive drift to the random walk becomes clearer as µ increases. 1.1.7 An empirical illustration To illustrate random walk-like behaviour in a real time series, we consider the exchange rate of the Swiss Franc (SWFR) against the UK £, with T = 7,347 daily observations from 2 January 1980. The data are graphed in Figure 1.6, with the mean of 2.59 superimposed on the ﬁgure. Note that the time axis has been scaled so that its range is from 0 to 1; in effect, each time division is represented as 1/T units of time. There are just 39 crossings of the sample mean during the observation period compared to an expected number of 68 for a sample of this size generated by a random walk with Gaussian inputs. The last 1,000 observations are plotted in Figure 1.7 as a scatter graph of yt and yt−1 ; this ﬁgure

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–20

Introduction to Random Walks and Brownian Motion

15

100

80

80

µ = 0.05

60

µ = 0.1

60

40

yt 40

yt

20

20

0 0

100

200

300

400

500

–20

0

100

200

t 150

400

500

300

400

500

150 µ = 0.15

µ = 0.2

100

100

yt 50

yt 50

0

0

–50

300 t

0

100

200

300

400

500

–50

0

100

200

t

t

Figure 1.5 Sample paths of drifted symmetric binomial random walks.

5 4.5 4 3.5 3 sample mean

2.5 2 1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

t/T Figure 1.6 Exchange rate (daily), SWFR:£.

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–20

0

16

Unit Root Tests in Time Series

2.8 2.7 2.6 2.5 yt 2.4

2.2 2.1 2

slope = 45° 2

2.1

2.2

2.3

2.4 yt–1

2.5

2.6

2.7

2.8

Figure 1.7 Scatter graph of daily, SWFR:£.

show the observations clustered around a line with a slope of 45◦ , which is suggestive of a random walk, although more formally testing would be required to assess this hypothesis. 1.1.8 Finer divisions within a ﬁxed interval: towards Brownian motion The next step in terms of obtaining the limiting process is to ﬁx the length of the time interval, and then divide it into smaller and smaller parts, so that in the limit as the size of the divisions tends to zero, the random walk becomes a continuous process. The random walk is then deﬁned on an interval on the real line with range zero to unity. The length of the walk T is ﬁxed and then divided into N small time steps, so that ∆t = T/N; N is then allowed to increase, so that the time divisions approach 0. There is no loss in setting T = 1 and, therefore, ∆t = 1/N. Within the unit interval an individual instant of time is denoted tj , which satisﬁes tj = tj−1 + ∆t, so that Θ = [t0 = 0, t1 , . . . , tN−1 , tN = 1], where tj = j/N. The other parameter in the random walk is the size of each step, or win/loss amount in √ a gamble, which is taken to be ∆yt = ( ∆t)εt . The variance of ∆yt is therefore (∆t)σε2 , a choice which ensures that if ∆t = 1 then the step size is εt , as in the standard random walk. The random walk is now: √ ytj = ytj−1 + ( ∆t)εt (1.19) The conditional variance of ytj is var(ytj |ytj−1 ) = ∆tσε2 , whereas the unconditional variance of ytj is var(ytj ) = tj σε2 ; and if σε2 = 1, then tj σε2 = tj .

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2.3

Introduction to Random Walks and Brownian Motion

17

2 N

N = 253 1.5 N = 252

1

0 N = 254 –0.5 N = 25 –1

0

0.1

0.2

0.3

0.4

0.5 t

0.6

0.7

0.8

0.9

1

Figure 1.8 Gaussian random walk as time divisions approach zero.

To illustrate the sample paths as the time division tend to zero, the Gaussian random walk was simulated over the unit interval, so that t ∈ Θ = [0, 1], εt ∼ N(0, 1), T = 1 and N = 25h , for h = 1, 2, 3, 4; with these values the unit interval is ﬁrst divided into 25 equal parts and ﬁnally into 390,625 equal parts, so that the grid of time divisions is at ﬁrst very coarse, but becomes ﬁner and ﬁner as N increases. The resulting sample paths are shown in Figure 1.8. An interesting question is whether the sample paths generated as ∆t → 0 have any characteristics of interest. The answer is yes, but we ﬁrst need a limiting result. Deﬁne a scaled version of ytj as follows: Yt j ≡

ytj √ σε N

(1.20)

If εt ∼ iid(0, σε2 ) and Ytj is generated as in (1.14) then as N → ∞, with T ﬁxed, so that ∆t → 0, it follows that: Ytj ⇒D N(0, tj ) = tj N(0, 1)

(1.21)

This result follows by application of the standard central limit theorem (CLT) and is an example of an invariance principle in the sense that although εt is not necessarily normally distributed, in the limit, as N → ∞, a suitably scaled version of ytj is normally distributed. In fact, the assumption that εt ∼ iid(0, σε2 ) is sufﬁcient rather than necessary for (1.21) to hold. The CLT still goes through

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yt 0.5

18

Unit Root Tests in Time Series

if {εt }Tt=1 is a martingale difference sequence (see Billingsley, 1995, p.475; and for generalisations and references, the interested reader is referred to Merlev`ede, Peligrad and Utev, 2006; Ibragimov and Linnik, 1971). √ Equation (1.21) states that ytj scaled by σε N has a limiting normal distri bution with variance var(ytj ) = tj ; thus, dividing the scaled partial sum by tj results in a random variable, denoted Ztj , which is distributed as N(0, 1). In summary: yt j √ ⇒D N(0, 1) σε tj N

(1.22)

These results, especially (1.22), and the division of a ﬁxed time interval into smaller and smaller parts, lead naturally to the concept of Brownian motion, which is considered next.

1.2 Deﬁnition of Brownian motion The stochastic process W(t) deﬁned in continuous time is said to be a Brownian motion (BM) process if the following three conditions are met: BM1: W(0) = 0. BM2: the increments are independent and stationary over time. BM3: W(t) ∼ N(0, tσ2 ); that is W(t) is normally distributed with mean zero and variance tσ2 . W(t) is a standard Brownian motion process if σ2 = 1, when it will be denoted B(t). If σ2 = 1 and W(0) = 0, then B(t) = W(t)/σ converts the process to have a unit variance and become standard BM. If W(0) = µ = 0, and σ2 = 1, then B(t) = [W(t) − µ]/σ is standard BM, which implies W(t) = µ + σB(t). A trended BM is obtained if W(t) = βt + σB(t), so that B(t) = [W(t) − βt]/σ is standard BM. In the case of standard BM, BM3 above is replaced by B(t) ∼ N(0, t). Given BM2 and BM3, and assume that we are dealing with standard BM, then two related results are of interest. First, the difference between BM at times t and s is normally distributed, thus: B(t) − B(s) ∼ N(0, t − s) = (t − s)N(0, 1)

(1.23)

where 0 ≤ s < t; this says that the increment of (standard) BM over the interval t − s is normally distributed with zero mean and variance t − s. A consequence of this result is that letting ∆t be an increment of time, then: √ B(t + ∆t) − B(t) ∼ ∆tN(0, 1) (1.24) The connection between the scaled random walk of equation (1.19) and BM should now be evident: the random walk is speciﬁed in discrete time and if

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Ztj ≡

19

εt ∼ iid(0, σε2 ), but is not niid, then Ytj is approximately normally distributed for ﬁnite N; BM is speciﬁed in continuous time and W(t) and B(t) are exactly normally distributed. Both of these differences disappear in the limit as ∆t → 0 a result formalised below as the functional central limit theorem (FCLT). Referring back to Figure 1.8, which graphs some sample paths of a random walk with increasingly ﬁne time divisions, ∆t → 0, the last of the sub-ﬁgures has ∆t = 1/254 = 0.00000256, and thus this could equally be taken to illustrate some sample paths of BM. BM provides a mathematical model of the diffusion, or motion over time, of erratic particles; for example, the biologist Robert Brown’s original observation in 1827 that pollen grains suspended in water exhibited a ceaseless erratic motion; being bombarded by water molecules, the pollen seemed to be the subject of myriad chance movements. A similar phenomenon can be observed with smoke particles colliding with air molecules. In both examples, the trajectory of the particle over any small period is spiky and seemingly chaotic, but observed over a longer period the particle traces out a smoother path that has local trends. In an economic context, it is evident that the behaviour of stock prices over time, particularly very short periods of time, can be quite erratic – or noisy; however, over a longer period, a direction is imparted to the level of the series. BM is used to model these phenomena: at any one point, or over a small interval, the movement, as captured by the ‘increments’, is erratic and seemingly without structure, whereas over a longer period, the individual erratic movements are slight relative to the whole path. Hence a key element of BM is the way that the erratic increments are built up into the level of the series. Whilst BM speciﬁes normal increments, it can be generalised to increments from other distributions, as might be appropriate for some ﬁnancial asset prices, whose distributions exhibit much greater kurtosis than is found in a normal distribution.

1.3 Functional central limit theorem (FCLT) A result of particular use in establishing the distribution of many unit root test statistics is the functional central limit theorem. Whereas the standard CLT applies to a suitably scaled random variable, the FCLT applies to a stochastic process, which deﬁnes a function rather than a single random variable. Below, for example, the simple random walk of length T, which is an example of a partial sum process, is written as a function of a variable r, such that 0 ≤ r ≤ 1. Allowing r to vary over this range emphasises that the random walk is a random function of r; this is evident from, for example, Figure 1.8, which plots some sample paths, or trajectories, from a random walk – the whole paths generally

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Introduction to Random Walks and Brownian Motion

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Unit Root Tests in Time Series

differ. As noted above, the device of letting ∆t → 0, used in plotting Figure 1.8, leads, in the limit, to Brownian motion. Assume, for simplicity that y0 = 0, then the simple random walk yt = ∑Tt=1 εt , t = 1, . . . , T, can be written equivalently as follows: [rT]

(1.25)

The notation yT (r) emphasises the ﬁxed length T of the sequence and the functional dependence on r. The notation [rT] indicates the integer part of rT; thus rT is exactly an integer for r = j/T, j = 1, . . . , T. (Note that j = 0 would follow if the lower limit of the summation in (1.25) was 0.) The virtue of (1.25) is that yT (r) can be considered as continuous function of r, albeit it will be a step function; however, the ‘steps’ become increasingly smaller as T → ∞, so that, in the limit, yT (r) is a continuous function of r. To consider this limit, yT (r) is ﬁrst normalised as follows: y (r) ZT (r) ≡ T√ (1.26) σε T Let εt ∼ iid(0, σε2 ), with σε2 < ∞, then the FCLT states that: ZT (r) ⇒D B(r)

(1.27)

This is sometimes stated in slightly abbreviated form as ZT ⇒D B (or with a variant of the ⇒D notation). Equation (1.27) states that a suitably normalised version of yT (r) converges to standard Brownian motion. If yT (r) is not √ normalised by σε2 , that is, deﬁne, say, vT (r) ≡ yT (r)/ T, then: vT (r) ⇒D W(r) = σε B(r)

(1.28)

The FCLT is another example of an invariance principle in that the convergence result is invariant to the distribution of the stochastic inputs that drive yT (r) and so ZT (r), in particular they do not have to be Gaussian. Of course, some assumptions have to be made about these inputs, but these assumptions, discussed below, are relatively weak, and the FCLT is simply extended to cover such cases. The nature of Brownian motion B(r) means that it is normally distributed for all r in its domain, its increments are normally distributed and it is jointly normally distributed for different values of r. The CLT is in this sense a by-product of the FCLT. The notation ⇒D is used here as it would be in the case of conventional asymptotic results, where it indicates convergence in distribution; however, here it refers here to the weak convergence of the probability measures (see Billingsley, 1995; Davidson, 1994, for more detail). The latter is more encompassing than convergence in distribution, which just compares the distribution of one random variable with another. In effect, the convergence relates to the convergence of one stochastic process to another, rather than of a single random variable to another.

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yT (r) = ∑t=1 εt

Introduction to Random Walks and Brownian Motion

21

1.4 Continuous mapping theorem (CMT) The FCLT is often used in combination with the CMT applied to function spaces to establish distributional results for unit root test statistics. We ﬁrst state the CMT and then its extension to function spaces. Consider the random variable xT and the continuous function f(.), then the CMT states that: and P(x ∈ Dg ) = 0,

then f(xT ) ⇒D f(x)

(1.29)

where Dg is the set of discontinuity points of f(x) and P(.) indicates probability, (for an elaboration, see Billingsley, 1995, Theorem 25.7; Davidson, 1994, Theorem 22.11, 2000, Theorem 3.1.3). A familiar case from elementary texts is when xT ⇒D x ∼ N(0, 1) and f(x) = x2 , then f(x) ⇒D χ2 (1); thus, if xT is asymptotically distributed as standard normal, then x2T is asymptotically χ2 (1). An example is provided by the standard regression t test, which has a small sample ‘t’ distribution but converges in distribution to N(0, 1), thus its square converges to χ2 (1). The next step is to extend the CMT to functionals, that is functions of stochastic processes which are themselves functions; in this case, interest is in a function of a stochastic process, ZT (r), such as g(ZT (r)) = ZT (r)2 , where g(.) is a continuous mapping, apart from a set of measure zero. The (extended) CMT for functionals of the stochastic process ZT (r) is as follows (where D is the domain of the argument of g(.)). Let g(.) be a functional that maps D to the real line, g: D → R, and which is continuous apart from a set of measure zero, if ZT (r) ⇒D B(r), then: g(ZT (r)) ⇒D g(B(r))

(1.30)

An application of the extended CMT for g(ZT (r)) = ZT (r)2 yields the following: if ZT (r) ⇒D B(r), then ZT (r)2 ⇒D B(r)2 . An application of the extended CMT, of particular importance in unit root tests, relates to the least squares (LS) estimator in the AR(1) model that nests the simple random walk. Consider estimating the following: yt = ρyt−1 + εt

(1.31)

where {εt }Tt=1 is assumed to be a sequence of iid random variables, with zero mean and constant variance, written εt ∼ iid(0, σε2 ). Clearly, ρ = 1 corresponds to the simple random walk of Equation (1.1), so that a natural hypothesis testing approach is to set H0 : ρ = 1 against HA : |ρ| < 1. In the context of (1.31) this is the unit root hypothesis, of which more in the next section and in particular a rationalisation of the word ‘root’ in this context. In this framework, equation (1.31) is the hypothesised data-generating process and the maintained regression, but more generally these may differ. Implicit in this set-up is that y0 is a

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if xT ⇒D x

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starting value and is either a constant – for example, y0 = 0 – or a draw from a distribution with a ﬁnite variance. One possible test statistic, suggested by Dickey and Fuller (see Fuller, 1976), is δˆ ≡ T(ˆρ − 1), where ρˆ is the LS estimator of ρ, large negative values of which will be inconsistent with H0 . The quantities needed to construct δˆ are given as follows: ∑Tt=1 yt yt−1 ∑Tt=1 y2t−1

ρˆ − 1 =

=

(1.32)

∑Tt=1 yt−1 (yt − yt−1 ) ∑Tt=1 y2t−1 ∑Tt=1 yt−1 εt ∑Tt=1 y2t−1

(1.33)

using yt − yt−1 = εt

δˆ ≡ T(ˆρ − 1) =T

=

∑Tt=1 yt−1 εt ∑Tt=1 y2t−1

∑Tt=1 yt−1 εt /T ∑Tt=1 y2t−1 /T2

(1.34)

ˆ which is the ratio Hypothesis testing requires the limiting distribution of δ, of two quantities whose limiting distributions are known (see, for example, Phillips, 1987; Banerjee et al., 1993; Patterson, 2010). In particular: 1

T−1 ∑t=1 yt−1 εt ⇒D T

T−2 ∑t=1 y2t−1 ⇒D T

0

1 0

1 W(r)dW(r) = σε2 [B(1)2 − 1] 2

W(r)2 dr = σε2

1 0

B(r)2 dr

(1.35) (1.36)

ˆ then follows from the extended CMT so that: The limiting distribution, F(δ), δˆ ⇒D

1 0

W(r)dW(r)

1 0

=

W(r)2 dr

ˆ ≡ F(δ)

(1.37)

1 [B(1)2 − 1] 2 1 B(r)2 dr 0

The second line uses (1.35) and (1.36), so that σε2 cancels from the numerator and the denominator. Note that equations (1.35), (1.36) and (1.37) involve integrals of Brownian motion; however, these are not integrals in the standard sense of Reimann or Reimann-Stieltjes integrals. Indeed, whilst BM is continuous it is nowhere differentiable and so these integrals do not exist. Rather, the integrals are deﬁned

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ρˆ =

Introduction to Random Walks and Brownian Motion

23

according to the Itˆ o calculus. To cover this topic here would require quite a lengthy digression; for an excellent introduction to the topic, the interested reader is referred to Mikosch (1998); and for an introduction tailored to unit root tests, see Patterson (2010).

The previous section introduced one of the ‘family’ of Dickey-Fuller (DF) test statistics for a unit root, which is just one of many tests for a unit root. Much has been written about the ‘unit root’ hypothesis, with a multiplicity of tests and a wide range of applications to be found in journal articles, textbooks and theses. In a selective survey published in 1990, Diebold and Nerlove (1990) noted then that ‘The unit root literature is vast . . . .’ It is two decades since that survey, with no abatement in the interest in unit roots, and the topic in some form is still one of the key areas of interest in journal articles. Additionally, many econometric software packages, those available both commercially and academically, include at least one and usually more such test statistics in their programmed options, and the results of such tests are routinely computed for inclusion in undergraduate and graduate project work, including doctoral theses, and in journal articles. A section on ‘unit root testing’ is now close to compulsory in all but the most elementary of econometric courses and textbooks. 1.5.1 What is a unit root? To gain some understanding of what is meant by a unit root, ﬁrst consider the simplest case where a sequence of random variables {yt }Tt=1 is generated by an AR(1) model so that, as in Equation (1.31), yt = ρyt−1 + εt , with εt , t = 1, . . . , T. If ρ = 1, then yt = yt−1 + εt ; that is, ∆yt = εt , where ∆yt ≡ yt − yt−1 , and there is said to be a unit root, strictly in the generating process, but often loosely referred to as in yt or in the time series of observations or realisations of yt . The next section considers how to generalise this idea. 1.5.1.i Generalising the random walk There is more than one way of representing the generalisation. In the ﬁrst representation the AR(1) model is extended directly with further lags on yt ; for example, the AR(2) model is written as yt = φ1 yt−1 + φ2 yt−2 + εt . For consistency, the AR(1) model would then be written with the coefﬁcient denoted φ1 rather than ρ. The AR(2) model could, potentially, have a single unit root, which corresponds to H0 : φ1 + φ2 = 1, or two unit roots, which corresponds to H0 : φ1 = 2 and φ2 = –1. In the latter case the model can be written as ∆2 yt = εt , so that ∆yt = ∆yt−1 + εt , where ∆2 yt ≡ ∆yt − ∆yt−1 . In this speciﬁcation, there is a unit root in the ﬁrst differences, which necessarily already have a unit root.

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1.5 Background to unit root and related tests

Unit Root Tests in Time Series

An alternative way of writing an AR model, which at the same time allows a simple but important generalisation, is to adopt a common factor interpretation (see Chapter 3, section 3.8.1.ii), in which the possible unit root is isolated, with any other dynamic effects originating from the error term. This model is, as is the simple AR model, more easily written with the use of the lag operator and lag polynomial, considered in detail in Appendix 2, introduced brieﬂy here with what is sufﬁcient for present purposes. First deﬁne the lag operator L, such that when applied to the variable yt , it induces the j-th order lag, that is Lj yt ≡ yt−j ; if j < 0, then Lj yt is a lead. The lag operator can be used to deﬁne a model with lags, such as an AR model or, when combined with a moving average (MA) error, an ARMA (autoregressive moving average) model. For example, the AR(2) model can be written as (1 − φ1 L − φ2 L2 )yt = εt ; and the ARMA(2, 1) model, which is of order 2 in its AR component and 1 in its MA component, is written as (1 − φ1 L − φ2 L2 )yt = (1 + θ1 L)εt . In general the ARMA(p, q) model is represented compactly as follows: φ(L)(yt − µt ) = θ(L)εt

(1.38a)

φ(L) = 1 − ∑

p φ Li j=1 i

(1.38b)

θ(L) = 1 + ∑j=1 θj Lj

(1.38c)

q

where µt comprises deterministic terms; for example, a constant or a constant and a linear trend, so that yt is adjusted for a nonzero long-run (deterministic) component by subtracting µt . With the beneﬁt of the lag operator and lag polynomial, the common factor form of the model is written as follows: yt = µt + zt

(1.39a)

φ(L)zt = θ(L)εt

(1.39b)

φ(L) = (1 − ρL)ϕ(L)

(1.39c)

From the perspective of this model, the AR polynomial φ(L) has been factored as the product of two polynomials (1 − ρL) and ϕ(L), one of which is a ﬁrstorder polynomial, with a unit root if ρ = 1, and the other is a polynomial of one lower order than φ(L). The unit root null hypothesis can then always be expressed as H0 : ρ = 1, whatever the order of φ(L). Contrast this with H0 in the p direct ARMA(p, q) model, which is ∑j=1 φj = 1. At this point note a convention that should be borne in mind (it is elaborated on in Appendix 2), which can be brieﬂy illustrated with the model (1−ρL)yt = εt . The deﬁnition adopted here is that the root of the lag polynomial (1 − ρL) is δ1 = ρ−1 , with reciprocal δ−1 1 = ρ; hence, strictly, it is δ1 not ρ that is the root. When ρ = 1 there is no contradiction in referring to ρ as the root, since δ1 = 1

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24

25

also. When ρ < 1 but close to 1, this situation is usually referred to as a ‘near’unit root; this is correct terminology and a usage we follow, but to be precise (on the deﬁnition adopted here), the root in such a case is δ1 = ρ−1 > 1. The existence of a unit root or roots generates a nonstationary process; that is, the probability structure is not constant over time. For example, in the ﬁrst example above, yt has a variance that increases linearly with time and autocovariances and autocorrelations that depend on t (that is they are not invariant to a translation of the time axis); nevertheless, taking the ﬁrst difference results in a stationary process. These properties were demonstrated in sections 1.1.4.iii and 1.1.4.iv. 1.5.1.ii Integrated of order d: the I(d) notation The idea that there are some nonstationary stochastic processes that can be made stationary by applying the differencing operator ∆ ≡ (1 − L) to the component random variables a sufﬁcient number of times leads to a commonly adopted deﬁnition of an I(d) series. The following deﬁnition was suggested by Engle and Granger (EG) (1987): ‘A series with no deterministic component which has a stationary, invertible, ARMA representation after differencing d times, is said to be integrated of order d, denoted yt ∼ I(d).’ (EG used the notation xt , whereas here yt is used throughout.) The reader is very likely to have encountered expressions such as yt ∼ I(1) or yt ∼ I(0). (For a detailed discussion of what constitutes an I(0) series, see Davidson, 2009.) Although EG focus on the integer cases d = 0 and d = 1, they note that their deﬁnition applies to fractional d. One could add, as a clariﬁcation, that d is the minimum number of differences needed to ensure stationarity. Of particular importance in empirical work is being able to distinguish between I(0) and I(1) series, and ﬁve properties of interest, due to Engle and Granger (1987), are summarised in Table 1.1. Given the critical nature of the differences between I(0) and I(1) series, and more generally, I(d) series with d ≥ 1, it is perhaps not a surprise that a number of tests have been suggested with this aim in mind. Most of the tests take the null hypothesis as yt ∼ I(d) with the alternative hypothesis as yt ∼ I(d – 1), the most frequent case being d = 1. However, it is also possible to reverse these roles so that the null hypothesis is yt ∼ I(d – 1) and the alternative is yt ∼ I(d). (More precisely, one should refer to the data-generating process as generating series, or observations, that are I(d).) A brief development of ‘unit root’ tests is considered in the next section. 1.5.2 The development of unit root tests In the applications including and immediately following the seminal contribution by Nelson and Plosser (1982), the unit root test statistics were usually those due to Dickey and Fuller (see Fuller, 1976), which have come to be known

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Introduction to Random Walks and Brownian Motion

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Unit Root Tests in Time Series

I(0)

I(1)

variance, var(yt )

var(yt ) is ﬁnite

var(yt ) → ∞ as t → ∞

autocorrelations, ρ(k)

invariant to translation of time axis ρ(k) → 0 for k large enough; ﬁnite sum

not invariant to translation of time axis ρ(k) → 1 for all k as t → ∞; not summable

innovation effect

transient

permanent

spectrum, f(λ), at zero, f(0)

0 < f(0) < ∞

f(0) → ∞ as λ → 0

expected time between crossings of E(yt )

ﬁnite

inﬁnite

Note: See Engle and Granger (1987); yt is either I(0) with zero mean or I(1) with y0 = 0.

as DF or, in their augmented form, ADF statistics. Subsequently, the Phillips and Perron (1988) developments of these statistics in a semi-parametric form, known collectively as PP tests, became popular and it was not unusual to see joint reporting of the ADF and PP tests. However, several other test statistics were suggested and the battery of such tests started to grow after Nelson and Plosser’s article. The development of further unit root test statistics continued in the 1990s, with a signiﬁcant contribution by Elliott, Rothenberg and Stock (ERS, 1996) and an allied paper by Elliott (1999); both papers were available in discussion paper form several years before their publication dates. These articles noted that whilst it was not possible to obtain a single test statistic that was uniformly most powerful across the entire parameter space of interest, it was possible to develop a test statistic that was most powerful against a particular point in the parameter space, hence the terminology of a ‘point-optimal’ test. The problem with this approach was that it seemed to require an inﬁnity of test statistics, one for each point in the parameter space under the alternative hypothesis. However, ERS were able to show that very little, if any, power was lost by, in effect, choosing just one value of the root in the stationary region as representative of all of those near to the unit root and then computing the test statistic using that value. Moreover, a variation of the approach led to the use of quasi-differenced data in standard tests such as the DF/ADF tests, so that ERS-type tests became easy to apply and popular – and familiar from the quasi-differencing approach to deal with weakly dependent errors in a regression model, as in the Cochrane-Orcutt procedure.

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Table 1.1 Contrasting properties of I(0) and I(1) series.

27

ERS-type tests joined the set of DF tests as those to which practitioners would most readily turn, partly because they were quickly incorporated into commercially available software. Indeed, they were often preferred because of their superior power under the assumptions of their derivation. At the risk of simpliﬁcation, the next important development came in exploiting the difference between the assumptions in ERS (1996) and Elliott (1999). In essence, the difference was quite simple: what was the nature of the starting point or initial condition in a time series process? For example, consider a trended series that is adequately modelled under the alternative hypothesis as stationary around a linear trend. Does it make much difference to the test results if the starting point is close to or far away from the trend (in some well-deﬁned measure)? For example, consider one of the data sets used in this book comprising US industrial production with 1,044 observations for 1919m1 to 2005m12: would it matter if the observations were taken as starting in 1925, which was relatively close to trend, or in 1935, which was relatively a long way from a linear trend? With about 1,000 observations one might be tempted into thinking that this would not make a difference. However, it transpires that it does make a difference (both in this case and in general) and the test results are markedly affected by the relative scale of the initial observation. What this means is that it is possible for contradictory test results to be obtained on the same data set either by using the same test statistic but with starting points with different characteristics, or by using different test statistics with different characteristics but at the same starting point. In a revival of use, it turns out that the DF tests, which were dominated by the ERS-type tests, in terms of power, for an initial observation that was close to trend (or a constant mean in the non-trended case), are more robust as the initial observation departs from trend. Given the importance of calibrating the trend to this result, the question of the appropriate speciﬁcation of the trend, an issue that had been largely sidelined, returns to have importance. However, to continue the pr´ecis of developments, the next task was to seek a unit root test statistic that was robust to the initial condition. One way of doing this was to combine test statistics with different characteristics. Simple linear combinations seem to work well, offering protection against an unknown initial condition at not too much cost in terms of power. Alternatively, as demonstrated by Elliott and M¨ uller (2006), it is possible to construct a unit root test that is robust to the scale of the initial condition. For a discussion of current issues in unit root testing, including the speciﬁcation of the trend, the role of initial conditions, see the special issue of Econometric Theory (2009), starting with Harvey et. al. (2009), and followed by a number of commentaries. Of course, there are many variations and complications that occur in practice and which have attracted attention. Perhaps the simplest practical consideration arises where a time series has a seasonal pattern, which characterises many

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Introduction to Random Walks and Brownian Motion

Unit Root Tests in Time Series

production, employment and consumption series. It is not surprising, therefore, that this area attracted attention not long after the Nelson and Plosser article. In some ways it is more natural to use the seasonally unadjusted rather than the seasonally adjusted data, as the latter necessarily involve some assumptions about the form of the seasonality. There is the risk in the latter of inducing patterns that are not present in the unadjusted series. However, it soon became clear that the presence of a seasonal period allowed the possibility of unit roots at frequencies other than at the zero frequency associated with the conventional unit root tests, and thus the tests became apparently quite complicated due to the need to consider the possibility of unit roots occurring at different seasonal frequencies. Another signiﬁcant development that affected the course of unit root testing was to swap the null and alternative hypotheses, so that the null hypotheses became that of stationarity whilst the alternative became that of nonstationarity. This development was not entirely straightforward because the null hypothesis is not now that of a point but of a region in the parameter space. One of the key contributions was a test due to Kwiatkowski, Phillips, Schmidt and Shin (1992), referred to as the KPSS test. This test exploited the duality between a structural model of a time series and its reduced form to solve the testing problem. In the former, the time series is viewed as being built up from components; for example, an unobservable level plus an irregular component. In turn, a number of other stationarity tests were suggested, including those for seasonally unadjusted time series. Notwithstanding the duality between conﬁdence interval construction and hypothesis testing, the emphasis in much of the early empirical literature was on hypothesis testing, that is, coming to a decision with two possible outcomes: either to reject or not to reject the null hypothesis of a unit root. Of course, it was well known that this dichotomy was often too simple: surely a conﬁdence interval would be more informative? To some extent reporting a p-value, as in the elementary textbook case, would help, but in fact it was rarely done, partly because the quantiles generally had to be obtained by simulation rather than reference to a standard set of tables. Two developments eased this case. The hypothesis testing/conﬁdence interval approaches were ‘re-connected’; for example, by Stock (1991), Hansen (1999) and Elliott and Stock (2001). Constructing a conﬁdence interval by inverting a test statistic is familiar from the standard ‘t’ statistic and the approach can be carried across to unit root tests. Indeed, as Elliott and Stock (2001) demonstrated, advantage can be taken of the more powerful unit root test statistics to invert one of these to get a shorter interval for a given conﬁdence level. Indeed, the circle was in a sense completed as a point-optimal test of the unit root hypothesis, along the lines of ERS (1996) and Elliott (1999), was equally a point-optimal test if the null and alternative hypotheses were swapped, so that

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1.6 Are unit root processes of interest? There have been a very large number of studies addressed to the issue of whether a particular series has been generated by a stochastic process with a unit root, and the question arises as to why there is such an (enduring) interest. This question is answered more fully in the next chapter, the present intention being to give an idea of some of the topics that have been studied. Nelson and Plosser (1982) considered 14 macroeconomic time series, such as GNP, industrial production, some price indices, and employment and bond yields. Subsequent research included a more detailed analysis of a number of these series, with particular interest focusing on aggregate measures of output, such as GDP or GNP, especially for industrialised countries (see, for example, Campbell and Mankiw, 1987a, 1987b; Cochrane, 1988; Rudebusch, 1992, 1993; Harvey, 1985). However, interest widened and many articles that involved the use of economic time series included a test of some form on the unit root hypothesis, in part because there was an underlying theoretical base for the distinction between unit root and non-unit root processes from an economic perspective (that is, it was not just a matter of the econometric aspects of the application). To give an idea of the underlying motivation, three areas of application are considered below. 1.6.1 Are there constant ‘great ratios’? An area of interest for the importance of unit roots relates to the implications of some growth models for the ratios of economic variables, sometimes referred to as the ‘great ratios’. In a seminal article, Klein and Kosobud (KK) (1961), suggested ﬁve celebrated ratios of economics, namely the savings-income (or consumption-income) ratio, the capital-income ratio, labour’s share of income, the income velocity of circulation and the capital-labour ratio (see also Kaldor, 1957), to which other, monetary ratios, such as the real money supply and the real interest rate (not strictly a ratio) have been added. KK constructed a small macroeconomic model which showed the connections between their ﬁve ratios. Later research developed the balanced growth implications of neoclassical growth models (see, for example, Brock and Mirman, 1972; King, Plosser and Rebelo, 1988a, 1988b; and King and Rebelo, 2000).

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a point-optimal stationary test, but of a null close to the unit root, could be obtained by using the other tail of the corresponding unit root test. The second development was encouraged by the increasing capacity of personal computers, so that large-scale simulations could be undertaken at little cost. This enabled bootstrapping to be applied to unit root tests and conﬁdence interval construction, and a straightforward outcome of the former was the p-value associated with a particular sample value of a test statistic.

Unit Root Tests in Time Series

Of course, whether such constancy of the great ratios, which would anyway only be approximate, held empirically is another matter, and on examining these ratios for the US economy, by regressing them on a constant and a time trend, KK concluded that only one of the ratios, that for labour’s share of income, could be considered approximately constant. On the other hand, at a descriptive level and with a more recent data set, King and Rebolo (2000) suggested that for US data the ratios of investment and labour income to output ﬂuctuated about a constant mean, and whereas there was an upward trend for the ratio of consumption to income (since 1952), the trend was relatively slight. We can interpret KK’s interest in the possible existence of approximately constant ratios, where the individual components are themselves trended, in the following way. Consider two time series, each with a stochastic trend of the form generated by the accumulation of shocks, as in Equation (1.2); then, in general, the stochastic trends will be unrelated, so that the ratio of the two series, or the logarithmic difference, will also have a stochastic trend. The exception to this rule is when the stochastic trends are annihilated, resulting in a trendless ratio; in such a case the time series are said to have a common trend and are cointegrated. For example, consider consumption, c, and output, q, on a per capita basis, and suppose each of these to have a stochastic trend, but this trend is common to each variable, such that the log difference c – q; that is, the log of the consumption-output ratio, is trendless. Similarly, extending the analysis to include per capita investment, i, so that each of c, i and q, have a stochastic trend, but balanced growth implies there is a single stochastic trend, such that the log ratios c – q and i – q are trendless. There are (at least) two ways to assess whether there is evidence to support the stability of the great ratios. The ﬁrst is to consider each (log) ratio separately and carry out a unit root test (or swap the null hypothesis and carry out a stationarity test); non-rejection of the null hypothesis would then be evidence against the stability of the ratio. Tests of this kind have been reported by Harvey et al. (2003) for the G7 countries; earlier work includes Kunst and Neusser (1990). An alternative is to consider a system approach in which several series are jointly modelled, and then tested for the extent of cointegration in the system. Both the references cited above also use this approach; additionally, Mills (2001) extends the analysis to consider whether there are not only common trends but also common cycles. 1.6.2 Purchasing power parity The theory of purchasing power parity (PPP) is fundamental to the theory of the real exchange rate and is a cornerstone of international economics. It is the macroeconomic analogue of the law of one price (LOOP). At the microeconomic level, the idea is that the price of a homogeneous good should be the same when converted to units of a common currency; in this case the nominal exchange

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31

rate is regarded as a variable exogenous to the ﬁrm’s decision, whereas at the macroeconomic level it is an endogenous variable determined by the ratio of (aggregate) price levels for the domestic and foreign economies. To consider this aspect, the following notation is adopted. The nominal exchange rate Et is deﬁned as the domestic currency price of a unit of foreign currency; and the prices of a homogeneous good in domestic and foreign currencies are denoted Pdg,t and Pfg,t , respectively. Then LOOP implies Pdg,t = Et × Pfg,t ; that is, the domestic price equals the foreign price expressed in units of the domestic currency; expressing the price in the units of the foreign currency would give the same result. From the perspective of an individual producer, who has no market power, the right-hand-side variables are exogenous. In a perfect, frictionless market (without tariffs or transaction costs), setting Pdg,t to be greater than Et × Pfg,t means that the domestically produced good is not competitive with its foreign counterpart and will not attract any market share. At the macroeconomic level, Et is the endogenous variable, determined by the aggregation of individual market decisions, rather than exogenous as at the microeconomic level. Let Pdt and Pft be suitably deﬁned aggregate price indices for the domestic and foreign countries, respectively, then PPP states that the following relationship should hold: Pdt Et = A (1.40) Pft where A is a constant. In the simplest version of PPP, A = 1, but A differing from unity is permissible within the general theory; for example, A = 1 could arise from the use of different base years in the construction of the indices Pdt and Pft . If A = 1, then PPP implies Pdt = Et Pft , so that the price levels are equalised in units of a common currency (here the domestic currency), which is the direct analogue of LOOP. In a weaker version of PPP, the elasticity of the nominal exchange rate with respect to relative prices is allowed to differ from unity. That is: δ Pdt Et = A (1.41) Pft Allowing δ = 1 is weaker in the sense that a 1% change in relative prices results in a δ% change in the nominal exchange rate, so that Pdt = Et Pft even if A = 1. The weak form of PPP results in: Pdt ln Et = ln A + δ ln Pft = ln A + δ(ln Pdt − ln Pft ) et = a + δ(pdt − pft )

(1.42) (1.43)

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where a lower-case letter denotes the logarithm of the corresponding upper-case variable. If δ = 1, so that the elasticity of the nominal exchange rate with respect to relative prices is unity, then taking the term in relative prices to the left-hand side gives the log of the real exchange rate: et − (pdt − pft ) = a

(1.44)

REt ≡ Et

Pft Pdt

=A

(1.45a) (1.45b)

In practice, the real exchange rate is not expected to be constant, but rather stochastic and mean-reverting. For convenience of notation, deﬁne ret ≡ et − (pft − pdt ), then the stochastic version of this equation is: ret ≡ et − (pft − pdt ) = a + ut

(1.46a) (1.46b)

where ut is I(0), hence some dependency is allowed in the structure of ut , but it must be weak in the sense of allowing the log of the real exchange rate to return to its mean given a shock; the lack of an immediate return to the PPP rate, following a shock, has been variously attributed to sticky prices, incomplete information and incomplete arbitrage. However, notwithstanding these ‘short run’ impediments, the argument goes, in the long run the real exchange rate reverts to the rate implied by PPP, although that reversion may be quite slow. One often-cited measure of the speed of return is the half-life of a shock; that is, when 50% of the overall adjustment to as shock has been reached. One way of testing this property is by way of a test for a unit root on ret , the presence of which is not compatible with mean reversion. Early studies used data from the post-Bretton Woods period of ﬂoating exchange rates for industrialised countries and one or more of the Dickey-Fuller tests or the Phillips and Perron (1988) semi-parametric versions of these tests, largely ﬁnding that the null hypothesis of a unit root in the generating process for ret could not be rejected. One of the difﬁculties in establishing the robustness of this ﬁnding was that measures of the persistence of shocks suggested that they had very long life, with estimated half-lives of ﬁve years or more (see, for example, Rogoff, 1996). Thus a key problem was seen as distinguishing a unit root from a very nearunit root, highlighting the problem of the power of unit root tests, which is the ability of a test to ﬁnd in favour of the alternative when the alternative is true.

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Thus, this equation states that the real exchange rate is constant. In terms of the level of the real exchange rate:

33

Subsequent research considered extending the data sets, incorporating data on ﬂoating and ﬁxed exchange rate periods, to improve the length of sampling from the underlying stochastic process. Whilst Et is endogenous under a freely ﬂoating exchange rate, this cannot be so under a ﬁxed exchange rate; nevertheless, provided that at least one of the domestic or foreign price level is free to move, a PPP equilibrium can be maintained and, in principle, data from ﬁxed and ﬂoating periods can be combined, although speeds of adjustment may well differ between the two regimes. A second route to improving power is to apply more powerful unit root tests, a candidate being the development of unit root tests due to ERS, which can improve power quite markedly (only ‘can’ because the power gain depends on the initial condition; see Chapter 7). A third possibility is to combine time series of exchange rates into a panel and exploit the resulting extension of the sample to obtain more efﬁcient estimates and more powerful tests. For an overview of the literature on PPP, see Sarno and Taylor (2002). 1.6.3 Asset returns A topic that has generated a great deal of interest is the time series behaviour of asset prices (suitably deﬁned to include dividends if payable); in particular, whether they follow a random walk, so that returns are serially random. There are several alternatives to this hypothesis. For example, according to the ‘chartists’ there are patterns in the previous behaviour of stock prices, so that there exist dependencies that can be exploited and used to predict future prices. Note that a stochastic process generating a random walk is a special case of a unit root process: it contains a unit root, but its increments, εt , are assumed to be uncorrelated, whereas a unit root process may exhibit weak dependency. Whilst it is common to associate the random walk hypothesis (RWH) with the efﬁcient markets hypothesis (EMH), as Lo and Mackinlay (1999) point out, these two hypotheses are not the same, although they coincide under certain restrictive conditions. Generally, “the Random Walk Hypothesis is neither a necessary nor a sufﬁcient condition for rationally determined security prices” (Lo and Mackinlay, 1999, p.5). Nevertheless, testing asset prices for a unit root has and continues to attract considerable interest in part because the rejection of the RWH implies a predictable component in stock market prices, notwithstanding the implications of such a ﬁnding for the EMH. (Indeed, the presence of a component that enables trading proﬁts is not inconsistent with what Lo and Mackinlay (1999) call the practical version of the EMH.) There are several versions of the RWH, depending upon what is assumed about the covariance between differently dated returns, and Campbell, Lo and Mackinlay (CLM) (1997) suggest a typology of three versions of the RWH. For simplicity, assume that an asset, with price Pt , that does not pay a dividend,

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evolves according to: Pt = µ + Pt−1 + εt

(1.47)

⇒ (1.48)

where µ, as in Equation (1.17), is drift interpreted in context as the expected price change. Motivated by Equation (1.48), in random walk terminology, the εt are often referred to as increments. RW1: εt ∼ iid(0, σε2 ), where iid indicates that {εt } is an independently and identically distributed sequence of random variables. RW2: εt ∼ inid(0, σε2 ); that is, the εt are independent but not identically distributed, which allows unconditional heteroscedasticity, such as may occur with long runs of data. RW3: cov(εt , εs ) = 0 for t = s; in this version, the increments are uncorrelated but may be dependent; this assumption allows conditional heteroscedasticity. Thus, one focus of the RWH has been to condition a test on the unit root implicit in Equation (1.48) by examining the time series properties of the increments as detailed in RW1–RW3. CLM (1997) provide a detailed analysis of a number of such tests. Rejection of the null hypothesis in these cases is not focussed on rejection of the unit root, but of the properties of the increments as detailed; even so, the implicit null hypothesis may well include the unit root, with rejection implying trend stationarity and, hence mean reversion, rather than a unit root with serially correlated increments. As noted in section 1.1.2, and summarised by the limiting distribution of Equation (1.4), there is some mean reversion in a random walk in the sense of returns to the origin in the form of sign changes and reﬂections. From that section, note that the expected percentage of mean reversions for large T in a √ random walk with symmetric binomial shocks is 100(0.7979/ T), whereas with √ Gaussian shocks it is 100(0.674/ T); for example if T = 5,000, a not unusual sample size for high frequency ﬁnancial data, then these percentages are 1.12% and 0.95% of the sample, respectively. If stock prices are mean-reverting in excess of these percentages, there is some dependency in successive random variables in the stochastic process that can be characterised as mean reversion. Thus it would be possible to undertake a test of the mean reversion of stock prices by direct reference to the number of sign changes and reﬂections. The general form of such a test has been suggested Burridge and Guerre (1996) and extended by Garc´ıa and Sanso´ (2006). Kim et al. (1991) summarise the evidence to that date as somewhat mixed; earlier studies tended to ﬁnd support for the random walk view, whilst later studies,

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∆Pt = µ + εt

35

using the variance ratio methodology (see Chapter 2, section 2.3) found against the lack of serial correlation. An often used test statistic is the q-th order variance ratio, V(q), which exploits the linear relationship between the variance of yt and yt + q (see Equation (1.10c)). In the context of asset prices, V(q) is the ratio of the variance of the q-period return to the variance of the one-period return divided by q. (A more detailed analysis is provided in Chapter 2, section 2.4.1.) In the case of a pure random walk V(q) = 1, whereas V(q) = 0 corresponds to a trend (or mean) stationary process; values of V(q) below one but above zero indicate persistence, but less than that of a random walk. Values of V(q) greater than one suggest that the impact of shock grows over time. The empirical evidence suggested typical estimates of V(q) that were greater than unity for small q, but less than unity for large q. This pattern means that the effects of shocks are ‘undone’ over time. Kim et al. (1991) drew a distinction between sample periods before and after World War II (WWII), with mean reversion typical of pre-WWII samples and mean aversion typical of post-WWII samples. Later views include those of Lo and MacKinlay (1999), who entitled their collection of journal articles A Non-Random Walk Down Wall Street, suggesting evidence against the random walk hypothesis. There are also some studies that focus directly on the unit root, since rejection of that part of Equation (1.47), implies predictability in stock prices. As examples of such studies, Chaudhuri and Wu (2003) consider stock prices for 17 emerging markets using Dickey-Fuller tests allowing for structural breaks in the time series and reject the unit root hypothesis at the 10% or 5% level for ten of these countries.

1.7 Concluding remarks One of the simplest processes that contains a unit root is the random walk, which generates random variables by the recursion yt = yt−1 + εt (a frequent alternative notation is St rather than yt ). Whilst usually too simple to characterise economic time series, this process is a useful place to start because it captures most of the key features of more complex processes with a unit root. Two of these relate to the nonstationarity and ‘mean aversion’ of a random walk. For example, if εt is white noise, with variance tσε2 , and y0 = 0 (or other ﬁxed number), then the variance of yt is tσε2 . Thus the variance increases linearly with t and so the process generating yt is not (weakly) stationary. The property of mean aversion, together with the increasing variance, gives the sample paths of random walks their typical pattern of rarely crossing E(yt ), wandering one side or the other of E(yt ) and getting increasingly further from E(yt ). By ‘mean reversion’ is meant either a reﬂection (that is, to ‘hit’ E(yt ) and return to the same side) or a sign change relative to E(yt ). Just how (in)frequently a random walk mean-reverts depends on the distribution of εt ; for example, with

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Introduction to Random Walks and Brownian Motion

Unit Root Tests in Time Series

Gaussian inputs, the mean number of reversions for a sample path of 1,000 realisations is just 20. The random walk is of interest because it seems to characterise some economic time series; for example, nominal, and perhaps real, exchange rates and asset prices more generally. The random walk can be generalised in a number of ways, two important ones being to allow for ‘drift’ and weakly dependent stochastic inputs. As to drift, if aggregate expenditure series such as GDP and consumption are to be considered to have random walk properties, the arbitrary nature of the ‘side’ of the walk must be controlled since, for example, GDP cannot become negative. The development to allow for weakly dependent stochastic inputs is one of the most important in practice and will be considered at several places in later chapters. The problem is that for many series it is unlikely that εt is white noise. A more general framework is to write the random walk as yt = yt−1 + zt , where the special case is zt = εt , otherwise E(zt zt−s ) = 0, for some s = 0; for example, an MA(1) process for zt is zt = εt + θ1 εt−1 . That the weak dependency might moderate the strength of the random walk-type behaviour can be seen from this simple extension by letting θ1 → –1, which negates the effect of the implicit coefﬁcient of unity on yt−1 in the random walk. A random walk is an example of a stochastic process, which is a collection of random variables with the particular feature that they are ordered in time, either discrete time or continuous time. Thus, whilst as far as history is concerned we only observe one sample path of a time series, that particular realisation or sample path is simply one of a number of possible sample paths. The limit of a random walk, as the step size tends to zero, is a process deﬁned in continuous time, that limit being Brownian motion. The sample paths of Brownian motion are known to be ‘spiky’ and non-differentiable. Brownian motion is of interest not just as a stochastic process that potentially characterises some economic time series, but because of its importance in distribution theory for unit root test statistics, which will be referenced in subsequent chapters. One feature of a random walk that is relevant to economic time series is the accumulation of ‘shocks’ leading to a stochastic trend, which can be considered as an alternative to a deterministic trend. The next chapter turns the focus onto these competing explanations of sustained movements in one direction, leading to one possible distinction of interest; that is, between processes that are difference stationary, as in the case of a unit root process, and those that are trend stationary, that is, stationary once a deterministic trend has been removed.

Questions Q1.1.i Derive a general expression for γ(2) for a random walk with white noise shocks.

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Q1.1.ii Obtain a general expression for γ(k) and show that it depends on t for given k. Q1.1.iii What difference does it make to γ(2) if y0 is a random variable with a bounded variance, say σ02 ? A1.1.i Start with some simple examples, where the time indices are two periods apart, to establish the general pattern.

= σε2 E(y2 −E(y1 ))(y4 − E(y4 )) = cov(ε1 + ε2 , ε1 + ε2 + ε3 + ε4 ) = 2σε2 In general: cov(yt−2 , yt ) = cov(∑j=1 εj , ∑j=1 εj ) t−2

t

= (t − 2)σε2 The general result follows because yt−2 and yt have (t – 2) terms in common of the form ε2j , j = 1, . . . , t – 2, with E(ε2j ) = σε2 ; and all cross-products εi εj , i = j, have an expectation of zero. The point to note is that not only does the variance of yt increase with time, but the autocovariances also increase with time and are not therefore invariant to a translation of the time index. A.1.1.ii Note that cov(yt−k , yt ) is given by: cov(yt−k , yt ) = cov(∑j=1 εj , ∑j=1 εj ) t−k

t

so that there are (t – k) terms in common of the form E(ε2i ) = σε2 , with crossproducts E(εi εj ) that are equal to zero; hence cov(yt−k , yt ) = (t − k)σε2 , which depends on t for given k. A1.1.iii The essential nature of the answer is unchanged. To illustrate,reconsider : cov(y1 , y3 ) = cov(y0 + ε1 , y0 + ε1 + ε2 + ε3 ) = σ02 + σε2 cov(y2 , y4 ) = σ02 + 2σε2 cov(yt−2 , yt ) = σ02 + (t − 2)σε2 .

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E(y1 −E(y1 ))(y3 − E(y3 )) = cov(ε1 , ε1 + ε2 + ε3 )

2

Introduction This chapter provides some motivation for the continued interest in assessing whether a stochastic process has a unit root, particularly for series that exhibit trended behaviour. In part, such interest lies in a critique of a procedure that models the trend component of a series as a deterministic function of time, usually as a simple low-order polynomial of time, a linear trend being a common choice. In essence, such a procedure is not meant to be taken as a structural explanation of the systematic movement in a series, but rather as a workable proxy for a large number of forces that could be reduced to a simple representation. In the case of some time series this view seems to ﬁnd support in the data, perhaps more so with slowly moving trended series such as population than with more erratic series such as the prices of ﬁnancial assets. Deviations from the trend are then viewed as transitory, the trend being the attractor to which the series reverts given sufﬁcient time for adjustment. This, of course, is the trend stationary view of the generation of time series. On the other hand, if the series is generated by a difference stationary process (see Chapter 1, section 1.1.4), removing what was erroneously thought to be the trend would induce an artiﬁcial periodicity into the resulting detrended series. After a brief introduction in section 2.1, this is the concern of section 2.2, which considers the consequences of the errors of spurious detrending and, alternately, spurious differencing. Section 2.3 takes up the importance of the distinction between trend stationary (TS) and difference stationary (DS) processes from the perspective of distinguishing the impact and permanence, or lack of it, of shocks. Whilst in the simplest of models the difference between a TS and a DS process is a clear dichotomy, the impact of shocks being transitory in the former and permanent in the latter, there are two related problems that have to be addressed. In the ﬁrst of these, in more realistic models, ﬂuctuations, or shocks, can be 38

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Why Distinguish Between Trend Stationary and Difference Stationary Processes?

modelled as generated by a combination of a stationary series and a random walk, in which case the relevant question is ‘How big is the random walk?’ (see Cochrane, 1988), implying that whilst there may be a random walk element, it is relatively unimportant. The second problem is endemic to data series of relatively ‘short’ span. This is part, but not all, of the argument of being able to distinguish a ‘near’-unit root process from a unit root process, with a necessarily ﬁnite sample of data. (For consideration of this issue in a more recent context, see M¨ uller, 2008.) As Rudebsuch (1993) notes, what is also of interest is whether it is possible to distinguish empirically between TS and DS representations that are plausible alternatives that imply very different adjustment patterns at realistic economic horizons; for example, less than ten years. For the latter problem what is of interest is the moving average representation relating the growth rate (or ﬁrst difference) to the history of shocks and the time proﬁle of the cumulative sum of the moving average coefﬁcients. Section 2.4 illustrates the key concepts with an empirical example. Section 2.5 continues the theme of considering problems that arise when a modelling process is invalidated by the use of integrated series. In that section the problem is that of spurious regressions using I(1) series. This is a problem that goes back at least to Yule’s (1926) observation of ‘nonsense’ correlations, which was put in a regression context by Granger and Newbold’s (1974) inﬂuential article, combined with Phillips, (1986) explanation of the underlying theory. Granger and Newbold (1974) showed that a regression of one I(1) variable on an unrelated I(1) variable, or variables, would produce standard goodness-of-ﬁt statistics indicating that there was a relationship beyond that of statistical chance. This observation gave rise to the practice of pre-testing candidate variables for their order of integration to ensure that only a regression that was ‘balanced’ in terms of its (mean) time series properties would be estimated.

2.1 What accounts for the trend? A concern about the presence of a unit root lies in the detrending procedure that was common practice prior to the early 1980s. This practice viewed a time series of observations on economic variables, such as employment and output, as comprising a secular, or trend, component and a cyclical component. The former captured the long-run tendency of the series, and the latter, deviations about the long run, which was viewed as being driven by capital accumulation and population growth, as suggested by simple growth theories of the time. On this basis the changes due to cyclical components were not long-lasting but ‘transitory’, whilst the trend captured permanent and rather slow-moving changes. As a result, time series were often detrended by regressing them on

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a linear time trend, usually ﬁrst taking the logarithm of the series, with the resulting residuals representing the relatively ‘fast-moving’ cyclical component, which was then the subject of business cycle analysis. This process is appropriate for a trend stationary series; that is, a series that is stationary once it has been detrended, usually by a prior regression on deterministic components, such as determined by a low-order polynomial trend. However, this deterministic trend/cycle decomposition could only at best be a sketch or stylisation of the components of a time series. An alternative can be developed by allowing the long-term movement to be determined by stochastic inputs: for example, inventions, discoveries and human catastrophes, such as wars. The trend would then be determined by the accumulation of these stochastic inputs: it would be a stochastic rather than a deterministic trend. A particular example of a process that generates a stochastic trend is a unit root process that becomes stationary on differencing. To regress such a series on a time trend and then obtain the residuals, identifying these as the cycle, would be fundamentally incorrect inducing a damped sine wave seemingly indicating periodic behaviour. These implications are considered further in the next section.

2.2 Inappropriate detrending and inappropriate ﬁrst differencing This section is concerned with the implications of spuriously detrending an integrated series, ﬁrst on the limiting distributions of the LS estimators from a trend regression and, second, on the induced periodicity of the detrended series. The following section considers the reverse error; that is, inappropriately assuming that the DGP generates an integrated series, when the series is, in fact, stationary. Consider a time series that is generated by a random walk process, but this is unknown to the researcher who, in order to remove a trend, regresses the series on a constant and a linear trend. The researcher believes the series to have been generated by a linear trend plus stationary deviations, so that it is trend stationary. There are a number of empirical examples of this detrending procedure in the literature. For example, suppose interest centres on examining a time series to isolate those components associated with cycles or periodicities of different frequencies. As noted in section 2.1, to avoid confusing long cycles with a trend component, one procedure is ﬁrst to remove the trend by a prior regression on a constant and a linear time trend and then examine the residuals from such a regression. If, however, the trend is the stochastic trend of a random walk, the procedure will be in error. To give an example from the economics literature, Durlauf and Phillips (1988) examine the excess volatility tests of Shiller (1979), where the nature of the DGP is critical to the analysis. In the Shiller framework, let pft be a forecast

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40

of pt ; for example, of stock prices, with associated forecast errors ut = pt – pft ; then an implication of rational forecasts is that the errors are orthogonal to the forecasts, implying that cov(ut , pft ) = 0, and hence var(ut ) = var(pt ) – var(pft ) > 0. In the general case, pt and pft are trended and a critical question in testing the non-negativity prediction is whether pt and pft are nonstationary. In Shiller’s original work, deterministic trends are removed before the tests are undertaken, but the balance of subsequent work is that it is more reasonable to assume that these series are stochastically trended. Durlauf and Phillips (1988) show that the properties of the test for non-negativity depend critically on whether it is correct to detrend the data. It might be thought that all one has to do is regress the variable on a trend and test its signiﬁcance; however, this is not the case. Consider Table 2.1, with column headings indicating the DGP as being either trend stationary or difference stationary, and row headings indicating the decision taken by the researcher. The off-diagonals indicate the two possible errors of: (i) choosing DS when the DGP is TS; and (ii) choosing TS when the DGP is DS. Both errors induce artiﬁcial, or spurious, characteristics into the resulting series. Table 2.1 Errors of speciﬁcation. Decision ↓

DGP→

Trend stationary Difference stationary

Trend stationary

Difference stationary

Correct Spurious differencing

Spurious detrending Correct

2.2.1 Spurious detrending 2.2.1.i Limit distributions and test statistics The simplest case is sufﬁcient to illustrate the problems. In this case the datagenerating process (DGP) is a random walk, with white noise inputs. The distributional results are the same without and with a drift (see Durlauf and Phillips, 1988, Theorems 2.1 and 3.1). In the former case, the DGP is ∆yt = εt . The regression model and the (spuriously) detrended series are, respectively: yt = β0 + β1 t + zt

(2.1)

yˆ˜ t ≡ yt − (βˆ 0 + βˆ 1 t)

(2.2)

Where ˆ above a coefﬁcient denotes the LS estimator; and let the F test of the null hypothesis βi = 0 be denoted Fβi . Durlauf and Phillips (1988) show that the limiting distributions of βˆ 0 and βˆ 1 are as follows: T−1/2 βˆ 0 ⇒D N(0, 2σε2 /15)

(2.3)

T1/2 βˆ 1 ⇒D N(0, 6σε2 /5)

(2.4)

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is signiﬁcantly different from zero and, hence, potentially justify a decision to detrend yt on the grounds that it is trend stationary. Durlauf and Phillips (1988, Theorems 2.2 and 3.2) show that both F tests (and by implication the t-test equivalents) diverge in that Fβi →p ∞. Hence, with probability one, the F tests will reject the null hypothesis of βi = 0 as the sample size increases, leading to the incorrect conclusion that the time trend is statistically signiﬁcant. The result on Fβ holds notwithstanding that βˆ 1 1

has a non-degenerate limiting distribution. The implications are better for the Durbin-Watson (DW) statistic, which Durlauf and Phillips (1988) show tends to zero in probability, and so will lead to rejection of a correct speciﬁcation. Moreover, the rate of convergence to zero is Op (T), so that this should be a useful diagnostic. However, the ﬁnding of a ‘low’ DW cannot be uniquely taken to imply that the sequence of yt has been generated by an integrated process (for example, misspecifying the error dynamics can also cause a low value for DW); nor should a low DW automatically imply that the way to proceed is to apply a Cochrane-Orcutt (quasi-differencing) correction procedure to eliminate the apparently dynamically related residuals, as that is not, in this case, the direction of the misspeciﬁcation. Nelson and Kang (1984) had earlier shown that R2 = 1 – RSS/TSS is misleading in the case of spuriously detrended regressions; (in the estimated version of Equation (2.1), RSS is the residual sum of squares, ∑Tt=1 zˆ 2t , and TSS is the ¯ 2 ). If the DGP is a simple random walk, then total sum of squares ∑Tt=1 (yˆ t − y) t zt = ∑i=1 εi , so that the zt are positively correlated, which inﬂates sample R2 relative to its population value (of zero in this case). The expectations of RSS and TSS are both of order T, so that the effect of the spurious detrending on R2 is not critically dependent on sample size. Nelson and Kang (1984) ﬁnd through simulation that the sample R2 will be about 0.44 when, in fact, there is no relationship; further, about 86% of the variation in the data is (spuriously) removed by the detrending procedure, so it seems to ‘work’. Durlauf and Phillips (1988) provide the theoretical background to these results, showing that R2 has a non-degenerate limiting distribution about an expected value of approximately 0.44. 2.2.1.ii Spurious periodicity Chan, Hayya and Ord (CHO) (1977) and Nelson and Kang (NK) (1981) showed that detrending a series generated by a random walk leads to artiﬁcial pseudoperiodic behaviour; thus, following such a procedure would lead to the

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The ﬁrst result implies that T1/2 βˆ 0 ⇒D N(0, 2Tσε2 /15) and thus βˆ 0 is not consistent and has a divergent distribution, which means that as T increases, so does the variance of the distribution of T1/2 βˆ 0 . In contrast, βˆ 1 is consistent and is √ weakly convergent under the usual T normalisation. However, this does not mean that an F or t test can be used to judge whether a particular value of βˆ 1

conclusion that there are cycles in the times series, but these cycles are spurious being an artefact of the detrending procedure. NK show that the expected value of the sample autocovariance function for given T is a ﬁfth-order polynomial in the lag length; this looks like a damped sine wave, which is indicative of periodic or cyclical behaviour, whereas none such exists in the original series. To illustrate these results by means of a simulation exercise, a simple random walk DGP, ∆yt = εt , is again sufﬁcient, with starting value y0 = 0 and εt ∼ niid(0, 1). (Adding drift to the random walk does not alter the results and so is not illustrated separately.) The theoretical autocovariances and autocorrelations are zero for a nonzero lag k = 0. The series was (spuriously) detrended as in Equation (2.2), with the detrended series denoted yˆ˜ t . The autocovariance function was estimated by: ˆ γ(k) = (T# )−1 ∑t = k + 1 yˆ˜ t yˆ˜ t−k T

for k = 1, . . . , T − 1

(2.5)

The divisor may be set as T# = T or, to reﬂect the number of terms in the summation, T# = T – k; the difference is critical as k increases; for example, when k = T – 1, there is only one term, yˆ˜ T yˆ˜ 1 , in the summation. In the simulations, ˆ T# = T – k and the autocorrelation function is estimated by ρˆ (k) = γ(k)/ γˆ (0). To estimate the expected value of the autocorrelation function under the misspeciﬁcation of inappropriate detrending, a random walk DGP of length (r) T was simulated R = 5,000 times; the output, yt for the r-th replication, was (r) regressed on a constant and a linear time trend, with detrended observation y˜ˆ ; t

(r) then ρˆ (k) was calculated using { yˆ˜ t }Tt=1 and denoted ρˆ (k)(r) ; ﬁnally, the simulated expected value of ρˆ (k) was obtained by averaging ρˆ (k)(r) over the number of replications to obtain ρ¯ˆ (k)(r) = R−1 ρˆ (k)(r) . The correlograms of the ρ¯ˆ (k)(r) are graphed in Figure 2.1 for k = 1, . . . , T – 1 and T = 100 and 200; these are plotted against the proportionate lag length, k/(T − 1), rather than k, and thus vary from 0 to 1, to show that what matters for the shape of each ρ¯ˆ (k)(r) function is the ratio of the lag length to the sample size. The ρ¯ˆ (k)(r) functions for different T are virtually indistinguishable; each has a minimum at approximately 0.34; that is, 34% of the length of the sample, with a subsequent local but fairly ﬂat peak between 0.8 and 0.9 of the length of the sample. The resulting damped sine wave indicates that the detrended series has a pseudo-periodic behaviour. NK show that the period implied by the theoretical expected autocorrelation function is 0.8T, which is associated with the frequency Fj = (1/0.8T) = 1.25T−1 . (Fj is the frequency interpreted as the inverse of the period of the cycle and has units of measurement that are the reciprocal of the time unit. See Chapter 13 for a brief introduction to spectral analysis and, for example, Chatﬁeld, (2004), provides an accessible introduction to the necessary concepts.)

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1 0.8

simulated autocorrelation functions

0.6 0.4

T = 100 T = 200

0 T = 200

–0.2

T = 100 –0.4

0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Proportionate lag length

0.8

0.9

1

Figure 2.1 AC functions for residuals of detrended series.

To examine this periodic behaviour further, consider the spectral density function, sdf, f(λj ), given by: f(λj ) =

1 1 ∞ γ(0) + ∑k=1 γ(k) cos(λj k) 2π π

(2.6)

where λj is the (angular) frequency (see Chapter 13, Equation (13.1)); the sample counterpart of which is: T−1 ˆ j ) = 1 γ(0) ˆ ˆ + 2 ∑k=1 γ(k) cos(λj k) (2.7) f(λ 2π ˆ j ), is plotThe corresponding average (over replications) of the sample sdf, f(λ ted in Figures 2.2a and 2.2b for T = 100 and T = 200, respectively. (The average sdf could alternatively be obtained by taking the Fourier transform of the autocorrelation function; see Nelson and Kang, 1981.) The sdfs show that most of the power is concentrated in the lower frequencies; in the case of T = 100, the spectral peak is at Fj = 0.012 (where Fj = λj /2π); and for T = 200, the spectral peak is at Fj = 0.006, both as anticipated from NK’s results. These sdfs show that as T increases, the frequency associated with the spectral peak tends to zero, indicating that a long (in the limit inﬁnite) cycle is spuriously induced into the detrended series. NK note that when, as is usually the case, fewer lags than the maximum are used in computing the autocorrelation function, the sdf peaks at the origin and resembles that of a ﬁrst-order AR process. The spurious periodicity results generalise, ﬁrst, as already noted, to the drifted random walk and, second, to extending zt to be serially correlated. On the latter, for example, NK specify zt = ϕ1 zt−1 + εt and observe that if ϕ1 > 0, the

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0.2

Why Distinguish Between TS and DS Processes? 45

4 maximum at freq = 0.012 3.5 3 2.5 ^

2 1.5 1 0.5 0

0

0.05

0.1

0.15

0.2 0.25 0.3 Frequency, Fj

0.35

0.4

0.45

0.5

Figure 2.2a Simulated spectral density function, T = 100.

20 18 maximum at freq = 0.006

16 14 12 ^

f(λj) 10 8 6 4 2 0

0

0.05

0.1 0.15 Frequency, Fj

0.2

0.25

Figure 2.2b Simulated spectral density function, T = 200.

spectral peak is shifted further towards the zero frequency origin, whereas if ϕ1 < 0, the spectral peak is shifted to the right. Whichever, the higher frequency component associated with the dynamic structure tends to be overpowered by the spurious periodicity, leaving the dominant low-frequency component.

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f(λj)

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Unit Root Tests in Time Series

2.2.2 Spurious ﬁrst differencing What of the reverse error, that of the other off-diagonal element in Table 2.1? Suppose that the time series is trend stationary, but it is detrended by taking the ﬁrst differences, whereas stationarity would be achieved by removing the trend. 2.2.2.i The over-differencing effect Consider the simplest case, where the DGP is: (2.8)

Then the ﬁrst difference is: ∆yt = β1 + ∆εt

(2.9)

Hence spurious ﬁrst differencing induces a negative unit root in the error term, a result that generalises quite straightforwardly to errors that are stationary although not white noise. This feature, sometimes referred to as over-differencing, suggests that a test of stationarity can be based on examining the residuals obtained from detrending for a negative unit root (see, for example, Tanaka, 1990); this topic is explored further in Chapter 11. In the case that εt is white noise and letting ωt = ∆εt , then E(ω2t ) = 2σε2 , E(ωt ωt−1 ) = −σε2 and E(ωt ωt−k ) = 0 for k > 1, so that ρω (1) = –0.5 and ρω (k) = 0 for k > 1. 2.2.2.ii Implications of over-differencing for the spectral density function The spectral density function of ωt multiplied by the constant 2πγ(0)−1 is 1 − cos(λj ), which compares to 1 for white noise inputs, where all frequency components have an equal weight in the spectrum; thus the estimated f(λj ) will exaggerate the high frequencies and attenuate the low frequencies. If zt = ϕ1 zt−1 + εt , then as ϕ1 → 1, so that in the limit there is a unit root, 2πγ(0)−1 f(λj )→ 1 (see Chan et al., 1977), which is the value for white noise, whereas a unit root should be indicated by an increasing concentration of power as λj → 0 + . What does the sdf look like when the DGP is trend stationary but the series is ﬁrst differenced? The deﬁnition of the sdf was given in Equation (2.6); however, note that there are some slight differences in deﬁnition of the sdf in different sources, but these amount to multiplying f(λj ) by a constant. One alternative is to use 2πf(λj ); another is to use the autocorrelations rather than the autocovariances, so that the sdf, f(λj ), is scaled by 2πγ(0)−1 , as in Chan et al., (1977). The revised deﬁnition using the autocorrelations is F(λj ) = 2πγ(0)−1 f(λj ), so that: ∞

F(λj ) = 1 + 2 ∑k=1 ρ(k) cos(λj k)

(2.10)

Thus, in the case of white noise, ρ(k) = 0 for k ≥ 1, so that F(λj ) = 1, whereas f(λj ) = (2π)−1 σε2 , which is constant; if σε2 = 1, then the constant is (2π)−1 = 0.159.

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yt = β0 + β1 t + εt

Why Distinguish Between TS and DS Processes? 47

2.5

2

1.5

simulated theoretical

0.5

0

0

0.5

1

1.5 2 Frequency, Fj

2.5

3

Figure 2.3 Power spectrum of over-differenced series.

In the case of spurious ﬁrst differencing of a white noise process, the induced autocorrelations are: ρ(1) = –0.5, ρ(k) = 0 for k ≥ 1. Hence, substituting these values into F(λj ) results in: F(λj ) = 1 + 2ρ(1) cos(λj )

(2.11)

= 1 − cos(λj ) The distortion relative to the white noise input is −cos(λj ). Note that F(λj ) is an increasing function of λj , 0 ≤ λj ≤ π, and cos(0) = 1, cos(π) = –1, so that the function starts from 0 increasing to 2. The effect of the spurious MA(1) ﬁlter is to reduce the contribution of the low frequencies and increase the contribution of the higher frequencies. As cos(0.5π) = 0, only the spectrum at the angular frequency of 0.5π is unchanged. To illustrate the typical spectrum in the case of overdifferencing, the theoretical spectrum given by (2.11) and the (average) simulated spectrum, with T = 500, are shown in Figure 2.3. This is a plot of F(λj ) against the (inverse time) frequency Fj . Note that the (average) simulated spectrum is very close to the theoretical representation.

2.3 Persistence and the impact of shocks One of the key differences between TS and DS processes is in the impact of a shock, both in its magnitude and its persistence, the latter sometimes being referred to as the ‘memory’ of the process. In a TS process, a shock has a transient

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effect, the series returning to trend (or ‘mean’ in a general sense) after a sufﬁcient period of time; in contrast, in the case of a DS process, at least some part of the shock is incorporated into the level of the series, hence a shock has a permanent rather than a transitory effect. 2.3.1 The role of shocks The technical issues can be illustrated with a simple model, which is a specialisation of Chapter 1, Equations (1.39a)–(1.39c), used at several points in this text: yt = µt + zt

(2.12a)

µt = β0 + β1 t

(2.12b)

(1 − ρL)zt = εt

(2.12c)

where, for example, yt is the log of real GNP and, hence, ∆yt is the growth rate; εt ∼ iid(0, σε2 ); and the speciﬁcation in (2.12c) captures any potential unit root element. If |ρ| < 1, then yt comprises a linear trend subject to deviations, zt , that are transient in their impact; this is a simple form of a TS model in which deviations about the trend are stationary. The closer ρ is to + 1, the more long-lasting are the shocks, but they will nevertheless die out eventually. On the TS representation, the growth rate is determined as: (1 − ρL)∆yt = β1 (1 − ρ) + vt vt = (1 + θ1 L)εt

(2.13a) where θ1 = − 1

(2.13b)

An important feature of this representation is that the error is a moving average (MA) of εt with a unit root. (Note that in moving from (2.12) to (2.13), ∆t = 1.) In contrast to the TS view, in which shocks are transient, with the deterministic linear trend providing the ‘attractor’ for yt , the DS view corresponds to setting ρ = 1, resulting in the growth rate determined as: ∆yt = β1 + εt

(2.14)

This is a random walk, with drift. As in the case of the random walk without drift, by considering y0 as an historical reference point, yt can be expressed as follows: yt = y0 + β1 t + υt

(2.15a)

υt = ∑

(2.15b)

t ε i=1 i

Superﬁcially, this looks like a TS model: it has an intercept, a linear trend and a ‘disturbance’; however, υt is the accumulation of shocks and will have a variance that increases over time, in contrast to the constant variance of the TS process.

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2.3.1.i A simple model

Why Distinguish Between TS and DS Processes? 49

These models are particularly simple representations of members of their respective classes, but they show that the role of shocks, and hence uncertainty in this framework, is treated very differently.

Discrimination between the DS and TS representations rests on the presence of a unit root in the error component (2.12c) of the levels of the series, yt , which indicates a DS process, or a unit root in the MA error component of the growth rate, ∆yt , indicating a TS process (assuming, as is usually the case, that yt is the log of the level). These implications carry across to models in which the errors are subject to a more complex ARMA structure. It is worth brieﬂy considering this extension. The model is as in Chapter 1, Equations (1.39a)– (1.39c), summarised here for convenience: yt = µt + zt

(2.16a)

φ(L)zt = θ(L)εt

(2.16b)

where φ(L) = (1 − ρL)ϕ(L) and µt represents the trend component using deterministic terms. Substituting for zt from (2.16a) and rearranging results in: A(L)(yt − µt ) = εt

(2.17a)

⇒ (yt − µt ) = A(L)−1 εt

(2.17b)

where A(L) = θ(L)−1 φ(L). Taking the ﬁrst differences throughout results in the following speciﬁcation: ∆(yt − µt ) = G(L)εt

(2.18a)

G(L) = A(L)−1 (1 − L) = (1 − ρL)

−1

ϕ(L)

(2.18b) −1

θ(L)(1 − L)

(2.18c)

This is the moving average representation of the growth rate from which the long-run impact of a one-unit (one-off) shock can be determined; this is obtained by evaluating the G(L) polynomial at L = 1; that is, G(1) = 1 + ∑∞ j=1 gj . This is a widely used measure of the persistence (or memory) of the underlying process. On setting L = 1 in (2.18b) it is evident that G(1) = 0, so that in the TS model the long-run impact of a shock is zero and the trend is unaffected – there is no long-run persistence to shocks. This feature contrasts with the impact of a shock in the DS model, where ρ = 1, implying that the unit roots in (2.18c) cancel and the MA polynomial is: G(L) = ϕ(L)−1 θ(L)

(2.19)

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2.3.1.ii More complex models

Unit Root Tests in Time Series

For example, Campbell and Mankiw (1987a) ﬁt a model where ϕ(L) = 1−ϕ1 L and ˆ θ(L) = 1 + θ1 L, with ϕˆ 1 = 0.522 and θˆ 1 = –0.179, so that G(1) = (1 − 0.179)/(1 − 0.522) = 1.717, implying that the immediate impact of a one unit shock is eventually magniﬁed to 1.717, so there is no meaningful trend to which yt returns given sufﬁcient time (see also McCallum, 1993). That the shock will not necessarily be magniﬁed can be illustrated in this simple model since G(1) < 1 if θ1 < −ϕ1 ; typically θ1 < 0 for economic time series; for example, let θ1 = –0.8 and ϕ1 = 0.2, then G(1) = 0.25. This example illustrates that as θ1 → –1 then G(1) → 0, reﬂecting the over-differencing implied by an MA unit root in the case of a TS process. Whether, in distinguishing the TS and DS speciﬁcations, one takes the route of testing for a unit root in the AR polynomial in the level of yt , for example, ρ = 1 in (2.18c), or a unit root in the MA polynomial in the growth rate ∆yt , for example, θ1 = –1 in (2.18c), it is evident that there will be a problem of distinguishing near-unit roots. For example, in the AR case one can expect little power in any test statistic to distinguish ρ = 0.99 from ρ = 1, especially in ﬁnite samples. There has therefore been a premium on improving the DF tests to gain additional power. The technical issue of whether or not to detrend a time series has relevance for the impact of macroeconomic ﬂuctuations and the design, and possible effectiveness, of counter-cyclical policy. This interest has led to a related debate about whether macroeconomic shocks have permanent effects. Given a shock, is the response – for example, to GNP and employment – a deviation from trend to which the series will revert, or does the shock persist in whole or in part? In the former case there is reversion to a trend (or mean) that can be interpreted as a natural or equilibrium level (for example of output and employment, as determined in a growth model of the Solow kind, or ‘natural’ rate of unemployment); and in the second, a lasting effect that permanently affects the subsequent level. In the simple random walk with drift model yt = µ + yt−1 + εt , with εt ∼ iid(0, σε2 ), a negative one-unit shock at time t, implies that Et (yt + h ) is lower by one unit for h = 1, . . . , ∞, whereas in a trend stationary model, the long-run effect of the shock is zero. A counter-cyclical policy by the ﬁscal and monetary authorities is relatively easy to design if there is trend reversion and the task is to ‘smooth’ out the effects of a temporary shock; on the other hand, the effects of a permanent shock must be evaluated over an inﬁnite horizon. The matter is complicated if a series admits a decomposition into a permanent and a transitory component, in which case it is not just a matter of the existence of the permanent component but its importance relative to the overall behaviour of the series. Cochrane (1988) phrased this as: ‘How big is the random walk in GNP?’ Cochrane’s proposed measure is based on the ratio of the variance of the q-th difference of yt to

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Why Distinguish Between TS and DS Processes? 51

the variance of the ﬁrst difference of yt ; this is related to the persistence measure G(1), but is not the same except in some special cases.

The title of Cochrane’s (1988) inﬂuential article highlighted the relevance not only of the existence of a unit root, implying random walk-type behaviour, but also the importance of the random walk element as, for example, characterised by the impact of shocks on long-term forecasts. In a simple random walk model the long-term forecast moves one-for-one with shock, so that G(1) = 1, whereas in a trend stationary model there is no long-run impact of a shock as the series simply reverts to trend, so that G(1) = 0. However, values of G(1) other than zero and one are possible, with partial reversion for 0 < G(1) < 1 and increasing divergence for G(1) > 1. In a development of this view, a series can be decomposed into a permanent component and a transitory component, known as a P-T decomposition, and the ‘importance’ question then relates to the relative contribution of each of these components. There are several ways of measuring the importance of the unit root component and four are brieﬂy outlined in the following subsections: Cochrane’s variance ratio; the spectral density at the zero frequency; the long-run variance; and a P-T decomposition. It turns out that these lead to equivalent measures, but each emphasises different aspects of the underlying structure. 2.4.1 The variance ratio for a simple random walk Cochrane introduced a variance ratio measure of persistence, which can be illustrated with the pure random walk, with or without drift (see also Lo and Mackinlay, 1999, for a development of the distribution theory associated with the variance ratio). For simplicity, consider the without drift case, with y0 = 0 and εt ∼ iid(0, σε2 ). The variance ratio is the ratio of the variance of the q-th difference of yt to q times the variance of the ﬁrst difference of yt . In a random walk with white noise εt , this ratio is unity for all q. The ﬁrst difference of a pure random walk is ∆yt = εt , hence the variance of 2 , is σ2 . A distinction that will prove to be of value later occurs when ∆yt , say σ∆y ε the structure of the noise exhibits weak dependence. For example, suppose that: ∆yt = εt + θ1 εt−1

(2.20)

2 then σε2 is the variance of ∆yt conditional on t – 1, say σ∆y|t−1 , whilst the uncon2 , is (1 + θ2 )σ2 . In the case of the pure random ditional variance of ∆yt , say σ∆y 1 ε walk, that is, θ1 = 0 in (2.20), these two variances coincide, but more generally they differ.

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2.4 Measures of the importance of the unit root

52

Unit Root Tests in Time Series

Continuing with the pure random walk, the q-th difference of yt and its variance are as follows: ∆q yt ≡ yt − yt−q ≡

(2.21)

∑i=0 ∆yt−i q−1

= ∑i=0 εt−i q−1

2 σ∆2q y = qσ∆y

(2.22)

= qσε2 Thus if the random walk yt = yt−1 + εt is the generating process, then a plot of q−1 σ∆2q y against q should be approximately constant. The q-th order variance ratio is deﬁned as follows: 1 σ∆q y 2 q σ∆y 2

V(q) ≡

(2.23)

Hence V(q) equals unity for a pure random walk. The limiting variance ratio as q → ∞ is also of interest, leading to the following deﬁnition: Vlim (q) ≡ lim V(q) q→∞

(2.24)

In contrast to the random walk case, if yt is generated by a simple TS process, then: yt = µt + εt

(2.25)

where µt comprises deterministic components. In this case: ∆q yt = µt − µt−q + εt − εt−q

(2.26)

Hence, noting that, by assumption, µt comprises deterministic components, then: σ∆2q y = Var(εt − εt−q ) = 2σε2 2 = σ2 , it then Notice that σ∆2q y does not increase with q, and as in this case σ∆y ε follows that V(q) is given by:

V(q) = 2/q Thus, if (2.25) is the generating process, then a plot of q−1 σ∆2q y against q should tend to zero as q increases, as should the variance ratio V(q) and Vlim (q). On

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Hence the variance of ∆q yt ≡ σ∆2q y is given by:

Why Distinguish Between TS and DS Processes? 53

this basis, Cochrane (1988) notes that if shocks to yt are partly permanent and partly temporary, it will only be the former that matter for the long run: to the extent that q−1 σ∆2q y differs from zero as q increases, then this will be the variance to the permanent – that is, the random walk – component. If, for large q, 0 < Vlim (q) < 1, then part of the one-period variance is temporary, not permanent.

The generating process given by (2.25) is too simple in general, but serves as a motivating example. The more general case was considered earlier in Equations (2.18a)–(2.18c), which, in slightly rearranged form, are: ∆yt = ∆µt + ξt

(2.27a)

ξt = G(L)εt

(2.27b)

For example, if ∆µt = µ, then (2.27a) is a random walk with drift and serially dependent shocks, ξt . Note that if ρ = 1, then G(L) = ϕ(L)−1 θ(L), as in (2.19). This generalisation allows ∆yt to exhibit serial dependence inherited from the structure of ξt , so that the autocorrelations of ∆yt are not necessarily zero. Writing G(L)εt more explicitly as G(L)εt = εt + ∑∞ j=1 gj εt−j , note that the unconditional 2 , of ∆y is: variance, σ∆y t ∞

2 σ∆y = E(εt + ∑j=1 gj εt−j )2

(2.28)

∞

= (1 + ∑j=1 g2j )σε2 a result that follows from E(εt εt−j ) = 0 for j = 0. Letting ρ(j) denote the j-th autocorrelation of ∆yt , an interpretation of the q-th order variance ratio in terms of the autocorrelations is as follows (see Cochrane, 1988, appendix): q−1 q − j V(q) = 1 + 2 ∑j=1 ρ(j) (2.29) q Moreover, in the limit as q → ∞, then: ∞

Vlim (q) = 1 + 2 ∑j=1 ρ(j)

(2.30)

From (2.30) it is evident that Vlim (q) < 1 if ∑∞ j=1 ρ(j) < 0, thus it is the sum of the autocorrelations that matters and it is possible to observe a variance ratio that is above unity for small q, but below unity as later negative autocorrelations come to dominate the sum; this pattern tends to characterise the empirical results on asset returns (see Chapter 1, section 1.6.3). 2 gives the spectral density funcMultiplying Vlim (q) of equation (2.30) by σ∆y tion (using the autocorrelations), sdf, of ∆yt , f∆y (λj ), evaluated at the zero

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2.4.2 Interpretation in terms of autocorrelations and spectral density

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Unit Root Tests in Time Series

frequency; that is, at λj = 0, which can be interpreted as the permanent (or longrun) component of yt since as λj → 0, the period associated with this frequency → ∞. Thus we have: (2.31)

Hence, if the process generating {yt }Tt=1 is trend stationary, then Vlim (q) = 0, which implies that f∆y (λj = 0) = 0, so that the spectral density is zero at frequency zero; if the process is difference stationary, then f∆y (λj = 0) = 0, and increases as the importance of the permanent component increases. (The deﬁnition of the sdf in (2.31) follows Cochrane, 1988, whereas the deﬁnition in Equation (2.6) uses the autocovariances scaled by (2π)−1 ; see also Chapter 13, Equation (13.1).) In the case of the generating process summarised by (2.27), it is possible to derive explicitly the ρ(j) in terms of the underlying MA coefﬁcients: ρ(j) =

∑∞ i=0 gi gi + j

(2.32)

2 ∑∞ j=0 gj

(see Cochrane, 1988, appendix) and where gj = 1. Substituting (2.32) into (2.30) results in: Vlim (q) =

=

G(1)2 2 (1 + ∑∞ j=1 gj )

(2.33)

2 (1 + ∑∞ j=1 gj ) 2 (1 + ∑∞ j=1 gj )

A question at the end of the chapter illustrates (2.32) and (2.33) for the MA(1) case. 2.4.3 Interpretation in terms of a permanent-temporary (P-T) decomposition The decomposition of a DS time series into a permanent component and a stationary component is an interesting line of research. There is more than one P-T decomposition, perhaps the most inﬂuential being due to Beveridge and Nelson (1981), generally referred to as the BN decomposition. This is considered at length in Chapter 3; here we summarise this decomposition of a series yt ∼ I(1) p into a permanent component denoted yt and a stationary component denoted p tr yt , respectively; the characteristics of each being that yt is I(1) and ytr t is I(0) . The BN decomposition is: p

yt = yt + ytr t

P-T decomposition

(2.34)

p p yt = µ + yt−1 + G(1)εt

permanent component

(2.35)

ytr t = D(L)εt

transitory component

(2.36)

lag weights

(2.37)

∞ g j=i+1 j

di = − ∑

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∞

2 f∆y (λj = 0) = [1 + 2 ∑j=1 ρ(j)]σ∆y

Why Distinguish Between TS and DS Processes? 55

p

By successive back-substitution, yt can also be expressed as: yt = µt + y0 + G(1) ∑j=1 εj p

t

(2.38)

Note that G(1) is G(L) evaluated at L = 1, with G(L) = ϕ(L)−1 θ(L), as in (2.19). p In the case of a simple random walk G(1) = 1 and therefore yt = yt and ytr t = 0, otherwise G(1) = 1 is a scaling factor on the cumulative shocks. p 2 , referred to as the innovation variance Let the variance of ∆yt be denoted σ∆yp of permanent component, then from (2.35) note that: 2 = G(1)2 σε2 σ∆yp

(2.39) p

Although the innovations to yt and ytr t are the same in the BN decomposition, this is not an essential feature of P-T decompositions. Fortunately, as Cochrane (1988, Fact 2) shows, in every P-T decomposition of yt , the innovation variance of the permanent component is the same. This feature is useful as we may then 2 2 as a measure of the importance of the unit root, or take the ratio of σ∆yp to σ∆y permanent, component of yt , thus: 2 σ∆yp 2 σ∆y

=

G(1)2 2 ∑∞ j=1 (1 + gj )

(2.40)

As we have already seen, this is just the limiting variance ratio (see Equation (2.33)). 2.4.4 Interpretation in terms of unconditional and conditional variances Another interpretation of persistence serves to introduce explicitly the con2 cept of the long-run variance, σ∆y,lr , which has an important role in unit root 2 tests. The quantity σ∆y,lr is the variance of the measure of long-run persistence associated with (2.18a), that is: 2 σ∆y,lr = var[G(1)εt ]

= G(1)2 σε2

(2.41)

The concept of the long-run variance is an important one in various unit root tests, it or its square root usually serving as a scale factor so that the test statistic has a limiting distribution that is a function of standard Brownian motion. 2 2 , the We may then observe that the long-run variance σ∆y,lr is the same as σ∆yp innovation variance of the permanent component, so that the ratio of the longrun variance to the unconditional variance of ∆yt is also the limiting variance

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p

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Unit Root Tests in Time Series

ratio, that is: 2 σ∆y,lr 2 σ∆y

=

=

G(1)2 σε2 2 2 (1 + ∑∞ j=1 gj )σε

(2.42)

2 (1 + ∑∞ j=1 gj ) 2 (1 + ∑∞ j=1 gj )

It is perhaps not surprising that the long-run variance, interpreted as the variance of the persistence measure, is also the innovation variance for the permanent component. The concept of the permanent component is that it is the part that persists after transient effects have died away. 2.4.5 The relationship between the persistence measure and the variance ratio It is evident from the previous subsections that the variance ratio, in its various guises, depends fundamentally on G(1), which is the moving average measure of persistence. The link can be interpreted as follows (see also Q2.2): Vlim (q) = G(1)2 (1 − R2 )

(2.43)

So that VRlim (q) is negatively related to an R2 measure deﬁned as: 2 R2 ≡ 1 − σε2 /σ∆y

=

2 − σ2 σ∆y ε 2 σ∆y

(2.44)

Writing R2 in this way emphasises that it is a measure of the goodness of ﬁt for ∆yt using lagged information from the moving average process (the ‘past’). For example, consider the MA(1) case, then: R2 =

g21 (1 + g21 )

(2.45)

The lagged information has an impact through g1 , with the magnitude of g21 indicating the importance of the past in assessing the predictability (measured by the variance reduction) of ∆yt ; if g1 is small, then, relatively, knowledge of g1 εt−1 is unimportant in determining the variance of ∆yt . 2 = σ2 , so that R2 = 0 and In the case of the pure random walk σ∆y ε Vlim (q) = G(1)2 , otherwise for R2 > 0, Vlim (q) > G(1)2 , with the difference between the two measures increasing as R2 increases.

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= Vlim (q)

Why Distinguish Between TS and DS Processes? 57

To illustrate some of the ideas and models of the preceding sections, we consider a sample of quarterly data (seasonally adjusted, s.a.) on US GNP (in constant $2005) for the overall period 1947q1 to 2008q4, comprising a total of 248 observations. The time series in levels and logs are graphed in Figure 2.4, from which the trended nature of the series is evident, and a log transformation is appropriate if the alternative to a unit root is a linearly trended series. A possible model therefore allows for a linear trend, µt = β0 + β1 t, with an ARMA model for the lag dynamics, as follows: (yt − µt ) = A(L)−1 εt A(L)−1 = φ(L)−1 θ(L) = (1 − ρL)−1 ϕ(L)−1 θ(L) where yt is the log of US GNP. The analogous form on taking ﬁrst differences, as in equation (2.18), is: ∆(yt − µt ) = G(L)εt G(L) = (1 − ρL)−1 ϕ(L)−1 θ(L)(1 − L) Note that G(L) = A(L)−1 (1−L). The estimation results are reported in Table 2.2, with 2.2a and 2.2b being sections for the model in logs, yt , and in ﬁrst differences, ∆yt . The roots of the component AR and MA polynomials are reported in Table 2.3. (An extensive discussion of ARMA models is contained in Chapter 3.) 15,000 GNP 10,000 5,000 1950

1960

1970

1980

1990

2000

2010

1980

1990

2000

2010

10 log GNP 9 8 7

1950

1960

1970

Figure 2.4 US GNP (quarterly, s.a.).

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2.5 Illustration: US GNP

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Unit Root Tests in Time Series

Table 2.2 Estimated ARMA models for US GNP (real). Table 2.2a ARMA(3, 2) model for yt yt constant coefﬁcient 7.606 ‘t’ 93.26

yt−1 2.278 18.93

yt−2 –2.005 –10.24

yt−3 0.714 7.80

εt−1 –1.052 –8.21

εt−1 –1.114 –10.67

εt−2 0.649 7.04

εt−2 0.607 6.04

trend 0.0077 16.54

∆yt constant coefﬁcient 0.008 ‘t’ 10.92

∆yt−1 1.350 15.16

∆yt−2 –0.776 –10.31

Table 2.3 Roots of the polynomials of the ARMA models. Table 2.3a ARMA(3, 2) model for yt Roots of AR polynomial root 1.031 modulus 1.031 reciprocal root 0.970 reciprocal modulus 0.970

0.889 ± 0.754i 1.166 0.654 ± 0.555i 0.858

Roots of MA polynomial 0.866 ± 0.948i 1.284 0.526 ± 0.575i 0.779

Table 2.3b ARMA(2, 2) model for ∆yt Roots of AR polynomial root 0.870 ± 0.729i modulus 1.135 reciprocal root 0.675 ± 0.566i reciprocal modulus 0.881

Roots of MA polynomial 0.875 ± 0.880i 1.241 0.568 ± 0.572i 0.806

Using the estimates from Tables 2.2a and 2.3a, the underlying AR polynomials for the model in terms of yt are obtained as follows: ˆ φ(L) = (1 − ρˆ L)ϕ(L) ˆ = (1 − 0.970L)(1 − (0.654 + 0.555i)L)(1 − (0.654 − 0.555i)L) = (1 − 0.970L)(1 − 1.308L + 0.736L2 ) Notice that there is a near-unit root, so that it is no surprise that ϕ(L) ˆ is very close to the AR lag polynomial of Table 2.2b; that is, (1 − 1.350L + 0.776L2 ), for the model with ∆yt as the dependent variable.

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Table 2.2b ARMA(2, 2) model for ∆yt

Why Distinguish Between TS and DS Processes? 59

(1 − 1.114 + 0.649) ˆ G(1) = (1 − 1.350 + 0.776) = 1.259 2 ˆ = 1.584 G(1)

ˆ lim (q) = V

ˆ G(1) ˆ2 (1 + ∑∞ j=1 gj )

= 1.369 These results suggest persistence in excess of that implied by a pure random walk and, at least in terms of general impact, accord with the earlier results of Campbell and Mankiw (1987a, 1987b). However, one should bear in mind Cochrane’s (1988) cautionary ﬁnding for US GNP that as the AR component was increased, the estimate of the G(1) tended to decline; for example, from 1.20 for an AR(1) model to 0.45 for an AR(15) model (where ∆yt is the dependent 1.7 1.6 1.5 1.4

estimated long-run response

Σgi 1.3 1.2 1.1 1

0

10

20

30

40

50

60

Lag Figure 2.5 Impulse responses based on ARMA(2, 2) model.

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Imposing a unit root results in the estimates reported in Table 2.2b and Table 2.3b. The roots of the AR and MA polynomials look similar, but if they cancelled exactly would imply that G(1) = 1, which is not the case here. The partial sums of the coefﬁcients of the G(L) polynomial are graphed in Figure 2.5; the ﬁgure shows that the estimated response to a shock is not monotonic, with the long run reached after about eight years. The persistence measures based on the estimates from Table 2.2b are as follows:

60

Unit Root Tests in Time Series

2.6 2.4 2.2 estimated Vlim(q)

2 1.8 Estimates 1.4

estimated G(1)

1.2 1 0.8

0

2

4

6

8

10

12

14

16

18

20

AR order Figure 2.6 Estimates of G(1) and variance ratio from AR(k) models.

variable), with a consequent decline in the variance ratio from 1.39 to 0.18. Cochrane suggested that a possible explanation for this ﬁnding is that as the variance ratio involves the inﬁnite sum of autocorrelations, so that apparently numerical small autocorrelations can be inﬂuential, the sample then available was insufﬁcient to provide enough occurrences of the ‘long run’. Recall that the ‘long run’ is determined as the calendar time when all short-run adjustments are completed, with an estimate of about eight years based on the responses graphed in Figure 2.5. However, to examine further the aspect highlighted by Cochrane, AR(k) models were estimated with k = 1, . . . , 20, over the common sample period 1952q2 to ˆ lim (q) are graphed in Figure 2.6 2008q4, and the various estimates of G(1) and V as a function of the AR order, k. Although there tends to be a decline in both of these persistence measures as k increases, it is nowhere near as extreme as in the ˆ lim (q) ≈ 1 results reported by Cochrane (1988, Table 3); for example, for k = 15, V ˆ lim (q) = 1.128. These estimates tend to support the importance and for k = 20, V of the random walk-type component in US GNP; however, it is not the intention here to provide a deﬁnitive answer to the question posed by Cochrane (‘How important is the unit root?’), which has been the subject of much enquiry, but to indicate the some of the techniques that have been used.

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1.6

Why Distinguish Between TS and DS Processes? 61

The previous section has highlighted, from the perspective of a univariate analysis, the nature of the errors involved in spuriously detrending and, alternately, over-differencing. An extension of these concerns arises when several variables are considered together, which gives rise to the possibility of spurious regressions. This development has seen the widespread use of unit root tests as part of a pre-test routine on the variables considered as candidates in a regression analysis. This pre-testing arises from two interests: ﬁrst, the possibility of spurious regressions; that is, regressions that indicate a relationship between the variables where there is none in the population; second, to ensure balance in a regression and hence the possibility of cointegration, as deﬁned in Engle and Granger (1987). The concept and implications of a spurious regression are considered below. 2.6.1 Spurious regression A spurious regression is one in which the regressand and the regressors have no relationship but diagnostic statistics, such as the R2 or the coefﬁcient t statistics, indicate that the regressors have a signiﬁcant effect on the regressand. Yule (1926, p.2) drew attention to the general problem in his famous article on ‘nonsense’ correlations. He noted: ‘It is fairly familiar knowledge that we sometimes obtain between quantities varying with time (time-variable) quite high correlations to which we cannot attach any physical signiﬁcance whatever . . . .’ In an inﬂuential article, Granger and Newbold (1974) showed that regressing one I(1) variable on an unrelated I(1) variable resulted in a spurious regression; that is, there would seem to be a relationship although none was actually present. Phillips (1986) established the theoretical insights behind this result and showed that the problem was present asymptotically, not just in ﬁnite samples. Granger et al. (2001) showed that a spurious regression problem occurs in ﬁnite (but large) samples with stationary variables that are each serially correlated. To illustrate the problem with I(1) variables, consider the following datageneration process for two unrelated variables, y1,t and y2,t : y1,t = y1,t−1 + ε1,t

(2.46a)

y2,t = y2,t−1 + ε2,t

(2.46b)

where ε1,t ∼ iid(0, σ12 ), ε2,t ∼ iid(0, σ22 ) and cov(ε1,t , ε2,t ) = 0 for all t. Thus y1,t and y2,t are generated by unrelated random walks. As y1,t and y2,t are unrelated, one might reasonably expect that this would be reﬂected in the following regression of y1,t on y2,t : y1,t = β1 + β2 y2,t + ut

(2.47)

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2.6 Unit root tests as part of a pre-test routine

Unit Root Tests in Time Series

Traditional diagnostics to assess a regression include R2 and the t statistic for β2 , denoted tβ2 . To illustrate what happens with a spurious regression, 10,000 replications were carried out for T = 100 and T = 1,000 with ε1,t and ε2,t drawn from independent N(0, 1) distributions. The distributions of R2 and tβ2 from these simulations are shown in Figures 2.7 and 2.8, respectively. Note that the distributions of R2 for the different sample sizes are virtually indistinguishable, suggesting that there is convergence in distribution, but not to the special case of a constant equal to zero. For each sample size, the average value of R2 is approximately 0.24. Note from Figure 2.8 that the distributions of tβ2 show no sign of converging. Using conventional critical values for a test of H0 :β2 = 0 against HA :β2 = 0 results in rejection rates of 76% for T = 100, 92% for T = 1,000, 98% for T = 10,000, and in the limit the rejection rate is 100%. This means that as the sample size increases, it is more rather than less likely that one would mistakenly conclude that there is a relationship between y1,t and y2,t . Phillips (1986) provides the theoretical basis to understand what is happening in such spurious regressions. On a weak set of assumptions for ε1,t and ε2,t , allowing some heteroscedasticity and serial dependence, he derived the following results: (i)(a) The distribution of the t statistic tβ2 diverges as T → ∞ (as does the t statistic on β1 ); this means that there is no asymptotically correct critical value for a standard hypothesis test using tβ2 . (i)(b) T−1/2 tβ2 has a convergent limiting distribution. Thus if a ‘conversion’ factor was required to ensure a test statistic that was convergent, then tβ2 √ should be scaled by 1/ T. Phillips (1986, Result c) provides the limiting distribution and, in principle, ﬁnite sample critical values could be simulated. (ii) The limiting distribution of βˆ 2 is non-degenerate; that is, it does not converge, let alone to β2 (and also the limiting distribution of βˆ 1 diverges). (iii) The limiting distribution of R2 is also non-degenerate; that is, it does not converge to zero; thus the problem does not ‘go away’ as the sample size increases. These results also hold for the multivariate case of a regression involving k > 2 unrelated random walks or, more generally, I(1) processes; let xt = (x1,t , . . . xk−1,t ), then the vector time series of interest is now (yt , xt ), comprising k variables. In some respects, such multivariate regressions are even more misleading with, for example, the average R2 increasing with k (see, for example, Ohanian, 1988, 1991). Moreover, with one exception, nothing of substance changes if the k I(1) processes are related, there being no requirement that the covariance matrix of the k variables is diagonal (as assumed in the speciﬁcation

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62

Why Distinguish Between TS and DS Processes? 63

6

T = 100 T = 1,000

5

4

2

1

0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2

R

Figure 2.7 Simulated density function of R2 .

0.07

T = 100 T = 1,000

0.06 0.05 0.04 f(tβ2) 0.03 0.02 0.01 0 –100

–80

–60

–40

–20

0 tβ2

20

40

60

80

100

Figure 2.8 Simulated density function of tβ2 .

of (2.46)). The important exception is where the vector time series is cointegrated (Engle and Granger, 1987) in the sense that there is a linear combination of (yt , xt ), that is, I(0). In this case a different theory applies (see, for example, Durlauf and Phillips, 1988).

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f(R2) 3

Unit Root Tests in Time Series

As evidenced in much of the practice of time series econometrics, unit root tests are used as part of a pre-testing methodology before a regression analysis. They can alert the user to the possibility of spurious regressions and help to establish whether the necessary condition of ‘balance’ in mean (Granger and Hallman, 1991) is satisﬁed for a potentially cointegrating set of variables to exist. As to the latter, the intuitive basis of the idea is that the time series properties of the dependent variable must be capable of reproduction by the chosen regressors; for example, if the dependent variable is I(1), there must exist a linear combination of the regressors with that property; this rules out the possibility of a single I(2) regressor, but not of a pair of such regressors, which themselves cointegrate.

2.7 Concluding Remarks It is appropriate here to look ahead at the next two chapters to see how they take forward some of the ideas that have been introduced in Chapters 1 and 2. These chapters have rather taken for granted some of the detail of the ARMA model, which is the cornerstone of linear modelling. So far the introductory nature of the arguments has meant that we can get by with an outline of the ARMA model – as needed, for example, in Chapter 1, section 1.5.1.i, in generalising the random walk, and in this chapter, section 2.3.1, on the role of shocks in models with a weak dependency structure in the errors. The further development of unit root tests requires a systematic introduction to, and development of, the ARMA (and ARIMA) model, and this is provided in Chapter 3. One of the problems in the estimation of ARMA models, particularly from the AR component of the model, is the ﬁnite sample bias of conventional estimators such as the least squares (LS) and Yule-Walker estimators. (This problem is addressed in Chapter 4.) Whilst there are some parts of the parameter space where the standard estimators are unbiased, this region does not characterise typical economic time series; thus one must generally expect that, for example, LS estimators of the AR coefﬁcients will have a ﬁnite sample bias, which can be quite substantial. Rudebusch (1993) was aware of this in assessing the robustness of his analysis of the distinction between estimating DS and TS speciﬁcations for US GNP and, in particular, in obtaining an estimate of the persistence measure G(1) (see Equation (2.18b)).

Questions Q2.1.i Consider the following specialisation of Equation (2.18a): ∆yt = G(L)εt

(2.48a)

G(L) = (1 + g1 L)

(2.48b)

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64

Why Distinguish Between TS and DS Processes? 65

Obtain the variance ratio ﬁrst for q = 2, then for the general case and for the limiting case as q → ∞ . Q2.1.ii Consider V(q) as a function of g1 . Q2.1.iii Graph the relationship between Vlim (q) and g1 . A2.1.i The relevant variances and covariances for the generating process comprising (2.48a) and (2.48b) are as follows: for s = 0, ±1, ±2, . . . ,

cov(∆yt ∆yt−1 ) = g1 σε2 cov(∆yt ∆yt−s ) = 0

for s = ± 2, ±3, . . . ,

Only the ﬁrst-order autocorrelation, ρ(1), is nonzero and is given by: cov(∆yt ∆yt−1 ) var(∆yt ) g1 = (1 + g21 )

ρ(1) =

Conﬁrm this using the general expression (2.32), specialised for this example (and bearing in mind that g0 = 1): ρ(1) =

∑1i=0 gi gi + 1

∑1i=0 g2i g1 = (1 + g21 )

The result is conﬁrmed. In the general case for q = 2 and σ22 ≡ var(∆2 yt ), we have: σ22 = var(∆yt + ∆yt−1 ) = var(∆yt ) + var(∆yt−1 ) + 2cov(∆yt ∆yt−1 ) 2 + 2cov(∆yt ∆yt−1 ) = 2σ∆y

V(2) =

=

σ22 2 2σ∆y 2 2σ∆y 2 2σ∆y

+

2cov(∆yt ∆yt−1 ) 2 2σ∆y

= 1 + ρ(1)

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var(∆yt−s ) = (1 + g21 )σε2

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Unit Root Tests in Time Series

Therefore, in the MA(1) case, we have: σ22 = 2(1 + g21 )σε2 + 2g1 σε2 V(2) = 1 +

g1 1 + g21

These results generalise for q ≥ 2 as follows:

V(q) = 1 + 2 =1+2

(2.49)

(q − 1) cov(∆yt ∆yt−1 ) 2 q σ∆y (q − 1) ρ(1) q

Vlim (q) = 1 + 2ρ(1)

(2.50) (2.51)

Then in the MA(1) case, with MA coefﬁcient g1 : σq2 = q(1 + g21 )σε2 + 2(q − 1)g1 σε2 V(q) = 1 + 2

(q − 1) g1 q (1 + g21 )

Vlim (q) = 1 + 2 =

g1 (1 + g21 )

(2.52) (2.53)

(2.54)

(1 + g1 )2 (1 + g21 )

A2.1.ii Note the following: V(q) = 1 for g1 = 0, which is the simple random walk case; V(q) → 0 as g1 → −1, which indicates over-differencing, so that the generating process is trend stationary; V(q) >1 for g1 > 1 for large q (such that (q − 1)/q ∼ =1), indicating that a shock is ampliﬁed. A2.1.iii Figure 2.9 graphs the relationship between Vlim (q) and g1 and, for comparison, the corresponding persistence measure G(1)2 based on G(1). Notice that G(1)2 = 0 for g1 = 0 and G(1)2 → 0 as g1 → −1, as in the case of the variance ratio; the two measures are reasonably close for the interval g1 ∈ [−1, 0], which is the likely parameter region for many economic time series, but depart for g1 > 1. The next question considers this relationship further. Q2.2 Show that the following holds, and interpret the equality: Vlim (q) G(1) = (1 − R2 )

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2 + 2(q − 1)cov(∆yt ∆yt−1 ) σq2 = qσ∆y

Why Distinguish Between TS and DS Processes? 67

4 3.5 3 2

G(1)

limiting variance ratio and 2.5 squared persistence measure 2

1 0.5 0 –1 –0.8 –0.6 –0.4 –0.2

0 g1

0.2

0.4

0.6

0.8

1

Figure 2.9 Measures of persistence for an MA(1) process.

A2.2 We have that: Vlim (q) = (1 − R2 ) =

G(1)2 σε2 2 σ∆y σε2 2 σ∆y

Hence, on substitution for (1 − R2 ): Vlim (q) = G(1)2 (1 − R2 ) From which we obtain: Vlim (q) G(1) = (1 − R2 ) Hence

Vlim (q) ≤ G(1) ≤

Vlim (q) 2 ); (1−Rmax

the lower bound holds for R2 = 0, implied

2 = σ2 ; R2 , the maximum value of R2 , is not generally unity, in contrast by σ∆y ε max

2 < to the usual R2 . Consider the MA(1) case, then R2 = g21 (1 + g21 )−1 and so Rmax 1 for ﬁnite g1 .

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Vlim(q)

1.5

3

Introduction The focus of this chapter is on autoregressive moving average (ARMA) models, which were introduced in a simple form in Chapter 1. These models are not only of interest in their own right, they serve to provide a background to interpret many of the issues arising in the context of unit roots and integrated time series. The ‘ARMA’ notation indicates that there are two components to the structure of these models. The ﬁrst part is the autoregressive (AR) component and the second the moving average (MA) component. The ARMA model is said to be integrated if a unit root, or roots, can be extracted from the AR component, in which case the appropriate notation is ARIMA, for an autoregressive, integrated moving average model. The kind of structure of interest in this chapter is where one of the roots of the AR polynomial might be a unit root whilst the others are outside the unit circle. The case of two unit roots is considered in Chapter 11 and the roots of a polynomial are considered extensively in Appendix 2. One of the aims of this chapter is to introduce the ARMA model, and some of its characteristics, in order to provide a background necessary to motivate a direct maximum likelihood (ML) approach to testing for a unit root. While Chapter 6 deals extensively with the some of the unit root test statistics suggested by Dickey and Fuller, it is often the case that there is an MA component to a time series model, in which case direct estimation, rather than an approximation of the MA component by extending the AR lag polynomial, may be an attractive option. The ﬁrst section, 3.1, speciﬁes the ARMA(p, q) model, whilst section 3.2 considers the roots of the AR and MA polynomials and the condition for stability. The inﬁnite MA representation of an ARMA is considered in section 3.3 and the AR approximation of a model with an MA component in section 3.4. Near-cancellation of roots is considered in section 3.5. Two important decompositions of lag polynomials, that is, the Beveridge-Nelson (BN) and the 68

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An Introduction to ARMA Models

An Introduction to ARMA Models

69

Dickey-Fuller (DF) are outlined in sections 3.6 and 3.7, respectively. Then, in section 3.8, three alternative ways of representing an ARMA model are outlined; the opportunity is taken here to introduce two important DF tests for a unit root, whereas section 3.9 considers estimation and testing by ML. These topics are taken up further in Chapters 6 and 7. Section 3.10 contains a detailed empirical example to illustrate some of the speciﬁcation and testing issues.

The ARMA model of order p in the AR component and q in the MA component for the univariate process generating yt , is: (1 − φ1 L − φ2 L2 − . . . − φp Lp )yt = (1 + θ1 L + θ2 L2 + . . . + θq Lq )εt

(3.1)

Note from Appendix 2 that the lag operator is deﬁned as Lj yt ≡ yt−j . The sequence {εt }Tt=1 comprises the random variables εt , assumed to be independently and identically distributed for all t, with zero mean and constant variance, σε2 , written as εt ∼ iid(0, σε2 ); if the (identical) distribution is normal, then εt ∼ niid(0, σε2 ). The convention in this book is that σε2 is the variance of εt ; where other variances are introduced, they will be subscripted to distinguish them from σε2 . The model in (3.1) does not yet include any deterministic components; this assumption will be relaxed shortly. The ARMA model can be written more succinctly by deﬁning the AR and MA polynomials, respectively, as: φ(L) = (1 − φ1 L − φ2 L2 − . . . − φp Lp )

AR(p) polynomial

(3.2a)

θ(L) = (1 + θ1 L + θ2 L2 + . . . + θq Lq )

MA(q) polynomial

(3.2b)

A note on convention is appropriate at this early stage. The coefﬁcients in the MA polynomial are written with a positive sign. This is not always the case in the literature and the convention in computer programs varies, so that you will also ﬁnd the MA polynomial written with a negative sign in front of each coefﬁcient. The ARMA model is then written as: φ(L)yt = θ(L)εt

(3.3)

Pure AR and pure MA models are clearly just the special cases corresponding to q = 0 and p = 0, respectively. That is: ARMA(p, 0) : φ(L)yt = εt

(3.4)

MA(0, q) : yt = θ(L)εt

(3.5)

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3.1 ARMA(p, q) models

70

Unit Root Tests in Time Series

φ(L = 1) = (1 − ∑i=1 φi )

sum of AR(p) coefﬁcients

(3.6a)

θ(L = 1) = (1 + ∑

sum of MA(q) coefﬁcients

(3.6b)

p

q θ) j=1 j

The ‘driving’ term in the ARMA model is the innovation εt . To emphasise that εt is the primary input to the process and yt is the output, the ARMA model can be rearranged with yt as the left-hand-side variable. From (3.1) and using the deﬁnition of the lag operator: yt = φ1 yt−1 + . . . + φp yt−p + εt + θ1 εt−1 + . . . + θq εt−q

(3.7)

A particular value of yt depends on lagged values of yt and a moving (weighted) average of the ‘noise’ εt ; by successive back-substitution yt can be expressed in terms of p initial values of yt and q – 1 initial values of εt . 3.1.1 Deterministic terms For simplicity, the speciﬁcation in (3.1) assumed E(yt ) = 0. If this is not the case, say E(yt ) = µt , then yt is replaced by yt − µt , and the ARMA model is: φ(L)(yt − µt ) = θ(L)εt

(3.8)

The term µt has the interpretation of a trend function, the simplest and most frequently occurring cases being where yt has a constant mean, so that µt = µ, and yt has a linear trend, so that µt = β0 + β1 t. The ARMA model can then be written is deviations form by ﬁrst deﬁning y˜ t ≡ yt − µt , with the interpretation that y˜ t is the detrended (or demeaned) data, so that (3.8) becomes: φ(L)y˜ t = θ(L)εt

(3.9)

ˆ t , replaces µt , so that In practice, µt is unknown and a consistent estimator, say µ ˆ t . Typically, in the constant mean case, the estimated detrended data is yˆ˜ t = yt − µ ˆ t , and in the linear trend case β0 the ‘global’ mean y¯ = T−1 ∑Tt=1 yt is used for µ and β1 are replaced by their LS estimators (denoted by ˆ over) from the prior ˆ t = βˆ 0 + βˆ 1 t. However, regression of yt on a constant and a time trend, so that µ these are not the only possible choices (see Chapter 7, section 7.4.8, on recursive mean adjustment, and Canjels and Watson, 1997, and Vogelsang, 1998). ˆ t replacing µt , (3.9) becomes: With µ φ(L)yˆ˜ t = θ(L)εt + ξt

(3.10)

ˆ t ). where ξt = φ(L)(µt − µ An alternative to writing the model as in (3.8) is to take the deterministic terms to the right-hand side, so that: φ(L)yt = µ∗t + θ(L)εt

(3.11)

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Note that the respective sums of the coefﬁcients in φ(L) and θ(L) are obtained by setting L = 1 in (3.2a) and (3.2b), respectively. That is:

An Introduction to ARMA Models

71

Where, for consistency with (3.8) µ∗t = φ(L)µt . For example, in the constant mean case and linear trend cases µ∗t is given, respectively, by: µ∗t = µ∗ = φ(1)µ

(3.12a)

µ∗t = φ(L)(β0 + β1 t)

(3.12b)

= β∗0 + β∗1 t p

p

where β∗0 = φ(1)β0 + β1 ∑j=1 jφj , β∗1 = φ(1)β1 and φ(1) = 1 − ∑j=1 φj . For example, if φ(L) = 1 − φ1 L, with linear trend µt = β0 + β1 t, then: (1 − φ1 L)(yt − (β0 + β1 t)) = θ(L)εt yt = β∗0 + β∗1 t + φ1 yt−1 + θ(L)εt where β∗0 = (1 − φ1 )β0 + φ1 β1 and β∗1 = (1 − φ1 )β1 . 3.1.2 The long run and persistence An ARMA(p, q) model is described as being causal if there exists an absolutely summable sequence of constants {ωj }∞ 0 , such that: ∞

y˜ t = ∑j=0 ωj Lj εt = ω(L)εt

(3.13)

The condition of absolute summability is ∑∞ j=0 |ωj | < ∞. The lag polynomial ω(L) is the casual linear ﬁlter governing the response of {yt } to {εt }. The representation in (3.13) is the MA form of the original model, which will be MA(∞) if φ(L) is not redundant; see also section 3.3. j −1 The MA polynomial is ω(L) = ∑∞ j=0 ωj L = φ(L) θ(L), with ω0 = 1; for this representation to exist the roots of φ(L) must lie outside the unit circle (see also section 3.2 and Appendix 2), so that φ(L)−1 is deﬁned. The MA form (3.13) provides the basis of a number of tools of interpretation of the original model. In particular, the concepts of the long-run solution, the long-run variance and a measure of persistence are based on (3.13). First, as to the long-run solution, note that on setting εt to its expected value of zero in (3.13), it follows that y˜ t = 0, which implies yt = µt , so that the trend function µt is the long-run value of yt if one exists. Moreover, by setting L = 1 in ω(L), we obtain the impact of a sustained oneunit shock on yt . First suppose that ω(L) = ∑Sj=0 ωj Lj is of ﬁnite order, S. Next consider a sustained shock so that εt++ s = εt + s + 1 for s ≥ 0, and let yt + s be the pre-shock value and yt++ s the post-shock value. Then, following through

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= φ(1)β0 + β1 φ(L)t

72

Unit Root Tests in Time Series

the pattern over time, we can establish that the difference between these two values is yt++ s − yt + s = ∑sj=0 ωj for s < S, and yt++ s − yt + s = ∑Sj=0 ωj for s ≥ S. The difference is the partial sum to s if s < S and the sum if s ≥ S; thus ω(1) = ∑Sj=0 ωj is a measure of persistence in that it shows how much yt++ S differs from yt + S . Increasing S, for example, letting S → ∞, does not change any aspect of principle outlined here. Thus, the limiting (or long run) effect of a sustained one-unit shock, where ω(L) is a lag polynomial of inﬁnite order, is: S→∞

∑j=0 ωj S

(3.14)

= ω(1) To put this another way, limS→∞ (yt++ S ) = yt + S + Λ(∞), so that in the limit yt++ S and yt + S differ by Λ(∞). It is for this reason that Λ(∞) can be interpreted as a measure of persistence; for example, suppose that Λ(∞) = 0, then the sustained unit shock has no effect in the long run – it is transitory. This calculation is simple in an ARMA model with φ(L) invertible as ω(1) = φ(1)−1 θ(1). As to the long-run variance of y˜ t , this is just the variance of ω(1)εt , which is the long-run variance of yt given that µt is assumed to be a deterministic 2 = var[ω(1)ε ] = ω(1)2 σ2 . function. Thus, σy,lr t ε As noted in the previous section, if no prior adjustment is undertaken then trend terms should be added to the regression speciﬁcation, in which case the implied long run can be retrieved by using (3.12a) for the constant mean case and (3.12b) for the with-trend case. In the former case, Equation (3.12a) is µ∗t = φ(1)µ, from which µ = φ(1)−1 µ∗ ; in the latter case, start with β1 = φ(1)−1 β∗1 p and solve β∗0 for β0 = φ(1)−1 (β∗0 − β1 ∑j=1 jφj ). In practice, estimates replace the unknown coefﬁcients. As an example, if φ(L) = (1 − 0.75L), θ(L) = (1 − 0.25L) and µ∗ = 0.5, then the implied long run is: yt =

(1 − 0.25) 0.5 + εt (1 − 0.75) (1 − 0.75)

= 2 + 3εt In evaluating the long run it has been assumed that εt takes its expected value 2 = 9σ2 . of zero, so that E(yt ) = 2 + 3E(εt ) = 2; also note that σy,lr ε

3.2 Roots and stability 3.2.1 The modulus of the roots of the AR polynomial must lie outside the unit circle The ARMA model given by (3.1) and (3.8) is a stochastic p-th order difference equation in yt . It will not be stable unless certain conditions on the roots of

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Λ(∞) ≡ lim

An Introduction to ARMA Models

73

(1 − φ1 L)yt = εt ⇒

(3.15)

yt = φ1 yt−1 + εt

(3.16)

Given a one-unit one-off shock to εt , yt will change by one unit, yt + 1 will change by φ1 , yt + 2 will change by (φ1 )2 , and so on. In this case, with no MA component, this just traces out an (inﬁnite) moving average polynomial, with h-th coefﬁcient given by (φ1 )h for yt + h . For the sum of the individual effects to be ﬁnite, even though the horizon is not, it is necessary that |φ1 | < 1. If this condition is not met, the inﬁnite sum does not converge and the (difference) equation is not stable. A detailed deﬁnition of the roots of a polynomial, with examples and extensions, which will be useful for this section, is contained in Appendix 2, especially sections A2.4 and A2.6. The general condition for stability in terms of the roots p is as follows. Consider the p-th order polynomial φ(z) = 1 − ∑i=1 φi zi , then the p solution to φ(z) = 0 implies ∏i=1 (z − δi ) = 0, where δi are the p roots or solutions of this equation. The general condition is that the modulus of the roots of φ(z) must lie outside the unit circle. In the case of the AR(1) lag polynomial, there is one root δ1 = φ−1 1 and |δ1 | > 1 for |φ1 | < 1; hence the stability condition is satisﬁed. A reminder on notation and convention is useful at this stage. The stability condition is sometimes stated in an apparently opposite form as the roots must lie inside the unit circle; however, this statement of the condition refers to the reciprocals of δi (see Appendix 2). 3.2.2 The special case of unit roots 3.2.2.i Unit roots in the AR polynomial A special but important case arises when one of the roots of the polynomial φ(L) is equal to one and the others have modulus greater than one. Starting from the simplest case consider the ARMA(1, 0) model given by (1 − φ1 L)yt = εt ; if φ1 = 1, then δ1 = φ−1 1 = 1, and the model can be written as: ∆yt = εt

(3.17)

The model in levels is not stable because the single root is on the unit circle but, trivially in this case, the model in the ﬁrst difference, ∆yt , is stable. If there is a single unit root, the original p-th order polynomial can be rewritten as the product of a ﬁrst-order polynomial, given by the ﬁrst difference operator (1 – L), and a polynomial of order p – 1. Multiplied together, these polynomials are of the same order, p, as the original polynomial. This is as

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the φ(L) polynomial are met. This can be seen intuitively in the simplest ﬁrstorder case, with no MA component (the MA polynomial is not relevant for this evaluation, it provides the stochastic input); that is:

74

Unit Root Tests in Time Series

follows: φ(L) = (1 − φ1 L − φ2 L2 − . . . − φp Lp )

(3.18)

= (1 − L)(1 − φ∗1 L − . . . − φ∗p−1 Lp−1 )

where φ∗ (L) is of one lower order than φ(L), and the model in terms of ∆yt is φ∗ (L)∆yt = εt . This is a principle that can be applied to multiple unit roots. For example, if there are two unit roots in φ(L), φ(L) can be factored into (1 − L)(1 − L)φ∗∗ (L) and the stable model in the second difference of yt is φ∗∗ (L)∆2 yt = εt , where φ∗∗ (L) is of order p – 2 and ∆2 yt ≡ ∆∆yt . This procedure is general enough to suggest a modiﬁcation to the ARMA(p, q) notation. If there are d ≥ 0 unit roots in the p-th order polynomial, and all the other roots meet the stability criterion, then there exists a stable ARMA(p – d, q) model in the d-th difference of yt . In discrete calculus, the inverse operation of taking a difference is summation. In a digression that is brief and useful for later work, note that the difference operator applied to yt is (1 − L)yt . Hence undoing this operation requires (1 − L)−1 (1 − L)yt = yt , and (1 − L)−1 yt = (1 + L + L2 + . . .)yt

(3.19)

∞ y j=0 t−j

=∑

∞

(1 − L)(1 − L)−1 yt = (1 − L) ∑j=0 yt−j ∞

(3.20)

∞

= ∑j=0 yt−j − ∑j=1 yt−j = yt However, by analogy with differential calculus, this process is referred to as ‘integration’ rather than summation, giving rise to the alternative notation for the ARMA(p – d, q) model as ARIMA(p∗ , d, q), where p∗ = p – d. This is read as an autoregressive integrated moving average model of orders p∗ , d and q. Although the notation p refers to the original order of the AR polynomial, it is common for the ARIMA model in the d-th difference to be referred to as ARIMA(p, d, q), in which case the ARMA model in the levels must be ARMA(p + d, q). Which convention is being used is generally clear from the context. 3.2.2.ii A notational convention The representation in (3.18) occurs sufﬁciently often in a general form throughout this book that a special notation is reserved for this way of writing the model. The idea is to reparameterise the polynomial φ(L) so that a possible unit root is isolated. The root that might be unity is designated ρ (more precisely in terms of the way that the roots have been deﬁned this is ρ−1 , but in the case of the unit

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= (1 − L)φ∗ (L)

An Introduction to ARMA Models

75

root there is no ambiguity), and the remaining polynomial and coefﬁcients are denoted ϕ(L) = (1 + ϕ1 L + . . . + ϕp−1 Lp−1 ). Thus: φ(L) = (1 − ρL)(1 − ϕ1 L + . . . − ϕp−1 Lp−1 )

(3.21)

The unit root then corresponds to ρ = 1, as in (3.18). The relationship between the coefﬁcients in φ(L) and ϕ(L) is the subject of Q3.1. From (3.21) note that p p−1 on setting L = 1, it follows that (1 − ∑j=1 φj ) = (1 − ρ)(1 − ∑j=1 ϕj ); hence an p

3.2.2.iii Unit roots in the MA polynomial Much of the relevant literature concentrates on unit roots in the AR polynomial, and that will generally be the case here; however, unit roots may be present in the MA polynomial and, whilst their presence does not have the same consequence, there are some features that should be noted (and tests for stationarity can be based on testing for a unit root in the MA polynomial; see Chapter 11). First, consider an MA(1) process: yt = (1 + θ1 L)εt

(3.22)

If θ1 = –1, then: yt = (1 − L)εt

(3.23)

The inverse operation implies that: (1 − L)−1 yt = εt

(3.24a)

∞ y i=0 t−i

(3.24b)

⇒

∑

= εt

Not that differencing results in a unit root in the MA polynomial; for example, if yt = εt then ∆yt = (1 − L)εt ; however, the differencing is unnecessary as the differencing operator is redundant, referred to as ‘over-differencing’. This observation also provides the basis of a unit root testing procedure as an MA process ﬁtted to ∆yt should indicate a unit root (see, for example, Tanaka, 1990). Cases of over-differencing are not usually as simple as this, but they are alerted by a unit root duplicated in each of the AR and MA polynomials. For example, consider an ARMA(1, 1) model with φ1 = 1 and θ1 = –1, then: (1 − L)yt = (1 − L)εt

(3.25a)

⇒ yt = εt

(3.25b)

The lag polynomials cancel, indicating that yt was inappropriately differenced. Over-differencing is a special case of cancellation of the roots in the AR and

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implication of ρ = 1 is φ(1) = 0, equivalently ∑j=1 φj = 1.

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MA polynomials, which is considered in section 3.3.3. This case turns out to be particularly problematic for unit root tests, because when θ1 is close to, although not actually equal to, –1, there is still a unit root in the AR polynomial, but realisations from the process will be very close to those from a stationary process.

The inﬁnite moving average representation of the ARMA model is considered in this section. If φ(L)−1 is well deﬁned, then implicit in the ARMA model, with a non-redundant AR polynomial, is an inﬁnite moving average representation of the process yt in terms of εt . If the AR polynomial is redundant, but the MA polynomial is not, the moving average representation is, of course, ﬁnite. 3.3.1 Examples Some simple examples will serve to show what φ(L)−1 looks like and what the conditions are for the MA representation to exist. Example 1: ARMA(1, 0) with µt = µ This is the case with p = 1 and q = 0, then: (1 − φ1 L)(yt − µ) = εt

(3.26)

Thus, φ(L) = (1 − φ1 L), and provided that |φ1 | < 1, then φ(L)−1 = 1/(1 − φ1 L) is well deﬁned. The latter is the sum of the following inﬁnite series, which is convergent if the stability condition on φ1 is satisﬁed: lim (1 + ∑s=1 φs1 Ls ) = 1/(1 − φ1 L) S

S→∞

(3.27)

The inﬁnite MA representation of (3.26) is: (yt − µ) = φ(L)−1 εt ∞

= εt + ∑i=1 φi1 Li εt

(3.28)

The weight on εt−i is φi1 , so the sequence will decline monotonically if 0 < φ1 < 1, and oscillate in sign, but monotonically decline in absolute value, if –1 < φ1 < 0. If φ1 > 1 or φ1 < –1, that is |φ1 | > 1, the MA representation of the AR process is not deﬁned because the relationship between the inﬁnite sum and the closed form (3.28) is no longer valid. Intuitively, if |φ1 | > 1 then the process is explosive. Example 2: ARMA(1, 1) What of more complex models involving convolutions of AR and MA polynomials? Consider the ARMA(1, 1) model, that is: (1 − φ1 L)(yt − µ) = (1 + θ1 L)εt

(3.29)

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3.3 Inﬁnite moving average representation of the ARMA model

An Introduction to ARMA Models

77

Then multiply through by (1 − φ1 L)−1 to obtain: (yt − µ) = (1 − φ1 L)−1 (1 + θ1 L)εt

(3.30)

Thus, w(L) = (1 − φ1 L)−1 (1 + θ1 L) and using (3.27) this can be expressed as: ∞

w(L) = (1 + ∑s=1 φi1 Li )(1 + θ1 L)

(3.31)

Multiplying out the two factors gives: j−1

j

j−1

For j ≥ 1 the general coefﬁcient is (φ1 + θ1 φ1 ), which will decline to zero provided, as in the simple ARMA(1, 0) case, |φ1 | < 1, irrespective of the value of θ1 . Notice that w(L) is generally an inﬁnite polynomial in L, w(L) = (1 + ∑∞ j=1 wj ). j

j−1

For the example in (3.32), the wj coefﬁcients are given by wj = (φ1 + θ1 φ1 ), j = 1, ..., ∞. To obtain the sum of the lag weights there is no need to obtain the (inﬁnite) series of wj directly; it is more convenient to use w(L) = φ(L)−1 θ(L) and w(L = 1) = φ(1)−1 θ(1). Consider the numerical ARMA(1, 1) example with coefﬁcients φ1 = 0.75 and θ1 = –0.25, then the weights in w(L) are: 1, 0.5, 0.375, 0.281, ... . The sum of this inﬁnite series is given by w(1) = (1 – 0.25)/(1 – 0.75) = 3, which exists because the inﬁnite series is convergent. 3.3.2 The general MA representation Larger values of p and q will inevitably lead to more complex expressions for the coefﬁcients in w(L). Fortunately, there is a simple relationship that can be exploited in calculating these coefﬁcients. In the general case w(L) = φ(L)−1 θ(L), hence φ(L)w(L) = θ(L). The method proceeds by expanding the polynomial product φ(L)w(L) and then equating coefﬁcients on like powers of L on the left-hand and right-hand sides of the identity. Thus, in terms of the coefﬁcients of the relevant polynomials: (1 − φ1 L − φ2 L2 − . . . −φp Lp )(1 + w1 L + w2 L2 + . . . + wj Lj + . . .) = (1 + θ1 L + θ2 L2 + . . . + θq Lq )

(3.33)

w1 = φ1 + θ1 ; w2 = φ1 w1 + φ2 + θ2 ; w3 = φ1 w2 + φ2 w1 + φ3 + θ3 ; w4 = φ1 w3 + φ2 w2 + φ3 w1 + φ4 + θ4 ; w5 = φ1 w4 + φ2 w3 + φ3 w2 + φ4 w1 + φ5 + θ5 ; .. .

.. .

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j

w(L) = 1 + (φ1 + θ1 )L + (φ21 + θ1 φ1 )L2 + (φ31 + θ1 φ21 )L3 + . . . + (φ1 + θ1 φ1 )Lj + . . . (3.32)

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Unit Root Tests in Time Series

and the general term is: wj = φ1 wj−1 + φ2 wj−2 + φ3 wj−3 + . . . + φp wj−(p−1) + φj (if j ≤ p) + θj (if j ≤ q) For j > p and j > q: (3.34)

It is evident that these relationships deﬁne a recursion, such that w1 is ﬁrst obtained, then w2 , and so on. As in the simple ARMA(1, 0) model, the inﬁnite moving average representation will not always be deﬁned. The required condition is the same as that for stability of the difference equation in yt ; namely, that the modulus of the roots of the AR polynomial φ(L) are outside the unit circle. 3.3.3 Inversion of the MA polynomial Another question of interest is whether an MA model can be inverted to obtain its implied AR representation. This requires that the following representation is well deﬁned: θ(L)−1 φ(L)(yt − µ) = εt

(3.35a)

A(L)(yt − µ) = εt

(3.35b)

where A(L) = θ(L)−1 φ(L). To see the essence of this transformation consider the simplest case, that is MA(1), then: (yt − µ) = (1 + θ1 L)εt (1 + θ1 L)−1 (yt − µ) = εt

if the inverse transformation exists

This is familiar ground since if |θ1 | < 1, then: lim (1 + ∑s=1 (−θ1 )s Ls ) = S

S→∞

1 (1 + θ1 L)

(3.36)

The AR representation of the ARMA(0, 1) model is therefore: ∞

[1 + ∑i=1 (−θ1 )i Li ](yt − µ) = εt

(3.37)

If θ1 is positive, then the inﬁnite AR model will have coefﬁcients on the lagged yt that alternate in sign; this is because (−θ1 )i = (−1)i θi1 , and (−1)i alternates in sign. However, ﬁtting an ARMA(0, 1) model to economic time series usually results in a negative value for θ1 , so that −θ1 is positive and the lag weights decline monotonically rather than oscillate. This implies some similarity between the ARMA(0, q) and ARMA(p, 0) models. To illustrate, the ARMA(0, 1) model with θ1 = –0.8 has an inﬁnite AR representation as: (1 + 0.8L + 0.64L2 + . . . + 0.8i Li + . . . + . . .)(yt − µ) = εt

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wj = φ1 wj−1 + φ2 wj−2 + φ3 wj−3 + . . . + φp wj−(p−1)

79

In this case, the condition required for the inversion of the MA(1) polynomial is |θ1 | < 1; if this condition is not satisﬁed, then the ARMA model is still valid, but it cannot be inverted. The general condition for inversion of the MA polynomial is straightforward and parallels that for the inversion of the AR model to an MA model; namely, that the modulus of the roots of θ(L) are outside the unit circle. However, to interpret the impact of this condition, note that if some of the MA roots are inside the unit circle, and hence the inversion condition is not satisﬁed, the model can be equivalently rewritten using the reciprocals of those roots that are inside the unit circle, resulting in a model with the same autocovariance function (Fuller, 1996, Theorem 2.6.4), hence the only practical problem arises when one or more of the MA roots is on the unit circle; that is, when there is a unit root in the MA polynomial.

3.4 Approximation of MA and ARMA models by a ‘pure’ AR model In practical terms, although the AR representation of an MA model is inﬁnite it has to be truncated to provide an approximation to the ARMA(0, 1) model; this is a procedure that is often used in unit root testing. However, some care has to be taken with this argument because convergence of the ARMA(p, 0) model as p → ∞ to the ARMA(0, 1) model is slow for |θ1 | ≈ 1, and this can have consequences for unit root tests that rely on this approximation. (We sidestep the question of the convergence metric as, at this stage, the point is deliberately intuitive.) The extent of agreement between the approximation and the true value can be gauged in the following way. Setting L = 1 in the inﬁnite series (3.37) results in i the sum of the coefﬁcients. That is, 1 + ∑∞ i=1 (−θ1 ) = 1/(1 + θ1 ) provided |θ1 | < 1; for example, if θ1 = –0.9 then 1/(1 + θ1 ) = 10. Suppose the inﬁnite series expansion is truncated to an upper summation limit of, say, r < ∞, what is the ratio of the resulting sum to the true limit, 1/(1 + θ1 ), expressed as a proportion of this limit? To examine this it is convenient to change notation so that λ = –θ1 , ∞ i i then 1 + ∑∞ i=1 (−θ1 ) = 1 + ∑i=1 λ . (It is assumed here that –1 < θ1 < 0, and the case where θ1 > 0 is considered as a question.) Then: ∞

1 + ∑i=1 λi = (1 − λ)−1 = 1 + λ + λ2 + . . . + λr−1 + λr (1 + λ + λ2 + ....) = 1 + λ + λ2 + . . . + λr−1 + λr (1 − λ)−1

(3.38)

The ‘remainder’ when the inﬁnite summation is truncated to r terms is λr (1−λ)−1 , so that the last term is λr−1 ; dividing by (1−λ)−1 to get a proportionate remainder the result is simply λr . To illustrate, consider θ1 = –0.9 and take

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An Introduction to ARMA Models

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r = 5, then λ = 0.9 and then the ‘remainder’ is λr (1 − λ)−1 = 0.95 (10) = 5.9049. Proportionate to the correct limit of 10, this remainder is 5.9049/10 = 0.59049. The calculations are: ∞

(1 + ∑i=1 0.9i )(1 − 0.9)−1 = [1 + 0.9 + 0.92 + 0.93 + 0.94 ]/10 + [0.95 (10)]/10 Setting r = 10 gives a proportionate error of 0.910 = 0.3487, which is still substantial, and even if r = 20, the proportionate error is 0.1216. The general message is that the approximation of an MA model by an AR model may well require a large value of p, especially if the dominant root of the MA polynomial is close to the unit circle.

3.5 Near-cancellation of roots In the ARMA(1, 1) case, w(L) = (1 − φ1 L)−1 (1 + θ1 L), and it is evident that the numerator and denominator polynomials will cancel each other out exactly if θ1 = –φ1 to leave w(L) = 1. This indicates that the original ARMA(1, 1) model was over-speciﬁed as there was no need for either the AR or the MA polynomial; it could be reduced to an ARMA(0, 0) model. This is an extreme case in order to make the point that care should be taken in specifying ARMA(p, q) models to ensure that the lag polynomials do not contain redundant elements, a case that was considered earlier when over-differencing resulted in a unit root in the MA polynomial. Exact cancellation is unusual; in the ARMA(1, 1) model, nearcancellation occurs if θ1 ≈ –φ1 , and the possibility of reducing the polynomials by one order should be considered. Consider the general (p, q) case. The AR and MA polynomials φ(L) and θ(L) can be factored as: φ(L) = (1 − a1 L)(1 − a2 L) · · · (1 − ap L) θ(L) = (1 − b1 L)(1 − b2 L) · · · (1 − bq L) ⇒ w(L) = φ(L)−1 θ(L) =

(1 − b1 L)(1 − b2 L) . . . (1 − bq L) (1 − a1 L)(1 − a2 L) . . . (1 − ap L)

(3.39)

It is convenient to order the roots in these factorisations, such that they decline in absolute value; that is, |a1 | ≥ |a2 | . . . ≥ |ap | and |b1 | ≥ |b2 | . . . ≥ |bq |. This way of writing w(L) makes it clear that cancellation of the roots can occur if any ai = bj , implying that p and q should each be reduced by one for each cancellation. The over-differencing case occurs if, say, a1 = b1 = 1, and p and q should therefore be p – 1 and q – 1. A useful diagnostic follows from noting that if there is a unit root in φ(L) then φ(1) = 0; this follows from setting L = 1 in (3.18). However, note that φ(1) = 0

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= [0.40951] + [0.59049]

An Introduction to ARMA Models

81

3.6 The Beveridge-Nelson (BN) decomposition of a lag polynomial Some decompositions of the ARMA lag polynomials occur quite frequently in practice and are often taken for granted. Here, and in the following section, two important decompositions are described. This section introduces the BN decomposition of a lag polynomial, due to Beveridge and Nelson (1981), and the next section introduces the DF decomposition. A separate, and probably its most frequent, use of the BN decomposition is to partition a time series into two components of different orders of integration, which is also of use in the analysis of partial sum processes (see Phillips and Solo, 1992). An ARIMA(p, 1, q) process generates a time series that can be decomposed into a permanent (P) component and a transitory (T) component, where the former is I(1) and the latter is I(0). Intuitively, the P-T decomposition is into a (stochastic) trend and a cyclical component, so that the long-run direction is provided by the trend, around which there are transitory short-run movements; however, in contrast to some deﬁnitions, in this decomposition there is no requirement that the cycle is periodic. The BN decomposition is one example of a P-T decomposition; references to other P-T decompositions are given at the end of this section.

3.6.1 The BN decomposition of a lag polynomial The BN decomposition looks similar to the DF decomposition of section 3.7 below, but it has a different use. It works with the MA representation of the model, which arises either where there is only an MA polynomial q with yt = θ(L)εt , and θ(L) = 1 + ∑i=1 θi Li , or where an AR model has been inverted, perhaps also with an MA component, so that yt = w(L)εt , where j w(L) = φ(L)−1 θ(L) = 1 + ∑∞ j=1 wj L . Initially consider the simplest case that of a ﬁnite order MA, then the BN decomposition is: θ(L) = θ(1) + D(L)(1 − L)

BN decomposition, pure MA

(3.40)

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will also follow if there is more than one unit root, so this feature alone will not identify the number of unit roots. Thus one should be aware of the unit root possibility if unconstrained estimation produces an estimate of φ(1) close to zero. In keeping with the discussion of the condition on the roots for stability to hold, note that one or more unit roots implies that w(1) is not deﬁned because it involves division by zero. Unit roots in the θ(L) polynomial also have interesting implications, since if any of the bj = 1, then θ(1) = w(1) = 0.

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q−1

q

where D(L) = ∑i=0 di Li and θ(1) = 1 + ∑j=1 θj . Using the BN decomposition yt is expressed as: yt = θ(1)εt + D(L)∆εt

(3.41)

To illustrate the connection between the MA and BN coefﬁcients di , consider the MA(2) = ARIMA( 0, 0, 2) model, then:

= 1 + θ1 + θ2 + d0 (1 − L) + d1 L(1 − L)

BN decomposition

= 1 + θ 1 + θ2 + d 0 − d 0 L + d 1 L − d 1 L 2 = 1 + θ1 + θ2 + d0 + (d1 − d0 )L − d1 L2 Hence, by equating coefﬁcients on like powers of L, the BN coefﬁcients can be obtained in terms of the original MA coefﬁcients, and these are as follows: d1 = − θ 2 (d1 − d0 ) = θ1 ⇒ d0 = − (θ1 + θ2 ) Note that 1 + θ1 + θ2 + d0 = 1 is implied by the last relationship. j Now consider the general ARMA case, where w(L) = 1 + ∑∞ j=1 wj L , then i D(L) = ∑∞ i=0 di L . The BN decomposition is as follows.

w(L) = w(1) + D(L)(1 − L) ∞ w + i=1 i ∞ w + i=1 i

(3.42)

∞ d Li (1 − L) i=0 i ∞ ∞ d Li − i=0 di Li + 1 i=0 i ∞ ∞ w + d0 + i=1 (di − di−1 )Li i=1 i

=1+ ∑ =1+ ∑

∑ ∑

=1+ ∑

∑

∑

Hence, equating terms, as before: (di − di−1 ) = wi ⇒

for i = 1, . . . , ∞

∞ w j=i+1 j

di = − ∑ ∞

1 + ∑i=1 wi + d0 = 1 ⇒

∞

d0 = − ∑i=1 wi In terms of the MA coefﬁcients, the BN decomposition is: ∞ ∞ w(L) = w(1) − ∑i=0 (∑j = i + 1 wj )Li (1 − L)

(3.43)

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θ(L) = θ(1) + D(L)(1 − L)

An Introduction to ARMA Models

83

A frequent use of the BN decomposition is to partition a time series generated by an ARMA process that has a unit root in the AR polynomial, into two mutually exclusive and exhaustive components: a trend, or permanent component and a cyclical, or transitory, component. In fact, Beveridge and Nelson (1981) used a slightly different, but equivalent, approach whereas we exploit the development in the earlier sections of this chapter (see also, for example, Newbold, 1990; Proietti and Harvey, 2000). The BN decomposition also has considerable value in the asymptotic analysis of unit root and related problems (see Phillips and Solo, 1992; Xiao and Phillips, 1998; Chang and Park, 2002). 3.6.2.i The BN decomposition in terms of the ARMA lag polynomials The starting point is a version of the ARMA model, with a unit root in the AR polynomial, which is isolated by adopting the notation of (3.21). The preferred notation is shown in the following representation: (1 − ρL)ϕ(L)yt = ϕ(1)µ + θ(L)εt

ARMA model

(3.44)

Next, to impose a unit root set ρ = 1: ϕ(L)∆yt = ϕ(1)µ + θ(L)εt ϕ(L)(∆yt − µ) = θ(L)εt

(3.45a) ARIMA model

(3.45b)

Notice that this version of the ARMA model is written to ensure that ∆yt has a potentially nonzero constant, µ, interpreted as drift in the random walk process. This is not always the case; for example, if the model is speciﬁed as φ(L)(yt − µ) = θ(L)εt and φ(L) is factorised as (1−ρL)ϕ(L), so that the term in the ‘constant’ is (1 − ρL)ϕ(L)µ, then this term disappears for ρ = 1. Returning to (3.45b), ﬁrst multiply through by the inverse of ϕ(L), rearrange and then apply the BN (lag polynomial) decomposition to the result. This gives: ψ(L) = ϕ(L)−1 θ(L)

∆yt = µ + ψ(L)εt = µ + [ψ(1) + D(L)(1 − L)]εt

use the BN decomposition applied to ψ (L)

= µ + ψ(1)εt + D(L)∆εt yt = ∆−1 µ + ψ(1)∆−1 εt + D(L)εt p = yt + ytr t

yt = ∆−1 µ + ψ(1)∆−1 εt p

p = y0 + tµ + ψ(1)

∑

t ε i=1 i

(3.46) to obtain the level, use ∆−1

(3.47)

permanent and transitory

(3.48)

permanent component

(3.49a)

equivalent representation

(3.49b)

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3.6.2 The BN decomposition for integrated processes (optional on ﬁrst reading)

Unit Root Tests in Time Series

ytr t = D(L)εt

transitory component

(3.50)

where ∆−1 is the summation operator (see Appendix 2). At this point note that (3.49b) is interpreted as the trend or permanent component of the series, which in turn comprises three components: the inip tial condition, y0 ; the deterministic trend contributed by the drift, tµ; and the stochastic trend comprising the scaled cumulative sum of the innovations, where the scaling factor is the ‘persistence’ parameter ψ(1), which is the long-run impact or persistence due to a unit change in εt . p To continue, note that yt follows a random walk with drift, so that: p

∆yt = µ + ψ(1)εt

(3.51) p

The permanent component yt is updated each period by µ, the deterministic change in yt , and by ψ(1)εt , interpreted as the persistence in the current shock. The transitory component has no persistence in the I(1) sense (but, as the examples below show, there is persistence in the sense that, in general, D(L) = 0). An alternative representation of the permanent and transitory components is also useful. The starting point is the same, that is: ϕ(L)(∆yt − µ) = θ(L)εt Now multiply through by θ(L)−1 ψ(1) to obtain: ϕ(L) θ(1) θ(1) ∆yt = µ + εt θ(L) ϕ(1) ϕ(1)

(3.52)

(3.53)

= µ + ψ(1)εt This is the same as the right-hand side of (3.51), so that: θ(1) ϕ(L) p yt + A yt = θ(L) ϕ(1)

(3.54)

where A is a constant of integration (strictly, summation), which is necessary because having the same ﬁrst difference does not imply the same level. An explicit expression for the constant is derived below. Apart from this constant, p which is zero if µ is zero, yt is a weighted sum of the current and lagged values of yt , with weights that sum to unity. p Multiplying yt in (3.54) through by θ(L) gives a recursive form of the relationship: θ(L)yt = [ϕ(L)ψ(1)]yt + A∗ p

(3.55)

where: A∗ = θ(1)A

(3.56)

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84

An Introduction to ARMA Models

85

p

Subtracting yt from yt gives the transitory component, ytr t : p

ytr t = yt − y t

= Ψ(L)yt − A

(3.57)

Where: θ(L)ϕ(1) − ϕ(L)θ(1) θ(L)ϕ(1)

Further, multiplying (3.57) through by θ(L) gives: θ(L)ytr t =

θ(L)ϕ(1) − ϕ(L)θ(1) y t − A∗ ϕ(1)

= (θ(L) − ψ(1)ϕ(L)) yt − A∗

(3.58)

Notice that Ψ(1) = 0 and, therefore, the (autoregressive) lag weights relating ytr t to yt sum to zero. This feature arises from the numerator polynomial in Ψ(L), which must have a unit root and so be expressible such that a term in (1 – L) can be factored out; say: θ(L)ϕ(1) − ϕ(L)θ(1) = λ(L)(1 − L)

(3.59)

where λ(L) = ∑ri=0 λi Li , with r = max(p, q) – 1, is deﬁned implicitly. The coefﬁcients in λ(L) can be obtained by expanding the left-hand side of (3.59) and equating coefﬁcients on like powers of L; an example below illustrates this procedure. The ‘circle’ of polynomial representations can now be completed. Rearranging (3.59), we have: ϕ(L)−1 θ(L) = ϕ(1)−1 θ(1) + λ(L)ϕ(L)−1 ϕ(1)−1 (1 − L)

(3.60)

Hence, on comparison with the BN decomposition, where ϕ(L)−1 θ(L) = ψ(L), see (3.46), then: D(L) = λ(L)ϕ(L)−1 ϕ(1)−1

(3.61)

In the case that the process is just MA, then D(L) = λ(L). The relationship in (3.61) is now used to obtain an explicit expression for the constant A. First, express εt by rearranging the ﬁrst line of (3.46), that is: εt = ψ(L)−1 (∆yt − µ) = ϕ(L)θ(L)−1 (∆yt − µ)

(3.62)

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Ψ(L) =

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Unit Root Tests in Time Series

Now substitute this into ytr t = D(L)εt and use (3.61) to obtain: −1 −1 −1 ytr t = λ(L)ϕ(L) ϕ(1) {ϕ(L)θ(L) (∆yt − µ)}

= λ(L)ϕ(1)−1 θ(L)−1 (∆yt − µ) = λ(L)ϕ(1)−1 θ(L)−1 ∆yt − λ(L)ϕ(1)−1 θ(L)−1 µ

(3.63)

Next multiply through by θ(L):

= λ(L)ϕ(1)−1 ∆yt − λ(1)ϕ(1)−1 µ

(3.64)

Therefore the ‘adjustment’ terms due to the non-zero mean are, respectively: A = λ(1)ϕ(1)−1 θ(1)−1 µ ∗

A = λ(1)ϕ(1)

−1

(3.65)

µ

(3.66) p

For example, in the case of an ARIMA(1, 1, 1) model, yt and ytr t can be expressed recursively as (see Q3.3 for the details): yt = − θ1 yPt−1 + ψ(1)(yt − ϕ1 yt−1 ) + (θ1 + ϕ1 ) (1 − ϕ1 )−1 µ

(3.67)

−1 −1 − θ1 ytr t−1 + λ(L)(1 − ϕ1 ) ∆yt − (θ1 + ϕ1 ) (1 − ϕ1 ) µ

(3.68)

p

ytr t =

p

This enables yt (or ytr t ) to be computed recursively given starting values and p estimates of the coefﬁcients; however, note that the I(1) nature of yt means that the choice of starting values is critical. Newbold’s computational method, which avoids an approximation is described in the next section. 3.6.2.ii Precise and efﬁcient computation of the BN decomposition Newbold (1990) developed an algorithm that avoids the problems of the p unknown initial condition, which implies arbitrary start-up values for yt or ytr t . tr The transitory component yt can be expressed entirely in terms of quantities that are known or estimated as part of the model estimation; that is: ˆ t + j + ϕ(1)−1 ∑j=1 ∑i = j ϕi xˆ t + (q−j + 1) ) ytr t = − (∑j=1 x q

p

p

(3.69)

where: xt = ∆yt − µ xˆ t + i = xt + i

for i ≤ 0

Note that xt is stationary, with drift subtracted and a unit root extracted. If the index i in xˆ t + i is not positive then ‘forecasts’ are just actual values. For p comparison note that the P-T decomposition is deﬁned as yt = yt + ytr t , whereas p Newbold (1990) deﬁnes the transitory component as ct = yt – yt and, hence, ytr t = –ct .

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θ(L)ytr = λ(L)ϕ(1)−1 (∆yt − µ)

An Introduction to ARMA Models

87

3.6.2.iii Illustrations of the BN decomposition Three examples are used here to illustrate the BN decomposition, with rather more detail in the ﬁrst case.

Example 1: ARIMA(0, 1, 2)

= µ + [(1 + θ1 + θ2 )εt − (θ1 + θ2 )∆εt − θ2 ∆εt−1 ] yt = ∆−1 µ + (1 + θ1 + θ2 )∆−1 εt p

∑

t p = y0 + tµ + (1 + θ1 + θ2 ) i=1 εi = µ + yPt−1 + (1 + θ1 + θ2 )εt

ytr t = − (θ1 + θ2 )εt − θ2 εt−1

permanent component p

yt viewed as ‘integrating’ the innovations p

yt follows a random walk with drift transitory component

Thus, d0 = −(θ1 + θ2 ), d1 = −θ2 and D(1) = −(θ1 + 2θ2 ) = 0 unless θ1 = −2θ2 . As this example is a pure MA process then λ(L) = D(L), where λ0 = −(θ1 + θ2 ), λ1 = −θ2 and λ(1) = −(θ1 + 2θ2 ); therefore: A = λ(1)ϕ(1)−1 θ(1)−1 µ =−

(θ1 + 2θ2 ) µ (1 + θ1 + θ2 )

Thus:

p

yt =

(1 + θ1 + θ2 ) (θ1 + 2θ2 ) µ yt − (1 + θ1 + θ2 ) (1 + θ1 L + θ2 L2 )

= − θ1 yPt−1 − θ2 yPt−2 + (1 + θ1 + θ1 )yt − (θ1 + 2θ2 )µ (1 + θ1 L + θ2 L2 ) − (1 + θ1 + θ2 ) (θ1 + 2θ2 ) µ yt + (1 + θ1 + θ2 ) (1 + θ1 L + θ2 L2 ) (λ1 + λ2 L)(1 − L) (θ1 + 2θ2 ) µ y = + t (1 + θ1 + θ2 ) (1 + θ1 L + θ2 L2 )

ytr t =

Multiplying through by θ(L) and rearranging gives: tr tr ytr t = − θ1 yt−1 − θ2 yt−2 + λ0 ∆yt + λ1 ∆yt−1 + (θ1 + 2θ2 )µ

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∆yt = µ + (1 + θ1 L + θ2 L2 )εt

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Unit Root Tests in Time Series

Example 2: ARIMA(0, 1, q) with q ﬁnite ∆yt = µ + (1 + ∑j=1 θj Lj )εt q

= µ + [(1 + ∑j=1 θj )εt − (∑j=1 θj )∆εt − (∑j=2 θj )∆εt−1 . . . − θq ∆εt−(q−1) ] q

q

q

yt = ∆−1 µ + ψ(1)∆−1 εt p

permanent component

= y0 + tµ + (1 + ∑j=1 θj ) ∑i=1 εi t

= µ + yPt−1 + (1 + ∑j=1 θj )εt q

p

yt follows a random walk with drift

ytr t = D(L)εt = − (∑

q θ )ε − ( j=1 j t

transitory component

∑

q θ )ε − . . . − θq εt−(q−1) j=2 j t−1

q di = −(∑j = i θj )

Note that and D(1) = −(θ1 + 2θ2 + 3θ3 + . . . + qθq ). Further, letting q → ∞, for the BN decomposition to be well deﬁned both θ(1) and ∑∞ j=1 jθj must be ﬁnite. The extension to the general case, which allows an invertible AR plus MA process, is then simple, since this will just require that ψ(1) and ∑∞ j=1 jψj are ﬁnite. Example 3: ARIMA(1, 1, 0) Prior to obtaining the decomposition, it is useful to note that for a stable AR(1) process: ∞

d0 = − ∑i=1 ϕ1i = − ϕ1 (1 − ϕ1 )−1 .. .

.. .

.. .

(3.70)

∞ j+1 dj = − ∑i = j + 1 ϕ1i = − ϕ1 (1 − ϕ1 )−1

With these simpliﬁcations in mind, the BN decomposition is obtained as: ∆yt = µ + ψ(L)εt = µ + (1 − ϕ1 L)−1 εt

in this case: ψ(L) = ϕ(L)−1

= µ + (1 + ϕ1 L + ϕ12 L2 + ϕ13 L3 + . . .)εt ∞

∞

= µ + (1 − ϕ1 )−1 εt − ∑i=0 (∑j = i + 1 ϕ1 )Li ∆εt j

∞ = µ + (1 − ϕ1 )−1 εt − ((1 − ϕ1 )−1 ∑i=0 ϕ1i + 1 ∆εt−i

Therefore, the permanent and transitory components in the BN decomposition are: yt = ∆−1 µ + ϕ(1)∆−1 εt p

permanent component

= y0 + tµ + (1 − ϕ1 )−1 ∑i=1 εi p

t

= µ + yPt−1 + (1 − ϕ1 )−1 εt

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q

p

An Introduction to ARMA Models ∞ −1 ytr ∑i=0 ϕ1i + 1 εt−i t = − (1 − ϕ1 )

89

transitory component

= (1 − ϕ1 )−1 (yt − ϕ1 yt−1 ) − ϕ1 (1 − ϕ1 )−1 µ (1 − ϕ1 L) (1 + θ1 ) = yt − ϕ1 (1 − ϕ1 )−1 µ (1 + θ1 L) (1 − ϕ1 ) (1 − ϕ1 ) − (1 − ϕ1 L) yt − λ0 (1 − ϕ1 )−1 µ ytr t = (1 − ϕ1 ) = − ϕ1 (1 − ϕ1 )−1 ∆yt + ϕ1 (1 − ϕ1 )−1 µ Also note that: D(L) = λ(L)ϕ(L)−1 ϕ(1)−1 =

−ϕ1 (1 + ϕ1 L + ϕ12 L2 + . . .) (1 − ϕ1 )

An empirical illustration in section 3.10.1.ii considers an ARIMA(1, 1, 1) model. The BN decomposition is not the only way to decompose a series into a permanent and a transitory component. One feature of the BN decomposition is that yPt and ytr t are both driven by the same innovation (compare (3.49) and (3.50)), implying a perfect negative correlation between the two components, whereas an alternative decomposition is into a mutually uncorrelated random walk and stationary components (see, for example, Lippi and Reichlin, 1992; Proietti and Harvey, 2000). (For a reconciliation of the BN and unobserved components approaches to the P-T decomposition, see Morley, Nelson and Zivot, 2003.)

3.7 The Dickey-Fuller (DF) decomposition of the lag polynomial and the ADF model The DF decomposition of a lag polynomial is used frequently in later chapters. Consider the lag polynomial A(L) = 1 – ∑ki=1 ai Li , if L = 1, then A(1) = 1 – α, where α = ∑ki=1 ai . This polynomial occurs in the speciﬁcation of an AR model of the form A(L)yt = εt , where A(L) is a lag polynomial that is not necessarily of ﬁnite order and εt is white noise. The order of the polynomial is denoted k, which will be equal to p if there is no MA component, otherwise generally k > p. Notice the use of A(L) as the autoregressive polynomial; this is a deliberate change of notation that distinguishes the AR polynomial, φ(L), in an

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i+1 In this case D(1) = −(1 − ϕ1 )−1 ∑∞ = – ϕ1 (1 − ϕ1 )−1 , which is ﬁnite as i=0 ϕ1 required; further, from (1 − ϕ1 ) − (1 − ϕ1 L) = −ϕ1 (1 − L), note that λ0 = −ϕ1 . In terms of equivalent representations, these are: 1 − ϕ1 L) p yt = yt + λ0 (1 − ϕ1 )−1 µ 1 − ϕ1

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Unit Root Tests in Time Series

AR model from an ARMA model with an invertible MA component, in which case A(L) = θ(L)−1 φ(L), which is an inﬁnite lag polynomial. (The DF decomposition was known before DF tests were in widespread use; for example, Anderson, 1971, p.175, equation (53), uses the transformation for p = 2 and is aware of the general context.) 3.7.1 The DF decomposition

A(L) = (1 − αL) − C(L)(1 − L)

(3.71)

k−1 where C(L) = ∑i=1 ci Li ; note that the C(L) polynomial is necessarily of one lower order than the A(L) polynomial. The ci coefﬁcients, i = 1, . . . , k – 1, are obtained as follows:

ci = − ∑j = i + 1 aj k

(3.72)

The reverse relationship may also be of interest; that is, the coefﬁcients in C(L) and α are available and the ai coefﬁcients need to be recovered. First, note the recursive relation between these coefﬁcients: ci – ci−1 = ai , for i = 2, . . . , k – 1, and ck−1 = −ak ; this recovers the ai coefﬁcients except a1 . For a1 note that a1 = α – ∑ki=2 ai , so all of the ai coefﬁcients are available. The DF decomposition of the lag polynomial is useful because an AR(k) model can be separated into one lagged level of yt and (k – 1) ﬁrst differences; thus, if yt is I(1), this is a separation into one I(1) component and (k – 1) I(0) components. In a common terminology, the original AR model is said to be augmented by the k – 1 ﬁrst difference components, giving rise to the shorthand ADF(k – 1), for the augmented Dickey-Fuller model of order k – 1. Consider the following application of the DF decomposition, which results in an ADF model: A(L)(yt − µ) = εt

AR representation

(3.73a)

[(1 − αL) − C(L)(1 − L)] (yt − µ) = εt

use the DF decomposition

(3.73b)

yt = A(1)µ + αyt−1 + C(L)∆yt + εt

then rearrange yt−1

(3.73c)

the ‘ADF regression’

(3.74)

⇒ ∆yt = A(1)µ + γyt−1 + ∑i=1 ci ∆yt−i + εt k−1

where γ = –A(1) = α – 1 and α = ∑ki=1 ai . Note that in the ADF regression, variables of potentially different orders of integration are separated. Also note that by letting k → ∞, this decomposition and regression are valid for the inﬁnite AR model, with summable coefﬁcients. In practice, the C(L) polynomial

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The DF decomposition of A(L), a lag polynomial of order k, is:

An Introduction to ARMA Models

91

is truncated so that: ∆yt = A(1)µ + γyt−1 + ∑i=1 ci ∆yt−i + εt,k k−1

(3.75)

Examples of the DF decomposition Some illustrations of the DF decomposition follow. First consider, k = 2 so that A(L) = 1 – (a1 L + a2 L2 ); hence, α = a1 + a2 , c1 = –a2 and a1 = α – a2 . As check note that: A(L) = (1 − αL) − C(L)(1 − L) = [1 − (a1 + a2 )L] − (−a2 L)(1 − L) = 1 − a1 L − a2 L + a2 L − a2 L2 = 1 − (a1 L + a2 L2 )

as required

Now consider the DF decomposition in the context of the AR(2) model, then: A(L)(yt − µ) = εt

general form

yt − a1 yt−1 − a2 yt−2 = A(1)µ + εt

AR(2) model

yt − (a1 + a2 )yt−1 + a2 ∆yt−1 = A(1)µ + εt

apply the DF decomposition

∆yt − (a1 + a2 − 1)yt−1 + a2 ∆yt−1 = A(1)µ + εt ∆yt = A(1)µ + γyt−1 + c1 ∆yt−1 + εt

subtract and add yt−1

the ‘ADF’ regression

where γ = − A(1) = a1 + a2 − 1, α = ∑i=1 ai , c1 = − a2 . 2

The questions at the end of this chapter include a further worked example. The end result of the use of the DF decomposition is the augmented DF regression, which is considered more extensively in Chapter 6.

3.8 Different ways of writing the ARMA model The earlier sections have introduced the ARMA and ARIMA models using a notation that is fairly standard in time series analysis, but for which there are some important alternatives. The notation so far can be summarised simply enough as follows (see section 3.1.1): φ(L)(yt − µt ) = θ(L)εt

(3.76)

j where µt = µ or a low order polynomial in time of the form µt = β0 + ∑h j=1 βj t .

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∞ where εt,k = (∑∞ j=k+1 aj )yt−1 + ∑i = k ci ∆yt−i + εt , and the terms in the summation are deemed negligible by some criterion.

92

Unit Root Tests in Time Series

This section introduces three ways of approaching the speciﬁcation of a set of dynamic models for unit root testing. Two of them, the error dynamics and common factor/components models, yield the same conclusions throughout, and the one to choose is a matter of convenience and emphasis. The third is the DF speciﬁcation, which does differ in its implications from the other two approaches; for example, in its implications for invariance of the test statistics to nuisance parameters. Whilst this topic is considered in Chapter 6, some understanding of the differences is necessary here to motivate the ML-based approach to testing for a unit root. One way of writing the ARMA model is to involve additional dynamics through the error term which, in essence, develops along the lines shown in Equation (3.21). In this form the possible unit root, ρ, is isolated as acting upon yt−1 , and further dynamics, if present, are taken to operate through an error term designated zt . If the error dynamics are pure AR of ﬁnite order, then the resulting ADF form of the model is also of ﬁnite order. This is the ﬁrst case illustrated below. If the error dynamics involve an MA component, then the ADF form is of inﬁnite order, and truncation is required for practical implementation. The error dynamics and common factor representations are important for estimation as they naturally suggest a generalised least squares (GLS) approach, a topic that is considered in Chapter 7. In this speciﬁcation, it is then natural to see the problem as the estimation of ρ (the possible unit root) subject to the nuisance parameters in the error. A different interpretation of the problem is referred to as the common factor (or components) model comprising (i) the deterministic component, plus (ii) a stochastic component that may not be weakly dependent, that is, it may contain a unit root. This way is also important in unit root testing (see, for example, Elliott, Rothenberg and Stock, 1996; Schmidt and Phillips, 1992; Canjels and Watson, 1997). Finally, the original statement of the testing framework for a unit root (see Fuller, 1976; Dickey and Fuller, 1979, 1981), uses a slightly different approach, which at ﬁrst glance looks to be, but is not, equivalent to the error dynamics and component approaches (see, for example, Schmidt and Phillips, 1992; DeJong et al., 1992).

3.8.1.i An error dynamics model AR(1) error dynamics In this example there is an AR(1) model for yt , and the errors zt are separated and subject to an AR(1) model. The end result is an ADF(1) = AR(2) model. The

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3.8.1 Alternative ways of representing the dynamics

An Introduction to ARMA Models

93

steps are as follows: (1 − ρL)(yt − µ) = zt

assume an AR(1) model for yt

(3.77a)

(1 − ϕ1 L)zt = εt

assume AR(1) model for the errors

(3.77b)

(1 − ϕ1 L)(1 − ρL)(yt − µ) = εt

combine the two components

(3.77c)

A(L)(yt − µ) = εt

where A(L) = (1 − ϕ1 L)(1 − ρL)

(3.77d)

∆yt = A(1)µ + γyt−1 + c1 ∆yt−1 + εt

the ‘ADF(1)’ regression

(3.78)

By equating coefﬁcients on like powers of L, the coefﬁcients in A(L) can be obtained as follows: 1 − a1 L − a2 L2 = 1 − (ϕ1 + ρ)L + ϕ1 ρL2 ⇒ a1 = (ϕ1 + ρ), a2 = − ϕ1 ρ

(3.79a)

c1 = ϕ1 ρ

(3.79b)

α = (ϕ1 + ρ) − ϕ1 ρ

(3.79c)

γ=α − 1 = (1 − ϕ1 )(ρ − 1)

(3.79d)

The end result is that if the error dynamics are generated by an AR(1) process, the resulting model is an ADF(1) model, with coefﬁcient restrictions as in (3.79a)–(3.79d). By extension, an AR(2) process for the errors results in an ADF(2) regression as the ﬁnal step and an AR(p) process for the errors results in an ADF(p) regression. MA(1) error dynamics In the next example, it is assumed that the errors are generated by an invertible MA(1) process. This gives rise to an inﬁnite ADF model, which will have to be truncated as a feasible practical representation of the process. (1 − ρL)(yt − µ) = zt

an AR(1) model for yt

zt = (1 + θ1 L)εt

an MA(1) model for the errors

(3.80a) (3.80b)

ϕ(L)zt = εt

invert the MA process, ϕ(L) = (1 + θ1 L)

(1 − ρL)ϕ(L)(yt − µ) = εt

substitute for zt

A(L)yt = A(1)µ + εt

A(L) = (1 − ρL)ϕ(L) = (1 − ρL)(1 + θ1 L)

−1

(3.80c) (3.80d)

−1

(3.80e)

⇒

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⇒

94

Unit Root Tests in Time Series ∞

∆yt = A(1)µ + γyt−1 + ∑i=1 ci ∆yt−i + εt

(3.81)

The last lines uses the DF decomposition on A(L). Notice that the ﬁnal ADF representation includes an inﬁnite number of lagged differences. This arises because the lag polynomial ϕ(L) is the inﬁnite-order polynomial given by: ϕ(L) = (1 + (−θ1 )L + (−θ1 )2 L2 + . . . + (−θ1 )j Lj + . . .) j

where ϕj = (−1)j θ1

The idea now is to truncate the inﬁnite ADF process, which is usually done by a rule that depends on T. We can view this truncation taking place either at the ﬁnal step, as in (3.75), or in the earlier step, (3.80c), following the inversion of the MA polynomial. ARMA error dynamics We are now in a position to consider an ARMA model for the errors. The model set-up is ﬁrst given and then the DF decomposition is applied to obtain the ADF representation: (1 − ρL)(yt − µ) = zt

an AR(1) model for yt

ϕ(L)zt = θ(L)εt

ARMA model for the errors

ψ(L)zt = εt

invert MA polynomial : ψ(L) = θ(L)

A(L)yt = A(1)µ + εt

A(L) = θ(L)−1 ϕ(L)(1 − ρL)

(3.83a) (3.83b) −1

∞

ϕ(L) = 1− ∑i=1 ψi Li (3.83c) (3.83d)

⇒ ∞

∆yt = A(1)µ + γyt−1 + ∑i=1 ci ∆yt−i + εt

(3.84)

Where the last line uses the DF decomposition on A(L). 3.8.1.ii Common factor model An alternative way of writing the model ‘moves’ the dynamics completely into the error process. In this, the common factor, set up there are three underlying structural equations: yt = µt + ut

yt is determined by a trend function plus error

(3.85a)

ut = ρut−1 + zt

a common factor on the error ut

(3.85b)

zt = ν(L)εt

speciﬁcation of the noise

(3.85c)

In the simplest case zt = εt , which we will use to illustrate the reduced form, which is obtained by substituting ut = yt − µt into the second equation and then

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∞

= 1 + ∑j=1 ϕj Lj

(3.82)

An Introduction to ARMA Models

95

rearranging the third equation (note that ut ≡ y˜ t , that is detrended yt ). In this example, with zt = εt , the result is: (yt − µt ) = ρ(yt−1 − µt−1 ) + εt

(3.86)

yt = µ∗t + ρyt−1 + εt

(3.87a)

µ∗t = (1 − ρL)µt

(3.87b)

This is, of course, the same as in the equivalent error dynamics model and is also the approach that was taken in extending the ARMA model to include deterministic components (see Equation (3.11)). Note that if ρ = 1, then the DGP reduces to ∆yt = ∆µt + εt ; for example if µt = µ, then ∆yt = εt , a pure random walk, and if µt = β0 + β1 t, then ∆yt = β1 + εt , which is a random walk with drift. To illustrate further, consider an example with zt = εt : (yt − µ) = ut

deterministics subject to an error

(3.88a)

(1 − ρL)ut = zt

structure of the error

(3.88b)

zt = (1 − ϕ1 L)−1 εt

zt = εt , (1 − ϕ1 L)−1 exists as |ϕ1 | < 1

(3.88c)

⇒ (1 − ϕ1 L)(1 − ρL)ut = εt

(3.88d)

⇒ (1 − ϕ1 L)(1 − ρL)(yt − µ) = εt

(3.88e)

The end result can also be interpreted as an AR(1) error dynamics model, thus one might ask the purpose of this way of writing the model. The answer is that it is sometimes useful to separate out estimation of the dynamic elements from estimation of the deterministic components. Given that ut has a dependent error structure, an alternative to (ordinary) least squares estimation is a GLS-based method that takes the error structure into account. The question of efﬁciency of LS applied to the trend components, relative to other methods of estimation, is the subject of the Grenander-Rosenblatt (1957) theorem and is exploited by Elliott, Rothenberg and Stock (1996) to improve the efﬁciency of trend estimation and obtain a more powerful test of the unit root for alternatives close to the unit circle.

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With some slight rearrangement, this can be expressed as:

96

Unit Root Tests in Time Series

The simplest version of the common factor model with a linear trend, µt = β0 + β1 t, is as follows: yt = β0 + β1 t + ut

linear trend subject to an error

(3.89a)

⇒ (yt − (β0 + β1 t)) = ut error structure

(3.89c)

zt = εt ⇒ (1 − ρL)(yt − (β0 + β1 t)) = εt

(3.89d)

⇒ yt = β∗0 + β∗1 t + ρyt−1 + εt

(3.89e)

β∗0 = (1 − ρ)β0 + ρβ1

(3.89f)

β∗1 = (1 − ρ)β1

(3.89g)

Note that the structural coefﬁcients, ρ, β0 and β1 , are identiﬁed from (3.89e), (3.89f) and (3.89g). Indeed, this approach should be familiar from Equation (3.11), with an appropriate change of notation, that is, ρ = φ1 . 3.8.1.iii Direct speciﬁcation There is a subtle but important difference in the error dynamic/common factor approach compared to the original direct (DF) speciﬁcation; (but see also Dickey, 1984 and Dickey, Bell and Miller, 1986, for a speciﬁcation in the spirit of the common factor approach). To see the difference, take the case where a constant is included in the maintained regression, then: yt = η0 + ρyt−1 + εt

(3.90)

The link with the previous notation has been broken deliberately. A difference in interpretation arises if η0 is considered a parameter that is unrelated to ρ, rather than related to ρ as in (3.87b). If (3.90) is considered a ‘stand-alone’ regression, its implications differ notably from those of the error dynamic/common factor interpretation. Consider the implications for H0 : ρ = 1. The speciﬁcation of (3.90) reduces to: ∆yt = η0 + εt

(3.91)

whereas in the case of (3.87), with µt = µ, the speciﬁcation under the null is: ∆yt = εt

(3.92)

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(1 − ρL)ut = zt

(3.89b)

97

Note that (3.91) includes a drift term, η0 , implying a deterministic trend in yt as well as the stochastic trend from the cumulated values of current and lagged εt , whereas (3.92) just implies the latter. Unless η0 = µ∗ = (1 − ρ)µ is imposed in (3.90) there will also be a difference under HA in the two formulations. For example, consider the difference in a simulation set-up in order to assess power. In the common factor approach, the long-run mean, µ, is kept constant as ρ varies, whereas in the DF speciﬁcation, the long-run mean varies because η0 is kept constant although ρ is varying. Schmidt and Phillips (1992) show that the power curves for a given value of ρ < 1 under HA from the two approaches cross. Extending the direct speciﬁcation to include a linear trend results in: yt = η0 + η1 t + ρyt−1 + εt

(3.93)

where neither η0 nor η1 are linked to ρ. The speciﬁcation in (3.90) can be tested by an F-type test that jointly tests ρ = 1 and η1 = 0, and that in (3.92) by an F-test that jointly tests ρ = 1, η0 = 0 and η1 = 0. Dickey and Fuller (1981) suggested a number of joint tests, and provided some critical values; these are considered in Chapter 6, section 6.3.8, along with a more detailed consideration of the implications of the different speciﬁcations for invariance of the proposed test statistics. Comparing (3.93) with its counterpart from the error dynamics/common factor approach, which in this case comprises equations (3.89e), (3.89f) and (3.89g), shows that the difference in interpretation appears again in the with trend case. Consider ρ = 1, then according to (3.93) the speciﬁcation is: ∆yt = η0 + η1 t + εt

(3.94)

If η0 and η1 are regarded as parameters unrelated to ρ, then η0 determines the linear trend in the levels solution and now, additionally, η1 will determine a quadratic trend in the levels solution. However, in the common factor set-up, the null speciﬁcation is: ∆yt = β1 + εt

(3.95)

and this is a random walk with drift. An important distinction that also arises is whether the distributions of the resulting test statistics are invariant to the nuisance parameters η0 and η1 in the case of the DF speciﬁcation, and µ, β0 and β1 in the case of the error dynamics/common factor approach. This is a topic taken up more extensively in Chapter 6, section 6.3. The default speciﬁcation throughout this book, and the one now generally favoured in the unit root literature, is the error dynamics/common factor approach. For a more detailed discussion of the issues, especially power comparisons, see Marsh (2007).

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An Introduction to ARMA Models

98

Unit Root Tests in Time Series

Whilst Chapter 6 is concerned extensively with estimation and inference in the ADF regression model, it is useful here to note some basic points. First, a test of the null hypothesis of a unit root in the ADF model can be formulated as a t-type test of the hypothesis that γ = 0. This type of test is probably the most widely used and takes the usual form of dividing the estimated value of γ by its estimated standard error. Such a test statistic does not have a ‘t’ distribution, but a nonstandard distribution that is a function of Brownian motion and is usually referred to as a DF distribution. To distinguish such tests from t tests, they are referred to as τˆ -type tests or pseudo t tests. For example, consider the following ADF(k – 1) model, where y˜ t ≡ yt − µt : ∆y˜ t = γy˜ t−1 + ∑i=1 ci ∆y˜ t−i + εt k−1

(3.96)

ˆ˜ t = yt − µ ˆ t , and y ˆ t is used in place of In practice, µt is replaced by an estimator, µ y˜ t . The test statistic has the following general form: τˆ =

γˆ ˆ σ(γ)

(3.97)

ˆ is the estimated standard error of γ. ˆ where γˆ is the LS estimator of γ and σ(γ) A subscript is usually added when the ADF regression includes a constant or a constant and a linear time trend, so that the ‘family’ of pseudo t tests is τˆ , τˆ µ and τˆ β , respectively.

3.8.3 DF n-bias tests: δˆ A second popular test statistic for the unit root null hypothesis, introduced by Dickey and Fuller, is the normalised bias of the general form δˆ = T(ˆρ − 1) (see ˆ Chapter 1, Equation (1.34)). Tests of this type are referred to as n-bias or δtype statistics. As in the case of the τˆ -type tests, a subscript indicates whether the ADF regression includes no deterministic terms, a constant or a constant ˆ δˆ µ and δˆ β , respectively. and a trend, leading to the ‘family’ of tests given by δ, However, unlike, the τˆ -type tests, the test statistic cannot be ‘read’ off from the ADF regression. ˆ The δ-type tests are considered in greater detail in the Chapter 6; however, for the purposes of this chapter, note the following two implications, which enable an estimator of (ˆρ − 1) to be obtained that is consistent under the null, and hence the test statistic can be computed. This test statistic is easier to motivate if the ARMA model is in error dynamics form (see section 3.8.1.i). In summary,

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3.8.2 A pseudo t test of the null hypothesis, γ = 0: τˆ

An Introduction to ARMA Models

99

this is: (1 − ρL)y˜ t = zt

where y˜ t ≡ yt − µt

ψ(L)zt = εt

where ψ(L) = θ(L)−1 ϕ(L)

⇒ where A(L) = θ(L)−1 ϕ(L)(1 − ρL)

A(L)y˜ t = εt

∞

∆y˜ t = γy˜ t−1 + ∑i=1 ci ∆y˜ t−i + εt

(3.98)

Two implications are as follows: (1)

Setting L = 1: A(1) = θ(1)−1 ϕ(1)(1 − ρ) = ψ(1)(1 − ρ) ⇒ γ = − A(1) = ψ(1)(ρ − 1)

(2)

(3.99)

From ψ(L)(1 − ρL) = (1 − αL) – C(L)(1 – L), which is the DF decomposition, note that if ρ = 1 then α = 1, and hence: ψ(L)(1 − L) = (1 − C(L))(1 − L)

⇒

ψ(L) = (1 − C(L))

⇒

∞ ψ Li = 1 − i=1 i

1−∑

∞ c Li i=1 i

∑

⇒

so that ψi = ci

i = 1, . . . , ∞

ψ(1) = (1 − C(1))

∞

where C(1) = ∑i=1 ci

(3.100)

Using these two results, speciﬁcally the ratio of (3.99) to (3.100), an estimator ˆ of (ˆρ − 1) that is consistent under the null and the corresponding δ-type test statistic, are provided, respectively, by: ρˆ − 1 =

γˆ ˆ (1 − C(1))

δˆ i = T(ˆρ − 1) =T

(3.101) (3.102)

γˆ ˆ (1 − C(1))

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using the DF decomposition ⇒

100 Unit Root Tests in Time Series

where i = µ, β, for the constant mean and linear trend cases, respectively; k−1 ˆ ˆ ci , and a ˆ above indicates the LS estimator. It has been assumed that C(1) = ∑i=1 C(L) has been truncated by a rule that ensures consistency of the estimators.

This section considers direct estimation of the ARMA model either via the conditional or unconditional likelihood function, and testing the unit root hypothesis within that approach. Least squares-based approaches are considered extensively in the Chapter 6. The focus in this chapter is not on the detail of estimation methods and algorithms, for which there are some excellent references, but more simply setting out a likelihood-based framework for a unit root test. 3.9.1 ML estimation In this section, we consider ML estimation of the ARMA model, with the object of formulating a test of the null hypothesis of a unit root. With this aim in mind, the error dynamics form of the ARMA model is the best point of approach as the unit root null can be represented simply as H0 : ρ – 1 = 0, the alternative being HA : ρ – 1 < 0, and further dynamics are placed in the error component. 3.9.1.i The ARMA model (Shin and Fuller, 1998) In this case, the model is: (1 − ρL)yt = zt

an AR(1) model for yt , ρ ∈ (−1, 1]

ϕ(L)zt = θ(L)εt

(3.103) (3.104)

p−1

q

where ϕ(L) = 1 − ∑i=1 ϕi Li , θ(L) = 1 + ∑j=1 θj Lj . The assumption in (3.104) is that zt follows an AR(p – 1 , q) process, given by: zt = ∑i=1 ϕi zt−i + εt + ∑j=1 θj εt−j p−1

q

(3.105)

The error process generating zt is assumed to be stationary, with no roots common to ϕ(z) and θ(z), and, for simplicity, no mean or trend term is present, but the model is easily amended (see below) for these cases. For maximum likelihood estimation it is assumed that εt ∼ niid(0, σε2 ), otherwise the normality assumption may be relaxed, so that εt ∼ iid(0, σε2 ). The assumption that there is not a second unit root is important (see Nielsen, 2001); if the AR component is second order or higher, there may be a second unit root, affecting the limiting distribution of the LR unit root test, which is therefore not pivotal.

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3.9 An outline of maximum likelihood (ML) estimation and testing for a unit root in an ARMA model

An Introduction to ARMA Models

101

The following notation will be required: Y = (y0 , y1 , . . . , yT )

(3.107)

Y1,−1 = (y0 , y1 , . . . , yT−1 ) z = (z1 , . . . , zT )

(3.108)

ϕ = (ϕ1 , . . . , ϕp−1 )

(3.109)

(3.110)

Ψ = (ϕ1 , . . . , ϕp−1 , θ1 , . . . , θq )

(3.111)

The model can then be written in vector form as: Y1 = ρY1,−1 + z

(3.112)

Subtracting Y1,−1 from both sides results in: ∆Y1 = γY1,−1 + z

(3.113)

The following variance-covariance (var) matrices are required: var(Y1 |Y1,−1 ) = E(zz ) = σε2 ΓT

conditional variance matrix of Y1

(3.114)

E(YY ) = ΣT + 1 = σε2 ΛT + 1

autocovariance matrix of Y

(3.115)

Note that for simplicity the dependence of the matrices ΓT and ΣT + 1 on the underlying parameters is left implicit; ΓT will depend on Ψ, whereas ΣT + 1 will additionally depend on ρ. 3.9.1.ii The log-likelihood functions The conditional log-likelihood (CLL) function is: CLL(Ψ, ρ, σε2 |{y}T1 ) = −

T T 1 1 ln 2π − ln σε2 − ln |ΓT | − 2 (Y1 − ρY1,−1 ) Γ−1 T (Y1 − ρY1,−1 ) 2 2 2 2σε

(3.116)

where |ΓT | is the determinant of ΓT . Maximisation of CLL(.) results in the conditional ML estimator of ρ denoted ρ . The unconditional log-likelihood (UCLL) function takes into account the generation of the initial observation, y0 ; the conditioning is now, therefore, explicitly on {y}T0 . That is: UCLL(Ψ, ρ, σε2 |{y}T0 ) = −

(T + 1) (T + 1) 1 1 ln 2π − ln σε2 − ln |ΛT + 1 | − 2 (Y Λ−1 Y) T+1 2 2 2 2σε

(3.117)

The ML estimator of ρ derived from (3.117) is denoted ρ˜ .

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Y1 = (y1 , . . . , yT )

(3.106)

The set-ups in (3.116) and (3.117) show the form of the conditional and unconditional likelihood functions in the general ARMA case. To obtain the maximum likelihood estimators, the general principle is to differentiate the log-likelihood with respect to the unknown parameters, Ψ, ρ and σε2 , and then set the derivatives to zero and solve for the ML estimators. The details of ML estimation are not the primary concern of this chapter; the interested reader may consult, for example, Box and Jenkins (1970), Anderson (1971), Nicholls and Hall (1979) and Fuller (1996). In the simplest case of no MA component and maximising the CLL function, the ML and LS estimators of the AR coefﬁcients are the same, otherwise the ML estimators are nonlinear, for example, if a nonzero mean µ is included and is known, ρ˜ is a root of a cubic equation, and if µ is unknown, ρ˜ is a root of a ﬁfth-order polynomial. 3.9.1.iii AR(1), stationary case with niid errors Zero mean case It may be helpful to illustrate the CLL and UCLL functions in the simplest of cases, where zt = εt . In this case the initial set-up is as follows: (1 − ρL)yt = zt

yt ∼ niid(0, σy2 ); t = 1, ..., T

(3.118a)

zt = εt

εt ∼ niid(0, σε2 )

(3.118b)

y0 = z0

initial condition

(3.118c)

In this example, ΣT + 1 is the autocovariance matrix of an AR(1) process, the common diagonal element of which is the unconditional variance of yt , σy2 = γ(0), obtained as follows: σy2 = E(yt )2 = ρ2 E(yt−1 )2 + E(εt )2 = ρ2 σy2 + σε2 = (1 − ρ2 )−1 σε2

(3.119)

Note that γ(0) is not deﬁned for ρ = 1; hence |ρ| < 1 is required for the UCLL function to be deﬁned. The off-diagonal elements of ΣT+1 are obtained on noting that: γ(k) = E(yt yt−k ) = ρE(yt−1 yt−k ) + E(εt yt−k ) ⇒ γ(k) = ργ(k − 1) as E(εt yt−k ) = 0

(3.120)

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102 Unit Root Tests in Time Series

An Introduction to ARMA Models

103

k = 1 : γ(1) = ργ(0) k = 2 : γ(2) = ρ2 γ(0) k = s : γ(s) = ρs γ(0)

ρ2 ρ .. . .. . ...

... ... .. . .. . ...

ρT + 1 ρT .. . .. . 1

         

(3.121)

= σε2 ΛT + 1 The conditional variance of yt is simply E[(yt − E(yt ))|yt−1 ]2 = E[yt |yt−1 ]2 = σε2 , thus: var(Y1 |Y1,−1 ) = E(zz ) = σε2 IT

(3.122)

The ‘initial’ condition focuses on how the AR process is started. In the case of the UCLL function the question arises as to how y0 was generated for which there are two basic possibilities; the ﬁrst is that y0 is a ﬁxed quantity, for example, zero. The second is that y0 is a random variable, and a secondary question arises as to the distribution from which y0 was drawn. As to the latter, there are two main possibilities: (i) y0 is a draw from the unconditional distribution of yt , that is, y0 ∼ niid(0, σy2 ); (ii) y0 is a draw from the conditional distribution of yt , that is, y0 ∼ niid(0, σε2 ). The former, generally preferred option, is illustrated here. In the case that y0 ∼ N(0, σy2 ), where σy2 = (1 − ρ2 )−1 σε2 , the UCLL function for the sequence {y}T0 is: 1 (T + 1) 1 ln 2π − (T + 1) ln σε2 + ln(1 − ρ2 )) 2 2 2 1 T − 2 ∑t=1 (yt − ρyt−1 )2 + (1 − ρ2 )y20 2σε

UCLL(ρ, σε2 |{y}T0 ) = −

(3.123)

The CLL function for the sequence {y}T1 is: CLL(ρ, σε2 |{y}T1 ) = −

T T 1 T ln 2π − ln σε2 − 2 ∑t=1 (yt − ρyt−1 )2 2 2 2σε

(3.124)

The estimator resulting from maximisation of (3.123) is the (unconditional) maximum likelihood estimator (MLE) whereas the corresponding estimator for (3.124) is the conditional (on the initial observation) ML estimator (CMLE).

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Hence, putting the pieces together:   1 ρ   ρ 1     .. ..   2 2 −1  ΣT + 1 = σε (1 − ρ )  . .   .. ..   . .   T + 1 ρT ρ

104 Unit Root Tests in Time Series

Nonzero mean case In practice, the case of a zero mean is exceptional, but the arguments are easily extended to the nonzero mean case and the trended case. We illustrate the former here; the latter is easily obtained by extension. In the case that yt has a nonzero mean µ, the revised model and conditional and unconditional likelihood functions are: t = 1, ..., T

(3.125a)

zt = εt

εt ∼ niid(0, σε2 )

(3.125b)

y0 = µ + z 0

initial condition

(3.125c)

UCLL(ρ, µ, σε2 |{y}T0 ) = (T + 1) 1 1 ln 2π − (T + 1) ln σε2 + ln(1 − ρ2 )) 2 2 2 1 T − 2 ∑t=1 [yt − µ − ρ(yt−1 − µ)]2 + (1 − ρ2 )(y0 − µ)2 2σε

−

(3.126)

CLL(ρ, µ, σε2 |{y}T1 ) = −

T T 1 T ln 2π − ln σε2 − 2 ∑t=1 [yt − µ − ρ(yt−1 − µ)]2 2 2 2σε

(3.127)

Conditional on σε2 , maximisation of CLL(ρ, µ, σε2 |{y}T0 ) follows from minimisation of the last term in (3.127), which is just the sum of squared residuals when estimators replace the unknowns, and hence, in this simple case, the CMLE of ρ is the same as the LS estimator, and the CMLE of µ is the same as the LS estimator. The CMLE of σε2 differs from the LS estimator in not making a correction for the degrees of freedom: 2 σ ε = T−1

∑t=2 (yt − µ − ρ yt−1 )2 T

(3.128)

Deriving the MLE from the UCLL function is more complex (for details, see Anderson, 1971; Fuller, 1996; Shin and Fuller, 1998). A simple practical method is considered below. As is usual, the log-likelihood is differentiated with respect to the unknown parameters, ρ, µ and σε2 , and the derivatives are set equal to zero. 3.9.2 Likelihood-based unit root test statistics The key to likelihood ratio (LR)-type or F-type tests of the unit root hypothesis in an ARMA model is that once reparameterised to isolate the possible unit root, the estimators of the remaining coefﬁcients; that is, those on the sta√ tionary series, have standard (asymptotically normal and standardised by T) distributions.

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yt − µ = ρ(yt−1 − µ) + zt

105

As an ARMA model can be reparameterised so that all the AR and MA terms except one, that is, the AR term on the lagged level yt−1 , are associated with stationary variables, the additional terms in an ARMA(p, q) model over and above a simple AR(1) model are asymptotically irrelevant as far as (LR and pivotal) test statistics associated with the potentially nonstationary term are concerned. In part, obtaining this result is why it is useful to write the ARMA model in error dynamics form, since in that case the nonstationarity is focused solely on the term ρyt−1 , with the null corresponding to ρ = 1; the remainder of the model involves the process generating zt , which is just a stationary AR(p – 1, q) process. It is important to draw out the implications of this result, separated into the pure AR case and the ARMA case. Take as a ‘base’ point the simplest case of an AR(1) process and the DF test statistic τˆ of section 3.8.2 (with limiting distribution given in Chapter 6). Now consider an AR(p) model with p > 1, then, following a reparameterisation by way of the DF decomposition (see section 3.7.1), all terms other than that on yt−1 involve stationary variables and the asymptotic distribution of the t-type test on the coefﬁcient on yt−1 is unchanged by the presence of the stationary variables. This will also be the case for a LR-type test and for an appropriately adjusted n-bias test. Now consider an ARMA(p, q) model with p ≥ 0 and q > 0. There are two ways to obtain the ‘asymptotic irrelevance’ result. First, provided the MA component is invertible, then the ARMA model can be approximated by a ‘long’ AR model, with the approximation error being made ‘negligible’ by an appropriate choice of order of the AR model (the required rate of increase in the order of the approximating AR model is considered in Chapter 6). Alternatively, don’t take the approximation route, but estimate the ‘complete’ ARMA model using either the conditional or unconditional likelihood function. The same result then holds. However, a word of caution is in order: this result relates to the asymptotic distribution, whereas some simulation evidence is reported below that shows the ﬁnite sample distributions are sensitive to the orders of the AR and MA components if the orders are not known.

3.9.2.i ML: conditional and unconditional approaches A unit root test statistic based on maximising the unrestricted and restricted residual sum of squares in an ARMA model was suggested by Yap and Reinsel (1995), and see also Dickey and Fuller (1981) for the AR model. This corresponds to a conditional maximum likelihood approach. Let εt be the innovation in the ARMA model and ε t the corresponding residuals from nonlinear estimation; for example, using an iterative Gauss-Newton approach. Denote the respective unrestricted and restricted residual sums of squares as SCUR and SCR , then

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An Introduction to ARMA Models

106 Unit Root Tests in Time Series

consider the following test statistic: S − SCUR Φ = T CR SCUR

If this test statistic was being applied in a situation uncomplicated by nonstationary variables, it would have a χ2 distribution under the null hypothesis; and if, in turn, the null was one-dimensional with a two-sided alternative, then Φ is the square of a corresponding normally distributed t-type test statistic. In the context of the unit root null, with a conditional likelihood or least squares approach, the correspondence is that the t-type test is the DF pseudo t; that is, τˆ in the simplest case, and τˆ 2 and Φ have the same asymptotic distribution. ˆ The three test statistics considered here are the τˆ -type test, the δ-type test and the LR test based on Φ, for each case corresponding to the ML estimators from maximising the CLLF and UCLLF. For illustrative purposes, a constant is assumed present in the regression model. (1)

(i) Unit root tests from the CLL function:

τ µ = ( ρ − 1)/ σ ( ρ )

(3.130)

δ µ = T( ρ − 1) 2

(3.131)

2

LRC,µ = CLL( ρ , µ, σ ε |{y}T1 ) − CLLR (1, µ R , σ ε,R |{y}T1 ) = 12 Φ

(3.132)

where σ ( ρ ) is the estimated standard error of ρ from the unrestricted CLL. The LR-type test, LRC,µ , is simply the difference between the unrestricted and restricted CLL functions, where the latter is distinguished by an R sub2

(2)

2

script and imposes the unit root; for example, σ ε,R is as σ ε in deﬁnition, but uses the residuals from the CLL restricted by the unit root. (ii) Unit root tests from the UCLL function, Shin and Fuller (1998): τ˜ µ = (˜ρ − 1)/˜σ(˜ρ)

(3.133)

δ˜ µ = T(˜ρ − 1)

(3.134)

˜ σ˜ ε2 |{y}T0 ) − CLL(ρµ , µ ˜ R , σ˜ R2 |{y}T0 ) if ρ˜ ≤ ρµ LRUC,µ = CLL(˜ρ, µ, =0

if ρ˜ > ρµ

(3.135a) (3.135b)

where ρµ = 1 − 4/T. Note that LRC,µ and LRUC,µ are ‘right-sided’ tests that reject for large positive values of the test statistic. ˆ Whilst in the case of the UCLL function, the τˆ -type and δ-type test statistics are, in construction, as in the CLL function case, the LRUC test statistic of Shin

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(3.129)

107

and Fuller (1998) has to overcome the problem that the UCLL function is only deﬁned for |ρ| < 1. The suggested solution (see (3.135a) and (3.135b)), recognises this problem and also addresses the ﬁnite sample bias in the ML and LS estimators of ρ in AR and ARMA models, a topic considered in Chapter 4. The ﬁrst-order bias of the LS estimator of ρ in an AR(1) model is −(1 + 3ρ)/T if an unknown mean is estimated and −(2 + 4ρ)/T if an unknown trend is estimated; at the unit root, ρ = 1, these expressions are −4/T and −6/T, respectively. Hence, in the former case an estimate with ρ˜ > 1 − 4/T can be viewed as consistent with ρ = 1, and in the latter case ρ˜ > 1 − 6/T is consistent with ρ = 1. Thus the strategy for hypothesis testing is to not reject the null by setting the LR-type test statistic LRUC equal to zero if the unrestricted estimate ρ˜ of ρ is greater than 1 − 4/T, or 1 − 6/T if the trend is estimated; otherwise, ρµ and not 1 is used in constructing the maximised value of UCLL(.) in the restricted case. 3.9.3 Estimation, critical values and ﬁnite sample performance 3.9.3.i Estimation There is a rather neat way of calculating the ML estimators in the unconditional case due to Cummins (1998), and implemented in TSP, but it can be applied in other software. Several statistical or econometric software packages have available a suite of Box-Jenkins routines for the estimation of seasonal ARMA models, which include a unconditional or an unconditional maximum likelihood option. The set-up for these routines can be reinterpreted into an error dynamics framework, which then enables estimation of the unrestricted and restricted unconditional LL functions. Consider the following ‘seasonal’ AR (SAR) model, which is set up for data of seasonal frequency denoted J: (1 − φ1J LJ )(1 − φ1 L)(yt − µ) = εt

(3.136)

This will be readily interpretable as an SAR(1×1) model with seasonal span of J; it has a ‘regular’ AR coefﬁcient of φ1 and a ‘seasonal’ AR coefﬁcient of φ1J . However, by instructing the program that the seasonal span is J = 1, this setup can be reinterpreted as an error dynamic AR model. Compare Equations (3.77a)–(3.77c) and, in particular, (3.77c) restated below: (1 − ϕ1 L)(1 − ρL)(yt − µ) = εt

(3.137)

Then these two representations are identical with ρ = φ1 and ϕ1 = φ11 , where we have set J = 1. This approach is now easily generalised to an ARMA set-up. The combination of (3.83a) and (3.83b) is: ϕ(L)(1 − ρL)yt = θ(L)εt

(3.138)

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108 Unit Root Tests in Time Series

3.9.3.ii Asymptotic distributions Shin and Fuller (1998) establish the following key result: if zt follows a stationary ARMA process, with εt ∼ iid(0, σε2 ), then the asymptotic distributions of the ˆ δ-type statistics for the CML and UCML estimators, which are of the general form δ = T( δ − 1) and δ˜ = T(δ˜ − 1), respectively, are the same as for the AR(1) model. In effect, the fact that zt = εt , does not affect the limiting distributions. A corollary is that the limiting distributions of the equivalent τ -type and τ˜ -type tests and tests based on comparing the unrestricted and restricted likelihoods are also unaffected. These results also apply to the cases where either a constant or a constant and a trend are estimated; the test statistics for these cases are distinguished by subscripts µ and β, respectively. The asymptotic distribution of LRC,µ was obtained by Yap and Reinsel (1995) and the asymptotic distribution of LRUC,µ by Shin and Fuller (1998); other sources for the relevant limiting distributions are Gonzalez-Farias (1992), Pantula et al. (1994) and Fuller (1996). There are similar but not completely analogous results for the ADF case; in this case it is necessary to distinguish whether the ADF regression arises from an underlying ﬁnite-order AR model or as an approximation to an ARMA model. In the latter case a condition is required on the expansion of the order of the ˆ ADF terms. Further, different results apply to the τˆ -type and δ-type tests, as the former are pivotal whereas the latter are not. See Chapters 6 and 7 for further details.

3.9.3.iii Critical values and estimation Shin and Fuller (1998, Table II) gives some critical values for LRUC,µ , which is the case where a mean is estimated. Note that there is very little variation in the critical values as T varies. Table 3.1 reports these and the critical values for LRC,µ (see note to the table). The unit root null hypothesis is rejected for large values of the latter test statistic, hence the upper quantiles are reported in Table 3.1.

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The lag polynomial ϕ(L) can be read into the software program as the ‘seasonal’ AR(p – 1) polynomial with (false) seasonal span of J = 1. The order of θ(L) is speciﬁed in the usual way as the order of the ‘regular’ MA polynomial, and the order of the ‘regular’ AR polynomial is set equal to 1. The estimation method is then set to (exact or unconditional) maximum likelihood; this is the TSP estimation routine used below in section 3.9.4. By setting the estimation method to conditional ML (NOEXACT in TSP), the conditional ML estimator can be obtained; this is the same as LS in the case of a pure AR speciﬁcation (but the ML pseudo t statistics will not make a degrees of freedom adjustment).

An Introduction to ARMA Models

109

90%

95%

99%

90%

95%

99%

T 100 250 500

LRC,µ 3.33 3.31 3.30

LRC,µ 4.14 4.11 4.11

LRC,µ 5.94 5.90 5.89

LRUC,µ 1.07 1.07 1.08

LRUC,µ 1.75 1.76 1.77

LRUC,µ 3.41 3.44 3.46

T 100 250 500

LRC,β 4.97 4.93 4.91

LRC,β 5.98 5.89 5.85

LRC,β 8.23 8.00 7.92

LRUC,β 1.97 1.98 2.00

LRUC,β 2.79 2.81 2.84

LRUC,β 4.67 4.72 4.70

T 100 250 500

1% τ˜ µ –3.26 –3.31 –3.36

5% τ˜ µ –2.67 –2.67 –2.70

10% τ˜ µ –2.37 –2.37 –2.40

1% δ˜ µ

5% δ˜ µ

10% δ˜ µ

–18.78 –19.20 –19.58

–12.67 –12.85 –13.16

–10.03 –10.13 –10.31

T 100 250 500

τ˜ β –3.93 –3.94 –3.88

τ˜ β –3.34 –3.31 –3.31

τ˜ β –3.05 –3.02 –3.02

δ˜ β –26.66 –27.57 –27.81

δ˜ β –19.93 –20.30 –20.64

δ˜ β –16.76 –16.93 –17.17

Source: LRUC,µ is from Shin and Fuller (1998); other values are from own simulations, which use an AR(1) model, with 100,000 replications. See also Yap and Reinsel (1995) for LRC,µ . The critical values

for the DF-type test statistics τ , δ , τ µ , δ µ , τ β and δ β are as their least squares counterparts; that is, standard DF, counterparts, and hence are not listed separately; see Chapter 6, Appendix, for DF critical values.

3.9.3.iv Finite sample performance Shin and Fuller (1998) report the results of a comparison of test statistics for the case where zt is generated by an MA(1) process, zt = εt + θ1 εt−1 . This case is visited again in Chapter 7 as it represents a challenge to the standard DF tests, especially as actual size becomes distorted for large negative values of θ1 (see Schwert, 1989). The Shin and Fuller results relate to the case where the order of the error dynamics in the ARMA model is assumed known, whereas the (A)DFtype tests have to rely on approximating the ARMA by a ‘long’ autoregression. In the case of the ‘long’ AR approach, the order of the approximation for the ADF tests was chosen by minimising the AIC (see Chapter 9 for a more extensive consideration of lag selection criteria). The simulation results summarised in Table 3.2 are extracted from Shin and Fuller (1998, Table I). The test statistics used in this comparison are τ µ , LRC,µ , and τ˜ µ and LRUC,µ , and the ‘DF’ statistic is τˆ µ . Note that the procedure for τˆ µ

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Table 3.1 Finite sample critical values for conditional and unconditional LR-type test statistics.

110 Unit Root Tests in Time Series

Table 3.2 LS, CML and ML: illustrative simulation results, MA(1) errors, T = 100. DF :ˆτµ

CML: τ µ

UCML:˜τµ

–0.8 0.0 0.8

33.9 4.7 5.8 –2.88

3.7 2.1 10.1 –2.88

2.5 1.9 5.2 –2.66

Power ρ = 0.95 ρ = 0.95 ρ = 0.95

–0.8 0.0 0.8

64.8 11.8 13.5

8.5 5.4 15.5

4.5 8.7 18.5

17.5 12.9 8.8

11.7 21.8 22.3

ρ = 0.90 ρ = 0.90 ρ = 0.90

–0.8 0.0 0.8

88.4 27.4 24.0

18.3 13.5 33.1

5.9 22.2 45.8

27.8 32.2 21.5

11.4 50.4 55.5

Size ρ = 1.0 ρ = 1.0 ρ = 1.0 Critical values

LRC,µ 8.0 5.3 3.8 4.14

LRUC,µ 6.0 6.1 5.4 1.75

Notes: There is a different notational convention here compared with Shin and Fuller, who have negative signs on their MA coefﬁcients. The critical values are also from Shin and Fuller, but for revised versions of some of these see Chapter 7; note that the limiting distributions of τˆ µ and τ˜ µ are the same. Source: Shin and Fuller (1998).

does not include any additional lags to ‘capture’ the serial correlation. More extensive simulation results, which include the normalised bias tests and allow for the selection of the orders of the error dynamics in the ARMA estimation, are reported in Chapter 7. As to size, Shin and Fuller (1998) ﬁnd that τˆ µ , LRC,µ and LRUC,µ are reason

ably accurately sized for θ1 = 0, whereas τ µ and τ˜ µ are undersized. The key area for differences amongst the tests is as θ1 becomes increasingly negative, then τˆ µ becomes considerably oversized, with an empirical size of about 34% for a nominal 5% size; however, LRC and LRUC maintain their size reasonably well even in this difﬁcult situation. (The case of positive θ1 is generally not a problem, except for τ µ , which becomes slightly oversized.) The reported power will be misleading for tests that are incorrectly sized since similar size is not being maintained in the comparison. The simplest case to compare is for tests when θ1 = 0; then we ﬁnd that for tests of similar size, LRUC,µ is the most powerful. This ordering is generally maintained as θ1 becomes more negative. The poor performance of τ µ and τ˜ µ is perhaps surprising and should be noted. Some of these issues are considered again in Chapter 7.

3.10 Illustration: UK car production The data in this example are (seasonally adjusted) quarterly observations on UK car production (000s) for the period 1977 q1 to 2004 q4, an overall sample of 112

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θ1

An Introduction to ARMA Models

111

13.1 13 12.9 12.8 12.7 12.6

12.4 12.3 12.2 12.1

1980

1985

1990

1995

2000

2005

Figure 3.1 UK car production (logs, p.a., s.a.).

observations; allowing for lags, there was common estimation sample of 108 observations. The data, yt , in logs, are graphed in Figure 3.1. Initial estimation suggested an error dynamics model of the form: (1 − ϕ1 L)(1 − ρL)(yt − µt ) = (1 + θ1 L)εt

(3.139)

The deterministic speciﬁcations were alternately µt = µ and µt = β0 + β1 t, and an ADF model was also estimated. 3.10.1 Estimated model and test statistics The estimated models of the form (3.139) are summarised in Table 3.3, with asymptotic ‘t’ statistics in (.) parentheses. The unit root test statistics are reported in Table 3.4. The estimates suggest one root close to unity, with the other root well into the stationary region; the MA(1) coefﬁcient indicates that this is an important component of the model, and the estimated coefﬁcient is in the invertible region. In the case of the demeaned data, there is broad agreement in the implications of the various test statistics; none of the test statistics are signiﬁcant at the 5% level, although there are some differences in the implied p-values. Of particular interest are the likelihood-based test statistics, LRC,µ and LRUC,µ , with

their respective estimates ρ = 0.988 and ρ˜ = 0.980, whereas the LS estimate is ρˆ = 0.928. Given that the threshold in the UCML procedure is 1 – 4/108 = 0.963, the test statistic LRUC,µ is set equal to zero, implying non-rejection of the null hypothesis.

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12.5

112 Unit Root Tests in Time Series

Table 3.3 ARMA models for UK car production, coefﬁcient estimates. coefﬁcient →

ϕ1

ρ

θ1

µt

CML ‘t’ statistic UCML ‘t’ statistic CML ‘t’ statistic UCML ‘t’ statistic

0.292 (1.51) 0.286 (1.78) 0.283 (1.03) 0.311 (1.67)

0.988 (56.16) 0.980 (50.25) 0.903 (11.91) 0.911 (14.56)

–0.665 (5.39) –0.677 (4.44) –0.613 (2.12) –0.643 (3.67)

µ µ β 0 + β1 t

Table 3.4 Test statistics for a unit root, car production data. ADF(2) tests

5%cv

τˆ µ –1.76 –2.88

5%cv

τˆ β –3.62 –3.44

CML tests

UCML tests

δˆ µ –7.78 –14.22 δˆ β

τµ

δµ

–0.59 –2.88

–1.26 –14.22

τ˜ µ –1.15 –2.67

τβ

δβ

LRC,µ 0.201 4.14

–27.94 –22.31

–1.28 –3.44

–10.51 –22.31

LRC,β 2.74 6.55

τ˜ β –1.42 –3.34

δ˜ µ –2.16 –12.67 δ˜ β –9.61 –19.93

LRUC,µ 0.00 1.86 LRUC,β 0.18 2.78

Notes: for critical values see Table 3.1. The critical values, as in Table 3.1, assume that ARMA and ADF orders are known; for the case where the order is not known, see Chapter 7.

The situation is more problematic in the case of detrended data. The LSbased tests indicate rejection at a nominal 5% level, whereas the ML-based tests do not; for example, LRUC,β = 0.18, which is not signiﬁcant. The difference is

in the estimates of ρ, which are: LS, ρˆ = 0.741, from ADF(2); CML, ρ = 0.903; UCML, ρ˜ = 0.911. Increasing the lag in the ADF(p) regression did not alter this difference. 3.10.2 The BN decomposition of the car production data This example is also used to illustrate the BN decomposition (see section 3.6) when a unit root has been imposed. The case considered is where drift is allowed. The general form, which is ARIMA(1, 1, 1), and estimated model are: (1 − ϕ1 L)∆yt = (1 − ϕ1 )µ + (1 + θ1 L)εt

ARIMA(1, 1, 1)

(1 − 0.312L)∆yt = 0.0020 + (1 − 0.703L)εt

estimated ARIMA(1, 1, 1)

∆yt = (1 − 0.312)−1 0.002 + (1 − 0.312L)−1 (1 − 0.703L)εt = 0.0029 + (1 − 0.312L)−1 (1 − 0.703L)εt

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β 0 + β1 t

An Introduction to ARMA Models

113

ˆ = 0.0029 is the estimated drift in the unit root process, not the The coefﬁcient µ mean of a stationary process. The idea is to decompose the series into a permanent component and a transitory component. Recall from section 3.6.2 that the underlying framework is: partition yt into permanent and temporary components

yPt = µ + yPt−1 + ψ(1)εt

yPt follows a random walk with drift ifµ = 0

ytr t = D(L)εt

transitory component

The quantities of interest are therefore ψ(L), ψ(1) and D(L), where: ψ(L) = ϕ(L)−1 θ(L) = (1 − 0.312L)−1 (1 − 0.703L) ψ(1) = (1 − 0.312)−1 (1 − 0.703) = 0.432 ∞

∞

(di − di−1 ) = ψi , di = − ∑j = i + 1 ψj for i = 1, . . . , ∞, d0 = − ∑i=1 ψi . In order to evaluate the (inﬁnite) sequence ψi note that this pattern is exactly as (3.32), hence: ψ(L) = 1 + (ϕ1 + θ1 )L + (ϕ12 + θ1 ϕ1 )L2 + . . . + (ϕ1i + θ1 ϕ1i−1 )Li + . . . = 1 + ψ1 L + ψ2 L2 + ψ3 L3 + . . . + ψi Li + . . . ∞

d0 = − ∑i=1 ψi = − [{(ϕ1 + θ1 )L + (ϕ12 + θ1 ϕ1 )L2 + . . . + (ϕ1i + θ1 ϕ1i−1 )Li + . . .] =−

(1 + θ1 ) ϕ (1 − ϕ1 ) 1

=−

(1 − 0.703) 0.312 = − 0.135 (1 − 0.312) ∞

di = − ∑j = i + 1 ψj =−

(1 + θ1 ) i + 1 ϕ 1 − ϕ1 1

(1 − 0.703) 0.312i + 1 (1 − 0.312) = − 0.432(0.312i + 1 ) =−

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yt = yPt + ytr t

Having obtained D(L), the transitory component is ytr t = D(L)εt , which, in principle, can be calculated directly using the residuals εˆ t for εt , and then the trend (permanent component) can be obtained by subtraction: yPt = yt – ytr t . However, note that D(L) is an inﬁnite-lag polynomial, whereas only a ﬁnite sequence of residuals εˆ t is available. In this case, the approximation error arising from (necessarily) truncating the residual sequence is slight; for example, the ﬁrst four terms of D(L) are –0.135, –0.042, –0.013, –0.004, but the truncation error, as noted in section 3.6.2.ii, however slight, can be avoided by using Newbold’s algorithm. For illustration, in terms of the equivalent representations, that is, yPt in (3.54) and ytr t in (3.63) (see also Q3.2), we have:

(1 + θ1 ) yt + λ(L)ϕ(1)−1 θ(1)−1 µ (1 − ϕ1 ) (1 + θ1 ) (θ1 + ϕ1 ) (1 − ϕ1 L) yt − µ = (1 + θ1 L) (1 − ϕ1 ) (1 + θ1 )(1 − ϕ1 )

p yt =

(1 − ϕ1 L) (1 + θ1 L)

Multiplying through by (1 + θ1 L) gives the recursive form: p yt =

p − θ 1 yt +

(1 + θ1 ) (θ1 + ϕ1 ) (1 − ϕ1 L)yt − µ (1 − ϕ1 ) (1 − ϕ1 )

= 0.703yPt−1 + 0.432(yt − 0.312yt−1 ) − (−0.703 + 0.312)(1 − 0.312)−1 0.002 = 0.703yPt−1 + 0.432(yt − 0.312yt−1 ) + 0.001 (1 + θ1 L)(1 − ϕ1 ) − (1 − ϕ1 L)(1 + θ1 ) (θ1 + ϕ1 ) tr yt = yt + µ (1 + θ1 L)(1 − ϕ1 ) (1 + θ1 )(1 − ϕ1 ) (θ1 + ϕ1 ) λ1 (1 − L) yt + µ = (1 + θ1 L)(1 − ϕ1 ) (1 + θ1 )(1 − ϕ1 ) where λ1 = −(ϕ1 + θ1 ). Multiplying through by (1 + θ1 L), simplifying and substituting for the numerical values of the coefﬁcients, gives: λ1 (θ1 + ϕ1 ) (1 − L)yt + µ (1 − ϕ1 ) (1 − ϕ1 ) (ϕ1 + θ1 ) (θ1 + ϕ1 ) ∆yt + µ = − θ1 ytr t−1 − (1 − ϕ1 ) (1 − ϕ1 )

ytr t =

− θ1 ytr t−1 +

tr ytr t = 0.703yt−1 + 0.568∆yt−1 − 0.001

Note that yPt + ytr t = yt , so that, in practice, it is only necessary to obtain one of yPt and ytr , the other being obtained by subtraction (but note that as the series t is modelled in the logs, this additive property will not transfer to the levels of the series). The estimated series for the (logs of) the permanent and transitory components, yPt and ytr t , respectively, are graphed in Figure 3.2 (left-hand scale P for yt , right-hand scale for ytr t ).

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115

0.1

permanent component

y pt

0.05 y tr t

12.5 0 transitory component

12 1975

1980

1985

1990

1995

2000

–0.05 2005

Figure 3.2 Permanent and transitory car production (logs).

3.11 Concluding remarks ARMA, or ARIMA, models are an important example of the class of linear models that is widely used in econometrics and statistics. It is often the default linear model that is used for illustration and empirical modeling. One of the aims of this chapter has been to establish a notation for this class of models, whilst recognising that there are alternative representations. One approach is to represent the model in its two distinct AR and MA components; the common notation in this case is to use φ(L) for the AR polynomial and θ(L) for the MA polynomial. However, it is sometimes convenient to write the model in its error dynamics form, which separates out the possibly nonstationary root from the φ(L) polynomial and places all further dynamics, whether due to AR or MA components, into an error. This is usually the more convenient form for unit root tests. This chapter has also introduced two lag polynomial decompositions that have been widely used for ARMA models. The BN decomposition arose in the context of decomposing a nonstationary series into a permanent component and a transitory component; however, it is used more widely than this and it is a useful part of the ‘toolkit’ for unit root testing. The ADF decomposition, which also serves to separate out the I(1) and I(0) components of a nonstationary series, is even more widely used, perhaps the most familiar context being the ADF regression that is the basis of many unit root tests. Given an ARMA model that might contain a unit root, perhaps the ﬁrst port of call in formulating a unit root test is to base such a test on maximum

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13

application of the BN decomposition

likelihood estimation, whether of the conditional or unconditional likelihood function. When the conditional likelihood function is used and there is no MA component, the resulting estimator is also the LS estimator; if there is an MA component, the corresponding LS-based estimator is a version of the GLS estimator, a connection that is explored further in Chapter 7. In the case of the unconditional likelihood function, Shin and Fuller (1998) suggested an approach that is easy to implement with econometric software that offers a (full) maximum likelihood procedure combined with a seasonal version of the familiar Box-Jenkins ARMA/ARIMA modelling suite. In all variations, a likelihood ˆ ratio test is an alternative to DF τˆ -type and δ-type tests; the latter, and some of their variations, are considered in detail in Chapter 6.

Questions Q3.1.i Consider equation (3.21), φ(L) = (1 − ρL)(1 − ϕ1 L + . . . − ϕp−1 Lp−1 ), and obtain the relationship between the coefﬁcients in φ(L) and ϕ(L). Q3.1.ii What are the implications of one or more unit roots in φ(L) for the sum φ(L)? A3.1.i The relationship between the coefﬁcients in φ(L) and ϕ(L) is given by expanding the right-hand side of (3.21) and equating coefﬁcients on like powers of L: φ1 = − (ρ + ϕ1 ) φ2 = (ρϕ1 − ϕ2 ) .. .

.. .

φj = (ρϕj−1 − ϕj ) .. .

.. .

φp−1 = (ρϕp−2 − ϕp−1 ) φp = ρϕp−1 A3.1.ii First consider: 1 − ∑j=1 φj Lj = (1 − ρL)(1 − ϕ1 L + . . . − ϕp−1 Lp−1 ) p

Then set L = 1, to obtain: 1 − ∑j=1 φj = (1 − ρ)(1 − ∑j=1 ϕj ) p

p−1

If there is a single unit root, then ρ = 1 and the right-hand side is zero, and p p hence φ(1) = 1 − ∑j=1 φj = 0 ⇒ ∑j=1 φj = 1. The right-hand-side polynomial could

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116 Unit Root Tests in Time Series

An Introduction to ARMA Models

117

be factored again to, say: 1 − ∑j=1 φj Lj = (1 − ρ1 L)(1 − ρ2 L)(1 − ψ1 L + . . . − ψp−2 Lp−2 ) p

p

and, following the same logic, if both ρ1 = 1 and ρ2 = 1 then ∑j=1 φj = 1. By extension, we conclude that the presence of one or more unit roots in φ(L) implies that φ(1) = 0.

Q3.2.ii Obtain the truncation error for θ1 = 0.9, with r = 5 and r = 4. A3.2.i Start from: ∞

1 + ∑i=1 (−θ1 )i = (1 + θ1 )−1 = 1 + (−θ1 ) + (−θ1 )2 + . . . + (−θ1 )r−1 + (−1)r (θ1 )r (1 + (−θ1 ) + (−θ1 )2 + ....) = 1 + (−θ1 ) + (−θ1 )2 + . . . + (−θ1 )r−1 + (−1)r (θ1 )r (1 + θ1 )−1 The remainder is therefore (−1)r (θ1 )r (1 + θ1 )−1 , which depends upon the sign of (−1)r , which is positive for r even and negative for r odd. Proportionately, the error is: (−1)r (θ1 )r . ∞ i i A3.2.ii For θ1 = 0.9, then 1 + ∑∞ i=1 (−θ1 ) = 1 + ∑i=1 (−0.9) = 1/(1 + 0.9) = 0.5263. For r = 5, the calculations are as follows:

= [1 − 0.9 + 0.92 + (−0.9)3 + 0.94 ] + [(−1)5 (0.9)5 (1 + 0.9)−1 ] = [0.8371] − [0.59049(0.5263)] = 0.5263 The proportionate error is (−1)5 (0.9)5 = –0.59049, that is –0.31078/0.5263. For r = 4, the calculations are as follows: = [1 − 0.9 + 0.92 + (−0.9)3 ] + [(−1)4 (0.9)4 (1 + 0.9)−1 ] = [0.181] + [0.6561(0.5263)] = 0.5263 The proportionate error is (−1)4 (0.9)4 = 0.6561; that is, 0.34530/0.5263. Q3.3 Obtain the recursive version of the BN decomposition for the example of section 3.6.2.iii. A3.3 The estimated model is AR(1, 1, 1), that is: (1 − ϕ1 L)(∆yt − µ) = (1 + θ1 L)εt

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Q3.2.i Consider the case where 0 < θ1 < 1, and obtain the approximation error i in truncating the inﬁnite sum 1 + ∑∞ i=1 (−θ1 ) to r – 1 terms.

118 Unit Root Tests in Time Series

which implies, on multiplying through by (1 + θ1 L)−1 ψ(1), that:

(1 − ϕ1 L) (1 + θ1 L)

(1 + θ1 ) (1 + θ1 ) ∆yt = µ + εt (1 − ϕ1 ) (1 − ϕ1 )

In general, the transitory component is given by: −1 −1 −1 −1 ytr t = λ(L)ϕ(1) θ(L) ∆yt − λ(L)ϕ(1) θ(L) µ

−1 −1 −1 −1 ytr t = λ0 (1 − ϕ1 ) (1 + θ1 L) ∆yt − λ0 (1 − ϕ1 ) (1 + θ1 ) µ

where λ0 is obtained from: (1 + θ1 L)(1 − ϕ1 ) − (1 − ϕ1 L)(1 + θ1 ) = λ0 (1 − L) ⇒ λ0 = − (θ1 + ϕ1 ) Substituting for λ0 gives: −1 −1 −1 −1 ytr t = − (θ1 + ϕ1 )(1 − ϕ1 ) (1 + θ1 L) ∆yt + (θ1 + ϕ1 )(1 − ϕ1 ) (1 + θ1 L) µ

Finally, multiplying through by θ(L) and rearranging gives: tr −1 −1 ytr t = − θ1 yt−1 − (θ1 + ϕ1 )(1 − ϕ1 ) ∆yt + (θ1 + ϕ1 )(1 − ϕ1 ) µ

The permanent component is obtained as a special case of: θ(1) yt + λ(L)ϕ(1)−1 θ(1)−1 µ ϕ(1) (1 + θ1 ) (1 − ϕ1 L) yt − (θ1 + ϕ1 )(1 − ϕ1 )−1 (1 + θ1 )−1 µ = (1 + θ1 L) (1 − ϕ1 )

p yt =

ϕ(L) θ(L)

Finally, multiplying through by θ(L) and rearranging gives: yt = − θ1 yt−1 + (1 + θ1 )−1 (1 − ϕ1 )−1 (1 − ϕ1 L)yt − (θ1 + ϕ1 )(1 − ϕ1 )−1 µ p

p

p

Note that yt + ytr t = yt as required. Q3.4 Using Newbold’s representation, obtain the BN decomposition for the AR(1, 1, 0) model, with µ = 0. A3.4 For convenience, the general case for ytr t is restated below: ˆ t + j + ϕ(1)−1 ∑j=1 ∑i = j ϕi xˆ t + (q−j + 1) ) ytr t = − (∑j=1 x q

p

p

where xt = ∆yt − µ and a ˆ above indicates an estimated value; then in the speciﬁc case to be considered, the ﬁrst component is zero (q = 0) and the second

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which specialises in this case to:

An Introduction to ARMA Models

119

component is: −1 ytr ∑j=1 ∑i=1 ϕi xˆ t + (1−j) t = − (1 − ϕ1 ) p

1

= − (1 − ϕ1 )−1 ϕ1 xˆ t

set p = 1

= − (1 − ϕ1 )−1 ϕ1 xt

using xˆ t = xt

for p = 1

Hence, the permanent component is: yPt = yt − ytr t = (1 − ϕ1 )−1 (1 − ϕ1 L)yt − (1 − ϕ1 )−1 ϕ1 µ Q3.5.i Conﬁrm the DF polynomial decomposition for p = 4. Q3.5.ii Given the following numerical values of the coefﬁcients from the DF decomposition: α = 0.9, c1 = –0.6, c2 = –0.4, c3 = –0.1. Recover the ai coefﬁcients. Q3.5.iii What are the roots of the A(L) polynomial given these coefﬁcients? A3.5.i A(L) = 1 – (a1 L + a2 L2 + a3 L3 + a4 L4 ); hence, α = a1 + a2 + a3 + a4 , c1 = − −(a2 + a3 + a4 ), c2 = –(a3 + a4 ) and c3 = –a4 . The DF decomposition is conﬁrmed as follows: A(L) = (1 − αL) − C(L)(1 − L) = [1 − (a1 + a2 + a3 + a4 )L]− [−(a2 + a3 + a4 )L − (a3 + a4 )L2 − a4 L3 )] − [(a2 + a3 + a4 )L2 + (a3 + a4 )L3 + a4 L4 )] = [1 − (a1 + a2 + a3 + a4 ) + (a2 + a3 + a4 )]L + [(a3 + a4 ) − (a2 + a3 + a4 )]L2 + [a4 − (a3 + a4 )]L3 − a4 L4 = 1 − (a1 L + a2 L2 + a3 L3 + a4 L4 ) as required. Further, it can be conﬁrmed that ci – ci−1 = ai , for i = 1, 2, as follows: c2 − c1 = − (a3 + a4 ) − [−(a2 + a3 + a4 )] = a2 c3 − c2 = − a4 − [−(a3 + a4 )] = a3 c3 = − a4

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= − (1 − ϕ1 )−1 ϕ1 (∆yt − µ) substituting for the deﬁnition of xt = − (1 − ϕ1 )−1 ϕ1 (1 − L)yt + (1 − ϕ1 )−1 ϕ1 µ

120 Unit Root Tests in Time Series

A3.5.ii Recover the ai coefﬁcients given α = 0.9, c1 = –0.6, c2 = –0.4, c3 = –0.1. Start with a4 = –c3 = 0.1; then a3 = c3 – c2 = –0.1 –(–0.4) = 0.3, a2 = c2 – c1 = –0.4 –(–0.6) = 0.2; ﬁnally a1 = α – (a2 + a3 + a4 ) = 0.9 – (0.2 + 0.3 + 0.1) = 0.3. A3.5.iii Find the roots of the A(z) polynomial, where:

Most computer routines (which would have to be used here) to obtain the roots normalise the coefﬁcient on the highest order of the polynomial (known then as a monic polynomial) and read in the coefﬁcients from the highest to the lowest order; thus, for example, in MATLAB, the polynomial is expressed as: z4 + 3z3 + 2z2 + 3z – 10, with coefﬁcients entered as the vector p = [1 3 2 3 –10], giving roots of [–3.0272, –0.5104 ± 1.7005i, 1.048]; the complex pair have an absolute value of 1.7754, so all roots have modulus outside the unit circle. Q3.6 Conﬁrm the general version of the DF decomposition of the lag polynop mial A(L) = 1 – ∑i=1 ai Li . A3.6 The proposition is: A(L) = 1 − ∑i=1 ai Li p

= (1 − αL) − C(L)(1 − L) = (1 − αL) − ∑i=1 ci Li (1 − L) m

deﬁne m = p − 1, so m + 1 = p

= 1 − αL − ∑i=1 ci Li + ∑i=1 ci Li + 1 m

m

= 1 − αL − (c1 L + c2 L2 + c3 L3 + . . . + cm Lm ) + (c1 L2 + c2 L3 + c3 L4 + . . . + cm Lm + 1 ) = 1 − (α + c1 )L − (c2 − c1 )L2 − (c3 − c2 )L3 − . . . − (cp−1 − cp−2 )Lp−1 + cp−1 Lp Equating coefﬁcients on like terms of Lj , we obtain: cp−1 = − ap cp−1 − cp−2 = ap−1 ⇒ cp−2 = − (ap−1 + ap ) cp−2 − cp−3 = ap−2 ⇒ cp−3 = − (ap−2 + ap−1 + ap ) .. .

.. .

.. .

c2 − c1 = a2 ⇒ c1 = − (a2 + . . . + ap−2 + ap−1 + ap ) (α + c1 ) = a1 ⇒ α = a1 + a2 + . . . + ap−2 + ap−1 + ap = A(1)

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1 − a1 z − a2 z2 − a3 z3 − a4 z4 = 1 − 0.3z − 0.2z2 − 0.3z3 − 0.1z4

An Introduction to ARMA Models

121

Q3.7.i Show that the following ARMA(0, 1) process does not satisfy the invertibility condition and obtain an alternative representation that does satisfy this condition. (yt − µ) = (1 − 4L)εt where εt ∼ (0, σε2 ) Q3.7.ii Assess the following ARMA(0, 2) model for invertibility:

A3.7.i By inspection, the root of the MA polynomial is δ1 = 1/4, which does not satisfy the condition that the roots lie outside the unit circle. However, we can write the model in terms of the inverse root as: (yt − µ) = (1 − (1/4)L)et where we take the properties of et to be determined. Note that the variance of εt is here denoted σε2 to distinguish it from the variance of et denoted σe2 . One way to do this is to note that γ(0) and γ(1), the only non-zero components of the autocovariance function are, in the case of the ﬁrst representation, σε2 (1 + θ21 ) and σε2 θ1 , respectively, and in the case of the second representation, σe2 [1 + (1/θ1 )2 ] and σε2 (1/θ1 )2 , respectively, where θ1 = –4.0. Writing σe2 = λσε2 , where λ is a constant to be determined, we ﬁnd that σε2 (1 + θ21 ) = λσε2 (1 + (1/θ1 )2 ) ⇒ λ = θ21 . So in this case, σe2 = 16σε2 . This is an example of a general result that 2 2 σe2 = (∏Rr=1 z−1 r ) σε , where R is the number of roots outside the invertibility condition (see Fuller, 1996, Theorem 2.6.4, but note the different convention on what are called the roots). A3.7.ii First note that the AR(0, 2) polynomial factors, so that: (yt − µ) = (1 − 4L)(1 − 0.5L)εt This MA polynomial has roots of δ1 = 1/4 and δ2 = 2 and appears not to be invertible because (1 − 4L)−1 is not well deﬁned. However, using the alternative representation, with the reciprocal of δ1 (see Fuller, 1996, Theorem 2.6.4), they are equivalent with et ∼ (0, (4σε )2 ). The alternative representation is: (yt − µ) = (1 − 0.25L)(1 − 0.5L)et = (1 − 0.75L + 0.125L2 )et The ﬁrst line implies the inversion to: (1 − 0.25L)−1 (1 − 0.5L)−1 (yt − µ) = et In the case of pure MA processes, the autocovariances are particularly simple to obtain from the following general representation (see, for example, Brockwell

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(yt − µ) = (1 − 4.5L + 2L2 )εt

122 Unit Root Tests in Time Series

and Davis, 1991, Theorem 3.2.1): γ(k) = σε2 ∑j=0 θj θj + |k q

for |k| ≤ q and γ(k) = 0 otherwise

Thus, in the case of θ1 = – 4.5, θ2 = 2.0, then: γ(0) = σε2 (1 + θ21 + θ22 ) = 22.25 γ(1) = σε2 (θ1 + θ1 θ2 ) = − 13.5

and, with θ∗1 = – 0.75, θ∗2 = 0.125: γ(0) = 16σε2 (1 + (θ∗1 )2 + (θ∗2 )2 ) = 22.25 γ(1) = 16σε2 (θ∗1 + θ∗1 θ∗2 ) = − 13.5 γ(2) = 16σε2 θ∗2 = 2.0 Thus, as expected, the autocovariance function is the same for both representations.

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γ(2) = σε2 θ2 = 2.0

4

Introduction The aim of this chapter is to introduce some of the problems that arise in estimating a simple AR model. At this stage an AR model is adequate to indicate the nature of the problems and because, even if an ARMA model is appropriate, an AR model of sufﬁcient order can, under some circumstances, be considered a reasonable approximation to the more general model. Indeed, at a pedagogical level, an AR(1) model can provide an adequate motivation for many of the problems that also occur for higher-order models. The problems considered in this chapter relate to estimation of the AR coefﬁcients and the use of conventional t statistics for inference. The latter naturally leads onto forming conﬁdence intervals for the coefﬁcients of interest, a topic that is continued in the next chapter. Inference, either through direct hypothesis testing or indirectly by means of conﬁdence interval construction, is fundamental to the question of assessing whether the process generating the data contains a unit root. The ﬁrst problem is that the LS and conditional maximum likelihood estimators in an AR model are biased in ﬁnite samples even when the sum of the AR coefﬁcients seems to be well inside the stationary region. This is clearly a concern in itself, especially as the bias is relatively quite large, with most of it being accounted for by terms of order T−1 . It is also of concern because other quantities of interest; for example, the long-run multiplier and forecasts, are quite sensitive nonlinear functions of the original parameters. The second problem is that of obtaining conﬁdence intervals of the key parameters with accurate coverage. For example, suppose in the simplest case of an AR(1) model, a unit root is suspected, but estimation delivers a point estimate below unity: does a, say, 90% conﬁdence interval include unity? Forming a two-sided conﬁdence interval in the usual way, assuming that the critical values

123

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Bias and Bias Reduction in AR Models

from the ‘t’ distribution are appropriate, will generally be invalid and substantially misleading. Interpreting the conﬁdence interval for the AR coefﬁcient, say φ1 , as the set of values for which the null hypothesis is not rejected (at, in this example, the 10% two-sided signiﬁcance level) indicates a further problem. There is a discontinuity (at least asymptotically) in the distribution of the LS estimator of φ1 at φ1 = 1; hence this must be reﬂected in the way that the conﬁdence interval is calculated if the two approaches are to be equivalent (see Chapter 5 for further discussion). This chapter ﬁrst considers, in section 4.1, analytical expressions for the ﬁnite sample bias of LS estimators of the coefﬁcients of an AR model; methods of bias reduction based on ﬁrst order or linear corrections are outlined in section 4.2, and these are followed up in section 4.3 with the use of recursively demeaned or detrended data and bootstrapping in section 4.4. Section 4.5 presents the results of a simulation study, and two empirical examples are used in section 4.6 to illustrate the methods. A word on notation at the beginning of this chapter: as noted in Chapter 3, especially section 3.8, the ARMA model can be written equivalently in several different ways. In testing for a unit root, the literature now tends to favour either the error dynamics or common factor approach, which separates out the root that is potentially unity. On the other hand, in this and the next chapter, the focus is on estimation and inference in AR models, for which the more convenient notation is the standard form φ(L)(yt −µt ) = εt , εt ∼ iid(0, σε2 ), which is a variant of equation (3.8) allowing for a deterministic mean or trend, µt .

4.1 Finite sample bias of the LS estimator 4.1.1 The bias to order T Consistency refers to an asymptotic property of an estimator; it does not imply that there is no ﬁnite sample bias. A general result is that if an estimator, say ˆ of ψ, is root-T consistent and asymptotically normal, then the bias of ψ ˆ will ψ generally be O(T−1 ). (Generally refers to the need for a bounded from above ˆ see MacKinnon and Smith, 1998.) If the ﬁrstsecond moment condition on ψ; ˆ then the bias of the resulting estimator will order bias can be removed from ψ, be of smaller order than T; hence it is often of practical use to consider obtaining a ﬁrst order bias adjusted estimator. However, a cost of bias reduction is usually an increase in the variance, so a root mean squared error criterion is usually relevant in evaluating competing estimators. The presence of bias also causes a problem for inference. For example, in using a conventional ‘t’ statistic for inference, we would need to be sure that a differˆ – ψ0 , where ψ0 is the hypothesised value of ψ ence in the numerator, say ψ ˆ is not consistent under the null, has arisen because the particular estimate ψ ˆ with a true value of ψ0 , rather than because ψ is a biased estimate of ψ resulting

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124 Unit Root Tests in Time Series

ˆ – ψ0 . The central concern of this chapter is in methods in a ‘large’ difference ψ of bias adjustment; problems arising from inference are considered further in the next chapter. The LS estimator of φ1 in an AR(1) model is biased in ﬁnite samples, with the extent of the bias depending on φ1 , increasing as |φ1 | → 1 and particularly as φ1 → 1, and decreasing with T. The bias is conveniently represented as an explicit term for the ﬁrst-order bias plus a remainder due to higher-order terms of smaller order. The magnitude of the bias depends upon the deterministic speciﬁcation in the AR model. Three cases of relevance are distinguished. The ﬁrst is when the mean of the process is assumed constant and known, a special case is the zero mean; the second is when the mean is constant but unknown; and the third case is when there is an unknown linear trend. In the AR(1) model, the bias in these three cases is: E(φˆ 1 ) − φ1 = − 2φ1 /T + o(T−1 )

(4.1)

E(φˆ 1 ) − φ1 = − (1 + 3φ1 )/T + o(T−1 ) E(φˆ 1 ) − φ1 = − (2 + 4φ1 )/T + o(T

−1

)

(4.2) (4.3)

This bias can be important for moderately sized samples, especially so as φ1 approaches 1. For example, if φ1 = 0.95 and T = 100 then, in the more usual case when the mean is unknown, E(φˆ 1 ) = 0.9115 + o(T−1 ) and the ﬁrst-order bias, is 0.0385 or 4.05% of φ1 . Whilst this may not seem large, it has implications for other quantities of interest. For example, a standard persistence measure, the long-run multiplier, is Λ ≡ φ(1)−1 = (1 − φ1 )−1 = 20 for φ1 = 0.95, but only 11.3 for φˆ 1 = 0.95 – 0.0385 = 0.9115; also σy2 = σε2 /(1 − φ21 )−1 = 10.256, assuming σε2 = 1, whereas using φˆ 1 = 0.9115 gives σˆ y2 = 5.911. Although illustrative, these calculations show that the LS estimates can be seriously misleading as to key parameters of interest and this is likely to be an important factor for typical economic time series, for which φ1 tends to be close to unity. Notice from (4.2) that there is a ‘ﬁxed’ point of no ﬁrst-order bias, this is where (1 + 3φ1 ) = 0 implying that only at φ1 = –1/3 is there no ﬁrst-order bias. If a linear time trend is included in the AR(1) regression, the ﬁrst-order bias is −(2 + 4φ1 )/T, with a ﬁxed point at φ1 = –0.5. The total ﬁrst-order bias is deﬁned as the bias in estimating φ by the sum p of the LS estimators φˆ ≡ ∑i=1 φˆ i ; this is usually the key parameter of interest, as it indicates how close the estimates are to the unit root and is the basis of the widely used persistence measure Λ. The ﬁrst-order bias for the individual coefﬁcients in AR(1) – AR(4) models is given in Table 4.1 and the total ﬁrstorder bias is given in Table 4.2, together with the coefﬁcient values that result in no ﬁrst-order bias. For both tables we distinguish whether a linear time trend is included in the AR(p) regression. The underlying details of how these biases are derived are dealt with in the appendix to this chapter.

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Bias and Bias Reduction in AR Models 125

126 Unit Root Tests in Time Series

Table 4.1 First-order least squares bias: limT→∞ T[E(φˆ i − φi )]. Table 4.1a Constant included in regression AR(p) φ1

φ2

φ3

φ4

AR(1) −(1 + 3φ1 ) −(2 + 4φ2 )

AR(3) −(1 + φ1 + 2φ3 ) −(2 − φ1 + 4φ2 + φ3 )

−(1 + 5φ3 )

AR(4) −(1 + φ1 + φ4 )

−(1 − 2φ1 + 5φ3 + φ4 )

−(2 + 6φ4 )

φ3

φ4

−(2 − φ1 + 2φ2 + φ3 + 2φ4 )

Table 4.1b Constant and trend included in regression AR(p) φ1

φ2

AR(1) −(2 + 4φ1 ) AR(2) −(2 + φ1 + 2φ2 ) −(2 + 5φ2 ) AR(3) −(2 + φ1 + 3φ3 ) −(3 − 2φ1 + 5φ2 + 2φ3 )

−(2 + 6φ3 )

AR(4) −(2 + φ1 + 2φ4 ) −(3 − 2φ1 + 2φ2 + 2φ3 + 3φ4 )

−(2 − 3φ1 + 6φ3 + 2φ4 )

−(3 + 7φ4 )

Sources: Table 4.1a based on Shaman and Stine (1988); Table 4.1b own calculations; Cordeiro and Klein (1994) give some bias expressions for low-order autoregressive moving average (ARMA) models; see appendix to this chapter for details on derivations.

Table 4.2 Total bias of LS estimators and ﬁxed points. Table 4.2a Constant included in regression p

Total ﬁrst-order bias in ∑i=1 φˆ i AR(1) AR(2) AR(3) AR(4)

−(1 + 3φ1 ) −(3 + φ1 + 5φ2 ) −(4 + 4φ2 + 8φ3 ) −(6 − 2φ1 + 2φ2 + 6φ3 + 10φ4 )

Zero ﬁrst-order bias coefﬁcients φ1 = –1 3 φ1 = –0.5, φ2 = –0.5 φ1 = –0.6, φ2 = –0.6, φ3 = –0.2 φ1 = –2 3, φ2 = –0.8, φ3 = –0.4, φ4 = –1 3

Table 4.2b Constant and trend included in regression Total ﬁrst-order bias in p ∑i=1 φˆ i

Zero ﬁrst-order bias coefﬁcients

Roots at zero ﬁrst-order bias; (|roots|)

AR(1) AR(2)

−(2 + 4φ1 ) −(4 + φ1 + 7φ2 )

φ1 = −0.5 φ1 = –1.2, φ2 = –0.4

AR(3)

−(7 − φ1 + 5φ2 + 11φ3 )

φ1 = –1.0, φ2 = –13 15, φ3 = –1 3

–2 −1.5 ± 0.5i (1.581) –1.855 −0.373 ± 1.216i (1.855, 1.272)

AR(4)

−(10 − 4φ1 + 2φ2 + 8φ3 + 14φ4 )

φ1 = –8 7, φ2 = –26 42, φ3 = –16 21, φ4 = –3 7

0.284 ± 1.196 − 1.173 ± 0.411i(1.229, 1.243)

Note: Obtaining the roots for the coefﬁcients in Table 4.2a is left as an exercise; see Q4.4 at the end of this chapter.

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AR(2) −(1 + φ1 + φ2 )

Bias and Bias Reduction in AR Models 127

To illustrate, with an unknown constant in the regression, the total ﬁrst-order bias in an AR(2) model estimated by LS is −(3 + φ1 + 5φ2 )/T; and with, for example, φ1 = 1.25 and φ2 = –0.35, then φ = 0.9 and Λ = 10, and the total ﬁrst-order bias is –0.05 for T = 50 and –0.025 for T = 100. With this bias, the estimates of φ and Λ are 0.85 and 6.66, respectively, for T = 50, and 0.875 and 8, respectively, for T = 100.

In this section we consider the impact of ﬁrst-order bias on the long-run multiplier Λ(∞) (see Chapter 3, section 3.1.2). As there is no ambiguity of reference ˆ = φ(1) ˆ −1 , where φ(1) ˆ ˆ Λ(∞) is referred to here as Λ, with its LS estimator Λ = 1 − φ, ˆφ ≡ ∑p φˆ j (and φ ≡ ∑p φj ). For simplicity, assume that there is only ﬁrst-order j=1 j=1 ˆ = φ + b1 , where b1 is the sum of ﬁrst-order biases in estimating bias, then E(φ) each φi (that is, the ﬁrst column in Table 4.2 divided by T). Evaluating Λ at the ¯ˆ = [1 − E(φ)] ˆ say Λ ˆ −1 , we obtain: expected value of φ, ¯ˆ = Λ =

1 (1 − φ − b1 ) 1 b1 + (1 − φ) (1 − φ)2 − b1 (1 − φ)

(4.4)

Therefore: ¯ˆ − Λ = Λ

b1 2 (1 − φ) − b

1 (1 − φ)

(4.5)

The implied error in estimating the long-run multiplier at the expected value of φˆ is very sensitive to (1 – φ), especially through (1 − φ)−2 for φ ‘close’ to 1. For ¯ˆ = 10, then with example, suppose in an AR(1) model φ = φ1 = 0.9, implying Λ ¯ ˆ − Λ = –0.037/(0.01 + 0.0037) = –2.70, that T = 100 there is a 3.7% bias in φ, but Λ is a –27% error in estimating Λ. ¯ˆ − Λ is sensible because it mirrors Evaluating the error in estimating Λ as Λ what is done in practice; that is obtain an estimate of φ and substitute that estimate into the expression for the calculation of Λ. The error could also be ˆ −1 ≡ (1 − φ) ˆ −1 , where the bias is evaluated by the bias in estimating Λ by Λ ¯ ˆ ˆ ˆ ˆ −1 because E(Λ) – Λ, rather than evaluating Λ − Λ. In general, E(Λ) = [1 − E(φ)] ˆ ˆ ˆ ˆ Λ is a nonlinear function of φ, say f(φ). Provided f(φ) is a convex function then, ˆ = E[(1 − (φ))] ˆ −1 ≥ from Jensen’s inequality (see, for example, Rao, 1973), E[f(φ)] ¯ −1 ˆ closer to Λ than Λ ˆ is to Λ; however, ˆ = [1 − E(φ)] ˆ f[E(φ)] . This would place E(Λ) ˆ is not everywhere guaranteed to be a convex function, especially close to f(φ) ˆ with respect to φˆ is 2(1 − φ) ˆ −3 the unit boundary. The second derivative of f(φ) and convexity requires φˆ < 1 (which implies a positive second derivative; see, for example, Sydsaeter and Hammond, 2008). Values of φˆ > 1 are possible even if φ < 1, although a negative ﬁrst-order bias b1 will counteract this; and when φˆ ˆ = f(φ) ˆ switches from a large positive value to a large crosses the unit boundary Λ negative value. (Yule-Walker estimation ensures that φˆ ≤ 1, but this method exhibits larger bias; see Shaman and Stine, 1988.)

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4.1.2 First-order bias and the long-run multiplier

This discussion shows that even if φˆ is unbiased, the LS estimator of the ˆ does not have that property because E[(1 − (φ))] ˆ −1 ≥ long-run multiplier, Λ, −1 ˆ [1 − E(φ)] ). Obtaining an unbiased estimator of Λ is not, therefore, just a matter of replacing the biased estimator φˆ with an unbiased estimator (where one is available). This is a problem in unbiased ratio estimation; a possible solution to this problem could be developed along the lines suggested by MacKinnon and Smith (1998), who suggested simulating the bias function and then adjusting the initial estimate. We consider this method further in section 4.4.2 in the context of linear bias correction, but the extension to nonlinear bias correction offers no new principles. 2 , The long-run multiplier is also closely related to the ‘long-run’ variance σy,lr which is a key parameter in adjusting some unit root test statistics for weakly 2 = Λ2 σ2 ; dependent errors. From Chapter 3, section 3.1.2, we have that σy,lr ε hence, a bias in estimating Λ is transmitted to the corresponding estimate 2 . of σy,lr

4.2 Bias reduction In the case of bias reduction, there are three practical questions of interest: 1. Can the bias in LS (and ML) estimation be reduced? 2. Does bias reduction come at the cost of increasing the variance? 3. If so, what is the effect on the mean squared error of the estimators? We set the general framework for bias reduction in this section, describing four related methods that are easy to implement. These are: bias reduction resulting from removing the ﬁrst-order bias; removing the linear bias; basing the estimates of the deterministic components on a recursive sample; and bootstrap bias adjustment. The next chapter extends the discussion to consider median, rather than mean, unbiased estimation. A summary of the terms used in the following sections is given in Table 4.3.

Table 4.3 A summary of the abbreviations used in the following sections. BCE FBCE FOBCE LBCE CBCE FLBCE RMALS BSBCE

Bias corrected estimator Feasible bias corrected estimator First-order bias corrected estimator Linear bias corrected estimator Constant bias corrected estimator Feasible linear bias corrected estimator Recursive mean adjusted least squares Bootstrap bias corrected estimator

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128 Unit Root Tests in Time Series

Bias and Bias Reduction in AR Models 129

4.2.1 Total bias and ﬁrst-order bias 4.2.1.i Total bias The total bias function for the scalar parameter φ and its corresponding LS ˆ is deﬁned by: estimator (or other consistent estimator) φ, ˆ −φ b(φ, T) ≡ E(φ)

(4.6)

Then if b(φ, T) is known, the revised bias corrected estimator given by: (4.7)

This correction ensures that φ˜ is unbiased. In the AR(1) case, typically φ1 > 0 and b(φ1 , T) < 0, therefore φ˜ 1 > φˆ 1 . Bias correction seeks to estimate the bias function, b(φ, T), with an estimator of this function evaluated at the unbiased ˜ T). This gives rise to an expression of the form φ˜ = φˆ – estimator denoted b(φ, ˜ T), which can be solved for φ. ˜ b(φ, 4.2.1.ii First-order bias A slightly lesser goal is to estimate the ﬁrst-order bias and obtain a ﬁrst-order unbiased estimator. To develop this approach, we split the bias function into two components; one that depends on terms in T−1 , b1 (φ, T), and the remainder that involves higher-order terms, br (φ, T), which are necessarily o(T). Thus: b(φ, T) = b(1) (φ, T) + b(r) (φ, T)

(4.8)

For example, in the AR(1) case: b(1) (φ1 , T) = − (1 + 3φ1 )/T

(4.9)

and b(r) (φ, T) = o(T). A ﬁrst-order bias corrected estimator (FOBC) is then obtained by removing the ﬁrst-order bias from the LS estimator: φ˜ (1) = φˆ − b(1) (φ, T)

(4.10)

A feasible version of this replaces b(1) (φ, T) by b(1) (φ˜ (1) , T), with superﬂuous parentheses on the subscript indicating a bias corrected estimator; an example of this method is given in the next section. Also in the next section, and the appendix to this chapter, we show that analytical expressions can be obtained for AR(p) models, which give the ﬁrst-order bias corrected estimator entirely in terms of the LS coefﬁcients. (The method can also be extended to ARMA models by analogy, using the bias expressions in Cordeiro and Klein, 1994.) 4.2.2 First-order unbiased estimators in the AR model In the AR(1) case, b(1) (φ1 , T) = −(1 + 3φ1 )/T (see Equation (4.2)) and therefore one possibility would be to estimate the ﬁrst-order bias function b(1) (φ1 , T) using φˆ 1 . However, whilst using φˆ 1 will reduce the bias, the bias will also be

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φ˜ = φˆ − b(φ, T)

130 Unit Root Tests in Time Series

underestimated because φˆ 1 tends to underestimate φ1 . This is not, therefore, the way to proceed. What is needed, as indicated in the previous section, is an evaluation of the bias at φ˜ 1(1) , where the notation φ˜ 1(1) indicates that this is the LS estimate φˆ 1 corrected for ﬁrst-order bias, hence the second subscript. To continue the example, the feasible ﬁrst-order bias corrected estimator is obtained from: φ˜ 1(1) = φˆ 1 − b(1) (φ˜ 1(1) , T) (4.11)

Solving for φ˜ 1(1) we obtain: φ˜ 1(1) =

1 T ˆ + φ (T − 3) (T − 3) 1

(4.12)

The solution was ﬁrst given by Orcutt and Winokur (1969). The variance of φ˜ 1(1) is simple enough to derive, but we postpone consideration of this until section 4.2.8, where it is put in a general context. The adjusted estimator φ˜ 1(1) has no ﬁrst-order bias, the remaining bias terms being o(T−1 ) by deﬁnition. The estimator is simple to implement being just a function of the LS estimator. Expressions for ﬁrst-order bias reduction in AR(p) models, p > 1, are more complex and are dealt with in the appendix to this chapter. To have conﬁdence in the method of ﬁrst-order bias correction, we need to consider whether ﬁrst-order bias accounts for most of the total bias. The simulated ﬁrst-order bias functions for the AR(1) model are plotted in Figure 4.1 for T = 100; the upper panel shows the case with a constant included in the regression and with-trend case is shown in the lower panel. The graphs are shown for values of φ1 likely to be of economic relevance. In both cases, the bias function from LS estimation of φ1 is also shown; for reference, this is also the total bias. The bias functions were obtained from a simulation experiment with 50,000 replications of the DGP yt = φ1 yt + εt , with εt ∼ niid(0, 1). These ﬁgures also include bias functions for other methods that are considered later in this chapter and which should be ignored for now. There are two key points to note from these ﬁgures: (1)

(2)

For most of the parameter space, the ﬁrst-order bias and the total bias are virtually identical, deviating only as φ1 approaches the (positive) unit root, (φ1 → 1). When a trend is included in the estimated regression, the total bias increases and the dip in the total bias function occurs slightly earlier than in the no trend case. However, generally, higher-order biases are small relative to ﬁrst-order bias. The FOBC estimator removes most of the bias, only failing to be so effective as φ1 → 1; even then the gain in bias reduction is substantial. A more detailed comparison is considered later.

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= φˆ 1 − (1 + 3φ˜ 1(1) )/T

Bias and Bias Reduction in AR Models 131

0

BSBC FOBC

bias adjusted methods

bias

–0.02

RMALS first-order bias LS

–0.04 No trend –0.06 0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0.05

RMALS

0 bias

FOBC

–0.05 With trend

–0.1

BSBC first-order bias

LS

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

φ1 Figure 4.1 Bias functions of estimators.

4.2.3 Simulating the bias MacKinnon and Smith (1998) have suggested an approach to bias reduction based on simulating the bias function, or sufﬁcient points on it to obtain the information required. As noted above, if the bias function b(φ, T) is known, then the bias corrected estimator, BCE, is simply φˆ – b(φ, T). However, b(φ, T) is generally not known, hence a method is sought to obtain a feasible bias corrected estimator (FBCE), which is just the LS estimator minus the bias, φ = φˆ – b( φ , T), where the bias function is evaluated at φ ; since φ appears on the left- and righthand sides of this expression, the FBCE has to be solved out. As we shall see, this is particularly simple in the linear case. At this point it is worth noting that there is a difference between the approach outlined in this section and the approximation involved in the previous section. In the latter we considered only terms of order T−1 in the bias function, with terms of higher order than T−1 considered negligible. MacKinnon and Smith (1998) consider a linear bias corrected (LBC) estimator, which takes a linear approximation to the bias function, thus the terms ignored in this approximation are second and higher powers of φ.

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φ1

132 Unit Root Tests in Time Series

4.2.3.i Illustration of linear bias correction using the AR(1) model To illustrate, consider the LBC estimator in an AR(1) model. The bias function for given T and AR(1) parameter φ1 is written simply as b(φ1 ). A ﬁrst-order Taylor series expansion of b(φ) about the LS estimator φˆ 1 is: b(φ1 ) = b(φˆ 1 ) + bf (φˆ 1 )(φ1 − φˆ 1 ) + R

Where b(φˆ 1 ) = b(φ1 = φˆ 1 ), bf (φˆ 1 ) ≡ ∂ b(φ1 = φˆ 1 )/∂ φ1 and R is the Taylor series remainder, which is assumed to be negligible. Now re-express the bias function so that the constant is separated, for given φˆ (and T), from the slope, which varies with φ1 : b(φ1 ) = [b(φˆ 1 ) − bf (φˆ 1 )φˆ 1 ] + bf (φˆ 1 )φ1

(4.14a)

= α + βφ1

(4.14b)

α = [b(φˆ 1 ) − bf (φˆ 1 )φˆ 1 ]

(4.15a)

β = bf (φˆ 1 )

(4.15b)

where:

The LBC estimator is obtained as the solution to φ 1 = φˆ 1 – b( φ 1 ). For b( φ 1 ) use

(4.14), but with φ1 replaced by φ. The solution is obtained as follows:

φ 1 = φˆ 1 − b( φ 1 )

= φˆ 1 − (α + β φ 1 ) =

φˆ 1 − α 1+β

=−

1 ˆ α + φ 1+β 1+β 1

(4.16)

The variance of φ 1 , with α and β known, is simply:

var( φ 1 ) =

1 var(φˆ 1 ) > var(φˆ 1 ) (1 + β)

for β < 0

(4.17)

The condition β < 0 is satisﬁed for most of the parameter space as the derivative ˆ ∂ b(φ1 = φˆ 1 )/∂ φ1 , is negative for all but of the bias function with respect to φ,

φ1 → − 1. However, both the estimator φ 1 and its variance are not yet feasible because, although φˆ is known, α and β are not. The feasible solution requires operational versions of α and β. From (4.14) note that if the bias function was known it, and its derivative, would need to be ˆ However, as the bias function will not generally be known, and evaluated at φ. in the case that the bias function is linear, one solution is to simulate the (linear) bias function b(φ1 ) = α + βφ1 for two distinct points in the parameter space

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(4.13)

Bias and Bias Reduction in AR Models 133

ˆ whereas a possible choice of φ1 . One of these could be the LS estimate φ1 = φ, ˜ for the other is φ1 = φ1(1) , since the latter is likely to be close to φ1 and can be obtained without simulation. Alternatively, MacKinnon and Smith (1998) suggested that the other point could be the constant bias corrected (CBC) estimate. This requires a slight but useful detour given its similarity to bootstrapping, which is introduced in section 4.5 and used in other chapters.

The CBC estimator is based on the idea that the bias can be approximated by a constant invariant to φ1 ; in effect this approach assumes that the bias function is horizontal with respect to φ1 . It is obtained as follows. Assume that the bias function is just a constant, denoted b, (given T), say: b(φ1 ) = E(φˆ 1 ) − φ1 =b

(4.18)

The CBC estimate, if b is known, is simply: = φˆ 1 − b φ˜ CBC 1

(4.19)

To estimate b, set the parameters of an AR(1) DGP equal to their LS estimates ˆ ∗ (the constant), and then simulate this DGP with say N draws of εt φˆ and µ from a standard normal distribution and estimate an AR(1) model by LS for each draw. Denote the resulting estimates by φˆ 1,j for j = 1, ..., N. The mean of these estimates is: N φ¯ˆ 1 = ∑j=1 φˆ 1,j /N

(4.20)

This is the simulation estimate of E(φˆ 1 ) and the consequent estimate of bias is bˆ = φ¯ˆ 1 – φˆ 1 . Hence the feasible CBC estimate is: = φˆ 1 − (φ¯ˆ 1 − φˆ 1 ) φ˜ CBC 1 = 2φˆ 1 − φ¯ˆ 1

(4.21)

The similarity of the CBC with the bootstrap bias corrected estimator will become apparent later in section 4.4. 4.2.5 Obtaining a linear bias corrected estimator Returning to the LBC estimate, but with two points in the parameter space of φ1 selected, the task is to estimate α and β. We assume that the chosen points are φˆ and φ˜ CBC (although, as suggested above, φ˜ 1(1) could be used in place of φ˜ CBC ). 1 1 We already know that the estimate of bias associated with φˆ is bˆ = φ¯ˆ 1 − φˆ 1 , so one pair of coordinates of the bias function has been estimated.

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4.2.4 Constant bias correction

134 Unit Root Tests in Time Series

To obtain the second pair of coordinates, run through the same simulation set-up using the same sequence of random numbers in the draws, but with φ˜ CBC 1 (or other estimate) in the DGP. Denote the resulting estimates φ˜ 1,j for j = 1, ..., ˜ N, and the mean of these estimates as φ¯˜ 1 = ∑N j=1 φ1,j /N, with the simulation esti¯ CBC ˆ φ) ˜ φ˜ CBC ) ˆ and (b, mate of bias now b˜ = φ˜ − φ˜ . This gives the coordinates (b, 1

1

to solve for α and β from bˆ = αˆ + βˆ φˆ 1 and b˜ = αˆ + βˆ φ˜ CBC . The solutions are: 1 ˜ (bˆ − b) CBC ˆ ˜ (φ1 − φ1 )

(4.22a)

αˆ = bˆ − βˆ φˆ 1

(4.22b)

These estimates are now used in (4.16) to provide a feasible LBC estimator. 4.2.6 Linear bias correction in AR(p) models, p ≥ 2 The expressions in Table 4.1 showed that in higher-order AR models, the general situation is that the bias of the LS estimators of one of the AR coefﬁcients depends upon the other AR coefﬁcients in the model. Thus bias correction should not be done one at a time on the AR coefﬁcients. However, this complication apart, the general principle is just as in the case of a single coefﬁcient. To analyse the general AR model, deﬁne the vector of AR coefﬁcients ˆ = (φˆ 1 , . . . , φˆ p ) Φ = (φ1 , . . . , φp ) and the vector of LS and LBC estimators by Φ LBC

LBC

LBC

and Φ = ( φ 1 , . . . , φ p ) , respectively. Correspondingly, deﬁne the vector ˆ as b(Φ) ˆ = [b1 (Φ), ˆ . . . , bp (Φ)] ˆ , where bi (Φ) ˆ is the bias bias function evaluated at Φ function for the i-th AR coefﬁcient. ˆ is: The ﬁrst-order Taylor series expansion of the bias function, evaluated at Φ, ˆ + Bf (Φ)(Φ ˆ ˆ +R b(Φ) = b(Φ) − Φ)

(4.23a)

ˆ Φ ˆ + Bf (Φ)Φ ˆ +R ˆ − Bf (Φ) = b(Φ)

(4.23b)

ˆ ≡ [∂ bˆ i (Φ = Φ)/ ˆ ∂ φj ], for i, j = 1, ..., p, which is the p × p matrix of Where Bf (Φ) ˆ ﬁrst-order derivatives of the bias vector with respect to φj , evaluated at Φ = Φ, ˆ ∂ φj , j = 1, ..., p; and R is the Taylor series with i-th row given by ∂ bˆ i (Φ = Φ)/ remainder, again assumed neglible in deriving the LBC estimator. LBC

The LBC estimator is the solution to Φ LBC

unknown and is replaced by b(Φ LBC

further notation, Φ LBC

Φ

ˆ – b(Φ); however, b(Φ) is =Φ

). On this understanding, and to avoid

is the solution to: LBC

ˆ − [b(Φ) ˆ − Bf (Φ) ˆ Φ ˆ + Bf (Φ) ˆ Φ =Φ

]

ˆ −1 b(Φ) ˆ ˆ − [I + Bf (Φ)] =Φ

(4.24a) (4.24b)

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βˆ =

1

Bias and Bias Reduction in AR Models 135

with variance-covariance matrix as follows: LBC

var(Φ

ˆ −1 var(Φ)[(I ˆ ˆ ]−1 + o(T−2 ) ) = [I + Bf (Φ)] + Bf (Φ))

(4.25)

FLBC

Φ

ˆ Φ) ˆ ˆ − [I + Bˆ f (Φ)] ˆ −1 b( =Φ

(4.26)

with variance-covariance matrix as follows: LBC

var(Φ

ˆ −1 var(Φ)[(I ˆ ˆ ]−1 + o(T−2 ) ) = [I + Bˆ f (Φ)] + Bˆ f (Φ))

(4.27)

4.2.7 The connection We can use the MacKinnon and Smith (1998) approach to motivate and derive the ﬁrst-order biased corrected estimator in (4.10). The approximation in the case of the FOBC estimator involves considering only terms of order T; terms of higher order (whether or not they are linear functions of Φ) are not included. ˆ T) = b(1) (Φ, ˆ T) + o(T−1 ) where, for example, in the AR(1) The bias function is b(Φ, case, b(1) (φˆ 1 , T) = −(1 + 3φ1 )/T. Denote the ﬁrst-order LS bias function for the iˆ Now, in the Taylor series expansion of (4.23) replace th coefﬁcient by bi(1) (Φ). f ˆ ˆ ˆ b(Φ) and B (Φ) = [∂ bi (Φ)/∂ φj ], respectively, by: ˆ = [b1(1) (Φ), ˆ . . . , bp(1) (Φ)] ˆ b(1) (Φ)

and

ˆ = [∂ bi(1) (Φ)/ ˆ ∂ φj ] Bf(1) (Φ) The resulting differences go into a revised remainder RR, and thus (4.23b) becomes: ˆ + Bf (Φ) ˆ (Φ − Φ) ˆ + RR b(Φ) = b(1) (Φ) (1)

(4.28) LBC

Following the same procedure used to derive Φ obtained as:

, the FOBC estimator is

ˆ − [I + Bf (Φ) ˆ ]−1 b(1) (Φ) ˆ Φ(1) = Φ (1)

where Φ(1) = ( φ 1(1)

φ 2(1)

...

(4.29)

φ p(1) ) ; some examples follow.

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ˆ and B(Φ), ˆ To complete the solution it is necessary to obtain estimates of b(Φ) f ˆ and hence B (Φ). MacKinnon and Smith (1998) suggest obtaining these by simulation which, although slightly more complex in practice, is no different in principle from the illustrative AR(1) case. Denoting the simulation estimates as ˆ Φ) ˆ and Bˆ f (Φ), ˆ respectively, then the feasible version of the LBC estimator is: b(

136 Unit Root Tests in Time Series

Example: AR(1) model, constant, no trend In the AR(1) model, the AR coefﬁcient is φ1 with LS estimator φˆ 1 and the ﬁrstorder bias function evaluated at φˆ 1 , is given by b(1) (φˆ 1 ) = −(1 + 3φˆ 1 )/T; the ˆ = ∂ b(1) (φˆ 1 )/∂ φ1 = −3/T. Hence derivative of the latter evaluated at φˆ 1 is Bf(1) (Φ) (4.29) is:

φ 1(1) = φˆ 1 − [1 − 3/T]−1 [−(1 + 3φˆ 1 )/T] 1 T ˆ + φ (T − 3) (T − 3) 1

(4.30)

This is, of course, the same as (4.12). In other cases the derivatives are easy to obtain; for example, they can be obtained using Table 4.1 for AR(1) to AR(4) models. Example: AR(2) model, constant, no trend ˆ = (b1(1) (Φ), ˆ Using the results in Table 4.1, the ﬁrst-order bias function is b(1) (Φ) ˆ , where b1(1) (Φ) ˆ = −(1 + φˆ 1 + φˆ 2 )/T and b2(1) (Φ) ˆ = −(2 + 4φˆ 2 )/T. The b2(1) (Φ)) ˆ is: required matrix of derivatives Bf(1) (Φ) ˆ = Bf(1) (Φ)

∂ b1 /∂ φ1 ∂ b2 /∂ φ1

∂ b1 /∂ φ2 ∂ b2 /∂ φ2

=

−1/T 0

−1/T −4/T

(4.31)

ˆ into Φ(1) = Φ ˆ −[I + Bf (Φ)] ˆ −1 b(1) (Φ), ˆ we obtain: Substituting into B(1) (Φ) (1) 

 φ 1 −1 (T − 1) φˆ 1  1(1)  = + T 0 φˆ 2 φ 2(1)

=

φˆ 1 φˆ 2

+

1 (T − 4)(T − 1)

−1 (T − 4)

(T − 4) 0

−1

1 + φˆ 1 + φˆ 2 2 + 4φˆ 2

1 (T − 1)

1 T

(4.32) 1 + φˆ 1 + φˆ 2 2 + 4φˆ 2 (4.33)

Collecting terms, the FOBC estimators are:

φ 1(1) =

φ 2(1) =

1 T−1

2 + T−4

T−2 T−4

+

T T ˆφ1 + φˆ 2 (T − 1) (T − 4)(T − 1)

T φˆ 2 (T − 4)

(4.34) (4.35)

For solutions with AR(3) and AR(4) models see Patterson (2007) and also Kim (2003).

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=

Bias and Bias Reduction in AR Models 137

4.2.8 The variance and mean squared error of the FOBC estimators The solutions in equations (4.34) and (4.35) suggest that the general structure ˆ where κ is a constant of the FOBC estimator is of the form Φ(1) = κ + Ψ(T)Φ, ˆ Hence the variance-covariance matrix of and ψ(T) depends on T, but not on Φ. Φ(1)

is:

ˆ var(Φ(1) ) = Ψ(T)[var(Φ)]Ψ(T)

(4.36)

onal elements of var(Φ(1) ), σˆ ( φ i(1) ), are used to form the FOBC ‘t’ statistics,

tFOBC ( φ i ) = ( φ i(1) − φi )/ˆσ( φ i(1) ). We consider these expressions further for the AR(1) and AR(2) cases. Example 1: AR(1) model, unknown constant, no trend

From φ 1(1) in (4.30) note that κ = (T − 3)−1 and Ψ(T) = T(T − 3)−1 ; hence

var( φ 1(1) ) is:

var( φ 1(1) ) =

T (T − 3)

2 var(φˆ 1 ) > var(φˆ 1 )

(4.37)

Thus the variance of the bias adjusted estimator will exceed that of the unadjusted estimator; notice that despite their similarity this is not always the case for the LBC estimator, for which the equivalent condition is that the bias function is negatively sloped. Example 2: AR(2) model, unknown constant, no trend From (4.34) and (4.35) we have: κ1 (T − 1)−1 (1 + 2/(T − 4)) κ= = 2/(T − 4) κ2 T/(T − 1) T/(T − 4)(T − 1) Ψ(T) = 0 T/(T − 4) Hence, taking the simpler variance ﬁrst:

var( φ 2(1) ) = [T2 /(T − 4)2 ]var(φˆ 2 ) > var(φˆ 2 )

var( φ 1(1) ) = [T2 /(T − 1)2 ]var(φˆ 1 ) + [T4 /(T − 4)4 (T − 1)2 ]var(φˆ 2 ) + [2T2 /(T − 1)2 (T − 4)] cov(φˆ 1 , φˆ 2 )

The ﬁrst two terms in var( φ 1(1) ) unambiguously indicate var( φ 1(1) ) > var(φˆ 1 ). The last term has an ambiguous effect depending on cov(φˆ 1 , φˆ 2 ), but is generally unlikely to lead to a reduction in the overall variance.

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ˆ is the LS variance-covariance matrix. The square roots of the diagwhere var(Φ)

138 Unit Root Tests in Time Series

4.2.8.i Mean squared error comparison Example 3: AR(1) model, unknown constant, no trend In the AR(1) model, with constant and no trend, the squared ﬁrst-order bias of the LS estimator φˆ 1 is (1 + 3φ1 )2 /T2 ; hence the mean squared error (mse) may be approximated by (1 + 3φ1 )2 /T2 + var(φˆ 1 ). The mse for the LS and FOBC estimators are: mse(φˆ 1 ) = (1 + 6φ1 + 9φ21 )/T2 + var(φˆ 1 ) + o(T−2 ) mse( φ 1 ) = var(φˆ 1 )T2 /(T − 3)2 + o(T−2 )

(4.39)

Ignoring the o(T−2 ) terms, the condition for mse( φ 1 ) < mse(φˆ 1 ) is: var(φˆ 1 )(

(1 + 6φ1 + 9φ21 )(T − 3)2 T2 (6T − 9)

For example, suppose T = 100 and φ1 = 0.5, then the condition is var(φˆ 1 ) < 0.0099; that is, var(φˆ 1 ) < 0.099. The scope for mse reduction increases as φ1 ˆ approaches 1; for example, if φ1 = 0.9, then the condition is var(φ1 ) < 0.022, that is var(φˆ 1 ) < 0.148. The AR(1) case with trend is considered as a question.

4.3 Recursive mean adjustment Recursive mean adjustment (RMA) has been suggested by So and Shin (1999), see also Shin and So (2002), as a method to reduce the bias induced by the correlation between the regressor(s) and the error term in an AR model. The method is considered here as a means of reducing the bias in the LS estimator of the coefﬁcients of a stationary AR model; however, it also has potential uses in other areas – for example, in improving the coverage of conﬁdence intervals, a topic considered in the next chapter, and as the basis for a unit root test (see Chapter 7). The mean and other deterministic components of yt are nuisance parameters as far as inference on the persistence of the process is concerned; however, although secondary, the way that these parameters are accounted for has implications for inference on the primary parameters of interest. The standard way of dealing with the deterministic components can be viewed, through the FrischWaugh theorem, as a prior demeaning or detrending of the data by a regression of the variable of interest on the deterministic components. This uses all, or nearly all, of the sample data to concentrate out the deterministic components. However, recursive mean, or trend, adjustment (for simplicity, both are referred to as recursive mean adjustment) uses just part of the sample data. The method is ﬁrst outlined for the AR(1) model, where the deterministic function comprises a constant mean, extended to the with-trend case and then to the more general AR model.

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(4.38)

Bias and Bias Reduction in AR Models 139

4.3.1 AR(1) model 4.3.1.i Constant mean Consider the basic AR(1) model with a nonzero mean (the more general AR model is considered in section 4.3.2). The model is: yt − µ = φ1 (yt−1 − µ) + zt

(4.40)

and assume that zt = εt . (For consistency with the AR notation of this chapter, φ1 is used; however, this could equally be written using ρ in place of φ1 ). In the LS solution µ is, in effect, estimated by the sample mean (over T – 1 observations); that is, y¯ = ∑Tt=2 yt /(T − 1), which results in the familiar LS estimator φˆ 1 of φ1 given by: ¯ t−1 − y) ¯ ∑T (yt − y)(y φˆ 1 = t=2 T 2 ¯ (y − y) ∑t=2 t−1

(4.41)

This estimator can be viewed as being obtained by applying LS to the regression: ¯ + υt yt − y¯ = φ1 (yt−1 − y)

(4.42)

¯ + εt . A slight variation on this procedure is to use all where υt = (1 − φ1 )(µ − y) T sample observations and calculate the ‘global’ mean y¯ G = ∑Tt=1 yt /T; and then ¯ using yt − y¯ G in the LS regression (4.42). ‘demean’ yt by y¯ G , rather than y, Whether the mean is deﬁned over T or T – 1 observations, the regressor is correlated with the error. For example, yt−1 − y¯ = yt−1 – (y2 + y3 + . . . yt + . . . + yT )/(T− 1) is correlated with the error υt for two reasons: ﬁrst, the presence of yt in the regressor leads to a direct correlation with υt , since the latter in part determines the former; second, because of the autoregressive structure of the regression model, the random variables yt + j for j ≥ 1 are also correlated with υt , the more so as φ1 → 1. Removing, or reducing, this should result in a reduction in the bias of the LS estimator of φ1 . In general terms, what is needed is an instrument for yt−1 − y¯ (or yt−1 − y¯ G ); this should be correlated with yt−1 − y¯ (or yt−1 − y¯ G ), but uncorrelated with υt (that is, satisfying the standard conditions of instrument relevance and instrument orthogonality). So and Shin (1999) propose using the recursively adjusted mean in the demeaning procedure. That is, instead of estimating the mean by ¯ just use the observations up to and including period t – 1 to calculate the y, mean; this avoids including yt + j for j ≥ 0, and thus an important source of the correlation with the error, υt , is removed. This procedure gives a sequence of estimators of the mean, say {y¯ rt }T1 : y¯ r1 = y1 ; y¯ r2 = 2−1 ∑i=1 yi ; y¯ r3 = 3−1 ∑i=1 yi ; ...; y¯ rt = t−1 ∑i=1 yi , ...; y¯ rT = T−1 ∑i=1 yi 2

3

t

T

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= (1 − φ1 )µ + φ1 yt−1 + zt

140 Unit Root Tests in Time Series

These means are given by the recursion: (4.43)

hence the term ‘recursive mean adjustment’. The dependent variable in the LS regression is now yt – y¯ rt−1 , whereas the regressor is yt−1 – y¯ rt−1 . The problem of correlation of the regressor with υt has been removed because the recursive mean estimator of µ does not include current or future dated yt . The LS regression using RMA data and the corresponding estimator φˆ 1,rma are: yt − y¯ rt−1 = φ1 (yt−1 − y¯ rt−1 ) + ζt φˆ 1,rma =

∑Tt=3 (yt − y¯ rt−1 )(yt−1 − y¯ rt−1 ) ∑Tt=3 (yt−1 − y¯ rt−1 )2

(4.44) (4.45)

Where ζt = (1−ρ)(µ− y¯ rt−1 ) + εt . Notice that summation is from t = 3 to T, because yt−1 − y¯ rt−1 = 0 for t = 2. 4.3.1.ii Trend in the mean Recursive mean adjustment can be extended to the inclusion of a trend or more general deterministic terms. For example, if the AR(1) model is: (yt − µt ) = φ1 (yt−1 − µt−1 ) + zt

(4.46)

with µt = β0 + β1 t and zt = εt , then the RMA regression model is: ˆ rt−1 ) = φ1 (yt−1 − µ ˆ rt−1 ) + ζt (yt − µ

(4.47)

ˆ rt = βˆ 0,t + βˆ 1,t t, and βˆ 0,t and βˆ 1,t are the LS estimators based on a regression where µ ˆ rt is then lagged of yt on a constant and t, including observations to period t; µ once. (Sul et. al., 2005, point out that recursive detrending leaves E(ζt ) = 0, so ˆ rt−1 is just the recursive the problem is not solved entirely.) The term yt−1 − µ residual at t – 1 from the preliminary detrending regression of yt on a constant and t; this regression requires three observations to generate nonzero residuals, so the RMA estimator is: φˆ 1 =

ˆ rt−1 )(yt−1 − µ ˆ rt−1 ) ∑Tt=4 (yt − µ ˆ rt−1 )2 ∑Tt=4 (yt−1 − µ

(4.48)

The least squares estimators obtained by this method are referred to generically as recursive mean adjusted least squares, or RMALS estimators.

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y¯ rt = y¯ rt−1 (t − 1)/t + yt /t

Bias and Bias Reduction in AR Models 141

4.3.2 Extension to AR(p) models 4.3.2.i Constant mean, AR(p) models To consider the extension to AR(p) models, p > 1, ﬁrst take the case of µt = µ, with p = 2, then the AR(2) model is: (yt − µ) = φ1 (yt−1 − µ) + φ1 (yt−2 − µ) + εt

(4.49a)

= − (φ1 + φ2 )µ + (φ1 + φ2 )yt−1 − φ2 (yt−1 − yt−2 ) + εt (4.49b)

Where ϕ = φ1 + φ2 and c1 = −φ2 , and, as before, the RMA estimator of the mean is given by the sequence {y¯ rt }T2 rather than a single estimator, so that the model to be estimated is: yt − y¯ rt−1 = φ(yt−1 − y¯ rt−1 ) + c1 (yt−1 − yt−2 ) + ζt

(4.50)

This is the scheme suggested by Shin and So (2001, equation 3.1); and, in general, the AR(p) model, with recursive mean estimation of a constant population mean µ, is: yt − y¯ rt−1 = φ(yt−1 − y¯ rt−1 ) + ∑j=1 cj (yt−j − yt−j−1 ) + ζt p−1

(4.51)

There is a variation on the resulting empirical model depending upon the stage at which the recursive mean estimator is used. Suppose in (4.49a), µ is replaced by its respective RMA estimator in each of the p = 2 regressors yt−j − µ, then (4.49a) and (4.49b) are, respectively: yt − y¯ rt−1 = φ1 (yt−1 − y¯ rt−1 ) + φ1 (yt−2 − y¯ rt−2 ) + ξt = φ(yt−1 − y¯ rt−1 ) + c1 [(yt−1 − yt−2 ) − (y¯ rt−1 − y¯ rt−2 )] + ξt

(4.52a) (4.52b)

In effect, the recursive residuals from the preliminary regression are used for all regressors. The difference between (4.49b) and (4.52b) is in the updating in the RMA estimation of the mean, which comes from the presence of the term y¯ rt−1 − y¯ rt−2 in square brackets [.] in (4.52b); using the relationship between the recursive estimates, y¯ rt−1 − y¯ rt−2 = (t−1)−1 (yt−1 − y¯ rt−2 ), which will not be equal to zero unless the ‘updating’ observation yt−1 is equal to the most recent recursive mean y¯ rt−2 . The corresponding AR(p) model can be written in ADF-type form as: (yt − y¯ rt−1 ) = φ(yt−1 − y¯ rt−1 ) + ∑j=1 cj [(yt−j − yt−j−1 ) − (y¯ rt−j − y¯ rt−j−1 )] + ξt p−1

(4.53) p

p

where ϕ ≡ ∑i=1 φi and cj = − ∑i = j φi + 1 . This speciﬁcation can be viewed as an example of a more general scheme in which the observations on yt are replaced by the residuals from a recursive regression on the variables in the deterministic function, µt , which in this case is just a constant.

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= φ(yt−1 − µ) + c1 (yt−1 − yt−2 ) + εt

142 Unit Root Tests in Time Series

4.3.2.ii Trend in the mean, AR(p) models Now consider the case of a linear trend, so that µt = β0 + β1 t. Starting from the AR(p) representation of the model: (yt − µt ) = ∑i=1 φi (yt−i − µt−i ) + εt p

(4.54a)

= φ(yt−1 − µt−1 ) + ∑j=1 cj [(yt−j − µt−j ) − (yt−j−1 − µt−j−1 )] + εt p−1

The simplest way to proceed is to form the sequence of recursive trend estimaˆ rt }Tt=2 , where, as before, µ ˆ rt is the ﬁtted value from a regression of yt on tors, {µ a constant and a deterministic time trend only using observations s = 1, . . . , t. The recursive residuals from these regressions are: uˆ rt = yt − µ ˆ rt

(4.55)

ˆ rt−j for The required lagged values of the recursive residuals are uˆ rt−j = yt−j – µ j = 1, ...., p. The regressors in (4.54a) can now be formed; then with regressand ˆ rt−1 ), the AR(p) model in ADF form is: (yt − µ ˆ rt−1 ) = φuˆ rt−1 + ∑j=1 cj (uˆ rt−j − uˆ rt−j−1 ) + ξt (yt − µ p−1

(4.56)

Then, as in the standard case, the method of least squares is used to estimate the coefﬁcients.

4.4 Bootstrapping Bootstrapping is an important principle in econometrics and statistics, and we use it here to obtain bias adjusted least squares estimates; bootstrapped conﬁdence intervals are considered in Chapter 5 and bootstrapped unit root tests are considered in Chapter 8. 4.4.1 Bootstrapping to reduce bias Bootstrapping to reduce bias is one of the primary uses of the technique. In the present context, Stine (1987) and Stine and Shaman (1989) suggested bootstrapping an autoregression to reduce the order of the ﬁnite sample bias. In general, the basic idea behind bootstrapping is that after an initial regression, obtaining, for example, an LS estimate φˆ 1 of φ1 and the corresponding LS residuals, the estimate φˆ 1 is treated as the ‘true’ – or population – coefﬁcient and the LS residuals are treated as the ‘true’ errors for the purpose of subsequent simulations. (Note that following convention the same notation is used for an estimate as an estimator, but the meaning should be apparent from the context.) The subsequent simulations, sometimes referred to as ‘replications’ in the bootstrap context, take draws from the initial LS residuals and the DGP is constructed using φˆ 1 .

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(4.54b)

Bias and Bias Reduction in AR Models 143

Use the observed realisations of the sequence {yt }T1 to obtain the LS estiˆ respectively; this is referred to as mates of φ1 and µ, denoted φˆ 1 and µ, the ‘initial’ regression. (ii)(a) From this initial estimation obtain the regression residuals {ˆεt }T2 , these residuals deﬁne the empirical distribution function (EDF) denoted F(ˆε), from which random ‘draws’ with replacement are subsequently taken. Each draw denoted eˆ t from this EDF is equi-probable, with probability (T − 1)−1 , and note that the residuals are automatically centred since the inclusion of a constant in the maintained regression ensures the mean of εˆ t is zero. If a constant is not included, the errors for each replication should be centred by subtracting the mean error. (Several different resampling schemes are considered in Chapter 8.) (ii)(b) Rescale the residuals by the factor [(T− − p)/(T− − 2p)]0.5 , where p = 1 for the AR(1) case and T− = T − 1 is the effective sample size (T − p in general). The rescaling factor is greater than 1 (marginally so for ‘large’ T) and has been suggested by Thombs and Schucany (1990) to counter the ‘deﬂation’ observed in ﬁtted (LS) residuals. To avoid introducing further notation, interpret F(ˆε) as the EDF for the rescaled residuals. (iii)(a) Generate further data by using φˆ 1 and µ ˆ as the ‘true’ coefﬁcients, with errors, eˆ t , obtained as random draws from the EDF given by F(ˆε), with replacement, as deﬁned in step ii. The bootstrap data is generated recursively, given an initial value y1 , as:

(i)

ˆ + φˆ 1 y1 + eˆ 2 , yb3 = µ ˆ + φˆ 1 yb2 + eˆ 3 , ..., ybT = µ ˆ + φˆ 1 ybT−1 + eˆ T yb2 = µ

(4.57)

where eˆ t is a random draw from {ˆεt }T2 . This gives rise to the bootstrap sequence {ybt }T2 . (iii)(b) Repeat the procedure in (iii)(a) B times. The choice of using the actual initial value y1 seems natural, but any of the {yt }T1 would be appropriate (or block of p contiguous values for the AR(p) model). Note that each bootstrap replicate uses the same initial value (or p initial values for the AR(p) model), so that the conditioning is kept constant. (iv) With each of the B data sets generated by (4.59), estimate the coefﬁcients by least squares, denote these as φˆ b1 and µ ˆ b for b = 1, ..., B. This

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For each replication there is a bootstrapped estimate denoted φˆ b1 . The bootstrap analogy is that φˆ b1 relates to φˆ 1 as φˆ 1 relates to φ1 , so that we can infer something about the latter from the former. In particular, the bias relating φˆ 1 to φ1 should be mimicked in the bias relating φˆ b1 to φˆ 1 . Greater conﬁdence can be put in this analogy as the number of bootstrap ‘replications’ is increased. The steps of a bootstrap are illustrated with an AR(1) model. Some variations and qualiﬁcations follow an outline of the basic procedure; and comments relating to the extension of the bootstrap to AR(p) models then follow.

144 Unit Root Tests in Time Series

(v)

ˆ b }B1 ; hence, deﬁne the bootstrap deﬁnes the sequences {φˆ b1 }B1 and {µ ¯ B B b b b ¯ˆ = ∑ ˆ b /B. means φˆ 1 = ∑b=1 φˆ 1 /B and µ b=1 µ From the B replications, estimate the bootstrap bias as biasBS (φˆ 1 ) = φ¯ˆ b − φˆ 1 1

and obtain the bootstrapped bias corrected (BSBC) estimate, which is the original estimate minus the bootstrap estimate of bias. The idea is that φ¯ˆ b1 mimics E(φˆ 1 ) whereas φˆ 1 mimics φ1 . The bootstrap bias is:

= 2φˆ 1 − φ¯ˆ b1

(4.58)

Two variations on this procedure have been suggested (see Thombs and Schucany, 1990; Kilian, 1998; Kim, 2003). One suggestion is to use the backward AR(1) model to generate the data. Kim (2003) found that use of the backward rather than usual (forward) AR model resulted in better forecasting performance. It is legitimate to use the backward version of the AR model as the correlation structure of the forward and backward processes; for example, correlation(yt , yt−s ) = correlation(yt , yt + s ), is the same for a stationary process. Also, forecasting performance is likely to be improved based on a process starting from the last (p) observation(s). A second qualiﬁcation is that if the BSBC estimate of φ1 results in |φˆ BS 1 | > 1, then the estimator is adjusted back to stationarity by setting φˆ BS = 0.99 (or –0.99 if that case occurs). 1 The extension of this process to AR(p) models is straightforward, with due attention to starting values and the recursions as indicated at step (iii)(a). In particular the recursions now need to be conditioned on p initial values of yt , for example y1 , . . . , yp , if the usual forward AR(p) model is being used. It is also usually more convenient to use the AR(p) model in ADF(p – 1) form, which isolates the coefﬁcient of interest, φ, on the lagged value of yt−1 . Given that the bias function for the LS estimator is very close to being linear over most of the relevant range of φ1 , the bootstrap may be more effective where a root close to but not actually unity is suspected. In that case, it is an alternative to the nonlinear bias adjustment procedure suggested by MacKinnon and Smith (1998). A bootstrap replication could also be deﬁned by starting with ﬁrstorder bias adjusted estimator, so that the bootstrap was effectively being used to correct biases of higher order than ﬁrst. The bias adjusted estimator can then serve as the base from which to generate bootstrapped prediction intervals (for example, Kilian, 1998; Gospodinov, 2002). Bias adjustment is likely to be important in this context since seemingly relatively small differences between the biased LS estimate φˆ 1 and φ1 magnify into large differences in h-step predictions through the difference h between φˆ h 1 and φ1 . For example, with T = 100 and φ1 = 0.95, then to ﬁrst-order

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ˆ ˆ φˆ BS 1 = φ1 − biasBS (φ1 )

Bias and Bias Reduction in AR Models 145

E(φˆ 1 ) = 0.9115, and at a modest prediction horizon of h = 4, (0.9115)4 = 0.69 compared to (0.95)4 = 0.815, a ratio of 84.7%; thus even partial correction to the bias offers a large improvement in predictive ability.

Four methods of bias reduction have been suggested in the preceding sections: ﬁrst-order bias correction, linear bias correction, recursive mean adjustment and bootstrap bias correction. The abbreviated references to the resulting estimators are as follows: FOBC, LBC, RMALS and BSBC. In this section we report an illustrative simulation study using an AR(1) set-up (see also Patterson, 2007); very similar results were obtained with higher-order AR models and are not reported here. Also the results for the FOBC and LBC were close enough to consider reporting only one of them and, as the FOBC estimator can be calculated without simulation, details on the results with the LBC estimator are omitted. The simulation set-up is as described in section 4.5.2. Of the details not referred to there, the bootstrap estimator was based on 999 bootstrap replications within each of the bootstrap runs. Referring again to Figure 4.1, ﬁrst consider the upper panel, which concerns the case where a constant is included in the regression. All of the bias adjusted methods deliver a considerable improvement in bias, with practically very little to choose amongst them, for most of the parameter space; however, as the unit root is approached the BSBC and FOBC estimators dominate the RMALS estimator. As the region close to the unit root is likely to be of particular economic importance, some details are extracted in Table 4.4. For example when φ1 = 0.99, the mean LS estimate is 0.94, whereas the means of the bias adjusted estimates are: FOBC, 0.98; BSBC, 0.977; RMALS 0.973. The ﬁrst-order bias for φ1 = 0.99 and T = 100 is calculated from the expression in Table 4.1a: b(1) (φ1 = 0.99, T = 100) = − (1 + 3(0.99))/100 = − 0.0397 This gives an estimate of the higher-order bias of (0.94 – 0.99) + 0.0397 = –0.0103; that is, a total bias of b(0.99,100) = –0.0397 – 0.0103 = –0.05. Moving away slightly from the unit root, when φ1 = 0.9, the corresponding means are: LS, 0.861; FOBC, 0.898; BSBC, 0.900; RMALS 0.894. Note that the mean of the BSBC estimator is ‘spot on’, with the FOBC estimator subject only to a small bias; the ﬁrst-order LS bias is b(1) (0.9,100) = –0.037, giving an estimate of higher-order bias of (0.861 – 0.9) + 0.037 = –0.002, indicating how little of the bias is now of smaller order than T−1 . The with-trend case is graphed in the lower part of Figure 4.1 and details are again extracted for φ1 close to the unit root. It is evident from Figure 4.1

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4.5 Results of a simulation: estimation by least squares, ﬁrst-order bias correction and recursive mean adjustment

146 Unit Root Tests in Time Series

No trend

LS mean rmse

FOBC mean rmse

BSBC mean rmse

RMALS mean rmse

φ1 = 0.99 φ1 = 0.96 φ1 = 0.93 φ1 = 0.90 φ1 = 0.87 φ1 = 0.84 φ1 = 0.81

0.940 0.0668 0.914 0.0667 0.888 0.0676 0.861 0.0685 0.832 0.0702 0.804 0.0736 0.772 0.0760

0.980 0.0471 0.953 0.0506 0.926 0.0548 0.898 0.0578 0.869 0.0611 0.839 0.0659 0.807 0.0682

0.977 0.0502 0.956 0.0525 0.928 0.0565 0.900 0.0592 0.870 0.0622 0.840 0.0668 0.808 0.0688

0.973 0.0435 0.947 0.0471 0.921 0.0512 0.894 0.0543 0.866 0.0577 0.837 0.0628 0.806 0.0657

With trend φ1 = 0.99 φ1 = 0.96 φ1 = 0.93 φ1 = 0.90 φ1 = 0.87 φ1 = 0.84 φ1 = 0.81

0.899 0.1072 0.883 0.0969 0.859 0.0935 0.834 0.0913 0.808 0.0902 0.782 0.0901 0.752 0.0913

0.958 0.0674 0.941 0.0636 0.912 0.0654 0.890 0.0667 0.863 0.0686 0.836 0.0721 0.805 0.0736

0.953 0.0748 0.942 0.0667 0.920 0.0681 0.893 0.0690 0.865 0.0706 0.838 0.0739 0.806 0.0749

0.977 0.0525 0.962 0.0527 0.941 0.0585 0.918 0.0633 0.894 0.0684 0.868 0.0741 0.838 0.0763

Notes: Emboldened entries indicate closest, in absolute value, to the true value for mean entries, and smallest root mean squared error (rmse) in the case of the rmse entries.

that RMALS tends to over-correct the bias adjustment, whereas the pattern of improvement for FOBC and BSBC is very similar to the no-trend case. A comparison is presented in Figure 4.2 and in the second sub-column for each of the estimators in Table 4.4. In the no-trend case, all of the bias adjustment methods deliver a reduction in rmse for φ1 >0.5 and are virtually indistinguishable for 0.4< φ1 <0.5. Whilst there is only a small margin of choice amongst them, the rmse of the RMALS estimator tends to be smallest, with the proportionate reduction increasing markedly as the unit root is approached; for example, a reduction of 21% at φ1 = 0.9 and a reduction of 35% at φ1 = 0.99. In the with-trend case, bias reduction by FOBC and BSBC is generally best, but for φ1 ≥ 0.87, RMALS dominates in terms of rmse, notwithstanding its relatively poor performance in terms of over-correcting the bias adjustment. The proportionate reductions in rmse are even greater in this case, with a reduction of 31% at φ1 = 0.9 and a reduction of 51% at φ1 = 0.99. Whilst these simulation results are illustrative, more extensive simulations conﬁrm these general results and indicate that bias reductions and rmse reductions are available for relatively little effort over that part of the parameter space which is likely to be relevant for typical economic time series.

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Table 4.4 Bias correction in an AR(1) model.

Bias and Bias Reduction in AR Models 147

No trend case

0.1

rmse

0.08 LS

all the bias adjusted methods dominate LS

0.06 0.04 0.2

0.3

0.4

0.5

0.6

0.7

BSBC FOBC RMALS

0.8

0.9

1

1.1

0.12 With trend

LS

0.1 rmse 0.08 BSBC FOBC and BSBC dominate LS

0.06

FOBC RMALS

0.04 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

φ1 Figure 4.2 RMSE functions of estimators.

4.6 Illustrations In order to illustrate the various methods of bias reduction, two empirical examples are considered in this section.

4.6.1 An AR(1) model for US ﬁve-year T-bond rate The ﬁrst illustration is an AR(1) model, with constant, for data on the US ﬁve-year T-bond rate. The data is annual for the period 1950–2000, giving an effective sample of 50 observations. An AR(1) model appeared adequate in terms of standard equation diagnostics and, as shown by a graph of the data in Figure 4.3, there was no role for a linear time trend; this was conﬁrmed from a supplementary regression. The results for the AR(1) model, with constant, estimated by various methods are presented in Table 4.5. The FOBC estimate can be calculated from some simple calculations, which are now illustrated. From (4.31), with T = 51, the FOBC estimate is:

φ 1(1) =

1 T ˆ + φ (T − 3) (T − 3) 1

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φ1

16 15 14 13 12 11 10 9 % p.a. 8 7 6 5 4 3 2 1 0 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 Figure 4.3 US ﬁve-year T-bond rate.

Table 4.5 LS and bias adjusted estimates of an AR(2) model for US ﬁve-year T-bond. φˆ 1 LS FOBC RMALS BSBC B = 999 B = 9,999 B = 99,999

ˆ Λ

ˆ µ

0.908 0.986 0.967

10.880 69.272 30.300

0.670 0.670 n.r

0.955 0.954 0.954

22.103 21.530 21.840

0.369 0.376 0.373

Notes: n.r. indicates not relevant as there is a sequence of estimates of the mean using recursive mean adjustment; Λ is the long-run multiplier.

=

51 1 + 0.90809 (51 − 3) (51 − 3)

= 0.98567 ≈ 0.986 The RMALS estimate of φ1 is not quite as large as the FOBC estimate, but still much closer to the unit root compared to the LS estimate. Note that the consequent increase in the estimate of the long-run multiplier is substantial.

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148 Unit Root Tests in Time Series

Bias and Bias Reduction in AR Models 149

9.5 9 8.5 8

7 6.5 6

1930

1940

1950

1960

1970

1980

1990

2000

Figure 4.4 US GNP (in logs, constant prices).

The bootstrap bias correction (BSBC) is undertaken for all of the estimated coefﬁcients in the model and, to assess the sensitivity of the results to the number of bootstrap replications, the results are given for B = 999, B = 9,999 and B = 99,999. (Obtaining the corrected estimates is almost costless in computing time; even a run with 99,999 replications only took a minute or so on a PC with 3.0 Mhz processor.) The estimates of φ1 stabilise quickly and show little variation as B increases (for example, the estimate of φ1 is virtually unchanged even if only 99 bootstrap replications are used). 4.6.2 An AR(2) model for US GNP An AR(2) model with time trend was estimated for the log of real US GNP over the period 1929–2000, giving 72 observations in all and an effective sample of 70 observations. The series is graphed in Figure 4.4, from which a positive trend is apparent. The question of whether the trend is better accounted for by a unit root or deterministic trend is deferred until later chapters; for the moment it is clear that if the data is generated by a stationary AR(2) model, a trend should also be included in the regression. The results for the AR(2) model, with constant and trend, estimated by various methods are presented in Table 4.6. The LS estimates are presented in the ﬁrst row of Table 4.6. As expected, the effect of the bias adjustments is to increase the estimate of the sum of the AR coefﬁcients, from 0.858 to 0.876 with the FOBC estimates and to 0.883 with the RMALS estimates; this is in keeping with the simulation results, which show a tendency for the RMALS estimate to be larger than the FOBC estimate. The FOBC

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data clearly trended

7.5

150 Unit Root Tests in Time Series

LS FOBC RMALS BSBC B = 999 B = 9,999 B = 99,999

φˆ 1

φˆ 2

φˆ

Λ

β0

β1

1.446 1.479 1.558

–0.588 –0.602 –0.675

0.858 0.876 0.883

7.043 8.094 8.568

0.943 0.943 n.r

0.0051 0.0051 n.r

1.480 1.483 1.482

–0.586 –0.587 –0.587

0.894 0.896 0.895

9.423 9.600 9.520

0.714 0.699 0.706

0.0038 0.0037 0.0038

Notes: n.r. indicates not relevant as there is a sequence of estimates of the trend coefﬁcients using recursive mean adjustment.

bias adjustments to the estimates of φ1 and φ2 tend to be partly offsetting, with the former increasing and the latter decreasing (since it becomes more negative). Also, the ﬁrst-order bias adjustment is fairly slight on φˆ 2 because it is close to the zero bias ﬁxed point of φ2 = –0.4. Turning to the bootstrap bias correction, the estimate of the primary coefﬁcient of interest; that is φ, shows little sensitivity to B, suggesting φˆ BS = 0.895, ˆ BS = 9.52 (for B = 99,999). with an estimated long-run multiplier of Λ

4.7 Concluding remarks Standard estimators of the coefﬁcients of AR (and ARMA) models are biased in ﬁnite samples; these biases can be quite critical in an evaluation of key issues such as the proximity of the dominant root to the unit root and the estimated persistence of shocks. Because of its nonlinear nature, the standard persistence measure, that is, the long-run multiplier Λ, is particularly vulnerable to small biases in the estimation of the component AR coefﬁcients. If the bias function was known, obtaining a bias corrected estimator would be simple: just remove the bias from the original estimator. It is not surprising, therefore, that the focus of attention in obtaining a bias corrected estimator is to estimate the bias function, or some relevant part of, or approximation to, the bias function. The ﬁrst-order bias corrected estimator concentrates on removing an estimate of the ﬁrst-order bias, where ﬁrst-order is interpreted in the sense of terms of order T. This works well, except when close to the unit root, where terms of smaller order than T become important. Nevertheless, at very little practical cost, the FOBC estimator removes a substantial part of the bias. Very close in approach are the bias corrected estimators based on an assumption of either a constant bias function or a linear bias function in the sense of linearity in the AR coefﬁcients. Both methods become feasible through a simulation of their respective bias functions. They also bear a similarity to the method of bootstrapping the bias, which was one of the original motivations

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Table 4.6 LS and bias adjusted estimates of an AR(2) model for US GNP.

for the bootstrap. The bootstrap analogy is simple to follow in this case. First estimate the regression of interest and obtain the AR coefﬁcients using LS or any consistent method of estimation; these coefﬁcients then become the DGP – or ‘population’ – coefﬁcients, as far as the bootstrap replications (or more loosely, simulations) are concerned, which take draws from the LS residuals. For a particular coefﬁcient, the bootstrap estimate of the bias is the mean of the estimated coefﬁcient using the bootstrap replications minus the original LS estimate. This estimates the bias, which is then used to correct the bias in the original LS estimate. A similar approach or approaches could be used for alternative deﬁnitions of estimator bias. For example, Andrews, (1993) simulation-based method of median, rather than mean, bias adjustment is considered in the next chapter. The method of recursive mean adjustment directly removes one of the sources of the bias. For example, in the case that the deterministic part of the AR model is just a constant, the standard approach is to demean the data in some form, either directly by subtracting an estimate of the mean from the original data, or by including a constant in the regression. However, these methods induce a correlation between the estimator of the mean and the error in the AR regression, which, as is well known, is undesirable for LS estimation. By conditioning the estimator of the mean only on data that lags period t, the correlation is removed. The suggested estimator of the mean is updated as additional observations become available; this creates a simple recursion relating the estimators of the mean. Note that the mean is not estimated by a single quantity but by a sequence that depends on t. A similar procedure can be used to detrend the data prior to estimation. An illustrative simulation study suggested that bias correction works both in terms of its primary aim, that of bias reduction, and in terms of mean squared error reduction, at least for that part of the parameter space likely to be of relevance for economic time series. The next chapter takes up some related issues primarily concerned with inference.

Questions Q4.1 Consider the AR(1) case, with constant but no trend. Can the variance of the FOBC estimator be less than the variance of the LS estimator? A4.1 The relation between the two variances is:

var( φ 1(1) ) =

1 1+ω

2 var(φˆ 1 )

The condition for var( φ 1(1) ) < var(φˆ 1 ), is ω > 0, where ω = ∂ (−(1 + 3φ1 )/T)/∂ φ1 is the slope of the bias function w.r.t φ1 ; in this case ω = −3/T, and hence the condition is not satisﬁed.

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Bias and Bias Reduction in AR Models 151

152 Unit Root Tests in Time Series

Q4.2 Consider the AR(1) model with trend. Obtain and compare the (approximate) mean squared errors of the LS and FOBC estimators of φ1 for φ1 = 0.75 and T = 100. A4.2 The ﬁrst-order bias of the LS estimator φˆ 1 is −(2 + 4φ1 )/T; hence the mean squared errors of the LS and FOBC estimators are: 2 + 4φ1 2 + var(φˆ 1 ) + o(T−2 ) T 2

T mse( φ 1(1) ) = var(φˆ 1 ) + o(T−2 ) T−4

The squared bias function for the LS estimator has a minimum of zero at φ1 = – 0.5, therefore mse(φˆ 1 ) must be less than mse( φ 1(1) ) close to this point since [T/(T − 4)]2 > 1; but this is sufﬁciently far from likely economic values to suggest that mse reduction is a real possibility. Ignoring o(T−2 ) terms, the mean square errors of the two estimators are equal for:

(T − 2) 1 T − φ1 = σ(φˆ 1 ) (T − 4) 2 2 where σ(φˆ 1 ) = var(φˆ 1 ); for example, if φ1 = 0.75 and T = 100, then mse( φ 1(1) ) is less than mse(φˆ 1 ), ignoring o(T−2 ) terms, for σ(φˆ 1 ) < 0.171 and greater than otherwise. Q4.3 Derive the ﬁrst-order bias vector for an AR(4) model with linear time trend. (See the appendix to this chapter.) A4.3 The appropriate expression for the bias vector is −(B1p + B2p + 2B3p )SΦ (see the appendix to this chapter). Hence: (B1p + B2p + 2B3p )Φ = Bp Φ    0 0 0 0 0 0  0 1 0 0 0   0        0 0 2 0 0  +  −1     0 0 0 3 0   0 −1 0 0 0 0 4     +  

0 −2 −2 −2 −2

0 0 −2 −2 0

0 0 0 0 0

0 0 2 2 0

0 2 2 2 2

             

0 0 0 −1 0 1 φ1 φ2 φ3 φ4



0 0 0 0 0

0 0 0 1 0 

      =    

0 0 1 0 1

0 −2 −3 −2 −3

      

0 1 −2 −3 0

0 0 2 0 0

0 0 2 6 0

0 2 3 2 7

      

1 φ1 φ2 φ3 φ4

      

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mse(φˆ 1 ) =

Bias and Bias Reduction in AR Models 153

   Bp SΦ = −   

1 2 + φ1 + 2φ4 3 − 2φ1 + 2φ2 + 2φ3 + 3φ4 2 − 3φ1 + 6φ3 + 2φ4 3 + 7φ4

      

T times the ﬁrst-order bias vector is therefore:   2 + φ1 + 2φ4  3 − 2φ + 2φ + 2φ + 3φ  4  1 2 3 ˆ = − T × b(1) (Φ)     2 − 3φ1 + 6φ3 + 2φ4 3 + 7φ4 ˆ are those given in Consult Table 4.1b to conﬁrm that the elements of b(1) (Φ) the row corresponding to the AR(4) model. Q4.4 Obtain the roots of the AR(p) models, p = 1, . . . , 4, with an unknown constant but no trend, at the zero ﬁrst-order bias ﬁxed points. Comment on the values that you obtain. A4.4 The roots of the polynomials are given in Table 4.7 for the coefﬁcients set at the ﬁrst-order bias coefﬁcients. Table 4.7 Roots for the zero bias case: constant included in regression.

AR(1) AR(2) AR(3) AR(4)

Zero ﬁrst-order bias coefﬁcients φ1 = –1 3 φ1 = –0.5, φ2 = –0.5 φ1 = –0.6, φ2 = –0.6, φ3 = –0.2 φ1 = –2 3, φ2 = –0.8, φ3 = –0.4, φ4 = –1 3

roots; [|roots|] –3; [3] –0.5 ±1.3229i; [1.414] –2.5874; [2.587] –0.2063 ±1.3747i, [1.390] –0.897 ±0.980i, 0.297 [1.329], [1.304]

±1.269i

Notes: All the roots have modulus greater than one, so that the stability (and stationary) conditions are satisﬁed; and the roots, (obviously apart from the AR(1) model) include at least one complex pair. Together these conditions imply damped adjustment cycles.

A practical point to note in obtaining the roots is that some programs require the coefﬁcients to be entered as those in a monic polynomial, starting with a coefﬁcient of 1 on the highest lag order. Consider, for example, using MATLAB to obtain the roots of the third-order lag polynomial: 1 − φ1 L − φ2 L2 − φ3 L3 ; in the way that we have deﬁned the roots, this would be read into MATLAB as L3 + a2 L2 + a3 L + a4 , so the aj coefﬁcients are input as: a2 = φ2 /φ3 , a3 = φ1 /φ3 and a4 = −(1/φ3 ). The MATLAB input is then a = [1 a2 a3 a4 ] followed by the instruction r = roots(a). Alternatively, the polynomial coefﬁcients could be read in as is conventionally written; that is, say, phi = [1 −φ1 −φ2 −φ4 ], and then take the reciprocals of the output roots, remembering that these are ordered as a column vector.

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

154 Unit Root Tests in Time Series

Q4.5 Obtain the FOBC estimators, and their variances, for an AR(2) model with constant and linear time trend.

ˆ into Φ = Φ ˆ −[I + Bf (Φ)] ˆ −1 b(1) (Φ), ˆ we obtain: Substituting Bf(1) (Φ) (1)   −1 −1 ˆ 1 + 2φˆ 2 ˆ1 φ 1 1 (T − 1) −2 2 + φ φ 1(1) =  + T T 0 (T − 5) 2 + 5φˆ 2 φˆ 2 φ 2(1)

=

φˆ 1 φˆ 2

+

1 (T − 5)(T − 1)

(T − 5) 0

2 (T − 1)

2 + φˆ 1 + 2φˆ 2 2 + 5φˆ 2

Collecting terms, the FOBC estimators are:

2(T − 3) T 1 2T + φˆ 1 + φˆ 2 φ 1(1) = T−1 (T − 5) (T − 1) (T − 5)(T − 1)

2 T + φˆ 2 φ 2(1) = T−5 (T − 5) To obtain the variances of these estimators it is convenient to exploit their linear nature in terms of the LS estimators. Thus, ﬁrst rewrite the estimators to exploit this linearity:   ˆ1 φ ψ ψ φ κ 1 11 12 1(1)  = + κ2 ψ21 ψ22 φˆ 2 φ 2(1)

where: κ1 = 2(T − 3)(T − 5)−1 (T − 1)−1 ; κ2 = 2(T − 5)−1 ; and ψ11 = T(T − 1)−1 ; ψ12 = 2T(T − 5)−1 (T − 1)−1 ; ψ21 = 0; ψ22 = T(T − 5)−1 . Hence: 

 φ ψ11 var  1(1)  = 0 φ 2(1)

ψ12 ψ22

var

φˆ 1 φˆ 2

ψ11 ψ12

0 ψ22

Q4.6 The bootstrap bias corrected estimator provides an estimate of the bias of the LS estimate biasBS (φˆ 1 ), such that the bootstrap bias corrected estimate ¯ˆ b ˆ ˆ ˆ is φˆ BS 1 = φ1 – biasBS (φ1 ) = 2φ1 − φ1 (see Equation (4.58)). Suggest a scheme that ‘bootstraps the bootstrap’ in order to obtain a bootstrap reﬁnement of the bias correction.

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A4.5 From Table 4.1b, the ﬁrst-order LS biases are b(1) (φˆ 1 ) = −(2 + φ1 + 2φ2 )/T for φˆ 1 and b(1) (φˆ 2 ) = −(2 + 5φ2 )/T for φˆ 2 ; hence the required matrix of derivatives, ˆ is: Bf(1) (Φ), ∂ b1 /∂ φ1 ∂ b1 /∂ φ2 −1/T −2/T f ˆ = B(1) (Φ) = 0 −5/T ∂ b2 /∂ φ1 ∂ b2 /∂ φ2

Bias and Bias Reduction in AR Models 155

4.8 Appendix: First-order bias in the constant and linear trend cases This appendix presents explicit expressions for ﬁrst order bias. Relevant references are: Shaman and Stine (1988), Stine and Shaman (1989), Cordeiro and Klein (1994) and Patterson (2007). A4.1 First-order bias correction: a general framework In AR(p) models, with the i-th AR coefﬁcient denoted φi , the two cases of greatest practical relevance are for the deterministic function with µt = µ and the linear trend case with µt = β0 + β1 t. The former case is considered ﬁrst and the linear trend case then follows. Higher-order polynomial trends are also easily accommodated within this framework. Deﬁne the vectors of parameters of interest as: Φ = (1, φ1 , φ2 , . . . , φp ) = (1, Φ), ˆ ˆ where Φ = (φ1 , . . . , φp ) and Φ ˆ = (φˆ 1 , . . . , φˆ p ) . Φ = (1, φˆ 1 , φˆ 2 , . . . , φˆ p ) = (1, Φ), Then: ˆ = Φ − T−1 Bp SΦ + o(T−1 ) E(Φ) = Ip+I − T−1 Bp S Φ + o(T−1 )

(4.59) (4.60)

where Bp is a (p + 1) × (p + 1) matrix of known coefﬁcients; S is the (p + 1)-th order identity matrix apart from the (1, 1) element which is –1, which serves to change the sign of the ﬁrst column of Bp . The (p + 1) vector –T−1 Bp SΦ is the ˆ . ﬁrst-order bias vector (which has a zero in the ﬁrst row), that is (0, b(1) (Φ)) The matrix Bp comprises three matrices: Bp = B1p + B2p + B3p ; B1p = diag(0, 1, ..., p); B2p has columns which are based on the column vectors ej and dj . The elements of ej and dj are 0 apart from a 1 in rows j + 3, j + 5, p + 1 – j for ej , ˜ e , ..., e , e ] and rows j + 2, j + 4, p + 1 – j for dj . Then B2p = [–e0 ,–e1 , ..., –ek ,0, k 1 0 ˜ when p is even and B = [–d , –d , ..., –d , 0 , d , ..., d , d , d ] when p is odd, 2p

1

2

k

k

2

1

0

where k = (p/2) – 1 and 0˜ is a (p + 1) x 1 vector of 0s. The (i, j)-th element of B3p is –1 for j < i ≤ r and 1 for r < i ≤ j, where r = p – j +2, and 0 elsewhere. k−1 If the deterministic part of the AR(p) regression is µt = ∑j=0 βj tj , k ≥ 1, then Bp = B1p + B2p + kB3p (see Shaman and Stine, 1989); thus, with inclusion of a linear time trend, so that k = 2, then Bp = B1p + B2p + 2B3p . (If the mean is known then there is no term in B3p .) Tables 4.1a and 4.1b, in the text, gave

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A4.6 The point of this question is that the estimate of bias by the bootstrap procedure is just that, an estimate; hence it can itself be bootstrapped. In the second bootstrap round, the (ﬁrst round) bootstrap bias corrected estimates are used to deﬁne a new set of residuals that form the empirical distribution function; the (second round) bootstrap replicates can be formed from the (ﬁrst round) bootstrap BSBC coefﬁcients together with draws from this revised EDF. Otherwise, the iterative sequence is as before.

limT→∞ T[E(φˆ i − φi )] and limT→∞ T[E(φˆ − φ)] for AR(p) models, p = 1, ..., 4, when µt = µ and µt = β0 + β1 t, respectively. The general effect of the inclusion of a trend, for economic time series that exhibit persistence, is to increase the bias. For example, the ﬁrst order bias in an AR(1) model with trend is −(2 + 4φ1 )/T compared to −(1 + 3φ1 )/T with just a constant. Note also that the zero bias point is moved further into the negative part of the parameter region, from φ1 = −1/3 to φ1 = −0.5. To illustrate the construction of the bias expressions, consider the AR(4) regression, including an unknown constant but without a trend (so k = 2), then Bp Φ and the vector −Bp SΦ are: (B1p + B2p + B3p )Φ = Bp Φ      0 0 0 0 0 0 0 0 0 0   0 1 0 0 0   0 0 0 0 0            0 0 0 1 +  0 0 2 0 0  +  −1       0 0 0 3 0   0 −1 0 1 0   −1 0 0 0 1 0 0 0 0 4      1 1 0 0 0 0 0  φ   −1   1 0 0 1   1     φ1        φ2  =  −2 −1 2 1 2   φ2        φ   −1 −2 0 5 1   φ  3 3 −2 0 0 0 6 φ4 φ4   −1 0 0 0 0 0 0 0 0   −1 1 0 0 1   0 1 0 0    − Bp SΦ = −  −2 −1 2 1 2   0 0 1 0    −1 −2 0 5 1   0 0 0 1 0 0 0 0 −2 0 0 0 6   0   1 + φ1 + φ4     = −  2 − φ1 + 2φ2 + φ3 + 2φ4      1 − 2φ2 + 5φ3 + φ6 2 + 6φ4   1 + φ 1 + φ4  2 − φ + 2φ + φ + 2φ  4  1 2 3 ˆ = − T × b(1) (Φ)     1 − 2φ2 + 5φ3 + φ6 2 + 6φ4

0 −1 −1 −1 −1

0 0 0 0 1

0 0 −1 −1 0

      

1 φ1 φ2 φ3 φ4

0 0 0 0 0

0 0 1 1 0

0 1 1 1 1

     ×  

      

(4.61)

ˆ can now be matched up with the row corresponding The ﬁnal vector T × b(1) (Φ) to the AR(4) model in Table 4.1a. The ﬁrst-order bias expressions in Table 4.1 are linear in φi , with a constant. Exploiting this structure, then T times the ﬁrst-order bias vector can be written ˆ = π + ΠΦ where π and Π are deﬁned implicitly. For example, as T × b(1) (Φ)

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in the AR(1) model in the ‘with-trend’ case, π = −2 and Π = −4, with Φ = φ1 ; corresponding expressions for the AR(2) model and the AR(3) models are as follows:

ˆ =− T × b(1) (Φ) AR(3) 

2 2

−

1 0

2 5

  2 1   ˆ = − 3  T × b(1) (Φ)  −  −2 2 0

0 5 0

φ1 φ2

  3 φ1   2   φ2  6 φ3

Also, note that as the bias function is linear in φi , thus the partial derivative of the bias function with respect to φi does not involve φi . Knowing the O(T−1 ) bias is the ﬁrst step in constructing the FOBC estimator, ˆ is then evaluated φ(1) . The ﬁrst-order bias function for the LS estimator, b(1) (φ),

at φ(1) , resulting in the ﬁrst-order bias corrected estimator denoted φ(1) in the text.

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AR(2)

5

Introduction This chapter is a continuation of the last. There were two central messages in that chapter, the ﬁrst being that standard estimators of the coefﬁcients in an AR(p) model have a ﬁnite sample bias. In response, and addressing the solution as obtaining an estimator with better mean bias, four methods of bias adjustment were considered; these were ﬁrst-order bias correction, linear bias correction, recursive mean adjustment and bootstrap bias correction. The second and related problem was that inference using normal or t distributions in AR models is likely to be misleading, increasingly so as the dominant root of the AR model approached unity. Both of these features serve to complicate the apparently simple question of assessing whether, in a particular application, the data have been generated by a nonstationary process and in obtaining a robust estimate of persistence and, hence, the contribution of the unit root. This chapter considers in greater detail some of the problems associated with inference. Although much of the emphasis in the unit root literature is on direct tests, it is also of interest, and indeed often somewhat more enlightening, to approach the problem of inference from the viewpoint of constructing conﬁdence intervals. Of course, there is a well known duality between conﬁdence interval construction and hypothesis testing, but care has to be taken in ensuring this duality holds when the quantiles of the distribution of a particular test statistic are not constant across the parameter space. This is particularly relevant when a conﬁdence interval is constructed by the percentile-t method; that is, a statement about the distribution of the t statistic, for example, is inverted to obtain an interval that contains the true value of the parameter with a prescribed probability. Given that the quantiles of interest are not constant, we cannot construct accurate conﬁdence intervals where the actual coverage matches the nominal coverage by conventional methods. For example, to simplify assume that there 158

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Conﬁdence Intervals in AR Models 159

However, what do we do if the quantiles zα and z1−α depend on ψ and if, further, zα = −z1−α , the equality being assumed in the standard approach? To give a motivating example from the text, see section 5.4.2, with T = 100 and ψ = φ1 = 0.75, in an AR(1) model with trend, the correct quantiles are z0.05 = –2.25 and z0.95 = 0.99, not ±1.67; as φ1 increases to 0.96, these became z0.05 = –2.95 and z0.95 = –0.052. In this case, inference assuming that tφ1 has a standard t distribution is very misleading. The solution to the inference problem is composed of two stages, the ﬁrst with two parts. In the ﬁrst stage, (i) recognise that the standard quantiles are generally inappropriate; for example, z0.05 = –1.96 may be very inaccurate even when the dominant root of the AR(p) model is well inside the stationary region; and (ii) the assumption that zα = −z1−α , is also likely to be inaccurate. In the second stage, build the nonconstancy of the quantiles into the process of constructing a conﬁdence interval throughout the relevant part of the parameter space, not just at one particular point. Solutions along these lines involve simulation or bootstrapping to obtain the correct (up to a simulation error) quantiles. Since the underlying problem that these solutions address relates to the bias in the LS (and ML) estimator, an interesting question, which potentially offers some economy of purpose, is whether conﬁdence intervals constructed using a bias adjusted estimator also correct for the inaccurate coverage probability that results from using standard methods. If so, at least the non-simulation-based bias adjusted estimators offer something other than pure bias adjustment at no extra cost. In fact, we ﬁnd that these methods do offer quite a substantial improvement but these are not the complete solution offered by other methods; nevertheless, they offer a more accurate ‘ﬁrst sight’ of how close the estimated root is to the unit root. The bias adjusted methods considered in the previous chapter were concerned with reducing the bias in the ‘mean’ sense; that is, seeking an estimator ˆ = ψ, at least to ﬁrst order. An alternative criterion is to consuch that E(ψ) struct an estimator that is median unbiased, in the sense that the probability of underestimation is exactly matched by the probability of overestimation. When an estimator has a symmetric distribution, these criteria amount to the same thing; however, the distribution of the LS estimator of the sum of the AR coefﬁcients is asymmetric for near-unit root processes, hence the median unbiased

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is a large enough sample to use the normal distribution for the t statistic given ˆ – ψ)/ˆσ(ψ), ˆ where ψ is a scalar parameter of interest and σˆ (ψ) ˆ is its by tψ = (ψ estimated standard error; then a standard (1 – 2α)% two-sided conﬁdence interˆ ± zα σˆ (ψ), ˆ where zα is the α% quantile from the normal val is constructed as ψ distribution. For example, if zα = −z1−α = –1.96, for α = 0.05, then the resulting ˆ ± 1.96ˆσ(ψ). ˆ 90% conﬁdence interval is ψ

criterion offers an alternative estimator. Andrews (1993) has shown how to construct a median unbiased estimator in this case and how to construct associated conﬁdence intervals (which are not based on the percentile-t method). The conﬁdence intervals in this chapter are constructed using standard t-type statistics to show the principles at work. At a later stage, various other test statistics for a unit root – which are, under some circumstances, more powerful than the t test – are introduced, and these may also be inverted to obtain conﬁdence intervals; hence the topic of conﬁdence intervals is revisited in Chapter 12 with the beneﬁt of the material covered in later chapters. The chapter progresses by ﬁrst reviewing conﬁdence intervals and hypothesis testing in section 5.1. This is followed in section 5.2 by an introduction to the quantile function, a special case of which is the median function. This leads on in section 5.3 to median unbiased estimation and in section 5.4 to quantiles using some of the bias adjusted estimators of Chapter 4. The central message from these sections is that the quantiles, especially those used for conﬁdence interval construction and hypothesis testing, are generally not constant. A way of incorporating this nonconstancy into the methodology is by using a bootstrap technique, which is the topic of section 5.5.

5.1 Conﬁdence intervals and hypothesis testing This section brieﬂy reviews some key concepts, such as conﬁdence intervals and hypothesis testing, that are central to this and later chapters. 5.1.1 Conﬁdence intervals ˆ is an estimator of a scalar parameter ψ and the problem is to form Suppose that ψ a conﬁdence interval for ψ. One such basis is to construct an interval based on a probability statement about the t statistic tψ for ψ. Let zα and z1−α be the lower and upper quantiles of the distribution of tψ under the null hypothesis, then we start with the following probability statement: P[zα ≤ tψ ≤ z1−α ] = 1 − 2α tψ =

ˆ − ψ) (ψ ˆ σˆ (ψ)

(5.1) (5.2)

The statement (5.1) says that the probability that the random variable tψ lies between zα and z1−α is 1 − 2α; this, of course, follows from the deﬁnition of the quantiles zα and z1−α . The probability statement is then rearranged using the deﬁnition of tψ to become a statement about ψ, as follows: ˆ ≤ψ≤ψ ˆ − zα σˆ (ψ)] ˆ = 1 − 2α ˆ − z1−α σˆ (ψ) P[ψ

(5.3)

This statement says that with probability (1 − 2α), the interval formed by the ˆ − z1−α σˆ (ψ) ˆ and the upper limit ψ ˆ − zα σˆ (ψ), ˆ will contain ψ. Thus, lower limit ψ

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forming such an interval in repeated samples, the interval will contain ψ in 100(1 − 2α)% of the samples. In a particular sample only one such interval can be formed and that is referred to as the conﬁdence interval with conﬁdence level 100(1 − 2α)%; that interval may – or may not – actually contain ψ, the underlying probability basis of the statement (the ‘conﬁdence’ level) refers to the probability statement (5.3). The general notation adopted for the conﬁdence interval so obtained is: (5.4)

Note that this notation allows for an asymmetry in the chosen quantiles; that is, whilst the usual case is α1 = α2 = α, the asymmetric case α1 = α2 , will also be of interest, and will be illustrated below. ˆ = 0.1, As an example of a conﬁdence interval, consider the estimates ψ ˆ = 0.02, with test statistic tψ distributed as N(0, 1) and α1 = α2 = α = 0.05; σˆ (ψ) then z0.05 = –1.645 and z0.95 = 1.645. The two-sided conﬁdence interval is: CI0.05 0.05 = [0.1 − 1.645(0.02), 0.1 − (−1.645)(0.02)] = [0.0671, 0.1329] thus, with 90% ‘conﬁdence’, ψ ∈ [0.0671, 0.1329]. Next, consider the variation with α1 = α2 , say α1 = 0.09 and α2 = 0.01, then z0.09 = –1.3408 and z0.99 = 2.3263 and the asymmetric 90% conﬁdence interval is: CI0.01 0.09 = [0.1 − 2.3263(0.02), 0.1 − (−1.3408)(0.02)] = [0.0535, 0.1268] A one-sided conﬁdence interval can be viewed as the limit of an asymmetric α two-sided interval. The one-sided intervals are obtained as the limits of CIα21 as α2 → 0 and, alternately, as α1 → 0. Referring back to the underlying probability statement in (5.3), ﬁrst modify it to allow an asymmetric setting of the quantiles, so that: ˆ − z1−α σˆ (ψ) ˆ ≤ψ≤ψ ˆ − zα σˆ (ψ)] ˆ = 1 − (α1 + α2 ) P[ψ 1 2

(5.5)

Now let α2 → 0, in which case limα2→0 z1−α2 = ∞ and (5.5) and (5.4) become, respectively: ˆ − zα σˆ (ψ)] ˆ = 1 − α1 P[−∞ ≤ ψ ≤ ψ 1

(5.6)

ˆ ˆ ˆ (ψ)] CI0.00 α1 = [−∞, ψ − zα1 σ

(5.7)

This is referred to as the lower 100(1 − α1 )% conﬁdence interval, because the conﬁdence region is to the left of the ﬁnite limit (and because α1 refers to the

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α

ˆ − z1−α σˆ (ψ), ˆ ψ ˆ − zα σˆ (ψ)] ˆ CIα21 = [ψ 1 2

162 Unit Root Tests in Time Series

lower quantile of the distribution of tψ ). Alternately, let α1 → 0, in which case limα1→0 zα1 = –∞, and (5.5) and (5.4) become, respectively: ˆ ≤ ψ ≤ + ∞] = 1 − α2 ˆ − z1−α σˆ (ψ) P[ψ 2

(5.8)

α2 ˆ − z1−α σˆ (ψ), ˆ + ∞) = [ψ CI0.00 2

(5.9)

ˆ − (−1.645)(0.02)] = (−∞, 0.1329] CI00.05 = (−∞, ψ ˆ − 1.645(0.02), + ∞) = [0.0671, + ∞) = [ψ CI0.05 0 is the upper oneCI00.05 is the lower one-sided conﬁdence interval, and CI0.05 0 sided conﬁdence interval. We may now observe that the intersection of the two one-sided conﬁdence intervals is the two-sided conﬁdence interval; that is: α

α

2 CIα21 = CI0.00 α1 ∩ CI0.00

(5.10)

Continuing with the numerical example: CI0.05 0.05 = (−∞, 0.1329] ∩ [0.0671, + ∞) = [0.0671, 0.01329] this is, of course, just as we found directly. In the case of unit root tests, the focus is usually on the lower one-sided conﬁdence interval, CI0.00 α1 , the rationale being that values of γ = ρ − 1 < 0 correspond to stationary processes, which are of interest, whereas values of γ > 0 correspond to explosive stationary processes. Interest therefore centres on whether γ = 0 is in CI0.00 α1 . The next subsection deals explicitly with the link between hypothesis tests and the construction of conﬁdence intervals. 5.1.2 The link with hypothesis testing The conﬁdence interval also offers a way to test the null hypothesis H0 : ψ = c against either a one-sided or a two-sided alternative hypothesis. In the standard ˆ and hence a particular realisation of CIαα , case, given a particular realisation of ψ, then if the interval includes c, the null hypothesis is not rejected, otherwise it is rejected. This approach is dual to using tψ as a test statistic for H0 : ψ = c; in the case of a two-sided alternative hypothesis, the null hypothesis is not rejected if |tˆψ | < |zα |, where tˆψ is the sample value of tψ , and rejected otherwise. (In this standard case, the t distribution will substitute for the normal distribution in ﬁnite samples.)

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α

2 is referred to as the upper 100(1 – α1 )% conﬁdence interval, because the CI0.00 conﬁdence region is to the right of the ﬁnite limit (and because α2 refers to the upper quantile of the distribution of tψ ). To continue the numerical example, with α1 = 0.05 and α2 = 0.0 and, alternately, with α1 = 0.0 and α2 = 0.05, then (5.7) and (5.9) are, respectively:

Conﬁdence Intervals in AR Models 163

Consider the case where the rejection region comprises ‘large’ positive values of the test statistic tψ for the null hypothesis H0 : ψ = c against the onesided alternative hypothesis HA : ψ > c. As before, for simplicity, assume that tψ ∼ N(0, 1), with quantiles zα1 and z1−α2 , where: ˆ − c) (ψ ˆ σˆ (ψ)

(5.11)

For a one-sided test of size α2 , P[(tψ |H0 ) > z1−α2 ] = α2 , where z1−α2 is the (1 – α2 ) quantile of the distribution of tψ under the null and P[(tψ |H0 ) ≤ z1−α2 ] = (1 – α2 ). For example, if α2 = 0.05, then z0.95 = 1.645 so that P[(tψ |H0 ) >1.645] = 0.05 and Pr[z|H0 ≤ 1.645] = 0.95. Given a sample value of tˆψ of tψ , then H0 is rejected if tˆψ >1.645, and not rejected otherwise. The duality with hypothesis testing arises by considering whether ψ = c is in the conﬁdence interval where, in this case, the interval is (upper or right) onesided and given by (5.9); that is: α2 ˆ − z1−α σˆ (ψ), ˆ ∞) CI0.00 = [ψ 2

ˆ − 1.645ˆσ(ψ), ˆ ∞) = [ψ For example, to continue the numerical example CI0.05 0.00 = [0.0671, + ∞]; thus, any H0 : ψ = c against HA : ψ > c would be rejected for c ≤ 0.0671 at the 100α2 % signiﬁcance level; for example, H0 : ψ = 0 is rejected in favour of ψ > 0 because c = 0 is not in the conﬁdence interval; H0 : ψ = 0.1 is rejected in favour of ψ > 0.1 because c = 0.1 is not in the conﬁdence interval, and so on, until c = 0.0671; for example, H0 : ψ = 0.7 is not rejected because c = 0.7 is in the conﬁdence interval. Carrying out the test of H0 : ψ = 0 against HA : ψ > 0, the test statistic is tˆψ = (0.1 – 0)/0.02 = 5, hence this H0 is rejected; on the other hand, taking c = 0.0671, then tˆψ = (0.1 – 0.0671)/0.02 = 1.645, and hence the decision is one of ‘indifference’ at the 5% signiﬁcance level, because tˆψ = z0.95 . Next, consider the case where the alternative is HA : ψ < c; thus, under H0 , P[(tψ |H0 ) < zα1 ] = α1 and P[(tψ |H0 ) ≥ zα1 ] = (1 – α1 ), where zα1 = –1.645 for α1 = 0.05. The decision rule is: given a sample value of tˆψ of tψ then H0 is rejected if tˆψ < zα1 and not rejected otherwise. Continuing the numerical example, the lower conﬁdence interval is CI0.00 0.05 = (–∞, 0.1329]. Thus any H0 : ψ = c against HA : ψ < c would be rejected for c ≥ 0.1329 at the α1 signiﬁcance level; for example, H0 : ψ = 0 is not rejected in favour of ψ < 0 because c = 0 is in the conﬁdence interval; one could argue intuitively that this much is evident since tˆψ is wrongsigned for rejection in favour of such an HA that ψ is negative. The test statistic for c = 0 is as before; that is, tˆψ = (0.1 – 0)/0.02 = 5, but now the point of comparison is z0.05 = –1.645 and, as tˆψ > z0.05 , H0 is not rejected. The null H0 : ψ = 0.15 would be rejected in favour of HA : ψ <0.15, because c = 0.15 is not in the conﬁdence interval; the test statistic is now tˆψ = (0.1 – 0.15)/0.02 = –2.5 and therefore tˆψ < z0.05 , so this H0 is rejected.

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tψ =

164 Unit Root Tests in Time Series

So far the analysis has assumed that the distribution of tψ is centred on 0 and symmetric, with the implication that, for example, the lower 5% quantile, z0.05 , is the negative of the upper 95% quantile, z0.95 . Further, it also assumes that the quantiles of the distribution of tψ under the null are invariant to ψ. Neither of these, as we shall see, is necessary to the analysis or likely to be the case in AR models; nevertheless, with care, the duality between hypothesis testing and conﬁdence interval construction can be maintained.

5.2.1 Constant quantiles Consider a random variable x with distribution function F(X), such that 0 ≤ F(X) = P(x ≤ X) ≤ 1; the range of F(X) is the interval [0, 1]. For a random variable x, if F(X) is continuous in X, it has an inverse function, Qx (.), which recovers X for a given value of F(X); that is, Qx (F(X)) = X. (An alternative notation, emphasising the inverse nature of this function, is F−1 (X).) From Qx (.) we ﬁnd the quantile; that is, the value of X that corresponds to a particular value of F(X); for example, let x = z, where z ∼ N(0, 1), then F(–1.96) = 0.025 and F(1.96) = 0.975; therefore, from the inverse function, Qx (0.025) = –1.96 and Qx (0.975) = 1.96. Note that, by deﬁnition, F(Qx (0.025)) = 0.025. Also, because of the symmetry of the N(0, 1) distribution, in this case the median is Qx (0.5) = 0, so that F(0) = 0.5. The notation zα and z1−α has been used to refer to the α and 1 – α quantiles of the normal distribution (or tα and t1−α for the t distribution); in terms of the notation of the previous paragraph, these are the quantiles Qx (α) and Qx (1 – α ), respectively, which correspond to F(zα ) = α and F(z1−α ) = 1 – α. If the precise nature of F(.) is not known – for example, because it refers to a ﬁnite sample distribution – it can be approximated by simulation. So far the possible dependence of the functions F(.) and Qx (.) on the underlying parameters has been suppressed. Now suppose that the random variable x may be a function of the scalar parameter ψ. If F(X) and, hence, Qx [F(X)] are unaffected by ψ, a plot of Qx [F(X)], for a particular choice of F(X) ∈ [0, 1], against ψ is just a horizontal line, with the quantile on the vertical axis. For example, choose F(X) = 0.95, then if x is distributed as standard normal, that is x = z, then Qx (0.95) = + 1.645. Invariance to ψ gives a horizontal line of + 1.645 for the 95% quantile when plotted against ψ. More generally, there will be variation in the value of the quantile reﬂecting the dependence of the distribution function for x on ψ. As an illustration, suppose we are interested in F(X) = 0.5, then Qx (0.5) is the median; that is, the 50% percentile. We now seek a convenient notation to indicate the functional dependence of Qx (0.5) on ψ. The notation that is used below is Qx (ψ|0.5); this indicates that the median function for the random variable x is a function

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5.2 The quantile function

Conﬁdence Intervals in AR Models 165

of the parameter ψ. Generally, Qx (ψ|ϑ) refers to the ϑ-quantile; for example, Qx (ψ|0.95) indicates the 0.95 quantile function as a function of ψ. The median function plays an important part in median unbiased estimation, and the quantile functions more generally are important in bootstrapping; both are considered below.

Inherent in the standard approach to conﬁdence interval and hypothesis testing is the idea that the quantiles do not vary with the coefﬁcient of interest. Consider this in the context of forming a (1 – 2α)% conﬁdence interval for ψ, ˆ We will need the t-function; that is, tψ as a function with point estimate ψ. of ψ, tψ (ψ). For ﬁnite samples, tψ has a t distribution, with the α-quantile and the (1 – α)-quantile denoted tα and t1−α , respectively. If these quantiles do not depend on ψ, a graph of tα and t1−α against ψ gives two horizontal lines equidisˆ The graphical interpretation is tant about zero by tα ( = –t1−α ) in units of σˆ (ψ). ˆ aided by plotting the t-function tψ (ψ); for a particular sample, that is given ψ ˆ ˆ and σˆ (ψ), the slope of this function is −1/ˆσ(ψ). The (1 – 2α)% conﬁdence interval is then obtained by projecting the points of intersection of the upper and lower quantiles with the t-function onto the ψ axis; this gives the left-hand and right-hand points of the conﬁdence interval, which is symmetric about the ˆ This is illustrated in Figure 5.1, with ψ ˆ = 0.1 and σˆ (ψ) ˆ = 0.02. point estimate ψ. The 90% conﬁdence interval is constructed by projecting the points of intersection of the t-function with the two quantiles; this gives CI0.05 0.05 = [0.671, 1.329] assuming, for simplicity, that the sample is large enough for tψ to have a normal distribution. 5.2.3 Complications in the AR case The standard approach does not apply generally to AR models. To see this it is sufﬁcient to focus on the AR(1) case. From Chapter 3 we saw that there was a ﬁnite sample bias in the estimation of φ1 and a bias in the estimation of σε2 . For typical economic time series, these factors result in a negative bias for the t statistic, with the implication that, even if constant, the quantiles are centred below the zero axis (quantiles are measured on the vertical axis). However, because the bias is not constant across the parameter space as φ1 varies, the quantiles are not constant either. We also know that for the null hypothesis H0 : φ1 – 1 = 0 in the p AR(1) model (more generally ∑i=1 φi – 1 = 0 in the AR(p) model), the t statistic, tφ1 , does not have the t distribution. Another aspect of this problem is that the actual coverage probability of the conﬁdence interval does not match the nominal conﬁdence level. For example, let σˆ (φˆ 1 ) denote the estimated standard error of the LS estimator of φ1 , then the (1 – 2α)% = 90% two-sided conﬁdence interval for large samples is φˆ 1 ±1.645ˆσ(φˆ 1 ). In repeated samples this conﬁdence interval should include the

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5.2.2 Nonconstancy of the quantiles

166 Unit Root Tests in Time Series

5 4

t-function

assuming constant quantiles

3 95% quantile

2 1 quantile 0

–2

5% quantile

–3 90% confidence interval

–4 –5

0

0.02

0.04

0.06

0.08

0.1 ψ

0.12

0.14

0.16

0.18

0.2

Figure 5.1 5% and 95% quantiles and t-function.

true value of φ1 in 90% of the cases. Further, since it is usually the case that symmetric, two-sided conﬁdence intervals are usually constructed, in 5% of the samples the true value should lie to the left of the lower bound of the conﬁdence interval, and in 5% of the samples the true value should lie to the right of the upper-bound conﬁdence interval. These statements will not hold generally for AR models even for quite large samples. 5.2.3.i Simulation of the LS quantiles To illustrate some key aspects of forming conﬁdence intervals for an AR(1) model, simulated data were generated from yt = φ1 y + εt with εt ∼ niid(0, 1), T = 100; the estimated model was yt = βˆ ∗0 + φˆ 1 y + βˆ ∗1 t + εˆ t , and φ1 was varied over the range φ1 ∈ [–1, 1.05] by intervals of 0.05; the 5% and 95% quantiles of tφ1 were obtained for each value of φ1 in the interval using 20,000 simulations. As φ1 increases, the lower and upper quantiles fall, dipping substantially as φ1 → 1. This is illustrated in Figure 5.2, which plots: (i) the constant lines ±1.650 corresponding to the quantiles of the t distribution for T = 100 with 2α = 0.1; and (ii) the upper and lower simulated quantiles for 2α = 0.1. Taking the nonconstancy of the quantiles into account, a (1 – 2α)% conﬁdence interval, with approximately correct coverage, is obtained by projecting the points of intersection of the t-function with the simulated upper and lower quantiles onto the φ1 axis. Note that this conﬁdence interval will not be symmetric about φˆ 1 ; in particular, for the relevant part of the parameter space, the left-hand limit is closer to φˆ 1 than the right-hand limit because of the dip in

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–1

Conﬁdence Intervals in AR Models 167

3 unit root

theoretical 95% quantile 2 1

simulated 95% quantile 0 quantile theoretical 5% quantile

–2 –3 –4

simulated 5% quantile –0.8 –0.6 –0.4 –0.2

0

0.2

0.4

0.6

0.8

1

φ1 Figure 5.2 Quantiles for LS estimates.

the lower quantile as φ1 → 1. For later reference, a convenient description of these quantiles is that they are the ‘grid’ quantiles, since they relate to varying φ1 over a grid of values, here [–1, 1.05], in the parameter space; this is a development taken up again later in section 5.5, which considers bootstrap conﬁdence intervals. 5.2.3.ii Empirical example: an AR(2) model for US GNP To illustrate the variation in the quantiles, the example of Chapter 3 is continued, which used an estimated AR(2) model, with time trend, for the log of real US GNP over the period 1929–2000, giving 72 observations in all and an effective sample of 70 observations. In this case the coefﬁcient of interest is φ = φ1 + φ2 . The estimated model (in ADF form) is: ∆yt = 0.943 − 0.142yt−1 + 0.588∆yt−1 + 0.005t + εˆ t

(5.12)

ˆ = 0.041; and Thus φˆ = 1 – 0.142 = 0.858; the estimated standard error is σˆ (φ) ˆ = 0.858 ± ˆ the conventional (1 – 2α)% = 90% conﬁdence interval is φ ± tα σˆ (φ) 1.667(0.041) = [0.790, 0.926]. This is illustrated in Figure 5.3 by plotting constant lines at ±1.667 together with the t-function given by (0.858 – φ )/0.041. The projection of the intersection of the t-function and the constant lines onto the φ axis gives the conventional but inaccurate conﬁdence interval. The simulated quantiles are also shown in Figure 5.3 and illustrate the nonconstancy of the quantiles, taking

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–1

168 Unit Root Tests in Time Series

3 t-function based on LS estimates

2 1

simulated 95% quantile

0

–2 simulated 5% quantile confidence interval based on constant quantiles simulated 90% confidence interval

–3 –4 –5 0.75

0.8

0.85

0.9 φ1

0.95

1

1.05

Figure 5.3 Quantiles for LS estimates: US GNP.

into account their dependence on the value of φ. As anticipated from Figure 5.2, both quantiles dip substantially as the unit root is approached. The grid 5% and 95% quantiles are –3.439 and 0.533, respectively, which differ markedly from –1.667 and + 1.667, resulting in a 90% conﬁdence interval of [0.836, 0.999]. This can be viewed graphically by projecting the two points of intersection of the t-function with the simulated quantiles onto the φˆ axis, which places much more of the interval to the right of the estimated value, and nearly includes the unit root; this contrasts with the ‘conventional’ 90% conﬁdence interval that incorrectly assumes constant quantiles. We can now return brieﬂy to the ‘ﬁdelity’ of the conﬁdence intervals, and note that it is the other side of the coin relating to the incorrect size of a test statistic; that is, a test undertaken at size α should result in 100α% rejections of the null hypothesis when the null hypothesis is correct. Consider the situation for the unit root hypothesis. In the AR(2) model above, if no account was taken of the variability in the quantiles, the t test for null that H0 : φ – 1 = 0 against HA : φ – 1 < 0 would be rejected for values of the sample ‘t’ that are larger negative than – 1.667; however, we know that the 5% critical value under the null hypothesis is about –3.3 (see Figure 5.3 where a vertical line from φ = 1 cuts the simulated 5% quantile at about –3.3) so that the null hypothesis would be spuriously rejected (that is, a test statistic between –1.645 and –3.3 should not lead to rejection, but it does); this means that the actual size of the t test is greater than the nominal

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quantile –1

Conﬁdence Intervals in AR Models 169

size and the test is said to be oversized. Using the wrong critical value, the size of the test is not ‘faithful’ to its nominal size – it lacks ‘ﬁdelity’.

5.3 Constructing conﬁdence intervals from median unbiased estimation

5.3.1 Inverse mapping and median unbiased estimation The picture so far is that there is a systematic bias in the LS estimator of the AR coefﬁcients and this bias becomes accentuated as the unit root is approached. Chapter 3 was concerned with obtaining bias adjusted estimators; for example, the ﬁrst-order bias corrected estimator exploits the knowledge that a substantial part of the bias arises from terms of order T. However, constructive use can be made of the knowledge that there is a systematic bias even without knowing how that arises. Provided that there is a monotonic relationship between the bias and φ1 (or φ more generally), this can be exploited to provide an unbiased estimator. So far we have only considered mean bias adjustment through the ˆ T) ≡ E(φ) ˆ − φ, but the same principles apply to median bias bias function b(φ, (see Andrews, 1993; Andrews and Chen, 1994). The ideas are initially illustrated in the context of an AR(1) model. We consider φ1 in the set given by φ1 ∈ Ω = (–1, 1]. An estimator of φ1 , denoted φˆ MU 1 , ˆ is said to be median unbiased if φ1 is the median of the distribution of φMU 1 for each φ1 ∈ Ω. The deﬁnition of median bias for the general estimator φ˜ 1 , and sample size T, then follows as: bmed (φ˜ 1 , T) ≡ med(φ˜ 1 ) − φ1 . By deﬁnition, ˆ MU bmed (φˆ MU 1 , T) = 0. Thus, to emphasise, since φ1 is the median of φ1 , the probabilities of over- and underestimation are exactly matched at 1/2. This is not generally the case for a mean unbiased estimator in the AR(1) model; for example, the distribution of the ﬁrst-order unbiased estimator is generally asymmetric. The central idea of obtaining a median unbiased estimator in the AR(1) model, based on the LS estimator, is due to Andrews (1993) and was extended to AR(p) models by Andrews and Chen (1994). The median unbiased estimator can be motivated by the following illustration. Suppose an estimate of φˆ 1 = 0.8 is obtained in a particular sample, then what is sought is the value of φ1 that would give a median for the LS estimator φˆ 1 of 0.8. If, for example, this median is 0.9, then 0.9 is taken to be the estimate of φ1 not 0.8. To develop this idea more formally we will need the quantile function of section 5.2 specialised to the median function. Recall that in the general notation introduced earlier, Qx (0.5) is the median of the distribution of x, and Qx (ψ|0.5) is the median of the random variable x as a

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This and the next section continue the theme of reducing the bias in estimation of AR models and constructing accurate conﬁdence intervals. In this section we consider Andrews’ median unbiased estimator.

170 Unit Root Tests in Time Series

mapped out. It may be useful to think of this in a simulation context since Qφˆ (.) and 1 the median unbiased estimator will usually have to be found by simulaˆ ˆ (.) indicates the simulation counterpart of Q ˆ (.). The tion; the notation Q φ1 φ1 median is the value of φ1 , such that Pr(φˆ 1 ≤ Q ˆ (φ1 |0.5)) = 0.5 and Pr(φˆ 1 ≥ φ1

Qφˆ (φ1 |0.5)) = 0.5. Let φˆ m 1 be the estimated values of φ1 , referred to as realisa1 tions, from m = 1, ..., M simulations for a given φ1 , and, for convenience, assume that M is odd; then order (sort) the φˆ m 1 from lowest to highest, and the median is the 1/2(M + 1)th element in the sorted, or ranked, list. To ﬁx ideas consider a simple numerical example. Suppose that in the AR(1) DGP, φ1 = 0.9 and, for T = 60, this value of φ1 generates a median for the LS estimator φˆ 1 of 0.8; that is, Qφˆ (φ1 = 0.9|0.5) = 0.8; the estimate is below the 1 true value reﬂecting the negative bias in the LS estimator. Then turning the question around, suppose we have the simulation distribution of φˆ 1 from which we obtain a median of 0.8: what value of φ1 does this correspond to? Of course, we know that the answer is φ1 = 0.9. Now perform a set of simulation experiments for each value of φ1 and, hence, ˆ ˆ (φ1 |0.5), for each permissible value of φ1 ∈ [0, 1]; that is, obtain the obtain Q φ1 median of the distribution of φˆ 1 for each value of φ1 . Graphically, this can be ˆ ˆ (φ1 |0.5) plotted on the vertical axis as a function of φ1 , represented with Q φ1

which is plotted on the horizontal axis. However, given the intention of the simulations, what is of interest is the inverse relationship that obtains φ1 from ˆ −1 a given median of the distribution of φˆ 1 . This relationship is denoted φ1 = Q ˆ φ1

(φ1 |0.5), which recovers the value of φ1 that corresponds to a particular value of the median of φˆ 1 ; its theoretical counterpart is Q −1 ˆ (φ1 |0.5). Note that this is the φ1

inverse relationship in the sense that it concerns the mapping φˆ 1 → φ1 . It is not the mapping between φˆ 1 and the quantiles of the distribution of φˆ 1 ; for example, inverting the median just obtains the distribution function F(.) = 0.5. The ˆ −1 estimator rule that deﬁnes the median unbiased estimator is φˆ MU 1 = Q ˆ (φ1 |0.5); φ1

that is, given the LS estimator φˆ 1 , use the simulated inverse median function to obtain an estimator of φ1 , denoted φˆ MU 1 . (An alternative notation, sometimes used in the literature, is to refer to the median function Qφˆ (φ1 |0.5) as m(φˆ 1 ), and the inverse median function 1 Q −1 (φ1 |0.5) as m−1 (φˆ 1 ); this is clearly a more economical notation, but the φˆ 1

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function of ψ. More generally, Qx (ψ|ϑ) is the ϑ-quantile function. This notation is now specialised to the present context. In particular, the random variable of interest is the LS estimator φˆ 1 , with distribution function F(φˆ 1 |φ1 ) ∈ [0, 1], where this function is written to indicate its dependence on φ1 . The median is that value of φˆ 1 such that F(φˆ 1 |φ1 ) = 0.5. The median function Qφˆ (φ1 |0.5) 1 plots the median of φˆ 1 against φ1 , so that, as φ1 varies, the median function is

Conﬁdence Intervals in AR Models 171

more extensive notation enables a general treatment of quantile functions and the parameters on which they depend.) The required sequence or algorithm is then as follows: (i)

for each φ1 in a grid of values in the interval (–0.999, 1.0], use the simˆ φˆ 1 |φ1 ) = 0.5 to determine the median of ulated distribution function F( ˆ ˆ (φ1 |0.5), which gives one ‘observation’ from ˆφ1 ; this is one point of Q φ1 ˆ −1 (φ1 |0.5); φ1 = Q

(ii)

repeat step (i) for different values of φ1 and so map out the function ˆ −1 (φ1 |0.5); φ1 = Q

(iii)

for a particular application, take the sample estimate φˆ 1 and use the ˆ −1 (φ1 |0.5) to obtain the corresponding estiinverse median function Q ˆ

φˆ 1

φ1

mate of φ1 ; this provides the median unbiased estimate φˆ MU 1 . If the LS estimate falls outside the set (–0.999, 1.0], the median unbiased estimate φˆ MU is taken to be –1 or + 1, respectively. 1 Clearly, this idea generalises to the inverse ϑ-quantile function Q −1 ˆ (φ1 |ϑ), φ1

which gives the value of φ1 corresponding to the ϑ-quantile of φˆ 1 ; the unknown ϑ-quantile functions are replaced by their simulation counterparts and enable conﬁdence intervals to be constructed. To illustrate the use of the quantile functions consider Figure 5.4, which graphs the functions for the median and the 5% and 95% quantiles; it is based on simulations of the AR(1) model, with a constant, for T = 100. This ﬁgure illustrates the case where φ1 = 0.82 (see the horizontal axis); then the median of the simulation distribution of the LS estimates is 0.791, which ˆ ˆ (φ1 = 0.82|0.5) = 0.791. The corresponding 5% and 95% quantiles are is Q φ1 ˆ ˆ (φ1 = 0.82|0.05) = 0.661 and Q ˆ ˆ (φ1 = 0.82|0.95) = 0.877, respectively. These Q φ1

φ1

are obtained by projecting the point corresponding to φ1 = 0.82 vertically until it cuts, respectively, the 5% quantile function, the median function and the 95% quantile function; these are then projected across to the φˆ 1 axis. The quantile functions are monotonic and they narrow as the unit root is approached, both at + 1 and –1, where the distribution of estimated values of φ1 effectively collapses. (Practically, the quantiles are not quite symmetric about φ1 = 0, with, for example, the absolute value of the median slightly greater for φ1 < 0 compared to φ1 > 0; but symmetry holds asymptotically.) The simulated inverse functions for the 5% quantile, the median and the 95% ˆ −1 (φ1 |0.05), Q ˆ −1 (φ1 |0.5) and Q ˆ −1 (φ1 |0.95), respectively, are quantile, that is, Q ˆ ˆ ˆ φ1

φ1

φ1

then graphed in Figure 5.5. Now φˆ 1 is on the horizontal axis and φ1 is on the vertical axis; as a result, the ordering of the quantile functions, other than the median, is reversed with the inverse 5% quantile function above the inverse 95% quantile function. Consider the inverse median function: in this case the

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φˆ 1

172 Unit Root Tests in Time Series

1 0.8 0.6

95% quantile = 0.877 median = 0.791 5% quantile = 0.661

0.4 0.2 0 –0.2

median function

95% quantile function

5% quantile function

–0.4 –0.6

actual φ1 = 0.82

–0.8 –1 –1 –0.8 –0.6 –0.4 –0.2

0

0.2 φ1

0.4

0.6

0.8

1

Figure 5.4 Quantile functions.

1 0.8 0.6 0.4

estimate of 0.791 implies a median of 0.82 inverse median

0.2 φ1

0 –0.2 –0.4 –0.6 –0.8

inverse 5% quantile inverse 95% quantile estimate of 0.791

–1 –1 –0.8 –0.6 –0.4 –0.2 0 0.2 estimated φ1

0.4

0.6

0.8

1

Figure 5.5 Inverse quantile functions.

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estimated φ1

Conﬁdence Intervals in AR Models 173

Table 5.1 Quantiles to obtain a median unbiased estimator.

φ1 1.00 0.98 0.96 0.94 0.92 0.90 0.88 0.86 0.84 0.82 0.80 0.78 0.76 0.74 0.72 0.70

0.05 quantile

0.5 quantile (median)

0.95 quantile

0.866 0.846 0.823 0.803 0.778 0.755 0.733 0.711 0.685 0.661 0.643 0.620 0.596 0.575 0.555 0.532

0.957 0.942 0.925 0.907 0.889 0.869 0.850 0.831 0.812 0.791 0.773 0.754 0.733 0.714 0.695 0.675

0.999 0.988 0.975 0.962 0.949 0.934 0.922 0.907 0.891 0.877 0.862 0.846 0.830 0.813 0.799 0.782

input is φˆ 1 ; to continue the example of the previous paragraph, let φˆ 1 = 0.791, ˆ −1 (φ1 |0.5) and across to the vertical axis, then projecting up to the function Q ˆ φ1

we see (as of course we know from Figure 5.4) that this corresponds to φ1 = 0.82. Hence in this case the median unbiased estimate, φˆ MU 1 , associated with the LS estimate of 0.791 is φˆ MU = 0.82. An extract from the data underlying Figures 5.4 1 and 5.5 is given in Table 5.1, and will be used to show how the more extensive tables in Andrews (1993) can be used. To conﬁrm understanding, note that when φ1 = 0.82, the median of the LS estimates is 0.791, and these numbers are shown in bold in Table 5.1.

5.3.2 Conﬁdence intervals based on median unbiased estimation Andrews (1993) has also suggested a method for obtaining conﬁdence intervals based on median unbiased estimation. Given a point estimate, the idea of a conﬁdence interval is to bracket the true value φ1 with 100(1 – 2α)% conﬁdence; that is, in 100(1 – 2α)% of cases in which the interval is constructed in this manner, the interval will contain φ1 . The proportion of cases in which the interval includes φ1 is the coverage probability. A 90% conﬁdence interval can be obtained from the data in Table 5.1 as follows. The upper end of the interval is found from the 5% quantile, where the φˆ 1 entry is 0.791; linear interpolation is accurate for these purposes. The value 0.791 is between the 5% quantile for φ1 = 0.94 and 0.92, and by linear interpolation we obtain 0.933. The lower end

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quantiles of LS estimator φˆ 1 of φ1

of the interval is between the 0.95 quantile for 0.72 and 0.7, and by interpolation we obtain 0.709. The 90% conﬁdence interval by this method is therefore [0.709, 0.933]. The conﬁdence interval can be obtained from either Figure 5.4 or 5.5, provided some care is taken to remember that the input is φˆ 1 = 0.791. We will use Figure 5.4 leaving, the use of Figure 5.5 to an exercise and, to assist, the part of Figure 5.4 around φˆ 1 = 0.791 is magniﬁed and shown in Figure 5.6. (One way to view the use of Figure 5.4 is that, as in the case of inference, we are in a sense going ‘backwards’ to infer something about φ1 from knowledge of one sample only, where φˆ 1 = 0.791 in that sample.) Given φˆ 1 = 0.791, a horizontal line is projected across the quantile functions; where this line cuts the 5% and 95% quantiles, perpendicular lines are dropped to the horizontal axis to obtain the range of φ1 values consistent with φˆ 1 = 0.791. (Of course, the line projected from φˆ 1 also cuts the median function at φ1 = 0.82.) This range is the 90% conﬁdence interval [0.709, 0.933]; note that the interval is not symmetric about the LS estimate. Also note that the conﬁdence interval is not constructed in the conventional manner by means of critical values from the t distribution and an estimated standard error.

1 0.95 95% quantile 0.9 0.85 estimated φ1 0.8

5% quantile

point estimate φ1 = 0.791

0.75 0.7 90% confidence interval φ1 = (0.709, 0.933)

0.65 0.6 0.6

0.65

0.7

0.75

0.8 φ1

0.85

0.9

0.95

1

Figure 5.6 90% conﬁdence interval.

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Conﬁdence Intervals in AR Models 175

5.3.3 Extension of the median unbiased method to general AR(p) models Andrews and Chen (1994) deal with the extension of the median unbiased method to general AR(p) models. The principle is the same but the complication is the presence of some ‘nuisance’ parameters. The starting point is the AR(p) model, which it is convenient to reparameterise as: p−1

(5.13)

(This reparameterisation is a variant of the ADF(p – 1) form of an AR(p) model; see Chapter 3, section 3.8.1.i.) The coefﬁcients on the ∆yt−j variables are a nuisance to estimation and inference on φ. Andrews and Chen (1994) proposed an iterative scheme to obtain a median unbiased estimator of φ: ﬁrst estimate (5.13) by least squares, and obtain (φˆ (1) , ϕˆ (1) ) where ϕˆ (1) = (β∗0 , cˆ 1 , . . . , cˆ p−1 , βˆ ∗1 ), and deﬁne zt = (1, ∆yt−1 , . . . , ∆yt−(p−1) , t); next, with the same procedure as in the AR(1) case, use the inverse median function to obtain a starting value for the median unbiased estimate and call this φˆ MU(1) . Then estimate the constrained regression with dependent variable yt −φˆ MU(1) yt−1 and regressors zt ; denote the resulting LS estimates ϕˆ (2) . Next, by regressing yt −ϕˆ (2) zt on yt−1 , obtain the second LS estimate φˆ (2) and hence obtain φˆ MU(2) . Deﬁne yt −φˆ MU(2) yt−1 and proceed with iterations as before, either for a ﬁxed number of iterations or convergence on φˆ MU(j) . With the ﬁnal value denoted φˆ MU(J) , the ﬁnal values of the nuisance parameters, ϕˆ (J + 1) , are obtained in a last regression conditional on this value.

5.4 Quantiles using bias adjusted estimators Another strategy for obtaining conﬁdence intervals that have a closer correspondence between the nominal coverage probability and the actual coverage probability is to use one of the bias adjusted estimators described in Chapter 3. A bias adjusted estimator recentres the LS estimator, and its t statistic, to counter the variation induced as φ → 1 (and, of lesser practical importance, as φ → –1). The bias adjusted estimators considered in Chapter 3 were the ﬁrstorder bias corrected (FOBC) estimator, the recursive mean (or trend) adjusted least squares (RMALS) estimator, and the bootstrapped bias corrected (BSBC) estimator. To illustrate the effect on the quantiles of recentring the estimator, the next section describes the results of a small simulation experiment based on the AR(1) model, with and without a trend (the set-up is as described in Chapter 4, section 4.5, and the reported results are illustrative of a much larger set of simulations that considered AR models to order 4). 5.4.1 Quantiles The results for T = 100 (to illustrate the general effects) are reported in Figures 5.7 and 5.8, with each ﬁgure comprising two panels. Figure 5.7 refers to the 5%

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yt = β∗0 + φyt−1 + ∑j=1 cj ∆yt−j + β∗1 t + εt

176 Unit Root Tests in Time Series

–1.5

RMALS

–2

FOBC BSBC

actual quantile –2.5

LS

No trend –3 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

–1 RMA

–1.5 –2 actual quantile

FOBC

–2.5 –3 –3.5 0.2

BSBC

With trend LS

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

φ1 Figure 5.7 5% quantiles for different estimators.

quantile, with the upper panel referring to the no-trend case and the lower panel to the with-trend case. In a similar way, Figure 5.8 refers to the 95% quantile. The 5% quantile is usually the more relevant quantile given interest in whether a particular estimate is close to the unit root. First, consider the 5% quantile in the case where the estimated regression includes a constant, which is illustrated in Figure 5.7, upper panel. Notice that all of the bias adjusted estimators deliver quantiles that show much less variation across the parameter space and are closer to the constant quantile (which is –1.66 in this case). The dip, associated with the move to the unit root, is now postponed until later, at around φ1 ≈ 0.95, and is less pronounced than for the LS estimator. Although there is little to choose among the different bias adjustment methods, the quantiles based on the BSBC and FOBC estimators are generally closest to the constant quantile, although the former suffers as φ1 → 1. Although the general picture is the same when a trend is included, the advantage is rather more in favour of the FOBC and BSBC estimators (see the lower panel in Figure 5.7). Figure 5.8 graphs the 95% quantile. In the case of no trend, illustrated in the upper panel, the quantiles associated with the RMALS estimator are best, whereas in the with-trend case (see the lower panel), the best method results from bias adjustment using the FOBC estimator.

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φ1

Conﬁdence Intervals in AR Models 177

3 BSBC

2

FOBC

actual quantile

RMALS

1 No trend 0 0.2

0.3

0.4

LS

0.5

0.6

0.7

0.8

0.9

1

1.1

3 RMALS

BSBC

2 FOBC

actual quantile 1 0

With trend LS

–1 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

φ1 Figure 5.8 95% quantiles for different estimators.

5.4.2 Coverage probabilities Another way of assessing the effect of bias reduction is to consider coverage of the conﬁdence intervals based alternatively on the estimators resulting from least squares (LS) and the bias adjusted methods; that is, in each simulation form the 90% conﬁdence interval using tα and t1−α , and count the number of times that the interval includes the true value of φ1 , then express this relative to the number of simulations. Table 5.2 summarises the simulated coverage probabilities for the AR(1) case for some selected values of φ1 (similar results in terms of numerical accuracy and ranking and were obtained for higher-order AR models). Note that the table entries distinguish between the two one-sided conﬁdence intervals and the (combined) two-sided conﬁdence interval in order to assess whether errors are more likely on one side. The notation used in Table 5.2 was deﬁned in section 5.1.1 and has the following meaning: CI0.05 0.05 is the two-sided conﬁdence interval, with a nominal 5% 0.00 in each tail; and CI0.05 0.00 and CI0.05 are, respectively, the one-sided nominal 95% conﬁdence intervals. The conﬁdence interval of particular interest in testing for a unit root is CI0.00 0.05 . As an example of interpreting the table entries, consider the column entry for φ1 = 0.75 in the no-trend case. At ﬁrst sight, the actual LS coverage probability

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φ1

178 Unit Root Tests in Time Series

Table 5.2 Simulated coverage probabilities.

95%

95%

φ1 = 0.75

Nominal level No trend

0.00 0.05 0.05 0.00 0.05 0.05 0.00 0.05 CI0.05 0.00 CI0.05 CI0.05 CI0.00 CI0.05 CI0.05 CI0.00 CI0.05 CI0.05

LS FOBC BSBC RMALS

98.7 94.1 94.0 94.6

With trend

0.00 0.05 0.05 0.00 0.05 0.05 0.00 0.05 CI0.05 0.00 CI0.05 CI0.05 CI0.00 CI0.05 CI0.05 CI0.00 CI0.05 CI0.05

LS FOBC BSBC RMALS

98.7 94.7 94.1 89.7

93.3 96.0 95.5 96.3

90.4 96.0 95.2 97.8

90%

91.0 90.1 89.5 90.9

89.1 90.7 89.3 87.5

95%

97.8 93.4 92.5 93.9

99.8 93.3 91.9 85.8

95%

φ1 = 0.96

90.9 95.6 95.1 95.6

85.4 95.1 94.4 98.1

90%

88.7 89.0 87.6 89.9

87.5 88.4 86.2 83.8

95%

99.7 89.8 86.2 95.6

100.0 93.4 90.5 91.0

95%

78.9 94.3 93.4 94.5

58.2 90.7 88.8 96.7

90%

78.6 84.1 79.7 90.0

58.1 84.5 79.4 87.8

Notes: Table entries are actual coverage from simulations using quantiles from the t distribution; for perfect ﬁdelity, simulated values should equal nominal values. The entries relate to the two one-sided 0.00 conﬁdence intervals CI0.05 0.00 and CI0.05 , with 95% nominal coverage and then the corresponding two, with 90% nominal coverage. sided conﬁdence interval, CI0.05 0.05

of 88.7% for CI0.05 0.05 looks accurate, or at least quite close to the nominal probability of 90%; however, as the two one-sided conﬁdence intervals reveal, there are compensating errors of over- and under-coverage with respect to the two 0.00 one-sided intervals CI0.05 0.00 and CI0.05 , where these are 97.8% and 90.9%, respectively. The situation is dramatically worsened as the unit root is approached. However, for example, in the case of bias adjustment by the method of recursive mean adjustment, the ﬁdelity of the nominal and actual coverage is very 0.05 well maintained even for φ1 = 0.96, where CI0.05 0.05 = 90.0%, CI0.00 = 95.6% and 0.00 CI0.05 = 94.5%. The ability of the bias adjustment methods to maintain the ﬁdelity of the nominal probability as the unit root is approached is not quite as good in the with-trend case. Even so, there are gains to be had from bias adjustment using FOBC and RMALS.

5.4.3 Section summary In summary, it is possible to adjust the LS estimator in a reasonably economical way to enable a reduction in the bias and, provided that the (dominant) root is not too close to the unit root, ensure that the nominal coverage of conﬁdence intervals, formed in the conventional way, have a much closer ﬁdelity to the nominal level of coverage. However, notwithstanding this economy, which in a sense is a by-product of getting better estimates of the autoregressive parameters

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φ1 = 0.51

Conﬁdence Intervals in AR Models 179

than is provided by LS in ﬁnite samples, the situation becomes increasingly challenging as the unit root is approached. Median unbiased estimation offers one solution, and bootstrapping, considered in the next section (see also Chapters 4 and 8), offers another.

The problem with the usual approach to forming conﬁdence intervals, and hence also for hypothesis testing, is that when φ1 , or φ more generally, is close to the unit root, the corresponding ‘t’ statistic does not have a t distribution, and even for a substantial sample size the ‘t’ statistic is not distributed as N(0, 1). As we have seen, this means that conﬁdence intervals formed in the standard way do not have correct coverage, and the actual size of a hypothesis test will not match the nominal size (or signiﬁcance level) of the test. This section describes methods of solving this problem based on bootstrapping, which can be viewed as an extension of the grid quantile method described in section 5.2.3.i and applied there to the LS estimates. The use of the bootstrap here differs in motivation from that in Chapter 4, where the concern was to provide a bias adjusted estimator; here, the focus is directly on forming conﬁdence intervals, in which case the central concern is to establish the ‘actual’ or simulated distribution of the bootstrapped ‘t’ statistic and then use that distribution, rather than an assumed distribution, to form conﬁdence intervals. Section 5.5.1 describes the percentile-t bootstrap conﬁdence interval, which is then extended in section 5.5.2 to a ‘grid’ version of the same general principle. 5.5.1 Bootstrap conﬁdence intervals The standard bootstrap percentile-t conﬁdence interval (to be described) is not a complete solution to the problem, but it is useful to start with it for two reasons: ﬁrst, it is part of the grid-bootstrap procedure to be described in section 5.5.1.ii, which does solve the problem; second, it is the bootstrap extension of simply simulating the LS quantiles at the LS point estimate to avoid using the assumption that the quantiles are correctly obtained from the t distribution, which is considered ﬁrst. 5.5.1.i The bootstrap percentile-t conﬁdence interval The procedure described brieﬂy here is a special case of the grid bootstrap, which is considered in greater detail in the next section. The aim is to obtain the (bootˆ where tφ = (φˆ − φ)/ˆσ(φ); ˆ the strap) sampling distribution of the t statistic for φ, notation now explicitly recognises that the statistic forming the basis of the conﬁdence interval is the t statistic. As before, the parameter of interest is φ; that is, the sum of the AR(p) coefﬁcients if p > 1, and the AR(1) coefﬁcient otherwise. In the bootstrap percentile-t method, the ﬁrst step is to estimate the model by

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5.5 Bootstrap conﬁdence intervals

180 Unit Root Tests in Time Series

from the sequence of LS residuals {ˆεt }Tt= p + 1 . For each of B replications, a regression is estimated that mimics the initial LS regression using the bootstrap data ybt in place of the actual data; starting values are usually taken as a block of length p from the sequence of actual data {yt }Tt=1 , a usual choice being the p initial values. ˆ σ(φ), ˆ The t statistic is calculated for each replication and denoted tbφ = (φˆ b − φ)/ˆ where a b superscript indicates the value from the b-th replication; then the bootstrap t distribution corresponds to sorted values of tbφ for b = 1, ..., B. (In a variation, it is valid to use the estimated standard error, σˆ (φˆ b ), from the bootˆ from the initial LS estimation.) If the empirical strap replication rather than σˆ (φ) model upon which the bootstrap is based is AR(p), p > 1, the simulations use the LS estimates of all AR coefﬁcients. Denote the lower and upper quantiles of the bootstrap distribution of tbφ

bs as tbs φ,α and tφ,1−α , respectively; then for an overall 2α signiﬁcance level, the 100(1 − 2α)% bootstrap conﬁdence interval (CI) is formed as: bs ˆ ˆ φˆ − tbs ˆ CIα,bs ˆ (φ), ˆ (φ)] α = [φ − tφ,1−α σ φ,α σ

(5.14)

To illustrate, we return to the AR(2) model for US GNP (see section 5.2.3.i), which is formulated to highlight the parameter of interest, in this case the sum of the AR coefﬁcients φ. The model is: yt = β∗0 + φyt−1 + c1 ∆yt−1 + β∗1 t + εt

(5.15)

LS estimation resulted in: yˆ t = 0.943 + 0.858yt−1 + 0.588∆yt−1 + 0.005t

(5.16)

As before, interest lies in forming a conﬁdence interval for the coefﬁcient on yt−1 . The bootstrap t distribution of tbφ was simulated using B = 24,999 replica-

tions, with the cumulative distribution function of tbφ , denoted F(tbφ ). A plot of this function is given in Figure 5.9a, with the quantiles on the horizontal axis and F(tbφ ) on vertical axis. Note that by construction 0 ≤ F(tbφ ) ≤ 1 and about 75% of the distribution lies to the left of zero. Also of interest, in the case of forming conﬁdence intervals, is the inverse function, denoted F−1 (tbφ ),

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LS and save the estimated coefﬁcients and residuals; the model is most conveniently estimated in ADF form, with yt as the dependent variable, so that the estimated coefﬁcient on yt−1 is φˆ and the coefﬁcients on the lagged ∆yt terms are cˆ j , j = 1, . . . , p – 1. This regression is conveniently referred to as the initial LS regression. The bootstrap recursion is based on the LS estimates, with draws with replacement from the LS residuals; this generates the bootstrap values of yt , denoted ˆ b + ∑p−1 cˆ j ∆yb + eˆ t , where the eˆ t are drawn with replacement ybt , from ybt = φy t−j t−1 j=1

Conﬁdence Intervals in AR Models 181

1 bootstrap F(tb)

0.8 0.6 F(tb)

75% of distribution to left of zero

0.4 0.2

–2.5

–2

–1.5

–1

–0.5 0 quantiles

0.5

1

1.5

2

Figure 5.9a Simulated cumulative distribution function.

2 95% quantile = 0.895

1 0 quantiles

5% quantile = –2.288

–1 –2 –3

0.05 0

0.1

0.95

bootstrap F–1(tb)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

F(tb) Figure 5.9b Simulated inverse cumulative distribution function.

which enables the quantiles to be determined for a given value of F(tbφ ); this inverse function is graphed in Figure 5.9b. For example, the 5% quantile is obtained by projecting a vertical line from the horizontal axis at F(tbφ ) = 0.05

until it intersects F−1 (tbφ ), and then projecting horizontally until the line intersects the vertical axis to give the corresponding 5% quantile. The 5% and 95% bs quantiles are tbs φ,0.05 = –2.288 and tφ,0.95 = 0.895, hence the 90% bootstrap CI

is CI0.05,bs 0.05 = [0.858 –0.895(0.041), 0.858 – (–2.288)(0.041)] = [0.821, 0.952], an interval length of 0.131. Recall that the 90% CI using the simulated LS distribution was [0.833, 0.999], an interval length of 0.164. Moreover, the right limit of the bootstrap conﬁdence interval is further away from a unit root. However, the problem with the standard percentile-t bootstrap CI is that it just plots the quantiles at the ﬁxed point corresponding to the estimate φˆ = 0.858. It takes account of the particular nature of the sample residuals on which to base the draws for the bootstrap replications, but that is all. The implicit assumption in this method is that the bootstrap quantiles are constant across the range of φ, and we have already seen with the simulated LS quantiles that this is not the case. What is needed is the bootstrap extension of the method described in

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0 –3

182 Unit Root Tests in Time Series

section 5.2.3.i, which was based on simulation of the relevant quantiles for a grid of values for φ.

The grid-bootstrap conﬁdence interval was suggested by Hansen (1999); this is valid for a DGP with a unit root and offers accurate coverage probabilities across the complete parameter space. As Hansen (1999) reports, other methods also do well, but not across the whole parameter space. For example, Stock’s (1991) nonsimulation method gets accurate coverage when the dominant root is close to 1, but not such good coverage with a clearly stationary root. Hansen (1999) suggests repeating the bootstrap/simulation process for a grid of values for φ, obtaining the quantiles as a function of φ; obviously, at the point ˆ the quantiles will be the same as the percentile-t bootstrap quantiles. This φ = φ, is why the percentile-t conﬁdence interval of the previous section is a special case of the grid procedure. By noting the likely range of the grid from initial LS estimation, the computational burden in a particular application can be eased. For example, with φˆ = 0.858, a grid from 0.75 to 1 in increments of 0.05 results in 50 points. Since this is not a simulation of a bootstrap, it is not a lengthy procedure to use a ﬁne grid. If the grid is not particularly ﬁne, one might also consider smoothing the bootstrapped quantiles using a nonparametric estimator, see Hansen (1999, p.597). We omit this step, and assume that a ﬁne grid, based on prior inspection of the likely range, is feasible. The simulation procedure for the grid bootstrap is outlined in the following nine steps for the AR(2) model ( = ADF(1)) that is relevant to the example of US GNP. As before, we assume that the goal of the process is to obtain the 100(1–2α)% grid-bootstrap conﬁdence interval for φ. 1.

The maintained ADF(1) regression is: yt = µ∗ + φyt−1 + c1 ∆yt−1 + βt + εt

2.

The corresponding (initial) regression estimated by LS is: ˆ + εˆ t ˆ t−1 + cˆ 1 ∆yt−1 + βt ˆ ∗ + φy yt = µ

3. 4.

(5.17)

(5.18)

ˆ for φ; ˆ and {ˆεt }T is the (sample) sequence with estimated standard error σˆ (φ) 3 of (unrestricted) LS residuals. ˆ that is, φr ∈ Gφ = [φˆ ± kˆσ(φ)]; ˆ Form a grid, or set Gφ , of points based on φ; k = 5 should be sufﬁcient. The restricted regression model for each φr ∈ Gφ is: yrt = µ∗r + cr1 ∆yt−1 + βr t + εrt

(5.19)

where yrt ≡ yt − φr yt−1 .

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5.5.1.ii The grid-bootstrap percentile-t conﬁdence interval

Conﬁdence Intervals in AR Models 183

The estimated restricted regression for each value of φr is given by: ˆ ∗r + cˆ r1 ∆yt−1 + βˆ r t + εˆ rt yrt = µ

6.

(5.20)

This regression imposes φ = φr and generates a sequence of restricted residuals {ˆεrt }Tt=3 and empirical distribution function (EDF), denoted F(ˆεr ), for each φr ∈ Gφ . Generate replicates (simulated data) from the bootstrap recursion using φr of step 3 and cˆ r of step 5: ybt = φr ybt−1 + cr1 ∆ybt−1 + eˆ rt

7.

(5.21) F(ˆεr )

eˆ rt

The are drawn, with replacement, from of step 5, for each value of φr ∈ Gφ . Denote the bootstrap replication index as b, b = 1, ..., B. The constant and time trend can be omitted in the replication step as the sampling distribution of the t statistic is invariant to the values of µ and β. Estimate the following regression model using the bootstrap data for b = 1, ..., B: ybt = µ∗ + φybT−1 + c1 ∆ybT−1 + βt + eˆ t

(5.22)

For each replication, compute the t statistic t(φr )G,b = (φˆ b − φr )/ˆσ(φˆ b ) and sort the B values to obtain an empirical distribution of t(φr )G,b for each φr ∈ Gφ ; thus there will be as many distributions as there are points in the ˆ grid Gφ . An alternative to using σˆ (φˆ b ) in the denominator is to use σˆ (φ) 8.

9.

from step 2 above. Test inversion step: obtain the quantiles of interest as in the percentile bootstrap, but for each φr ∈ Gφ . For example, if B = 1,999, the ϑ-quantile is the [(B + 1)ϑ]-th element of the sorted t-values; for the 0.9 quantile (90%-tile) and B = 1,999, this is the 1,800th ordered element of the simulated distribution of t(φr )G,b . Typically, the quantiles of interest are those for ϑ = α, and ϑ = 1 – α; for example, ϑ = 0.05 and 0.95; denote the gridbootstrap quantiles by tG,bs , where the G superscript indicates that the ϑ quantile is allowed to vary for each value of φr ∈ Gφ . Form the grid-bootstrap (1 – 2α)% conﬁdence interval: G,bs ˆ ˆ ˆ ˆ CIα,bs ˆ (φ), φ − tG,bs σˆ (φ)] α α,G = [φ − t1−α σ

for each φr ∈ Gφ

(This conﬁdence interval assumes that the size of the interval is set symmetrically, but this is not necessary.) The US GNP example of section 5.2.3.i is used to illustrate this procedure. The ˆ = 0.041, initially suggesting a grid relevant LS estimates are φˆ = 0.858 and σˆ (φ) over the range [0.66, 1.06]. However, a grid search conﬁrmed the anticipation that the left end-point of the CI would be much closer to φˆ than the right endpoint, and a relatively ﬁne grid of 80 evenly spaced points on the interval [0.74, 1.06] was used, with B = 9,999 bootstrap replications.

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5.

184 Unit Root Tests in Time Series

Our interest is in obtaining the grid-bootstrap quantiles, tG,bs and tG,bs α 1−α , in order to form the 90% conﬁdence interval. From the bootstrap replications, the 95% quantile was straightforward and gave tG,bs 0.95 = 0.975. The 5% quantile has a tangent close to the t-function (a situation Hansen observed in one of his examples), but a ﬁner grid search gave tG,bs 0.05 = –3.268. Hence the 90% gridbootstrap conﬁdence interval is:

= [0.818, 0.992] All of this is shown in Figure 5.10, which plots the quantiles and the t-function; the interval CI0.05,bs 0.05,G is obtained by projecting the intersection of the quantiles with the t-function onto the horizontal axis. The difﬁculty in obtaining the right end-point is evident on the ﬁgure, given that the t-function and the quantile function are virtually tangential close to the unit root. Note that this interval is larger than the percentile-t bootstrap interval; this occurs because in that method the quantiles are those at φr = 0.858, but the quantiles are not constant and, in particular, tG,bs 0.05 becomes noticeably more negative as φ → 1. This serves to emphasise the point that an appropriate method of forming a conﬁdence interval must take into account the variation in the quantiles. The difference that bootstrapping makes in this example can be seen by comparG,bs ing tG,bs 0.05 = 0.975 and t0.95 = –3.268, resulting in a 90% grid-bootstrap conﬁdence

interval of CI0.05,bs 0.05,G = [0.818, 0.992], with the grid LS quantiles of 0.533 and – 3.439, respectively. The latter were used in Figure 5.3 and resulted in a 90% grid LS conﬁdence interval of CI0.05 0.05 = [0.836, 0.999]. In this case, the difference occurs primarily to the left limit of the conﬁdence interval, with a relatively minor change to the right limit. In some situations a one-sided conﬁdence interval may be of greater interest than the two-sided interval. The 100(1 – α)% one-sided conﬁdence interval corresponds to the set of non-rejected null hypotheses against the appropriate one-sided alternative hypothesis at an 100α% (one-sided) signiﬁcance level. However, it is important to take into account any variation that may exist in the quantiles of the test statistic over the range of φ. The relevant conﬁdence interval G,bs ˆ ˆ to see if the unit root is included is CIα,bs σˆ (φ)]. For exam0.00,G = (–∞, φ – tα ple, continuing the empirical example, the 5% one-sided conﬁdence interval G,bs ˆ ˆ of interest is CI0.00,bs ˆ (φ)] = (–∞, 0.992]. 0.05,G = (–∞, φ – t0.05 σ

5.6 Concluding remarks In the case of AR models, and viewing them as approximations to more general models, caution should be exercised in transferring across standard assumptions about constant quantiles and the construction of conﬁdence intervals. There are

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CI0.05,bs 0.05,G = [0.858 − 0.975(0.041), 0.858 − (−3.268)(0.041)]

Conﬁdence Intervals in AR Models 185

3 t-function 2 bootstrap 95% quantile

1

–1 bootstrap 5% quantile

–2

–3 –4 0.75

bootstrap 90% confidence interval 0.8

0.85

0.9 φ

0.95

1

1.05

Figure 5.10 Bootstrap quantiles: US GNP.

three possible approaches to the problem of incorrect inference. A simple, but only proximate, solution is to base a conﬁdence interval on a bias adjusted estimator; apart from the bootstrap bias corrected estimator, the methods illustrated here do not require a simulation stage and can be calculated in one step. They offer an improvement in the accuracy of the coverage probability of conﬁdence intervals at little cost. The two other approaches are just variations on a single theme, that of simulation in order to obtain better estimates of the relevant quantiles. First, in the case of the median unbiased estimator and associated conﬁdence intervals, simulation is used in the ﬁrst step to obtain a function that maps the LS estimate into the median; similarly, the LS estimate can be mapped into other quantiles in order to provide a simulation estimate of the conﬁdence interval. The method suggested by Andrews (1993) uses draws from N(0, 1) in the simulation but, equally, a bootstrap version could be constructed, which takes draws with replacement from the empirical distribution function based on the LS residuals from initial estimation. The median unbiased method is not explicitly based on a percentile-t approach to constructing a conﬁdence interval. However, this is the basis of the other simulation methods described in this chapter. Simulation is used over a grid of values of the parameter of interest – here φ, the sum of the AR(p) coefﬁcients – to obtain G simulated quantile functions of the t statistic, one for each

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0

value of φ in the grid. Simulating the quantile functions in this way enables the quantiles to be plotted over the range of the parameter space that includes all possible values of φ. Graphically, the intersection of the t-function with the relevant quantile functions provides the estimated conﬁdence interval. Two variants on this method were suggested; both were grid based, differing only in whether the random input was drawn from N(0, 1) or on a bootstrap basis from the empirical distribution function deﬁned by the LS residuals. Simulating the quantiles just at the estimate φˆ takes into account the difference from the standard quantiles, but only at one point; simulation across the grid captures the variation necessary to construct a conﬁdence interval with correct coverage. The bootstrap principle has wide application in econometrics and its use in obtaining explicit bootstrap unit root tests is considered in Chapter 8; in principle, these focus on one point in the grid of the grid bootstrap, which is the bootstrap generated under the null of a unit root. However, the opportunity is taken to develop a number of practical complications in constructing a bootstrap unit root test. Chapter 12 considers the construction of conﬁdence intervals with the beneﬁt of test statistics which are, under some circumstances, more powerful than the simple t-type test used in this chapter.

Questions Q5.1 Why is the assumption of constant quantiles, underlying the conventional approach to the construction of conﬁdence intervals, inappropriate for an AR model? A5.1 In the case of an AR model, the LS quantiles are not constant across the parameter space even for large samples and for values of the LS coefﬁcients well into the stationary region; for example, as illustrated in Figure 5.2, the 5% quantile varies from –1.40 at φ1 = –0.99 through to –3.30 at φ1 = 0.99; it comes closest to the constant quantile of –1.645 around φ1 = –0.56, where it is –1.648. This occurs because the zero bias ﬁxed point for an AR(1) model with estimated constant is at φ1 = –0.5. If constancy is incorrectly assumed, inference is misleading. For example, the conﬁdence intervals will not have coverage that matches their nominal level of conﬁdence. The simulations reported in Table 5.2 show that if φ1 = 0.75 in an AR(1) model, which is a clearly stationary value, then whilst the overall % coverage appears quite accurate; for example, 87.5% for a nominal 90% conﬁdence interval, this is a result of unbalanced coverage of the two component onesided conﬁdence intervals that comprise the two-sided interval; the one-sided 0.00 conﬁdence interval CI0.05 0.00 has coverage of 99.8% whilst the CI0.05 one-sided conﬁdence interval has coverage of 85.4%.

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186 Unit Root Tests in Time Series

Conﬁdence Intervals in AR Models 187

A5.2 The conceptualisation of the problem is actually somewhat easier using Figure 5.5 since the input, φˆ 1 = 0.791, is on the ‘x’ axis, which is traditionally associated with the input to a function. Part of Figure 5.5 is extracted in Figure 5.11; given the input φˆ 1 = 0.791, a vertical line is projected up until it cuts the inverse 95% and 5% quantile functions, respectively, from where it is projected across horizontally to meet the vertical axis φ1 to give the 90% conﬁdence interval. The important point to remember in obtaining the conﬁdence interval is that information from the sample via φˆ 1 is used to infer an interval for φ1 , and not the other way round. Q5.3 What is a median unbiased estimator and when will it differ from an ‘unbiased’ estimator? A5.3 An estimator is median unbiased if the median of its distribution is equal to the true value of the coefﬁcient being estimated. The usual sense in which the description unbiased is used is that the expectation of the estimator is equal to the DGP value, φ1 , that is E(φˆ 1 ) = φ1 . In the case of the LS estimator of φ1 , we know that this property is not satisﬁed in ﬁnite samples, and for φ1 > 0, there is a negative bias.

0.95 inverse 5% quantile

0.933 0.9

inverse median

0.85 φ1

90% confidence interval 0.8 inverse 95% quantile

0.75 0.709 0.7

0.65 0.65

estimate of 0.791 0.7

0.75

0.8

0.85

0.9

0.95

estimated φ1 Figure 5.11 Obtaining a conﬁdence interval.

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Q5.2 In the text, the conﬁdence interval based on simulated quantiles was obtained from Figure 5.4. Show how to obtain the conﬁdence interval from Figure 5.5.

188 Unit Root Tests in Time Series

A5.4 First, recap how the median unbiased estimator is obtained. The problem is to obtain a median unbiased estimate given the LS estimate φˆ 1 of φ1 . The solution is ﬁrst to simulate the distribution of the LS estimator for a range of values of φ1 (or have reference to simulated distributions). This gives rise to a distribution of estimates from which the median estimate can be located. For example, from Table 5.1, it is possible to infer that if φ1 = 0.82, then the median of the resulting LS estimates would be 0.791; hence, given a single LS estimate of φˆ 1 = 0.791, the corresponding median unbiased estimate is φˆ MU 1 = 0.82. Now the mean unbiased estimate can be obtained in the same way; note in the simulations the mean of the distribution in addition to the median of the distribution. In effect, add a column to Table 5.1, which records the mean of φˆ 1 for each value of φ1 , and then ‘invert’ the estimate using the mean column to obtain the mean unbiased estimate.

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Q5.4 Explain how the principle underlying Andrews’ (1993) median unbiased estimator could be used to obtain a mean unbiased estimator.

6

Introduction This chapter is concerned with some of the basic issues underlying testing for a unit root using least squares (LS) and related estimation methods. An understanding of these issues is necessary to interpret developments subsequent to the seminal work of Dickey (1976), Fuller (1976) and Dickey and Fuller (1979, 1981). This chapter progresses through the standard tests associated with Dickey and Fuller, continuing with problems arising from the treatment of nuisance parameters and dependent errors. There are broadly three solutions to this complication of dependent errors. The ﬁrst is an application of the Dickey-Fuller (DF) decomposition of Chapter 3, section 3.7, which augments the DF regression by lagged values of the dependent variable. The second is the semi-parametric approach associated with Phillips and Perron, leading to tests referred to as PP tests. The third approach, which was encountered in Chapter 3 when the dependent errors arose from an ARMA model, relates to the solution of incorporating these directly into the parametric form of the model and estimating by an efﬁcient method, such as maximum likelihood. The ﬁrst two approaches are considered in this chapter and the third is considered further in Chapter 7. This chapter covers much of the basic framework of testing for a unit root using commonly applied tests. It starts in section 6.1 by setting up the underlying framework, following this in section 6.2 with some motivation for the nonstandard distributions induced as the dominant root approaches unity; following Chapter 3, section 3.8, section 6.3 of this chapter returns to some of the frequently used DF tests, with a consideration of issues of size and power in section 6.4. Whilst the standard testing framework allows for a linear trend under the alternative hypothesis of stationary, the alternative of a nonlinear trend is brieﬂy considered in section 6.5. Section 6.6 considers two ways of potentially increasing the power of DF tests; in this case by using the forward 189

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Dickey-Fuller and Related Tests

and reverse recursions. Section 6.7 considers the important topic of departures from iid errors, an issue that is taken up in greater detail in Chapter 7; this leads onto section 6.8 and the semi-parametric test statistics suggested by Phillips and Perron. A consideration of power, including that of the DF and PP tests, is provided in section 6.9. This chapter closes by noting that a number of problems are still to be addressed, amongst these are the importance of lag selection, especially in the case of dependent errors, and the implications of multiple testing for the cumulative type I error. The theme of dependent errors is continued in the Chapter 7, with the application of generalised least squares (GLS) principles to the estimation and inference problems.

6.1 The basic set-up This section outlines the basic framework and concepts required for subsequent developments in this chapter. 6.1.1 An AR(1) process To motivate the discussion, the DGP is a simple AR(1) model, which is ﬁrst introduced (as an error dynamics model) without any deterministic components. (1 − ρL)yt = zt

t = 1, ...., T

z t = εt

(6.1) (6.2)

Where {εt }Tt=1 is a sequence of zero mean, independent and identically distributed random shocks. Equation (6.2) is trivial at the moment, but it serves to emphasis that an assumption is being made concerning the properties of zt ; in this case, zt = εt . The variance of zt is denoted σz2 throughout. The initial value of yt is y0 . If |ρ| < 1 and yt has a nonzero (long-run) mean, µ, then the DGP is: yt − µ = ρ(yt−1 − µ) + zt

(6.3)

Alternatively, (6.3) may be written as: yt = µ∗ + ρyt−1 + zt

(6.4)

where µ∗ = µ(1 − ρ); note that the link between µ∗ and ρ serves to distinguish the error dynamics framework from the DF speciﬁcation (see Chapter 3, section 3.8.1). 6.1.1.i Back-substitution The AR(1) DGP can be written in terms of an initial value y0 and the weighted cumulative error by successive back-substitution for lagged values in (6.1). The

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190 Unit Root Tests in Time Series

Dickey-Fuller and Related Tests 191

following are obtained in this way: yt = ρyt−1 + εt = ρ2 yt−2 + εt + ρεt−1 = ρ3 yt−3 + εt + ρεt−1 + ρ2 εt−2 ⇒ yt = ρt y0 + εt + ρεt−1 + ρ2 εt−2 + . . . + ρt−1 ε1 = ρt y0 + ∑i=1 ρt−i εi t

Further, the expected value of yt , E(yt ), is: E(yt ) = ρt E(y0 ) + ∑i=1 ρt−i E(εi ) t

(6.6)

t

= ρ E(y0 ) Let AE(yt ) be the asymptotic expectation of E(yt ) in the sense of the limit of E(yt ) as t → ∞ in (6.6). Then, provided |ρ| < 1, and given E(εi ) = 0 for all i, the asymptotic expectation is zero: AE(yt ) = 0

(6.7)

If the data are generated by (6.3), then there is an additional term in (6.5) due to µ; yt , E(yt ) and AE(yt ) are then given, respectively, by: yt = ρt y0 + ∑i=1 ρt−i εi + µ∗ ∑i=0 ρi t

t−1

(6.8)

E(yt ) = ρt E(y0 ) + ∑i=1 ρt−i E(εi ) + µ∗ ∑i=0 ρi t

t−1

AE(yt ) = lim (µ∗ ∑ t→∞

= µ(1 − ρ)

t−1 i ρ) i=0

(6.9a) (6.9b)

1 (1 − ρ)

=µ Thus, given |ρ| < 1, µ is the ‘long-run’ mean of yt , and the impact of the starting value y0 and any particular shocks are ‘forgotten’ as t → ∞. If, however, ρ = 1, the starting value is not forgotten. By direct substitution or by setting ρ = 1 in (6.5) or (6.8), bearing in mind that µ∗ = µ(1 − ρ) = 0 for ρ = 1, then yt can be expressed as: yt = y0 + ∑i=1 εi t

(6.10)

It is evident from (6.10) that the distribution of yt now depends on the distribution of y0 as well as the distribution of {εt }Tt=1 .

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(6.5)

192 Unit Root Tests in Time Series

It is also helpful at this stage to derive the variances that are used later. The variance of yt conditional on yt−1 is simple enough, evidently it is just var(yt |yt−1 ) = var(εt ) = σε2 . Once the conditioning on yt−1 is removed, the (unconditional) variance of yt will depend upon y0 and, as it may also depend on time, 2 . The ﬁrst of two leading assumptions on the generit has a time subscript, σy,t ation of y0 is that it is a constant (often taken to be zero) with probability one, referred to as the ﬁxed initial condition, ﬁc. If y0 is a constant, it does not contribute anything to the variance of yt , which is then given by: 2 σy,t = var(εt + ρεt−1 + ρ2 εt−2 + . . . + ρt−1 ε1 )

= σε2 (1 + ρ2 + . . . + ρ2(t−1) )

(6.11)

The last equality uses E(εt εs ) = 0 for all t = s (so a white-noise assumption would be adequate here); only even powers of ρ appear in the variance. Notice that as this variance depends upon t, the generating process is not strictly stationary; however, as t → ∞ for |ρ| < 1, the asymptotic variance σy2 is constant and is given by: 2 → σy2 = σy,t

1 σ2 as t → ∞ (1 − ρ2 ) ε

(6.12)

To obtain (6.12) note that (1 + λ + λ2 + . . . + λt ) → 1/(1 − λ), with λ = ρ2 . The assumption |ρ| < 1, ensures that the inﬁnite series in powers of ρ2 is summable, 2 = tσ2 , which is not constant, said to be square summable. If ρ = 1, then σy,t ε conﬁrming the nonstationarity of the process. A slightly different approach leads to the same expression for the asymptotic variance. Rather than assume that the sequence starts at t = 1, with start-up value y0 , assume that the process generating yt started in the inﬁnite past and, for convenience, the sample period of interest is indexed as t = 1, ..., T. Thus, although, as in Equation (6.9), the asymptotic variance includes a term in ρi εt−i , as i → ∞ this term is negligible for |ρ| < 1 and can be ignored. A second possible assumption for y0 is that it is a draw from a speciﬁed distribution, one possibility being a draw from the unconditional distribution of yt , with mean µ and variance σy2 , so that y0 ∼ iid(µ, σy2 ). With or without a nonzero mean, the term var(ρt y0 ) = ρ2t

1 σ2 (1−ρ2 ) ε

needs to be added to the right-hand

side of (6.11), so that: 2 σy,t = σε2 (1 + ρ2 + . . . + ρ2(t−1) ) + ρ2t

1 σ2 (1 − ρ2 ) ε

(6.13)

The asymptotic variance σy2 is as before provided that |ρ| < 1, since the last term vanishes asymptotically. These results relate to asymptotic properties, but in small samples, and especially for |ρ| close to 1, the particular start-up assumption can be critical for the properties of estimators.

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6.1.1.ii Variances

Dickey-Fuller and Related Tests 193

The variance of a sum with an increasing number of terms, in this case the sum of the zt sequence, will also be required at a later stage. For example, the sum of zt from t = 1 to T is: ST = ∑t=1 zt T

(6.14)

2 is deﬁned as (see also Chapter 3, section 3.1.2): Then the long-run variance σz,lr

T→∞

(6.15)

In the case that zt = εt , the variance of ST is particularly simple, but it will serve 2 . The variance of S , given that E(S ) = 0, is E(S2 ) = Tσ2 as an illustration of σz,lr T T ε T assuming that the εt are white noise. Normalising E(S2T ) by dividing by T (which can be interpreted as giving the ‘average’ variance), then: T−1 E(S2T ) = σε2 2 , as T → ∞; however, this Thus, in this case, σε2 is also the limiting variance, σz,lr is a special case and more interesting cases arise when zt = εt , when the zt are referred to as errors.

6.1.1.iii Weakly dependent errors The sequence {zt }Tt=1 is assumed to be weakly stationary with a zero mean. Thus some forms of dependent errors are permitted provided that the following invariance principle for partial sums of weakly dependent errors is satisﬁed (see, for example, Phillips, 1987a; Ouliaris et al., 1989): S[rT] √ ⇒D W(r) = σz,lr B(r) T

(6.16)

[rT]

where S[rT] = ∑t=1 zt , 0 ≤ r ≤ 1, [rT] is the integer part of T×r and B(r) is standard Brownian motion. The limit being understood as T → ∞ and where ⇒D indicates 2 is the long-run variance. (When emphasis or clarity weak convergence and σz,lr 2 will be subscripted to indicate to which random variable it refers.) requires, σz,lr √ Let XT (r) denote S[rT] normalised by T and scaled by σz,lr , then:

S[rT] XT (r) = √ ⇒D B(r) Tσz,lr

(6.17)

For example, suppose that zt is generated by a stationary AR(1) process, such that: (1 − ϕ1 L)zt = εt

|ϕ1 | < 1

Then the quantity of interest is T−1 E(S2T ) = T−1 E(∑Tt=1 zt )2 as T → ∞.

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2 σz,lr ≡ lim T−1 E(S2T )

194 Unit Root Tests in Time Series

A simple way to obtain this variance is to use the ‘trick’ that setting L = 1 deﬁnes the long run in a distributed lag function. Thus, ﬁrst invert the AR(1) 2 : process to obtain the inﬁnite MA process and then obtain σz,lr zt = (1 − ϕ1 L)−1 εt

=

σε2 (1 − ϕ1 )2

Now consider the general stationary, invertible ARMA case for generation of the 2 are given by: zt sequence. Then zt and σz,lr zt = ϕ(L)−1 θ(L)εt 2 σz,lr = var(zt |L = 1) θ(1) 2 2 σε = ϕ(1) 2 was used in Chapter 3 (see section 3.1.2). In fact this expression for σz,lr 2 = 0 and ϕ(1) = 0 is necessary Note that θ(1)/ϕ(1) = 0 is necessary to avoid σz,lr to avoid division by zero; together these require that θ(1) = 0 and ϕ(1) = 0. In the MA case, the presence of a positive unit root would result in θ(1) = 0, whereas in the AR case the presence of a positive unit root would result in ϕ(1) = 0, so unit roots are ruled out or, if present, would need to be removed. 2 as the long-run variance Additional justiﬁcation for the description of σz,lr

is provided by noting that the limiting distribution of T−1/2 S[rT] when r = 1 is 2 so that: normal with variance σz,lr ∑Tt=1 zt 2 √ ) ⇒D W(1) ∼ N(0, σz,lr T

(6.18a)

where W(1) = σz,lr B(1) (see Phillips, 1987a; Ouliaris et al., 1989). Hence, note that: ∑Tt=1 zt √ ⇒D B(1) ∼ N(0, 1) σz,lr T

(6.18b)

The invariance principle is satisﬁed for zt generated by stationary ARMA models, but it is also satisﬁed for more general linear processes. A linear process has the following form: ∞

zt = ∑j=0 ψj ξt−j

(6.19)

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2 σz,lr = var(zt |L = 1) εt = var (1 − ϕ1 )

Dickey-Fuller and Related Tests 195

2 ∑∞ j=1 jψj < ∞, Wang et. al. (2002, Theorem 2.2) show that ξt may be generated by a martingale difference sequence (MDS) with heteroscedastic variances. Although such variations are of interest, for simplicity of exposition we adopt one of two speciﬁcations for zt : ﬁrst, zt = εt ; and, second, where zt = εt , then the zt are generated by a stationary ARMA process.

6.1.2 The LS estimator First, consider the simplest case where zt = εt , with standard assumptions: εt ∼ iid(0, σε2 ), σε2 < ∞. y0 is assumed to be a draw from the unconditional distribution of yt , y0 ∼ (µ0 , σy ), where µ0 = 0 in this case. Initially, assume that the maintained regression matches the DGP: yt = ρyt−1 + εt

(6.20)

The least squares (LS) estimators for ρ and σε2 are: ρˆ = σˆ ε2 =

T

T

t=2

t=2

∑ yt yt−1 / ∑ y2t−1

(6.21)

T

∑ (yt − ρˆ yt−1 )2 /(T − 2)

(6.22)

t=2

The adopted convention is that ˆ above a coefﬁcient indicates a LS estimator. These estimators are conditional in the sense that information from y0 is not directly incorporated into the LS minimisation function. The LS estimator of σ2 (ˆρ), the standard error of ρˆ , is: σˆ

2

(ˆρ) = σˆ ε2

T

∑

−1 y2t−1

(6.23)

t=2

If a constant is included in the maintained regression then the regression model is: yt = µ∗ + ρyt−1 + εt

(6.24)

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where {ξt }∞ 0 are stochastic inputs. If ξt = εt , that is the ξt are iid random variables with ﬁnite variance, then the usual condition for the invariance principle to be satisﬁed is the absolute summability condition, ∑∞ j=0 |ψj | < ∞. However, Wang et al. (2002, Theorem 2.1) show that this can be weakened to allow some processes that do not satisfy this condition; for example, a linear process with ψj = j−1 h(j) where ∑∞ j=0 h(j)/j = ∞, which cannot be expressed by a ﬁnite order ∞ ARMA process, is permissible. Also, maintaining ∑∞ j=0 ψj = 0, ∑j=0 |ψj | < ∞ and

196 Unit Root Tests in Time Series

where µ∗ = µ(1 − ρ). The LS estimators are: ρˆ =

T

T

t=2

t=2

∑ (yt − y¯ (0) )(yt−1 − y¯ (−1) )/ ∑ (yt−1 − y¯ (−1) )2

ˆ ∗ = y¯ (0) − ρˆ y¯ (−1) µ σˆ ε2 =

(6.25a) (6.25b)

T

∑ (yt − µˆ ∗ − ρˆ yt−1 )2 /(T − 3)

(6.25c)

t=2

∑ [(yt − y¯ (0) ) − ρˆ (yt−1 − y¯ (−1) )]2 /(T − 3)

t=2

σˆ

2

(ˆρ) = σˆ ε2

(6.25d)

−1

T

∑ (yt−1 − y¯ (−1) )

2

(6.25e)

t=2

T−1 where y¯ (0) = ∑Tt=2 yt /(T − 1) and y¯ (−1) = ∑Tt=2 yt−1 /(T − 1) = ∑t=1 yt /(T − 1). The estimator ρˆ will not necessarily lie in the unit interval. As noted in Chapter 3, an alternative approach is to construct the (estimated) ˆ and use that in a LS regression without a constant. demeaned data, yˆ˜ t = yt − µ, The estimator of ρ then has the same form as (6.21). There are several ways of demeaning the data, which generally lead to slightly different estimates in ﬁnite samples. The usual practical method is to demean the data using the ‘global’ ˆ = y¯ G = ∑Tt=1 yt /T, so that the demeaned data is yˆ˜ t = yt – y¯ G ; another mean, µ ˆ = y¯ (0) . The former is often used, and the resulting estimator is: possibility is µ

ρˆ˜ =

=

T

T

t=2

t=2

∑ yˆ˜ t yˆ˜ t−1 / ∑ yˆ˜ 2t−1

(6.26a)

T

T

t=2

t=2

∑ (yt − y¯ G )(yt−1 − y¯ G )/ ∑ (yt−1 − y¯ G )2

(6.26b)

The estimators ρˆ and ρˆ˜ will not be distinguished although they are in principle distinguishable; throughout ρˆ will be taken as referring to ‘the’ LS estimator. √ In the stationary case |ρ| < 1, ρˆ is T-consistent for ρ, hence an appropriate normalised (non-degenerate) sequence is T1/2 (ˆρ − ρ), which is asymptotically distributed as N(0, 1 − ρ2 ), (see, for example, Fuller, 1996). Thus: (ˆρ − ρ) T1/2 ⇒D N(0, 1) (1 − ρ2 )

(6.27)

and: σˆ ε2 →p σε2

(6.28)

√ If ρ = 0, so that yt = εt , then ρˆ has an asymptotic standard error σ(ˆρ) = 1/ T. The result in (6.27) is an asymptotic result and the distribution in ﬁnite samples

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T

=

Dickey-Fuller and Related Tests 197

can look quite different from the normal, particularly as ρ → 1; this point is considered further in section 6.2. The next section considers the situation when zt is weakly stationary process rather than white noise. The case of a trend is most easily dealt with, conceptually at least, through the general approach where the term µt represents the deterministic components, and the data is ﬁrst detrended. In the linear trend case µt = β0 + β1 t and the prior detrending step is to regress yt on a constant and a linear time trend, such that: t = 1, . . . , T

(6.29)

The resulting LS estimators are βˆ 0 and βˆ 1 , and the residual, yˆ˜ t , is detrended yt : yˆ˜ t = yt − (βˆ 0 + βˆ 1 t)

t = 1, . . . , T

(6.30)

As in the case of mean estimation, one variation analogous to the LS solution in (6.26a) is to detrend yt using the trend regression over t = 2, . . . , T and to detrend yt−1 using the trend regression over t = 1, . . . , T – 1. However, the usual approach is to use all available data in detrending (analogous to the ‘global’ mean). The regression then takes the same form as (6.20), but with the detrended data: yˆ˜ t = ρyˆ˜ t−1 + ε˜ t

t = 2, . . . , T

(6.31)

where ε˜ t = (1 − ρ)(β0 − βˆ 0 ) + ρ(β1 − βˆ 1 ) + (1 − ρ)(β1 − βˆ 1 )t + εt . In this set-up ε˜ t = εt ˆ t = µt , but ρ = 1 is not sufﬁcient in this case. As before, in the stationary if µ case, |ρ| < 1, ρˆ is consistent for ρ and the limiting distribution of T1/2 (ˆρ − ρ) is N(0, 1 − ρ2 ). 6.1.3 Higher-order AR models It is useful to extend the AR(1) model to higher-order cases and obtain their ADF representation. For example, suppose that {zt }Tt=1 is not an independent sequence, but is generated by an AR(1) process, then this implies that the process generating yt is an AR(2) process. The intermediate steps are reserved for an end of chapter question. (This simple example was also considered in Chapter 3, section 3.8.1.i.) The model to be considered is: yt = ρyt−1 + zt

(6.32a)

zt = ϕ1 zt−1 + εt

(6.32b)

⇒ yt = φ1 yt−1 + φ2 yt−2 + εt

(6.33a)

⇒ ∆yt = γyt−1 + c1 ∆yt−1 + εt

(6.33b)

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yt = β0 + β1 t + ut

198 Unit Root Tests in Time Series

where: φ1 = ρ + ϕ1 , φ2 = − ρϕ1

(6.34a)

γ = φ1 + φ2 − 1, c1 = − φ2

If yt is replaced by yt − µ in (6.32a), then µ∗ = µ(1 − ρ)(1 − ϕ1 ) is added to (6.33); the extension to the case where µt is a linear trend is straightforward (see Chapter 3, Equation (3.12b)). The roots of the polynomial φ(L) = (1−φ1 L−φ2 L2 ) are 1/ρ and 1/ϕ1 , which are greater than one in absolute value provided |ρ| < 1 and |ϕ1 | < 1. Note that φ(1) = (1 − ρ)(1 − ϕ1 ) = 0 if either ρ = 1 and/or ϕ1 = 1; that is, there is at least one unit root. The AR(2) model, or AR(p) in general, is sometimes not derived through the intermediate steps by combining (6.32a) and (6.32b), but simply stated as the natural extension of an AR(1) model. However, in the unit root literature, the error dynamics/common factor form is usually favoured (see Chapter 3, section 3.8.1). The generalisation of {zt }Tt=1 to an AR(p – 1) process implies an AR(p) process for yt . Consider the following model: (1 − ρL)yt = zt

(6.35a)

ϕ(L)zt = εt

(6.35b)

⇒ φ(L)yt = εt

(6.36a)

φ(L) = ϕ(L)(1 − ρL)

(6.36b)

p−1

p

where ϕ(L) = 1 − ∑i=1 ϕi Li and φ(L) = 1 − ∑i=1 φi Li , with φ(1) = ϕ(1)(1 − ρ). The ADF form of (6.35) can also be derived explicitly by what is an application of the DF decomposition of a lag polynomial (see Chapter 3, section 3.7), noting here that A(L) = φ(L), then: yt = ∑i=1 φi yt−i + εt p

= (∑

p φ )y − i=1 i t−1

(6.37a)

∑

p φ (y − yt−2 ) − i=2 i t−1

∑

p φ (y − yt−3 ) . . . i=3 i t−2

. . . − φp (yt−(p−1) − yt−p ) + εt

(6.37b)

= φyt−1 + ∑

p−1 c ∆yt−j + εt j=1 j

(6.37c)

⇒ ∆yt = γyt−1 + ∑j=1 cj ∆yt−j + εt p−1

p

(6.38) p

where φ = ∑i=1 φi , γ = –φ(1) = φ – 1 and cj = − ∑i = j + 1 φi . If yt is replaced by yt − µ in (6.35a), then µ∗ = µ(1 − ρ)ϕ(1) is added to (6.37a)–(6.37c) and (6.38).

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(6.34b)

Dickey-Fuller and Related Tests 199

The difference between φ(L) and A(L) of Chapter 3, section 3.8.1.i, is that the latter allows for an MA component in the error dynamics, which implies that C(L) is an inﬁnite order polynomial in L, whereas if there is just an AR component the C(L) polynomial is ﬁnite of order p – 1. 6.1.4 Properties of the LS estimator, stationary AR(p) case

Yp = (yp + 1 , . . . , yT ) ε = (εp + 1 , . . . , εT ) φ = (φ1 , . . . , φp )  yp yp−1   y yp  Xp =  p + 1 ..   . yT−1 yT−2

... .. . .. . ...

y1 y2 .. . yT−p

      

The model can then be written in vector-matrix form as: Yp = Xp φ + ε

(6.39)

Note that the p-th column of Xp is the p-th lag of Yp ; for example, if p = 2 then Y2 = (y3 , . . . , yT ) and the ﬁrst and second columns of Xp are (y2 , . . . , yT−1 ) and (y1 , . . . , yT−2 ) respectively. Conventions vary on the dating of observations in the sample for an AR(p) model; for example, one convention, by analogy with the designation of y0 as the ‘initial’ observation, is to extend this to p initial observations, y0 , . . . , y−(p−1) , which involves negative dating; to avoid this, the adopted convention is that the ﬁrst column of Xp starts with yp , implying that the last column (the p-th lag) starts with y1 . The consistency and asymptotic distribution of the LS estimator φˆ are usually derived under the assumptions that the starting values (y1 , . . . , yp ) are ﬁxed or, more realistically, if stochastic, then (y1 , . . . , yp ) is independent of εt for t > p; εt ∼ iid(0, σε2 ), σε2 < ∞; and the modulus of the roots of the AR lag polynomial lie outside the unit circle (see Appendix 2, A2.6; and see Fuller, 1996, chapter 8; Anderson, 1971, chapters 5 and 6, for further elaboration of the conditions and results). The usual LS estimators are: φˆ = (Xp Xp )−1 Xp Yp ˆ (Yp − Xp φ)/(T ˆ − 2p) σˆ ε2 = (Yp − Xp φ)

(6.40) (6.41)

The consistency and asymptotic normality results stated for the LS estimator in the stationary AR(1) carry over to the least squares estimators of each of the

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It is convenient to deﬁne the set of observations in the regression model (6.37a) in vector-matrix form. First deﬁne the notation to be used as follows:

200 Unit Root Tests in Time Series

T1/2 (φˆ − φ) ⇒D N(0, σε2 Σ−1 p )

(6.42)

σˆ ε2

(6.43)

→p σε2

where p limT→∞ (T−1 Xp Xp ) = Σp is the ﬁrst p × p block of the autocovariance matrix of an AR(p) process (see Amemiya, 1985, p. 173; Anderson, 1971, Theorem 5.5.7; Fuller, 1996, Theorem 8.2.1, where the latter also proves the distributional theorem for martingale difference errors; Amemiya, 1985, does not make a degrees of freedom adjustment for σˆ ε2 , whereas Fuller does). An illustration of Σp for p = 2 is provided in a question at the end of this chapter. If µt = 0, then Yp and Xp are adjusted accordingly through prior demeaning or detrending; and if a degrees of freedom adjustment is made, then it should also reﬂect the number of deterministic components. Note that in the stationary √ case φˆ is T-consistent.

6.2 Near-unit root case; simulated distribution of test statistics The results summarised in (6.27) for the AR(1) model and (6.42) for the AR(p) model are large sample results; they are, as illustrated in this section, compromised as ρ → 1. We concentrate here on the t statistic for ρˆ with a simple DGP given by yt = ρyt−1 + εt . Assume that ρ is estimated from a regression model of the following form: yt = µ∗t + ρyt−1 + εt

(6.44)

where µ∗t = (1 − ρL)µt , and µt contains the deterministic terms, which are here none, a constant, or a constant and a linear time trend. The t statistic for ρˆ is, as usual: ρˆ − ρ tρ = (6.45) σˆ (ˆρ) In a hypothesis testing situation, the value of ρ in (6.45) will be as speciﬁed by the null hypothesis. tρ should have a t distribution with T – 2 degrees of freedom for |ρ| < 1, and for moderately sized samples; and for example, for T > 50 the distribution of tρ should be close to normal. However, there are two reasons why this may not be the case, and hence why inference on ρ may be misleading. First, even if |ρ| < 1, the LS estimator is generally biased in ﬁnite samples (see Chapter 4); if 0 < ρ < 1 the bias is negative, which is then imparted to tρ , the distribution of which is shifted to the left compared to the t or normal distributions. Second, whilst for |ρ| < 1, the limiting distribution of tρ is normal, if ρ = 1 then the asymptotic distribution switches to a function of Brownian motion (see section 6.3), which again implies a leftward shift relative to a normal distribution centred on zero.

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coefﬁcients φi in the stationary AR(p) model.

One way of representing the distance of ρ from unity is to specify ρ as ρ = 1−c/T where c is a positive constant. For example, if T = 100, then c = 0, 1, 5, 10, corresponds to ρ = 1.00, 0.99, 0.95 and 0.90, respectively. Figures 6.1a–6.1c show the distribution of tρ for these four cases, as well as ρ = 0 in order to benchmark the differences. The ﬁgures are based on 50,000 replications, with εt ∼ niid(0, 1); in the stationary cases, y0 is a draw from N(0, σy2 ), σy2 = (1 − ρ2 )−1 σε2 , and the ﬁrst 100 observations are discarded. There are three ﬁgures, one for each of the following cases: the estimated regression matches the DGP – that is, there is no constant (see Figure 6.1a); a constant is included (see Figure 6.1b); and a constant and linear trend are included (see Figure 6.1c). The ﬁgures illustrate three points: the leftward shift in the distributions in each of the ﬁgures as ρ → 1; the shift is more pronounced as more (superﬂuous) deterministic terms are included in the estimated model; and not only is there a leftward shift, the distribution becomes more leptokurtic than the standard normal when deterministic terms are included in the regression. To indicate just how misleading conventional inference is, the proportion of cases in which the value of tρ is less than the 5% left-tailed quantile of –1.645 from the normal distribution, is reported in Table 6.1. The departure from the normal becomes more pronounced as ρ → 1 and as more deterministic terms are included; for example, when ρ = 0.85 the proportion increases from 6.45% to 12.20% and then to 20.21% in the three cases of no constant, constant, and constant and trend, respectively. These simulations make it evident that the ﬁnite sample distributions are changing gradually as ρ → 1. Another direction of change occurs as T → ∞, so that the limiting distribution is approached; in that case it is legitimate to 0.4 0.35 0.3 0.25

ρ = 0.95 ρ = 0.85

ρ = 0.90

ρ = 0.0

ρ = 1.0

0.2 0.15 0.1 0.05 0 –3

no constant in the estimated model –2

–1

0

1

2

3

Figure 6.1a Distribution of tρ (no constant in estimated model).

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Dickey-Fuller and Related Tests 201

202 Unit Root Tests in Time Series

0.5 constant in the estimated model

0.45 0.4 0.35

ρ = 0.0

ρ = 0.95

ρ = 0.85

0.3 0.25 ρ = 1.0

ρ = 0.90

0.2

0.1 0.05 0 –4

–3

–2

–1

0

1

2

3

Figure 6.1b Distribution of tρ (constant in estimated model).

0.7 0.6

constant + trend in the estimated model ρ = 0.95

0.5 0.4

ρ = 0.90 ρ = 0.85

ρ = 1.0 ρ = 0.0

0.3 0.2 0.1 0 –5

–4

–3

–2

–1

0

1

2

3

Figure 6.1c Distribution of tρ (constant and trend in estimated model).

talk of just three distributions: one for |ρ| < 1, one for ρ = 1 and one for |ρ| > 1. √ In the ﬁrst case, the LS estimator ρˆ is T-consistent with a limiting normal distribution for T1/2 (ˆρ − ρ); this is the most straightforward case in the sense that hypothesis testing then usually relates to H0 : ρ = 0, with the familiar normal distribution for the ‘t’ statistic in the large sample case. In the second case, ρˆ is also consistent, but at the rate T so that the appropriate normalised sequence is T(ˆρ − ρ); the limiting distribution of this quantity is not normal, but a function of Brownian motion often referred to as a DF distribution. This is the case considered at length below. In the third case, the rate of convergence is even faster at the rate T3/2 and the limiting distribution of T3/2 (ˆρ − ρ) is Cauchy. The explosive case is usually ruled out, as there seems little evidence of explosive

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0.15

Dickey-Fuller and Related Tests 203

Table 6.1 Percentage of times that tρ < –1.645.

No constant Constant Constant + trend

ρ=0

ρ = 0.85

ρ = 0.90

ρ = 0.95

ρ = 1.0

4.96 6.00 7.14

6.45 12.20 20.21

6.82 14.20 25.82

7.52 19.66 38.12

8.94 45.70 76.30

economic time series, although some characteristics of series with two unit roots may make such a series look explosive.

6.3 DF tests for a unit root The aim of this section is to draw together the implications of previous sections as they relate to the set of tests associated with Dickey and Fuller (see Fuller, 1976; Dickey and Fuller, 1979, 1981), which provide the foundation of many tests for a unit root. These tests are referred to generically as DF, or DF-type, tests. (See also Chapter 3, sections 3.8.2 and 3.8.3.) 6.3.1 First steps: formulation and interpretation of the maintained regression The maintained regression for the DF tests is the regression that with appropriate restrictions reduces to the speciﬁcation under the null, but otherwise represents the view under the alternative hypothesis. It can be arrived at from three equivalent ways depending on whether the starting point is the AR (or ARMA) or the error dynamics or the common factor version of the model (see Chapter 3, sections 3.7.1 and 3.8.1). Conceptually, the simplest way to view the maintained regression for unit root testing, where the focus is on the single dimensioned hypothesis H0 : ρ = 1 compared to the stationary alternative HA : |ρ| < 1, is through the approach that ﬁrst undertakes a prior demeaning or detrending of the data. The maintained regression then always has the same general form, which is k−1 ˜ ∆y˜ t = γy˜ t−1 + ∑j=1 cj yt−j + εt ; what differs is the deﬁnition of y˜ t . The null hypothesis is H0 : γ = 0 and the stationary alternative is HA : –2 < γ < 0, corresponding to ρ = 1 and |ρ| < 1, respectively. However, it is often the case that the maintained regression is speciﬁed in terms of yt , rather than y˜ t , and deterministic terms are included separately. Then the critical difference is whether or not the coefﬁcients on the deterministic variables are linked through the reduced form relations back to the structural coefﬁcients. This section deals with this approach. In effect it is mostly a summary or recap of previous results, but it does draw out the possibility, hitherto not encountered, that joint hypothesis tests may also be of interest.

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Note: Based on 50,000 simulations of the DGP yt = ρyt−1 + εt , with T = 100.

204 Unit Root Tests in Time Series

Some examples will provide a recap of the maintained regression. First, consider the AR(1) model resulting from the common factor representation: (yt − µ) = ut

(6.47)

(1 − ρL)ut = εt

(6.48)

The resulting DF model, which is the maintained regression, is in this case: (6.49)

where µ∗ = (1 − ρ)µ and γ = (ρ − 1). The null hypothesis is H0 : γ = 0 and HA : –2 < γ < 0. Note that if ρ = 1, then ∆yt = εt , a pure random walk, whereas if |ρ| < 1, then yt − µ = ρ(yt−1 − µ) + εt , so that yt has a long-run mean of µ about which there are stationary deviations. If the deterministic component allows for a linear trend, then: (yt − β0 − β1 t) = ut

(6.50)

(1 − ρL)ut = εt

(6.51)

The resulting maintained regression is: ∆yt = β∗0 + β∗1 t + γyt−1 + εt

(6.52)

where β∗0 = (1 − ρ)µ + ρβ1 and β∗1 = (1 − ρ)β1 . Note that if ρ = 1, then ∆yt = β1 + εt , whereas if |ρ| < 1 then yt – (β0 + β1 t) = ρ[yt−1 −(β0 + β1 (t − 1))] + εt , so that yt has a linear trend of (β0 + β1 t) about which there are stationary deviations. Now consider adding in dynamic structure that results in an ADF(1) model. This can be viewed either as specifying φ(L) in the AR model as a second order, that is, (1 − φ1 L − φ2 L2 ); or specifying the error dynamics polynomial as ϕ(L) = (1 − ϕ1 L); or specifying the common factor lag polynomial as (1 − ϕ1 L)(1 − ρL). The end result is the same, whichever route is taken. The resulting ADF(1) model, for simplicity ﬁrst just with a constant, is: ∆yt = µ∗ + γyt−1 + c1 ∆yt−1 + εt

(6.53)

where µ∗ = (1 − ϕ1 )(1 − ρ)µ, γ = (1 − ϕ1 )(ρ − 1) and c1 = ϕ1 ρ (see Chapter 3, section 3.8, equation (3.79)). If ρ = 1, then ∆yt = ϕ1 ∆yt−1 + εt ; that is, an autocorrelated random walk; whereas if |ρ| < 1, then yt exhibits stationary, but correlated, deviations about the long-run mean µ, that is, yt − µ = (ϕ1 + ρ)(yt−1 − µ) −ϕ1 ρ(yt−2 − µ) + εt . If the deterministic component allows for a linear trend, then the resulting ADF(1) model is: ∆yt = β∗0 + β∗1 t + γyt−1 + c1 ∆yt−1 + εt

(6.54)

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∆yt = µ∗ + γyt−1 + εt

Dickey-Fuller and Related Tests 205

where β∗0 = φ(1)β0 + β1 ∑2j=1 jφj , β∗1 = φ(1)β1 , φ(1) = 1 − ∑2j=1 φj and φ1 = φ1 + ρ, φ2 = −ϕ1 ρ. If ρ = 1, then: (6.55)

which is an autocorrelated random walk with drift. If the null hypothesis is correct then yt can be differenced to stationarity, referred to as DS, hence, a series with a single unit root is also said to be difference stationary; on the other hand if the alternative hypothesis is correct and β1 = 0, the series is said to be trend stationary, referred to as TS, that is it is stationary after removal of the trend; without the detrending, the series would not be stationary because the mean of yt is not constant. As a matter of terminology, the term ‘trend stationary’ tends to be used even if β1 = 0. 6.3.2 Three models Suppose it is clear that a particular time series is positively (negatively) trended, as is the case for many aggregate economic time series, then the issue at hand is whether this is better explained by stationary deviations around a deterministic trend, with a positive (negative) trend coefﬁcient, or a stochastic trend (that is, there is a unit root). Then, in the latter case, the upward movement is provided by a positive drift; without drift, the upward movement must occur by the chance direction of the random walk, which is an unsatisfactory general explanation for many economic time series. Thus a correct base from which to assess these two competing explanations is one that allows both explanations to arise as restrictions on the maintained regression. If the time series of interest displays no evident trend, but may depart in either direction from a ﬁxed point, then a drift term is not required according to the null hypothesis: positive or negative deviations (or ‘walks’) are both, in principle, possible, and there is no reason to impose a direction. Equally, under the alternative hypothesis, if the point to which the series tends to return, or ‘revert’, is nonzero, it will be necessary to allow a nonzero mean. The appropriate maintained regression is therefore that which includes a constant. Finally, if the point of reversion is zero, then the constant may be excluded from the maintained regression. The threefold distinction in the maintained regression occurs at a number of points throughout this and subsequent chapters, and has been captured in the alternative speciﬁcations µt = 0, µt = µ and µt = β0 + β1 t. These are most straightforwardly interpreted in the error dynamics/common factor approach as the previous examples illustrate. Recall, however, that the maintained regression may also be speciﬁed in ‘direct’ form (the direct speciﬁcation) as in Chapter 3, section 3.8.1.iii, which gives rise to a difference of interpretation of the coefﬁcients on the deterministic components, with implications for the

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∆yt = β1 + ϕ1 ∆yt−1 + εt

206 Unit Root Tests in Time Series

H0

DGP under H0

Maintained regression

M0

γ=0

∆yt = εt

∆yt = γyt−1 + εt

M1

γ=0

∆yt = εt

∆yt = µ∗ + γyt−1 + εt

M2,a

γ=0 β1 = 0

∆yt = εt

∆yt = β∗0 + β∗1 t + γyt−1 + εt

M2,b

γ=0

∆yt = β1 + εt

∆yt = β∗0 + β∗1 t + γyt−1 + εt

Notes: µ∗ = (1 − ρ)µ; β∗0 = (1 − ρ)β0 + ρβ1 ; β∗1 = (1 − ρ)β1 .

invariance of the distributions of DF test statistics, particularly under the null hypothesis. A summary for each approach is given in Tables 6.2 and 6.3 for the simplest DGPs and maintained regressions. Where the models are split into sub-parts, they are given two subscripts; when statements apply equally to all members of the class, it is sufﬁcient to refer to the ﬁrst subscript only. With exceptions that are noted, the preferred framework here is that of the error dynamics/common factor representations, as in Table 6.2. Table 6.2 lists three models, which are designated Mi . M0 and M1 are straightforward, just reducing the maintained regression to a pure random walk by imposition of ρ = 1 (that is, γ = 0); in the case of M1 , this occurs because µ∗ = (1 − ρ)µ. M2 is split into two sub-models under H0 ; in the case of M2,a , H0 is a joint hypothesis imposing γ = 0 and β1 = 0 to reduce the maintained regression to a random walk; and in the case of M2,b imposition of γ = 0 reduces the maintained regression to a random walk with drift. In a test of γ = 0 designed for M2,b , β1 is a nuisance parameter under H0 and similarity is achieved if the distribution of such a test is invariant to β1 (see section 6.3.5 for further details). The set-up in Table 6.3 refers to the direct speciﬁcation of Chapter 3, section 3.8.1.iii; as noted there, the notation deliberately differs from the error dynamics/common factor approach of Table 6.2. Where the models differ from those in Table 6.2, they are now designated MMi . The simplest versions of H0 are those that just restrict γ, such that γ = 0, saying nothing of the other parameters; however, similarity requires that these other parameters do not affect the distribution of the test statistic(s). For example, a test of γ = 0 in MM1,b implying the DGP ∆yt = η0 + εt , requires invariance with respect to η0 , which is not achieved by the DF-type tests, see section 6.3.5. The summary in Tables 6.2 and 6.3 is easily extended for ADF versions of these regressions. If the underlying AR(p) lag polynomial is ﬁnite, then p – 1

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Table 6.2 Summary of simple maintained regressions: error dynamics/common factor approach.

Dickey-Fuller and Related Tests 207

H0

DGP under H0

Maintained regression

MM0

γ=0

∆yt = εt

∆yt = γyt−1 + εt

MM1,a MM1,b

γ = 0, η0 = 0 γ=0

∆yt = εt ∆yt = η0 + εt

∆yt = η0 + γyt−1 + εt ∆yt = η0 + γyt−1 + εt

MM2,a MM2,b MM2,c

γ = 0, η0 = 0, η1 = 0 γ = 0, η1 = 0 γ=0

∆yt = εt ∆yt = η0 + εt ∆yt = η0 + η1 t + εt

∆yt = η0 + η1 t + γyt−1 + εt ∆yt = η0 + η1 t + γyt−1 + εt ∆yt = η0 + η1 t + γyt−1 + εt

Note: Tests of the joint hypotheses are considered in section 6.3.7.

lags of ∆yt are added to the right-hand side of the maintained regression; if the AR polynomial is inﬁnite, the number of lags to be added to the maintained regression has to be large enough by some criteria, and the question of how ‘large’ is large is considered in Chapter 9. 6.3.3 Limiting distributions of δˆ and τˆ test statistics The test statistics of interest here are the family of pseudo t tests τˆ , τˆ µ and τˆ β , and ˆ δˆ µ and δˆ β (see Chapter 3, sections 3.8.2 and 3.8.3). the family of n-bias tests δ, The limiting distribution of each test statistic has the same general form for variations in the speciﬁcation of µt . To introduce some of the issues, the simplest case, M0 , is considered ﬁrst. Initially, a fairly restrictive set of assumptions is adopted and these are then relaxed; thus, as in Fuller (1976), assume that zt = εt ∼ iid(0, σε2 ), and that y0 = 0. Then the limiting null distributions of δˆ and τˆ are given by:

1 B(r)dB(r) ˆ DF δˆ ⇒D 0 1 ≡ F(δ) 2 0 B(r) dr

=

1 (B(1)2 − 1) 2 1 B(r)2 dr

(6.57)

0

Notice that the second line uses the result that 1

τˆ ⇒D 0

1 0

B(r)dB(r) = 12 (B(1)2 − 1).

B(r)dB(r) 1/2 ≡ F(ˆτ)DF

1 2 0 B(r) dr

=

1 (B(1)2 − 1) 1/2 2 1 2 0 B(r) dr

(6.58)

where B(r) is standard Brownian motion. The DF subscript on the limit distributions indicates that these distributions are often referred to as the DF distributions. It is apparent from the results in section 6.2 that the left-sided

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Table 6.3 Summary of maintained regressions: DF representations.

208 Unit Root Tests in Time Series

critical values for the unit root null will be more negative than in the case of the standard normal distribution. Consideration of the selection of an appropriate critical value is taken up in section 6.4.3. In the case of estimation of M1 or M2 , standard Brownian motion is replaced by demeaned or detrended Brownian motion, respectively, which are given by: 1 0

B(s)ds

B(r)β = B(r) + (6r − 4)

1 0

(6.59) B(s)ds − (12r − 6)

1 0

sB(s)ds

(6.60)

The resulting limiting distributions are of the same general form as (6.57) and (6.58), but with B(r)i , i = µ, β, in place of B(r). The least squares estimators of ρˆ and σˆ ε are the ones appropriate to the models M0 , M1 or M2 ; that is, respectively, to no deterministics, a constant, and a constant and a linear trend. The limit distributions are then distinguished ˆ DF , F(δˆ µ )DF , F(δˆ β )DF and F(ˆτ)DF , F(ˆτµ )DF , F(ˆτβ )DF , respectively. Notice as: F(δ) that the sub-cases M2,a and M2,b are not distinguished; this is because in the error dynamics/common factor set-up, the limiting null distributions are invariant to β1 . In the case of the direct (DF) speciﬁcation, there are some differences of note. First, if the DGP under the null is ∆yt = εt , then the limiting distributions are each as in the corresponding cases listed above. However, in the Models MM1,b and MM2,c , the null DGPs permit a random walk with a drift and a trend, respectively, and the limiting distributions are not invariant to these parameters, a matter that is taken up in the next section. 6.3.4 Obtaining similar tests It is generally desirable to obtain tests that are at least asymptotically similar, so that the limiting null distribution of the test statistic does not depend upon nuisance parameters in the DGP; thus a level α test under one set of values of the nuisance parameters will be a level α test under another (arbitrary) set of values. Exact similarity is the case where the ﬁnite sample distributions are unaffected by nuisance parameters. Where there is no risk of confusion, similarity is taken to refer to exact similarity. Possible nuisance parameters include the initial condition and the parameters governing the deterministic components (the distributions are invariant to σε2 , the variance of εt , provided it is ﬁnite). The dynamic coefﬁcients in higherorder AR/ADF models are also potential nuisance parameters, but they are dealt with separately, usually by adjusting the form of the test statistic. The effect of the former differs between different representations of the DGP. As noted in Chapter 3, section 3.8.1, and distinguished in Tables 6.2 and 6.3, there are two approaches to be distinguished: on the one hand, the original direct (DF)

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B(r)µ = B(r) −

Dickey-Fuller and Related Tests 209

speciﬁcation; and, on the other, the error dynamics/common factor approach. These lead to slightly different DGPs if a deterministic function is included, which is usually of the form µt = µ or µt = β0 + β1 t. In this case and in the error dynamics/common factor approach, the situation is relatively simple and we consider this ﬁrst. 6.3.4.i Similarity and invariance in error dynamics/common factor approach

(yt − µt ) = ut

(6.61a)

(1 − ρL)ut = εt

(6.61b)

where µt = µ or µt = β0 + β1 t. The initial condition is u0 = y0 − µ0 , which is the deviation of the initial observation from the mean or trend. If µ0 = 0 then u0 = y0 , and if µt = µ or µt = β0 + β1 t, then u0 = y0 − µ or u0 = y0 − β0 , respectively. In this illustration, and more generally in this approach, the DF-type tests are similar with respect to µ and (β0 , β1 ) under H0 and HA provided that the maintained regression at least matches the deterministic terms in the null DGP. The DF-type tests in this framework are also similar with respect to the initial condition under H0 provided u0 = 0 or at least µt = µ; that is, if u0 = 0, then at least a constant is included in the deterministic components. However, the initial condition is an important parameter under HA that affects the distribution of the test statistics and hence power (see DeJong et al., 1992; and for a general discussion of the issues see Schmidt and Phillips, 1992). 6.3.4.ii Similarity and invariance in the direct (DF) speciﬁcation Achieving similarity if the DGP is as in the direct (DF) speciﬁcation of Chapter 3, section 3.8.1.iii and Table 6.3 differs from the previous section and is somewhat more complex. The key results are summarised below. Initial condition: under the null hypothesis, ρ = 1 If u0 = 0, then the maintained regressions for the three models, MM0 , MM1 and ˆ MM2 , each yield similar τˆ -type and δ-type tests. If u0 is a random variable, or if u0 = ξ is ﬁxed constant, then the test statistics from MM0 are not similar; for example, suppose that u0 = ξ1 and, alternately, u0 = ξ2 = ξ1 , all other realisations of ut being identical, then the test statistics from MM0 will differ, but will be asymptotically similar. However, the test statistics from MM1 and MM2 are each similar, including as a special case being similar to those with u0 = 0. Thus, to ensure that similar tests are used when u0 = 0, at least a constant is required in the maintained regression; in effect, the constant partials out the

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To illustrate the issues, consider the common factor DGP given by:

210 Unit Root Tests in Time Series

Initial condition: under the alternative hypothesis, |ρ| < 1 None of the tests are similar under any of the three models, MM0 , MM1 and MM2 . This means that the power functions of the DF test statistics will be a function of ρ and the initial condition, a problem observed by Evans and Savin (1981) (see also M¨ uller and Elliott, 2003; Elliott and M¨ uller, 2006; Harvey and Leybourne, 2005, 2006). Indeed, the power functions can differ substantially for different values of u0 . Drift The next case of relevance is when the random walk is drifted, so that the DGP under the null hypothesis yt is: ∆yt = β1 + εt

(6.62)

The question then is whether the test statistics are invariant to the nuisance parameter β1 . As might be anticipated from the previous discussion, it is now the test statistics obtained from MM0 and MM1 that are not similar; however, they are similar when obtained from the maintained regression of MM2 which, with a drifted random walk for the null, is referred to as MM2,b . To achieve invariance with respect to an arbitrary value for the drift, a linear trend should be included in the maintained regression. This line of argument can be extended in that to achieve invariance with respect to the nuisance parameters of a polynomial of order K in the DGP, the maintained regression should include a polynomial of order K + 1. The presence of drift, or a trend, in the null DGP alters the limiting distribution of the test statistics, which no longer have a DF-type distribution but a normal distribution (see, for example, West, 1988). The convergence of the DF distribution to a normal as the drift scaled by σε increases is, however, quite slow. To illustrate this case, a drift component, β1 , was added to the random walk, where β1 /σε = (0, 1) in increments of 0.1, so that drift is scaled in terms of the innovation standard error (as the latter is unity, β1 /σ = β1 in this case); note that in a typical context where yt is the log of variable, say on a quarterly basis, even β1 /σε = 0.1 is a substantial drift. The estimated model includes a trend since in a hypothesis testing context, the alternative to a drifted random walk is a trend stationary process (other simulation details are as for Figures 6.1a–6.1c). The slow rightward shift in the distribution of tρ as the drift parameter increases is illustrated in Figure 6.2.

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importance of the arbitrary u0 and the maintained regressions for the two models MM1 and MM2 will ensure similarity in this case. The question of asymptotic similarity for MM0 , with arbitrary u0 , is considered in section 6.3.5.

Dickey-Fuller and Related Tests 211

0.5 0.45

distribution shifts (slowly) to the right as β increases

β1 = 0.0

β1 = 0.1

0.4 0.35 0.3

0.2 0.15 0.1 0.05 0 –4

–3

–2

–1

0

1

2

3

Figure 6.2 Distribution of tρ with drift in DGP.

6.3.5 Continuous record asymptotics Continuous record asymptotics explains the importance of the initial condition in ﬁnite samples and its irrelevance in the limit, a concern that is pertinent to ˆ model MM0 (≡ M0 ). The limiting (T → ∞) null distributions of the DF δ-type and τˆ -type tests were initially derived with the starting condition u0 = 0 (see, for example, Fuller, 1976; Dickey and Fuller, 1981); however, Phillips (1987a) shows that the same limiting distributions result if the assumption on the initial condition is relaxed to u0 ‘has a certain speciﬁed distribution’, so that the DF distributions have much wider applicability. The Monte Carlo results of Evans and Savin (1981) illustrated the problem that changes in u0 lead to changes in the ﬁnite sample distributions of the DF test statistics, and the lack of similarity was noted above for M0 and MM0 . Phillips (1987a) uses continuous record asymptotics to show how the ﬁnite sample results can be related to the invariance found in the limiting distribution. The idea in continuous record asymptotics is that the overall time span of observations, denoted N, is ﬁxed, but this is divided into sampling intervals, h, that become smaller and smaller. There are T such observations sampled at an interval of h, such that T × h = N. The limit is now as h → 0, and in this limit the record is continuous. These concepts can be motivated by observations obtained from ﬁnancial markets where data are recorded at very high frequencies, such as daily and minute-by-minute.

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β1 = 0.7

0.25

In conventional large T asymptotics, h is ﬁxed (usually h = 1) and T → ∞; but as Phillips (1987a) shows, the role of the initial condition is important when N is ﬁxed, so that the sample is of ﬁnite length, and h → 0. The initial condition is measured in units of σε so that ξ0 = u0 /σε , which is y0 /σε in the case that µt = 0, as in M0 and MM0 . For example, letting τˆ h denote the τˆ -type test from M0 , but with ﬁxed N and h → 0, the limiting distribution is (Phillips, 1987a, Theorem 6.2):   √ B(1)2 − 1 + 2B(1)ξ0 / N 1   τˆ h ⇒ (6.63)  1/2  √ 1 2 1 2 2 B(r) dr + 2( ξ / N) B(r)dr + ξ /N 0 0 0 0 It is evident that a nonzero initial condition induces both a location shift and a scale change, which explains Evans and Savins’ observation that the distribution concentrates as ξ0 increases. However, as N → ∞, the distribution in (6.63) reduces to τˆ DF and the importance of the initial condition disappears. 6.3.6 The long-run variance as a nuisance parameter So far it has been assumed that zt = εt ∼ iid(0, σε2 ); however, in practice this will rarely be the case. A leading alternative is to assume that zt is generated by 2 = σ2 and σ2 , or more precisely an ARMA process, which will mean that σz,lr ε z,lr σε /σz,lr , is in effect a nuisance parameter as it affects the null distributions, both ﬁnite sample and limiting. This section, which is based on Xiao and Phillips (1998), focuses on the ˆ DF δ-type test to show that the limiting distribution in this case is a simple scalar multiple of the corresponding DF distribution. Thus, multiplying δˆ by the inverse of this scalar multiple returns a test statistic with a DF distribution. The procedure is made feasible with a consistent estimate of σε /σz,lr . A similar approach can be applied to the DF τˆ -type test, but consideration of that is delayed until section 6.7.1.ii. For simplicity consider the set-up as in M0 , but with zt generated by an ARMA process. The model is: (1 − ρL)yt = zt

(6.64a)

ϕ(L)zt = θ(L)εt ⇒ A(L) = θ(L)

−1

ϕ(L)(1 − ρL)

(6.64b)

Where y0 is a random variable with ﬁnite variance and zt is a stationary and invertible ARMA process; then, as shown in Chapter 3, Equation (3.75), this results in the ADF(∞) given by: ∞

∆yt = γyt−1 + ∑i=1 ci ∆yt−i + εt

(6.65)

Adopting a truncation rule results in the ﬁnite-order ADF(k – 1): ∆yt = γyt−1 + ∑i=1 ci ∆yt−i + εt,k k−1

(6.66)

∞ where εt,k = (∑∞ j=k+1 aj )yt−1 + ∑i = k ci ∆yt−i + εt .

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212 Unit Root Tests in Time Series

A condition is required on the expansion rate for k∗ = k – 1, so that k∗ increases with the sample size. A sufﬁcient condition in this case is given by Said and Dickey (1984), such that k∗ > T1/3 → 0 as T → ∞, and there exist constants, ζ and s, such that ζk∗ > T1/s (see also Xiao and Phillips, 1998, Assumption A4). Chang and Park (2002) explore the relaxation of this rate of expansion to less stringent rates as they depend on the structure of the innovations. ˆ where γˆ is the LS estimator from (6.66), it follows that the Letting δ˜ = Tγ, limiting distribution of δ˜ under the null ρ = 1 is (Xiao and Phillips, 1998, Theorem 1): 1 0 B(r)dB(r) ˜δ ⇒D σε (6.67) 1 2 σz,lr 0 B(r) dr σε ˆ DF = F(δ) σz,lr Hence, the ‘sigma’ ratio, σε /σz,lr , is a nuisance parameter that scales the DF ˆ DF . However, note that a simple transformation removes the distribution, F(δ) nuisance parameter from the asymptotic distribution. That is: 1 σz,lr B(r)dB(r) 0 (6.68) δ˜ ⇒D σε B(r)2 dr Now what is required is to replace σz,lr and σε by consistent estimators, denoted σˆ z,lr and σˆ ε , respectively. To obtain these, ﬁrst note that the long-run variance of 2 , in the ARMA case (see section 6.1.1.iii on weakly dependent errors), is: zt , σz,lr θ(1) 2 2 2 = σε (6.69) σz,lr ϕ(1) The parameters θ(1) and ϕ(1) can be consistently estimated from the ADF(k*) regression, (6.66), which is of ﬁnite order if the MA component is redundant. 2 and σ2 are then consistently estimated by: Given the expansion rate for k∗ , σz,lr ε 2 1 2 2 σˆ z,lr,AR(k σˆ ε,k (6.70) ∗) = ˆ (1 − C(1)) 2 σˆ ε,k = T−1 ∑t = k εˆ 2t,k T

(6.71) ˆ ˆ C(1) = ∑k∗ i=1 ci

and where εˆ t,k are the residuals from LS estimation of (6.66), i ,. Hence a consistent {cˆ i } are the estimates of the coefﬁcients of C(L) = ∑k∗ c L i=1 i (autoregressive) estimator of the inverse sigma ratio is given by: σˆ z,lr,AR(k∗ ) 1 1 σˆ ε,k = ˆ σˆ ε,k σˆ ε,k (1 − C(1)) 1 (6.72) = ˆ (1 − C(1))

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Dickey-Fuller and Related Tests 213

214 Unit Root Tests in Time Series

This provides the required scalar multiple for (6.68), so that: σˆ z,lr,AR(k∗ ) σˆ ε,k

δ˜

Tγˆ ˆ (1 − C(1)) 1 0 B(r)dB(r) ˆ DF = F(δ) ⇒D 1 2 0 B(r) dr

=

(6.73)

(6.74)

The reader may note that (6.73) has already been derived in Chapter 3, section 3.8.3, but by a different line of reasoning. The last result (6.74), conˆ This result generalises to models M1 and ﬁrms the form and distribution of δ. M2,a , either by including the deterministic terms directly in the ADF regression or demeaning the data under M1 or detrending the data under M2,a ; then standard Brownian motion is replaced by either B(r)µ or B(r)β as required. 6.3.7 Joint (F-type) tests in the DF representation ˆ The DF δ-type and τˆ -type tests are one-dimensional tests that focus on the null and alternative hypotheses given by H0 : ρ = 1 and HA : |ρ| < 1. However, several joint tests are also possible and Dickey and Fuller (1981) suggested three F-type tests. The test statistics are denoted Φi , i = 1, 2, 3, and are based on the conditional likelihood function, so the ML estimators of the regression coefﬁcients are also the LS estimators. The general form of the test statistic is: (T − p) − K RRSS − URSS Φi = (6.75) URSS g where RRSS is the residual sum of squares restricted by H0 ; URSS is the residual sum of squares in the maintained regression; the numerator degrees of freedom is the number of observations for estimation, T – p, minus K, the number of estimated regression coefﬁcients, and p is the order of the AR model, which is 1 in the case of the simple maintained regressions of Table 6.3; g is the dimension of the null hypothesis. The Φi test statistics are applicable for higher-order AR models, in which case p > 1, although the usual way of estimating such models is by means of the ADF representation. Referring back to Table 6.3, the joint hypotheses relate to models MM1,a , MM2,a and MM2,b ; the null hypotheses are summarised in Table 6.4. Note that in each case, HA is ‘not H0 ’, which, as in standard F-type tests, allows for two-sided alternatives. These tests are useful in principle to distinguish when the DGP does or does not have drift, but not equally so. For example, consider MM1,a , if the random walk has drift it will tend to stay on the side of the axis matching the sign of the

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δˆ ≡

Dickey-Fuller and Related Tests 215

Model

H0,i,j

DF test statistic

MM1,a MM2,a MM2,b

(γ, η0 ) = (0, 0) (γ, η0 , η1 ) = (0, 0, 0) (γ, η0 , η1 ) = (0, η0 , 0)

Φ1 Φ2 Φ3

drift and a trend will be present in the levels solution; however, the maintained regression for MM1,a excludes a trend and so is mismatched and is unlikely to be used in such a case. Either H0,1,a should be not rejected or the maintained regression is incorrect. In the case of MM2,a , non-rejection of H0,2,a using Φ2 reduces the DGP to a pure random walk, for which the maintained regression of MM1 would have been more suitable as it does not contain a superﬂuous trend. It is MM2,b that is likely to be of greatest interest, as the DGP under the null hypothesis could reasonably be a drifted random walk where the alternative is that the series is trend stationary with a non-redundant trend. In this context η0 is a nuisance parameter, but as noted by Dickey and Fuller (1981), the distribution of Φ3 does not depend on η0 so it may be used in both cases when η0 = 0 and η0 = 0. Non-rejection of H0,2,b using Φ3 allows the possibility of a drifted random walk as it implies non-rejection of the unit root, γ = 0 and implies that the DGP may contain drift η0 . A conditional approach may have merit to determine this last point, as Φ2 and Φ3 only differ on the inclusion of η0 in the former; for example, non-rejection using Φ3 , but rejection using Φ2 implies η0 = 0. Although useful in principle, the Φi statistics are little used in practice, in part due to their low power. Marsh (2007) shows that the powers of Φ1 , Φ2 and Φ3 are well below the power envelope, especially when the maintained regression includes a trend; that is, for Model MM2 , but that is the case likely to be of greatest interest. The reader is referred to Marsh (2007) for other tests that improve upon Φi . 6.3.8 Critical values: response surface approach Fuller (1976, 1996) tabulated critical values for the set of τˆ and δˆ tests for a number of values of the sample size; subsequently MacKinnon (1991) used a response surface approach to obtain the critical values for a particular sample size, and Cheung and Lai (1995a, 1995b) extended this to account for higherorder lags in the maintained regressions; Patterson and Heravi (2003) increased the number of lags and accuracy of the response surfaces.

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Table 6.4 Null hypotheses for DF joint F-type tests.

216 Unit Root Tests in Time Series

The general notation is C(τ, α, T∗ , k∗ ), which is the estimate of the 100α percentile of the distribution of test statistic τ for T∗ and k∗ = k − 1. The ‘observations’ in the response function are indexed by j = 1, ..., N, and T∗ ≡ T − k adjusts for the actual sample size. The general form of the response function for given α, from which the response surface coefﬁcients are estimated, is: I

H

i=1

h=1

∑ κi /(T∗ )i + ∑ ωh (k∗ /T∗ )h + ξj

(6.76)

for j = 1, . . . , N. The empirical percentiles that form the ‘observations’ on the dependent variable for estimation of the response functions are constructed by simulating ∆yt = εt , εt ∼ niid(0, 1) and y1 = ε1 , with N = 100,000 replications; setting σε2 = 1 is without loss as the distributions are invariant to σε2 . However, the ﬁnite sample distributions are not invariant to the distributional assumption on εt , although they are quite robust to fatter tails – as, for example, in the case that εt are draws from a t distribution with v degrees of freedom where v is small, for example, v = 3. A factorial experimental design was used over all different pairings of T and k∗ , where k∗ = 0, 1, ..., 12. The design was arranged so that for each k∗ the minimum effective number of observations in each sample was 20. Increments (in parentheses) were then as follows: 20(1), 51(3), 78(5), 148(10), 258(50), 308(100), 508(200), 908. In all, there were 66 × 13 = 858 sample points from which to determine the response functions for the 1%, 5% and 10% signiﬁcance levels. The estimated coefﬁcients are reported in the appendix to this chapter, Table 6.12. All the tests are left-sided in the sense that a realised value of the test statistic less than the α% critical value leads to rejection of the null hypothesis. To illustrate, part of Table 6.12 is extracted below in Table 6.5, from which the 5% critical values for τˆ µ are calculated for T = 101, k∗ = 0 and k∗ = 4, respectively; note that k∗ is the order of the ADF and k = k∗ + 1 is the implied order of the corresponding AR model, T is the overall sample size, so that T∗ = 100 and T∗ = 96, respectively.

Table 6.5 Illustration of response surface coefﬁcients. Model: M1 (constant and no trend) 5% critical values Test

κ∞

κˆ 1

κˆ 2

κˆ 3

ω ˆ1

ω ˆ2

ω ˆ3

R2

τˆ µ

–2.8633

–3.0799

–5.9769

0

0.9018

–1.4699

1.5274

0.9097

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Cj (τ, α, T∗ , k∗ ) = κ∞ +

Dickey-Fuller and Related Tests 217

Thus for the maintained regression given by: k∗

∆yt = µ∗ + γyt−1 + ∑j=1 cj ∆yt−j + εt,k the following 5% critical values are obtained: k∗ = 0 C(ˆτµ , 0.05, 100, 0) = κˆ ∞ +

2

∑ κˆ i /(T∗ )i

= − 2.8633 − (3.0799/100) − (5.9769/1002 ) = − 2.895 k∗ = 4 Cj (ˆτµ , 0.05, 96, 4) = κˆ ∞ +

2

3

i=1

j=1

∑ κˆ i /(T∗ )i + ∑ ωˆ j (k∗ /T∗ )j

= − 2.8633 − (3.0799/96) − (5.9769/962 ) + 0.90175(4/96) − 1.4699(4/96)2 + 1.5274(4/96)3 = − 2.861 For example, in an ADF(4) with T = 101 observations, a sample value of τˆ µ less than –2.861, leads to rejection of the null of a unit root. Notice that a quick estimate of the 100α percentile is provided by the asymptotic coefﬁcient κˆ ∞ .

6.4 Size and power Xiao and Phillips (1998, Theorem 2), and see also Phillips and Ouliaris (1990) and Nabeya and Tanaka (1990), show that under the alternative hypotheˆ sis, the divergence rate of the δ-type tests is faster than that of the τˆ -type tests. Speciﬁcally, if yt is generated according to HA , so that ρ < 1, then, for some c = 0: T−1 δˆ →p c T

−1/2

τˆ →p c

(6.77a) (6.77b)

Of course, a variation on the t-type test with faster divergence, whilst maintaining the directional nature of the test, could be obtained by taking an odd power of the test; for example, cubing the τˆ µ tests, but then size is likely to suffer. This result does not say that δˆ is more powerful than τˆ for all T, but for T sufﬁciently large, δˆ will be more powerful than τˆ ; in fact, as the simulations

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i=1

reported below indicate, the ‘more powerful’ result holds almost without qualiﬁcation, which raises the question of why use τˆ -type tests? The answer relates to how the tests perform on the criterion of size retention when the errors are not white noise; for example, if there is some dependency structure, then the ˆ δ-type tests tend not to be as robust as the τˆ -type tests. Although more extensive power comparisons are given elsewhere in this book, it may be helpful to ‘benchmark’ later results by giving some basic results here. The simulation set-up is that there are two alternative error structures for zt ; that is, MA(1) and AR(1), so that zt = (1 + θ1 L)εt and (1 − ϕ1 L)zt = εt , respectively, combined with the model structures given by M0 , M1 and M2,a . The stochastic input εt is niid(0, 1); the MA(1) coefﬁcient is θ1 = –0.5, 0.5 and the AR(1) coefﬁcient is ϕ1 = 0.0, 0.3, 0.6; T = 200 and there were 20,000 replications, with the critical values calculated using the response surface coefﬁcients from Table 6.12. In the case of power comparisons, the initial condition is a draw from the unconditional distribution of zt . The simulation results are reported in Table 6.6. In the case of ADF-based tests, some criterion for lag selection needs to be adopted. The rule that k∗ /T1/3 → 0 as T → ∞ and ζk∗ > T1/s is a simulation rather than a lag selection rule. In practice, a variety of rules, including a ﬁxed lag choice, have been used and some of the leading ones are considered in ˆ and Chapter 9. Here, since the evaluation is primarily to compare the δ-type τˆ -type tests, the lag length k∗ is set alternately at 0 and 8; both are simple rules of thumb; one makes no adjustment and the other sets a ‘reasonably’ long lag. (Note that power is not size-adjusted.)

Table 6.6 Illustrative size and power comparisons of DF-type tests. Table 6.6a MA(1) error structure: zt = (1 + θ1 L)εt δˆ δˆ µ δˆ β δˆ δˆ µ δˆ β τˆ τˆ µ τˆ β Lag

0

θ1 = –0.5 ρ = 1.0 ρ = 0.95 ρ = 0.90 ρ = 0.85 θ1 = 0.5 ρ = 1.0 ρ = 0.95 ρ = 0.90 ρ = 0.85

0

τˆ µ

τˆ β

8

8

8

0

0

0

8

8

8

44.4 66.6 84.8 100 100 100 100 100 100 100 100 100

16.4 78.1 99.4 99.8

25.6 87.9 99.5 99.3

28.6 68.7 97.5 99.5

43.2 100 100 100

59.7 99.0 95.8 100

81.8 99.8 72.7 100

5.1 67.3 97.3 99.9

4.9 25.4 65.6 85.9

4.7 14.3 41.7 63.4

0.7 72.8 90.5 99.7

18.2 94.7 98.3 98.1

27.5 82.5 98.2 98.6

29.4 61.7 91.7 98.3

1.1 38.9 89.1 99.6

2.7 4.2 27.5 67.7

0.3 1.5 7.4 28.1

5.3 66.7 92.8 98.2

5.4 22.5 56.1 74.8

5.8 13.9 33.3 51.3

0.5 5.9 41.7 86.1

0

τˆ

0.0 1.3 0.9 37.9

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218 Unit Root Tests in Time Series

Dickey-Fuller and Related Tests 219

δˆ

δˆ µ

δˆ β

δˆ

δˆ µ

δˆ β

τˆ

τˆ µ

τˆ β

τˆ

τˆ µ

τˆ β

Lag

0

0

0

8

8

8

0

0

0

8

8

8

ϕ1 = 0.0 ρ = 1.0 ρ = 0.95 ρ = 0.90 ρ = 0.85

4.7 79.6 99.9 100

5.2 46.9 95.0 99.9

5.0 23.1 73.0 98.0

4.6 75.6 99.8 99.9

4.6 40.6 90.3 99.7

3.6 16.2 56.8 90.8

5.1 79.9 99.8 100

5.1 33.0 86.7 99.9

5.0 19.2 64.8 95.8

5.0 65.3 93.8 99.0

5.3 24.3 55.7 78.0

5.0 14.4 34.7 55.1

ϕ1 = 0.3 ρ = 1.0 ρ = 0.95 ρ = 0.90 ρ = 0.85

0.8 29.2 89.4 99.9

0.7 5.8 40.4 86.4

0.1 1.5 9.5 40.7

5.0 75.6 99.4 100

4.8 40.1 87.6 99.4

3.9 15.5 52.0 86.2

1.4 29.3 90.1 99.7

2.5 5.8 85.4 70.8

0.9 1.1 82.4 31.6

5.4 64.4 92.5 98.4

5.3 23.3 54.3 74.2

4.9 14.6 33.0 56.7

ϕ1 = 0.6 ρ = 1.0 ρ = 0.95 ρ = 0.90 ρ = 0.85

0 2.1 18.1 61.0

0 0.1 0.8 67.0

0 0 0 0.2

6.1 71.0 99.8 99.9

6.5 38.2 95.9 97.0

5.5 16.4 71.4 86.1

0.4 6.2 24.2 62.3

4.5 0.7 1.4 4.1

1.2 0.2 0.3 0.6

5.0 62.3 90.0 97.0

5.1 22.0 47.7 67.7

5.0 13.2 28.8 44.9

A brief summary of the results follows. MA(1) error dynamics. (1)

(2) (3)

ˆ When θ1 = −0.5, the empirical size is very inaccurate for the δ-type tests even with a lag of 8; however, by comparison the τˆ -type tests with this lag maintain their nominal size. When θ1 = 0.5, it is again only the τˆ -type tests with a lag of 8 that maintain ˆ their nominal size, the equivalent δ-type tests now being undersized. A power comparison is misleading given that the other tests suffer severe size distortions.

AR(1) error dynamics (1) (2) (3) (4) (5)

If no lag is needed, as when ϕ1 = 0.0, but a long lag is used, the τˆ -type tests ˆ are better than the δ-type tests at maintaining nominal size. Some power is lost for each of the tests if the longer lag is unnecessary. When a longer lag is necessary but no lag is used, size deteriorates markˆ edly – both tests become undersized; more so in the case of the δ-type tests. ˆ Where a power comparison is valid, that is, size is correct, the δ-type tests are generally more powerful than the τˆ -type tests. Where size is comparable, the impact on power of over-specifying the deterministics in the maintained regression can be substantial; for example, when ρ = 0.95 and ϕ1 = 0.0, τˆ µ and τˆ β have empirical powers of 33.4% and 19.2%, respectively.

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Table 6.6b AR(1) error structure: (1 − ϕ1 L)zt = εt .

220 Unit Root Tests in Time Series

ˆ Overall, if the errors are white noise, then the δ-type tests are more powerful than the τˆ -type tests; however, in the case of AR(1) and MA(1) errors, the τˆ ˆ type tests are more robust than the δ-type tests, providing the lag length is long enough. The importance of getting the lag length ‘right’ and ‘long enough’ is clearly paramount to good inference, and this is considered more extensively in Chapter 9.

The standard trend stationary alternative assumes that the deterministic trend is linear (or trivially a constant); this could be extended in similar fashion to a higher-order polynomial in time, which could be viewed as approximating a nonlinear trend function by a low-order Taylor series approximation. Allowing nonlinear trends raises the question, what is a trend? Granger et al. (1997, p.67) suggest that a deterministic trend may have some, but not many, changes of direction within a sample, but should eventually have none outside the sample, which would rule out non-monotonic functions. For some developments of the trended alternative in the context of testing for a unit root see Ouliaris et al. (1989) and Bierens (1997). The trend stationary alternative is easily generalised to allow more general polynomial trends. For example, consider: (yt − µt ) = zt (1 − ρL)zt = εt where µt = ∑Jj=0 βj tj . This speciﬁcation results in: (1 − ρL)(yt − ∑j=0 βj tj ) = εt J

(6.78)

∆yt = γyt−1 + (1 − ρ)β0 + (1 − ρL) ∑j=1 βj tj + εt J

(6.79)

If the null of a unit root is correct, then: ∆yt = εt More generally ∆yt = zt , where zt is generated by a stationary process. Whilst the extension to a higher-order polynomial trend does not create any new problems of principle, in a particular application it would be important to assess the implications – for example, for forecasting – of ﬁtting such a trend. The DF framework can still be used, but the asymptotic distributions of the ˆ δ-type and τˆ -type tests will depend on the order of the ﬁtted trend polynomial; this raises no practical difﬁculties as the form of the asymptotic distribution is known and the relevant quantiles are easy enough to simulate.

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6.5 Nonlinear trends

Dickey-Fuller and Related Tests 221

Ouliaris et al. (1989) allow µt to be a general deterministic function, with the polynomial trend as a special case. They note that a test can be based on the 2 observation that under H0 , ∆yt is a stationary process with σ∆y,lr > 0, whereas under the alternative, ﬁrst differencing induces a unit root in residuals from the detrending regression, given by:

ˆ t = ∑Jj=0 βˆ j tj . The suggested test For example, in the polynomial trend case µ procedure is based on assessing whether ε˜ t has a MA(1) representation with 2 a unit root, where under the null hypothesis σ∆y,lr > 0, but under the alter2 2 native σ∆y,lr = 0. A test statistic is based on estimating the ratio σ∆y,lr /σz2 , say 2 vˆ 2 = σˆ ∆y,lr /ˆσz2 . Simulated percentiles are provided in Ouliaris et al. (1989) and the test is based on the bounds principle where the lower and upper bounds, vˆ 2L and vˆ 2U , are such that if the sample value of vˆ 2 < vˆ 2L , then the conclusion is in favour of the alternative hypothesis, whereas if vˆ 2 > vˆ 2U then the conclusion is in favour of the null hypothesis. Whilst Ouliaris et al. (1989) suggest 2 an estimator of σ∆y,lr based on a weighting the periodogram ordinates around the zero frequency, other methods, which may have better properties, are considered in section 6.8.2.ii. For an alternative procedure see Bierens (1997), who suggests using orthogonal Chebishev polynomials to approximate an unknown nonlinear deterministic time trend within the ADF framework and provides quantiles for testing and some empirical illustrations.

6.6 Exploiting the forward and reverse realisations A simple way of improving the power of the DF-type tests is to use the information in the sample twice! Two ways of doing this are outlined in this section. The ﬁrst (section 6.6.1) is based on the reversal of the order in time of the observations, which leads to an equally valid representation of the integrated (stochastic trend) nature of a unit root process. Each representation can then be used to obtain a test statistic for the unit root null hypothesis and a decision can be based on some combination of the two test statistics. A second method (section 6.6.2) of using information from the two representations is to obtain a single test statistic from a method which weights the two residual sums of squares. 6.6.1 DF-max tests (Leybourne, 1995) To start, consider the simplest version of the unit root model: yt = yt−1 + εt

(6.80)

then, by back-substitution, regarding y0 as given (see Equation (6.10)): yt = y0 + ∑i=1 εi t

(6.81)

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ˆt ε˜t = ∆yt − µ

222 Unit Root Tests in Time Series

In this representation, the dependence of yt is on yt−1 , implying a backward recursion (but note that this way of writing the difference equation is usually referred to as the forward realisation). A unit root test is then based on the following regression model:

The process can also be viewed as ‘starting’ from the last observation in the sample, yT , and unravelling the cumulative contributions of the innovations (see Leybourne, 1995). Note that (6.80), can be rearranged so that yt−1 = yt – εt ; then, by successive substitution starting at yT (regarding yT + 1 as given): yT = yT + 1 − εT + 1 yT−1 = yT − εT = yT + 1 − ε T + 1 − ε T yT−2 = yT + 1 − εT−1 = yT + 1 − εT + 1 − εT − εT−1 .. . yT−i = yT + 1 − ∑j=0 εT + 1−j i

.. . y1 = y 2 − ε 2 = yT + 1 − ∑j=0 εT + 1−j T−1

(6.82)

Clearly this is also an integrated process that has the same structure as the usual way of representing the stochastic trend. To exploit this more fully, deﬁne the variable rt = yT + 1−t that reverses the time order of the series, such that for t = 0, 1, . . . , T + 1, r0 = yT + 1 , r1 = yT , r2 = yT−1 , .., rT + 1 = y0 ; and also deﬁne ξt = −εT + 1−(t−1) , such that ξ1 = −εT + 1 , ξ2 = −εT , ξ3 = −εT−1 , . . . , ξT = −ε2 . Then the integrated process is: rt = r0 + ∑j=1 ξj t

(6.83)

This suggests that a unit root test may also be obtained from the reverse realisations: rt = µ∗r + ρr rt−1 + ηt

(6.84)

The DF tests using rt rather than yt will be distinguished by an r superscript; for example, τˆ rµ .

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yt = µ∗ + ρyt−1 + εt

Dickey-Fuller and Related Tests 223

1%

5%

10%

τˆ max µ 100 200

–2.14 –2.13

–2.45 –2.44

–3.11 –3.06

τˆ max β 100 200

–2.84 –2.83

–3.16 –3.12

–3.75 –3.72

Source: Extracted from Leybourne (1995, Tables 1 and 2).

Leybourne (1995) shows that the DF τˆ -tests and their counterparts using the reversed observations have the same asymptotic distributions, a result that also ˆ extends to the ADF versions of the test (and by extension to the DF δ-tests). There are several ways of combining the information in the forward and reverse realisations. One method, described below, is to combine the residual sums of squares deﬁned by the two representations, possibly with unequal weights. Another way is to obtain a single test statistic from the two tests, for example τˆ µ and τˆ rµ . Leybourne (1995) suggests taking the maximum value of the two tests; for example, τˆ max τµ , τˆ rµ ). Bearing in mind that sample realisations are µ = max(ˆ typically negative, this means choosing the test statistic that is more favourable to non-rejection of the null hypothesis. Some critical values for the τˆ max and µ τˆ max tests are given in Table 6.7. β The simulation evidence in Leybourne (1995) shows that the DF-max tests are more powerful, sometimes much more so, than their standard DF counterparts; they also are robust to near I(2) processes and some other (symmetric and nonsymmetric) distributions for the innovations. Leybourne et al. (2005) also show that the DF-max tests fare well more generally when compared to several other unit root tests. This is a comparison we return to later in Chapter 7. 6.6.2 Weighted symmetric tests An alternative way of combining the information from the forward and reverse realisations is to weight the residual sums of squares from the forward and reverse realisations. To start, assume for simplicity that µt = 0, then the weighted sum of squares is: WSS(ρ, wt ) = ∑t=2 wt (yt − ρyt−1 )2 + ∑t=1 (1 − wt + 1 )(yt − ρyt + 1 )2 T

T−1

(6.85)

The weights wt determine how much inﬂuence each of the sums of squares has in the determining the estimator of ρ. The simple symmetric estimator uses the weights wt = 0.5, whereas the weighted symmetric (WS) estimator uses the weights wt = (t − 1)/T (see Pantula et al., 1994; Park and Fuller, 1995). Cases

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Table 6.7 Critical values of the DF-max tests, and τˆ max τˆ max µ β .

224 Unit Root Tests in Time Series

WSS(ˆρws , wt ) = uˆ uˆ = uˆ b uˆ b + uˆ f uˆ f

(6.86)

Where uˆ = (uˆ b uˆ f ) is a 1×2(T – 1) vector of residuals from the backward and forward regressions. Hence the data for a double-length regression, which can be used in a standard OLS routine, where ydt is the dependent variable and ydt−1 is the regressor, are: √ √ √ √ √ √ ydt = ( w2 y2 , w3 y3 , ..., wT yT ; wT y1 , wT−1 y2 , ..., w2 yT−1 ) √ √ √ √ √ √ ydt−1 = ( w2 y1 , w3 y2 , ..., wT yT−1 ; wT y2 , wT−1 y3 , ..., w2 yT )

(6.87a) (6.87b)

Note that 1 − w2 + j = wT−j , which is used in the data construction to emphasise the symmetry of the weighting scheme. The estimated coefﬁcient from the double-length regression of ydt on ydt−1 is ˆ denoted ρˆ ws , which can be used to construct the δ-type test, δˆ ws = T(ˆρws − 1) and ws ws the τˆ test, denoted τˆ . Deterministic terms, as in Models M1 and M2 , can be accounted for by prior demeaning or detrending (and might then attract a further degrees of freedom correction if an LS-type estimator is being used), in which case yˆ˜ t replaces yt in (6.87) and throughout. These adjustments give rise ˆ ws ˆ ws ˆ ws to the test statistics denoted δˆ ws µ , δβ and τ µ ,τ β , respectively. Some care has to be taken in the construction of the WS τˆ -type tests when these are to be obtained from the double-length regression. Speciﬁcally, in the simple case, the double-length regression contains 2(T – 1) observations and 2(T – 1) – 1 degrees of freedom, and standard LS routines will use this in the calculation of the coefﬁcient standard error. However, this is incorrect for present purposes as there are really only T – 1 observations and (T – 1) – 1 = T – 2 degrees of freedom. Hence the estimated standard error for ρˆ ws , say σˆ (ˆρws ), should be multiplied by a degrees of freedom correction, so that: (2T − 3) 0.5 σˆˆ (ˆρws ) = σˆ (ˆρws ) (T − 2)

(6.88)

The adjusted standard error σˆˆ (ˆρws ) is then the denominator in the τˆ -type test where the numerator is (ˆρws − 1), so that τˆ ws = (ˆρws − 1)/σˆˆ (ˆρws ). In practice, the

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other than µt = 0, are dealt with in the usual way using yˆ˜ t , the (estimated) demeaned or detrended series; for example, in the case of demeaned data, ¯ where y¯ is the mean of yt over t = 1, . . . , Pantula et al. (1994) use yˆ˜ t = yt – y, T observations (the ‘global’ mean). As to the ease of computation, note that (6.85) is just the residual sum of squares that results from the T – 1 observations in the regression √ √ w y = ρ( wt yt−1 ) + ub,t , t = 2, ..., T and the T – 1 observations in the regression t t (1 − wt + 1 )yt = ρ( (1 − wt + 1 )yt + 1 ) + uf,t , t = 1, ..., T – 1; then

Dickey-Fuller and Related Tests 225

degrees of freedom correction will also have to allow for estimated deterministic terms and additional lags. When it is necessary to augment the maintained regression by lags of the dependent variable, by analogy with the ADF regression, the implications for the WS versions of the tests are as follows. The weighted sum of squares, which is written to exploit the link with the ADF regression, is now:

t=k+1

wt (∆yt − γyt−1 −

∑

∑ cj ∆yt−j )2 +

j=1

T−(k + 1)

+

k−1

(1 − wt + 1 ) −∆yt + 1 − γyt + 1 +

t=1

2

k−1

∑ cj ∆yt + 1 + j )

j=1

(6.89) By analogy, this is referred to as the AWSDF (augmented weighted symmetric Dickey-Fuller) regression. The weights are now deﬁned as follows (see also Park and Fuller, 1995; Fuller, 1996): wt = (t − (k − 1) − 1)/(T − 2(k − 1)); wt = 0 for 1 ≤ t ≤ k; wt = 1 for T ≥ t ≥ T – k. Note that the weights start from wk + 1 and that ∆yt + j = yt + j − yt + j−1 ; the dependent variable in the forward sum of squares is written using −∆yt + 1 = yt − yt + 1 . The double-length regression is:                

                

√ wk+1 ∆yk+1 √ wk+2 ∆yk+2 .. . √ wT ∆yT √ − wT ∆y2 √ − wT−1 ∆y3 .. . √ − wk+1 ∆yT−(k−1) √ wk+1 yk √ wk+2 yk+1 .. . √ wT yT−1 √ wT y 2 √ wT−1 y3 .. . √ wk+1 yT−(k−1)

        =       

√ wk+1 ∆yk √ wk+2 ∆yk+1 .. . √ wT ∆yT−1 √ − wT ∆y3 √ − wT−1 ∆y4 .. . √ − wk+1 ∆yT−(k−1)+1

··· ··· ··· ··· ··· ··· ··· ···

√ wk+1 ∆y2 √ wk+2 ∆y3 .. . √ wT ∆yT−(k−1) √ − wT ∆yk+1 √ − wT−1 ∆yk+2 .. . √ − wk+1 ∆yT

     γˆ ws     c1   .   ..    ck−1   

     

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T

∑

WSS(γ, wt ) =

226 Unit Root Tests in Time Series

       +       

ub,k+1 ub,k+2 .. . ub,T uf,1 uf,2 .. . uf,T−k

               

(6.90)

As in the simplest case, a degrees of freedom correction is required to the LS ˆ estimated standard error on γˆ ws . Fuller (1996) suggests σˆ u2 = uˆ u/(T−k−1), which implies the standard error adjustment [(2T – 3k)/(T – k – 1)]0.5 ; an asymptotic equivalent is the degrees of freedom motivated adjustment, so that: (2T − 3k) 0.5 σˆˆ (ˆρws ) = σˆ (ˆρws ) (T − 2k)

(6.91)

As before, further adjustments are necessary if either prior demeaning or detrending has take place. The power of the WS tests generally exceeds that of their DF counterparts, sometimes quite considerably. Pantula et al. (1994) suggest that to protect against the loss of size retention when there are dependent errors, the lag in the AWSDF should be chosen by the rule that adds two to the lag chosen by AIC, say lag(AIC) + 2. Further consideration of the size and power of the WS tests is reported in sections 6.7 and 6.8, and critical values can be obtained from the response surface method using the coefﬁcients reported in Table 6.12 in the appendix to this chapter.

6.7 Non-iid errors and the distribution of DF test statistics Standard assumptions on the derivation and proof of the limiting distributions for the DF tests start with the case that {zt } = {εt } where εt ∼ iid(0, σε2 ). However, this assumption can be substantially relaxed, and normality is not required. Phillips (1987a) showed that the relevant limit distributions can be established under very general conditions where the error sequence exhibits (a) weak dependence, (b) heterogeneity, (c) some nonstationarity. The frequently used (stationary) ﬁnite-order ARMA models are included by assumption (a). (See also Fuller, 1996, Theorem 10.1.1, for a proof based on the assumption that {εs }t1 is a martingale difference sequence.)

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

Dickey-Fuller and Related Tests 227

Further work by Gonzalez and Pitarakis (1998) on the effects of dependent errors established a number of results by theoretical means that had only previously been indicated by Monte Carlo simulations. This section takes up the case initially considered in section 6.3.6, where the zt are not iid, leading examples being where an MA, AR or ARMA process is generating the errors as a function of a white-noise input εt (see Phillips, 1987a; Phillips and Perron, 1988). The ADF approach arises exactly if zt is generated by a pure AR process and by approximation if there is an MA component to the generating process. The Xiao and Phillips (1998) approach to the derivation of the DF-type tests, considered in section 6.3.6, was an indication of how the asymptotic distribution of tests based on Tγˆ were ‘contaminated’ by the nuisance parameters in the error structure, but a simple transformation of Tγˆ combined with consistent estimation of the sigma ratio led to a feasible test statistic.

6.7.1.i Distinguishing variances When the zt are not iid, there are three variances that need to be distinguished, as follows (see also section 6.1.1): (i) (ii) (iii)

The variance of εt , denoted σε2 , which is also the conditional variance (of yt ). The variance of zt , denoted σz2 , which is the unconditional variance of yt , (which could also, therefore, be denoted σy2 ). 2 . The ‘long-run’ variance of zt , denoted σz,lr

The ﬁrst of these is straightforward as σε2 is the variance of εt and the variance of yt conditional on lagged yt . If {zt } = {εt } then, generally, σz2 = σε2 (only generally because, for example, setting {zt } = {−εt } results in a series with the same variance); examples of σz2 for some familiar processes are given below. The long2 , arises from the sum or partial sum of the errors where σ2 ≡ run variance, σz,lr z,lr limT→∞ E(S2T /T), ST = ∑Tt=1 zt where this limit is assumed to exist and is ﬁnite (see 2 consistently has been Equation (6.15)). The question of how to estimate σz,lr referenced in section 6.1.1 and will be considered further in section 6.8.2.

6.7.1.ii Effects on the limit distributions In the case of M0 , the effect of introducing error dynamics is to alter the asymptotic distributions of the test statistics δˆ and τˆ as follows:

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6.7.1 Departures from iid errors

228 Unit Root Tests in Time Series

1 1 σ2 B(1)2 − 2z 1 2 2 σz,lr 0 B(r) dr   2 1 1 σlr σ   τˆ ⇒D B(1)2 − 2z  1/2  2 σz σz,lr 1 2 0 B(r) dr δˆ ⇒D

(6.92)

In the simpler case previously considered where zt ∼ iid(0, σε2 ), then σε2 = σz2 = 2 , and: σz,lr ˆ DF δˆ ⇒D F(δ)

(6.94)

τˆ ⇒D F(ˆτ)DF

(6.95)

In models M1 or M2 then, as before, standard Brownian motion is replaced by demeaned or detrended Brownian motion as appropriate. It is nonzero autocovariances in {zt } that lead to differences between σε2 , σz2 2 . In an ARMA process, these arise from either or both the autoregressive and σz,lr or moving average dynamic components. Note that zt being iid is sufﬁcient but not necessary for σz2 = σε2 , as this equality it will also hold for a martingale difference sequence with some heterogeneity (see Phillips and Perron, 1988, p.339). Comparing (6.57) with (6.92) and (6.58) with (6.93), it is informative to write the limit distributions as those in the standard case plus the difference due to the non-iid errors: 2 σz,lr − σz2 1 1 1 1 δˆ ⇒D + B(1)2 − 1 1 1 2 2 2 2 2 σz,lr 0 B(r) dr 0 B(r) dr (6.96a)

2 σz,lr − σz2 1 ˆ DF + 1 = F(δ) 1 2 2 2 σz,lr 0 B(r) dr τˆ ⇒D

1 + 2

σz,lr σz

 1  B(1)2 − 1  1 2

2 − σ2 σz,lr z

σz σz,lr

0

  

1

1 2 0 B(r) dr

(6.96b) 

1 B(r)2 dr

 1/2 



 1/2 

(6.97a)

 2 2 σ − σ (σz,lr − σz ) 1 z  z,lr F(ˆτ)DF + = 1+  σz 2 σz,lr σz 1

0

  1/2 

1

(6.97b)

B(r)2 dr

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(6.93)

Dickey-Fuller and Related Tests 229

ˆ notice that the effect of the non-iid errors is a location displaceIn the case of δ, 2 > σ2 and to the left for ment, moving the limit distribution to the right for σz,lr z 2 < σ2 . In the case of τ ˆ the effects are twofold: a scale displacement and a σz,lr z 2 > σ2 and negative for σ2 < σ2 . location displacement; both are positive for σz,lr z z z,lr

Gonzalo and Pitarakis (1998) show by analytical means that the expectations ˆ = –1.781 and E(ˆτ) = –0.423 for iid of the limiting distributions of δˆ and τˆ are E(δ) errors, with asymptotic variances 10.11 and 0.9626, respectively. These results show the leftward shift in the asymptotic distributions of δˆ and τˆ when ρ = 1 compared to the stationary case. However, note that in the case of ﬁnite sample sizes, the shift is gradual as ρ → 1 rather than sudden at ρ = 1. ˆ and E(ˆτ) are: For an MA(1) process, E(δ) θ1 ˆ E(δ) = − 1.78143 + 5.56286 (6.98) (1 + θ1 )2 θ1 (1 + θ1 ) + 2.09211 (6.99) E(ˆτ) = − 0.42310 (1 + θ21 )1/2 (1 + θ1 )(1 + θ21 )1/2 ˆ and E(ˆτ) are: For an AR(1) process, E(δ) ϕ1 ˆ = − 1.78143 + 5.56286 E(δ) (1 + ϕ1 ) (1 + ϕ1 ) 1/2 ϕ1 2.09211 E(ˆτ) = − 0.42310 + (1 − ϕ1 ) (1 − ϕ1 )1/2 (1 + ϕ1 )1/2

(6.100) (6.101)

ˆ = –113.04 and E(ˆτ) = –6.60; and when ϕ1 = 0.8, For example, when θ1 = –0.8, E(δ) ˆ = 0.69 and E(ˆτ) = 1.52. The expectations E(δ) ˆ and E(ˆτ) for some values of E(δ) θ and ϕ1 for M0 , M1 and M2 are given in Table 6.8. The mean of both limit distributions shifts to the left as θ1 < 0 and ϕ1 < 0, respectively, and to the right as θ > 0 and ϕ1 > 0, respectively. Note that the expected value of the asymptotic distribution shifts further to the left as more deterministic terms are included in the maintained regression. 6.7.1.iv Effect of non-iid errors on the size of the tests Because large negative values of the test statistic lead to rejection of the unit root null, the effect of a leftward shift in the correct asymptotic distribution ˆ for δ-type tests, when the standard DF distribution is used, is to increase the actual size relative to the nominal size. The standard test, which does not take account of the non-iid errors, is thus oversized in such a case and there are spurious rejections of the unit root hypothesis. The effect of a rightward shift is to lead to undersizing. The effect on the τˆ -type tests is generally of the same kind, but can be complicated by the scale (as well as location) shift.

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6.7.1.iii Expectations of the limit distributions

230 Unit Root Tests in Time Series

Table 6.8 Effect of non-iid errors on means of limit distribution of test statistics.

θ1

ˆ E(δ)

E(δˆ µ )

E(δˆ β )

E(ˆτ)

E(ˆτµ )

E(ˆτβ )

–0.9 –0.7 –0.1 0 0.1 0.7 0.9

–502.44 –45.05 –2.49 –1.78 –1.32 –0.43 –0.39

–846.1 –88.0 –6.69 –5.34 –4.47 –2.75 –2.70

–1506.2 –166.0 –12.78 –10.18 –8.55 –5.28 –5.16

–14.03 –4.10 –0.61 –0.42 –0.27 0.12 0.14

–20.0 –6.23 –1.71 –1.53 –1.40 –1.09 –1.09

–27.93 –8.83 –2.44 –2.18 –1.99 –1.56 –1.55

AR(1) error structure ϕ1

ˆ E(δ)

E(δˆ µ )

E(δˆ β )

E(ˆτ)

E(ˆτµ )

E(ˆτβ )

–0.9 –0.7 –0.1 0 0.1 0.7 0.9

–51.85 –14.76 –2.34 –1.78 –1.28 0.52 0.85

–101.47 –30.43 –6.59 –5.34 –4.37 –0.96 –0.29

–191.67 –57.85 –12.51 –10.18 –8.35 –1.81 –0.53

–4.417 –2.228 –0.593 –0.42 –0.26 1.04 2.48

–6.68 –3.65 –1.70 –1.53 –1.38 –0.65 –0.36

–9.49 –5.19 –2.41 –2.18 –1.97 –0.96 –0.49

ˆ and E(ˆτ) from Gonzalo and Pitarakis (1998, Tables 1 and 2), these are Notes: E(δ) exact from analytical calculations; other table entries are from 50,000 replications with T = 10,000, and are subject to simulation error. The standard case is shown in bold.

In the MA(1) case, the incorrect sizing is substantial even for moderate values of θ1 . For example, the simulation results reported in Table 6.6 indicated that the actual size of the τˆ µ test for a 5% nominal size when θ1 = –0.5, and T = 200, is close to 60%. This explains the simulation results that ﬁnd the actual size approaches 100% as θ1 → –1 (see, for example, Schwert, 1987). When θ1 is positive, the limit distributions shift to the right, and the tests become undersized. Similar but less pronounced effects are found for the AR(1) case. As ϕ1 → –1, the distortions are less pronounced than when θ1 → –1; but this is empirically the more unlikely case. As ϕ1 → + 1 there is a slightly greater shift to the right, for each of the limit distributions, compared to θ1 → + 1. The shift to the right results in undersizing of the tests; for example, observe from Table 6.6 that the empirical size of τˆ µ when ϕ1 = 0.3 for a 5% nominal size was 2.5%. The theoretical results of Gonzalo and Pitarakis (1998) explain why this oversizing and undersizing occurs.

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MA(1) error structure

Dickey-Fuller and Related Tests 231

The presence of ARMA(1, 1) errors does not necessarily lead to greater displacements than in either the pure MA or pure AR cases. Indeed, near-cancellation of the roots leads to much less distortion than if one or other parameter was zero. 6.7.2 Illustrations Two examples are used here to illustrate the different variances and the impact on the limit distributions of some non-iid errors.

First, consider zt generated by an MA(1) process, then: zt = εt + θ1 εt−1 = (1 + θ1 L)εt σz2 = E(εt + θ1 εt−1 )2 = E(ε2t ) + θ21 E(εt−1 )2

θ(L) = (1 + θ1 L) unconditional variance of zt using E(εt εt−1 ) = 0

= (1 + θ21 )σε2 2 σz,lr = θ(1)2 var(εt )

long-run variance of zt

= (1 + θ1 )2 σε2 where εt ∼ iid(0, σε2 ). Consider the ratio: 2 σz,lr

σz2

=

(1 + θ1 )2 (1 + θ21 )

and a graph of this ratio is shown in Figure 6.3, from which it is evident that: θ1 > 0 ⇒

2 σz,lr

σz2

> 1 and θ1 < 0 ⇒

2 σz,lr

σz2

<1

A number of economic studies have suggested that θ1 < 0, hence the case where 2 < σ2 seems more likely. σz,lr z 2 in (6.96a) and (6.96b), the limit distributions are: Using σz2 and σz,lr 1 θ1 ˆδ ⇒D F(δ) ˆ DF + (6.102) 1 2 (1 + θ1 )2 0 B(r) dr (1 + θ1 ) θ1 1 F(ˆτ)DF + τˆ ⇒D 1/2 2 2 1/2 1/2 (1 + θ1 ) (1 + θ1 )(1 + θ1 ) 1 2 0 B(r) dr

(6.103) 2 > σ2 , and if θ < 0 then σ2 < σ2 , giving rightward Note that if θ1 > 0 then σz,lr 1 z z z,lr and leftward shifts, respectively, to the limiting distributions of δˆ and τˆ relative

to the iid case. In the case the limit distribution of τˆ , the scale factor is above unity if θ1 > 0 and below unity if θ1 < 0.

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6.7.2.i MA(1) errors

232 Unit Root Tests in Time Series

2 1.8 MA(1) errors

1.6

θ1 > 0 implies σ 2z,lr > σ 2z

1.4 1.2 σ 2z,lr / σ 2z

0.8 0.6 0.4 θ1 < 0 implies σ 2z,lr < σ 2z

0.2 0 –1

–0.8

–0.6

–0.4

–0.2

0 θ1

0.2

0.4

0.6

0.8

1

Figure 6.3 Ratio of variances: long-run/unconditional (MA(1) errors).

6.7.2.ii AR(1) errors In the AR(1) case: zt = ϕ1 zt−1 + εt (1 − ϕ1 L)zt = εt σz2 = E(ϕ1 zt−1 + εt )2 = ϕ12 E(z2t−1 ) + E(ε2t )

using E(zt−1 εt ) = 0

= σε2 /(1 − ϕ12 )

using E(z2t−1 ) = σz2

2 σz,lr = ϕ(1)−2 var(εt )

2 σz,lr

σz2

=

1 σ2 (1 − ϕ1 )2 ε

=

(1 − ϕ12 ) (1 − ϕ1 )2

A graph of this ratio is shown in Figure 6.4, from which the following are evident: ϕ1 > 0 ⇒

2 σz,lr

σz2

> 1 and ϕ1 < 0 ⇒

2 σz,lr

σz2

<1

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1

Dickey-Fuller and Related Tests 233

2 > σ2 . Using σ2 and σ2 in The case ϕ1 > 0 seems more likely implying σz,lr z z z,lr (6.96b) and (6.97b), the limit distributions are:

1 ϕ1 1 2 (1 + ϕ1 ) 0 B(r) dr (1 + ϕ1 )1/2 ϕ1 F(ˆτ)DF + τˆ ⇒D (1 − ϕ1 )1/2 (1 − ϕ1 )1/2 (1 + ϕ1 )1/2 1

ˆ DF + δˆ ⇒D F(δ)

1

1/2

B(r)2 dr

(6.105) The limit distributions of δˆ and τˆ are shifted to the right for ϕ1 > 0 and to the left for ϕ1 < 0; and the scale factor is above unity if ϕ1 > 0 and below unity if ϕ1 < 0. 6.7.2.iii Effect of non-iid errors on the size of DF tests The incorrect sizes of the DF tests are shown in Table 6.9, top panel, for MA(1) errors and in Table 6.9, lower panel, for AR(1) errors, both for a 5% nominal size. These results are from the asymptotic distribution and cannot be attributed to ﬁnite sample effects. In the case of MA(1) errors, the oversizing is quite substantial even for moderate values of θ1 ; for example, when θ1 = –0.1 the actual size is virtually double the nominal size, and actual size is 100% for θ1 = –0.9.

9 8 AR(1) errors 7 6

σ2 z,lr

2 /σz

ϕ 1 > 0 implies σ 2z,lr > σ 2z

5 4 3 2

ϕ 1 < 0 implies σ 2z,lr < σ 2z

1 0 –1

–0.8

–0.6

–0.4

–0.2

0 ϕ1

0.2

0.4

0.6

0.8

1

Figure 6.4 Ratio of variances: long-run/unconditional (AR(1) errors).

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0

(6.104)

234 Unit Root Tests in Time Series

Table 6.9 Effect on (asymptotic) size of DF tests with non-iid errors.

θ1

δˆ

δˆ µ

δˆ β

τˆ

τˆ µ

τˆ β

–0.9 –0.7 –0.5 –0.3 –0.1 0 0.1 0.3 0.5 0.7 0.9

100 83.3 47.0 21.4 8.2 5.0 3.1 1.4 0.8 0.6 0.5

100 97.6 67.9 23.1 9.4 5.0 2.6 0.9 0.4 0.3 0.2

100 100 88.7 44.4 11.5 5.00 2.0 0.5 0.1 0.0 0.0

100 81.6 45.1 20.5 8.0 5.0 3.2 1.6 0.9 0.8 0.8

100 96.2 62.4 26.5 8.3 5.0 3.3 2.5 2.1 2.3 2.3

100 99.9 86.3 40.8 10.7 5.0 2.4 1.0 0.7 0.8 0.7

AR(1) errors ϕ1

δˆ

δˆ µ

δˆ β

τˆ

τˆ µ

τˆ β

–0.9 –0.7 –0.5 –0.3 –0.1 0 0.1 0.3 0.5 0.7 0.9

86.6 51.4 30.6 16.9 7.8 5.0 3.0 0.7 0.1 0.0 0.0

98.5 73.7 45.2 23.9 9.4 5.0 2.3 0.3 0.0 0.0 0.0

99.9 92.6 64.9 33.2 11.1 5.0 1.8 0.1 0.0 0.0 0.0

85.1 49.7 29.3 16.2 7.8 5.0 3.1 0.9 0.2 0.1 0.3

97.4 68.2 39.4 19.9 8.3 5.0 3.2 2.2 3.2 5.7 14.2

100.0 90.6 61.2 30.6 10.5 5.0 2.2 0.7 0.8 1.6 6.2

Notes: Results obtained from 50,000 simulations with T = 10,000. Slight differences from the corresponding entries for Table 6.6 are due to the larger sample size used here.

In the case of AR(1) errors, when ϕ1 < 0 the tests are oversized (rejecting too frequently given the nominal size); the oversizing becomes more pronounced as ϕ1 → –1, and approaches 100%. When ϕ1 > 0, the more likely case in practice, the tests (generally) become undersized (rejecting too infrequently given ˆ the nominal size), approaching zero for the δ-type tests even at quite moderate values of ϕ1 (for example, ϕ1 = 0.3). As noted in section 6.4, in both the MA(1) and AR(1) cases the τˆ versions of the test are better than the equivalent δˆ test, in the sense that there is less size distortion, indicating a greater robustness to the dependent errors. However, there is not much comfort here as the τˆ tests still show severe size distortions. The MA(1) case for θ1 = –0.5, is also illustrated graphically in Figures 6.5a– 6.5c for the τˆ -type tests with different deterministic terms; in each case there is

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MA(1) errors

Dickey-Fuller and Related Tests 235

0.4 0.35 actual size = 45.1%

θ1 = 0.0

0.3 0.25 0.2 0.15

θ1 = –0.5

0.05 0 –6

–5

–4

–3

–2

–1

0

1

2

3

Figure 6.5a Distribution of τˆ with MA(1) errors. 0.5 0.45

θ1 = 0.0

0.4 0.35

actual size = 62.4%

0.3 0.25 0.2 0.15

θ1 = – 0.5

0.1 0.05 0 –8

–7

–6

–5

–4

–3

–2

–1

0

1

Figure 6.5b Distribution of τˆ µ with MA(1) errors.

a leftward shift in the limit distribution relative to the case with θ1 = 0. The effect on the actual size of a unit root test that uses the critical value for θ1 = 0 can be read off as the area under the distribution for θ1 = –0.5, which is to the left of the 5% quantile in the case that θ1 = 0 (shown as a vertical line on Figure 6.5a). As anticipated, if the ‘wrong’ 5% quantile is used then the actual size greatly exceeds the nominal size; this oversizing increases from 45.1% for τˆ to 86.3% for τˆ β . The AR(1) case is illustrated graphically in Figures 6.6a–6.6c, where ϕ1 = 0.7, which induces a rightward shift in the limit distributions relative to the case with ϕ1 = 0. As before, the effect on the actual size of a unit root test that uses the critical value for ϕ1 = 0 can be read off as the area under the distribution

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0.1

236 Unit Root Tests in Time Series

0.7 θ1 = 0.0

0.6 0.5 0.4

actual size = 86.3%

0.3

0.1 0 –9

–8

–7

–6

–5

–4

–3

–2

–1

0

Figure 6.5c Distribution of τˆ β with MA(1) errors. 0.4

ϕ1 = 0.0

0.35 0.3 0.25

ϕ1 = 0.7

0.2 0.15 0.1 actual size = 0.1%

0.05 0 –3

–2

–1

0

1

2

3

4

5

6

Figure 6.6a Distribution of τˆ with AR(1) errors.

for ϕ1 = 0.0, but which is to the left of the 5% quantile in the case that ϕ1 = 0 (shown as a vertical line on each ﬁgure). Both undersizing and oversizing are evident due the combined scale and location effects in the case of the τˆ -type test; this test is oversized at 18.9% and 9.4%, respectively, in the no constant and constant included cases; but undersized at 3.5% in the trend case.

6.8 Phillips-Perron (PP) semi-parametric unit root test statistics The previous section has shown that the limit distributions of δˆ and τˆ are not invariant to dependent errors, and that the error (or dynamic) structure can have a fundamental effect on inference. One way of approaching this problem

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θ1 = –0.5

0.2

Dickey-Fuller and Related Tests 237

0.7 0.6

ϕ1 = 0.0

0.5 0.4 0.3

ϕ1 = 0.7

actual size = 5.7%

0.1 0 –5

–4

–3

–2

–1

0

1

2

3

4

Figure 6.6b Distribution of τˆ µ with AR(1) errors.

0.7

ϕ1 = 0.0

0.6 0.5 0.4

ϕ1 = 0.7

0.3 0.2 0.1 0 –4

actual size = 0.1% –3

–2

–1

0

1

2

3

Figure 6.6c Distribution of τˆ β with AR(1) errors.

has already been considered; that is, to augment the DF maintained regression either as an exact solution (potentially) if the error structure is AR, or as an approximate solution if the errors contain an MA component. Then, with consistent estimators of the required variances, a simple transformation returns the asymptotic distribution of the test statistic to the standard (DF) case. The ADF approach assumes an autoregressive parametric structure to augment the DF maintained regression. The developments of the DF-test statistics due to Phillips and Perron adopt a similar approach, but assume less structure about the nature of the error dynamics than the ADF approach. The end result is a

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0.2

238 Unit Root Tests in Time Series

semi-parametric method of estimating the long-run variance. To see how this approach works, initially consider the simplest case where the model is M0 . 6.8.1 Adjusting the standard test statistics to obtain the same limit distribution First, consider the limit distribution of δˆ with dependent errors, which is: 2 2 σz − σz,lr 1 1 1 2 2 2 σz,lr 0 B(r) dr

(6.106)

The second term on the right is the difference due to the dependent errors, and taking this to the left-hand side results in: 2 σz,lr − σz2 1 ˆδ − 1 ˆ DF ⇒D F(δ) (6.107) 1 2 2 2 σz,lr 0 B(r) dr 2 and a variable Thus what are needed are consistent estimators of σz2 and σz,lr −1 2 1 B(r)2 dr . Denoting the former as that is asymptotically distributed as σz,lr 0

1 2 , respectively, and noting that T−2 T y2 2 2 σ˜ z2 and σ˜ z,lr ∑t=1 t−1 ⇒D σz,lr 0 B(r) dr (see Phillips, 1987, Theorem 3.1 (a)), a corrected test statistic that in the limit has the DF distribution is:   2 −σ ˜ z2 σ˜ z,lr 1   Zδˆ = δˆ − (6.108) 2 T−2 ∑T y2 t=1 t−1

ˆ DF ⇒D F(δ) A corrected τˆ statistic can be obtained in the same way. First note that:   2 2 σz,lr 1 1 σz,lr − σz   τˆ ⇒D F(ˆτ)DF + (6.109)  1/2  σz 2 σz,lr σz 1 2 0 B(r) dr The corrected τˆ statistic is therefore obtained as: 2 σ˜ z,lr − σ˜ z2 1 1 σ˜ z τˆ − Zˆτ = 1/2 σ˜ z,lr 2 σ˜ z,lr T−2 ∑Tt=1 y2t−1

(6.110)

⇒D F(ˆτ)DF By extension, if the appropriate model is either M1 or M2 , then the analogous 2 1 B(r)2 dr, i = µ, β, where y ˜ t is demeaned or result is that T−2 ∑Tt=1 y˜ 2t−1 ⇒ σz,lr 0 i detrended yt , respectively. Explicit adjustments to the DF tests for the cases of M1 or M2 , referred to as Zδˆ µ and Zˆτµ and Zδˆ β and Zˆτβ , respectively, are given in Phillips and Perron (1988, p.341).

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ˆ DF − δˆ ⇒D F(δ)

Finite sample critical values for the PP tests are also typically taken as the ﬁnite sample critical values for the corresponding DF test. There are two reasons why some degree of approximation is involved in this procedure. First, as Cheung and Lai (1995a, 1995b) demonstrate in the context of a pure AR process generating the errors, the ﬁnite sample critical values for the DF tests show some sensitivity to different lag lengths. The simulations (T = 100) in Phillips and Perron (1988) in the case of zt ∼ niid(0, 1), show that using Zˆτµ for a 5% nominal size results in an actual size between 6.3% and 6.9% for m = 2 to 12, where m is the bandwidth parameter in semi-parametric estimation (see the next section). However, the actual size of Zˆρµ varies only between 4.4% and 5% for the same range of m. Second, it is typically the case that lag lengths, or bandwidths in the context of kernel estimation for PP tests, are not known but chosen by a data-dependent procedure, which introduces a further random variable into the ﬁnite sample distributions. 6.8.2 Estimation of the variances 6.8.2.i The unconditional variance, σz2 Two consistent estimators of σz2 are available depending on whether the null hypothesis is imposed (again, M0 is considered explicitly, with analogous results for M1 or M2 obtained by substituting yˆ˜ t ). These are as follows: 2 σ˜ z,a = ∑t=1 (yt − ρˆ yt−1 )2 /T T

Null hypothesis not imposed

(6.111)

Null hypothesis imposed

(6.112)

= ∑t=1 zˆ 2t /T T

2 σ˜ z,b = ∑t=1 (yt −yt−1 )2 /T T

= ∑t=1 zt /T T

Both estimators are consistent under the null, and substitution of one for the 2 , other will not change the asymptotic distribution of the PP tests. However, σ˜ z,b which is based on ﬁrst differences, will not be consistent against stationary alternatives, and therefore the power characteristics of the PP tests will depend on which is used. 2 , which is consistent under the unit root null and stationThe estimator σ˜ z,a ary alternatives, seems to be the preferred choice (Phillips and Ouliaris, 1990); 2 is consistent for explosive alternatives, ρ > 1, (Phillips, 1987a). Howfurther, σ˜ z,a ever, Schwert (1989) notes that because of the negative bias in the least squares estimator of the unit root, the autocorrelations of the residuals and the autocorrelations of the ﬁrst differences are not similar (as they should be) when θ1 = –0.8, and as a result they are a poor basis for estimating σz2 . The dilemma 2 or σ 2 is resolved in more recent work, which shows of whether to choose σ˜ z,a ˜ z,b that this is not an issue if modiﬁed versions of the PP tests are used, and these developments are considered in section 6.8.3.

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Dickey-Fuller and Related Tests 239

240 Unit Root Tests in Time Series

6.8.2.ii Estimating the long-run variance Semiparametric methods T 2 = lim 2 Recall that σz,lr T→∞ E(ST /T), where ST = ∑t=1 zt , and it is assumed that

2 ), and deﬁne σ2 as the E(zt ) = 0 for all t. Consider T−1/2 ST which ⇒D N(0, σz,lr T

variance of T−1/2 ST , then: σT2 ≡ var[T−1/2 ST ]

(6.113)

= T−1 ∑t=1 E(z2t ) + 2T−1 ∑κ = 1 ∑t = κ + 1 E(zt zt−κ ) T−1

T

Then let the number of autocovariances γ(κ) = E(zt zt−κ ) in σT2 be truncated to 2 be the consequent approximation to σ2 : m < T, and let σTm T 2 σTm = T−1 ∑t=1 E(z2t ) + 2T−1 ∑κ = 1 ∑t = κ + 1 E(zt zt−κ ) T

m

T

(6.114)

2 will be ‘close’ to σ2 if the contribution of the omitted autoThe quantity σTm T covariances, γ(κ) = E(zt zt−κ ) for κ > m, is ‘small’. For example, it is known that if zt is generated by an MA(q) process, then γ(κ) = 0 for κ > q. More generally, Phillips (1987a) shows that, given certain regularity conditions, if m → ∞ as 2 T → ∞, then σT2 – σTm →p 0 as T → ∞ (see Phillips, 1987a, lemma 4.1; also M¨ uller, 2007). 2 it is necessary to replace E(z2t ) and To construct an estimator based on σTm E(zt zt−κ ) by respective estimators, for which there are again two possibilities depending on whether the null hypothesis is imposed. If the null is imposed, then zt = ∆yt ; whereas if the null is not imposed, then the LS residuals zˆ t are used; that is, zˆ t = yt – ρˆ yt−1 , where ρˆ is the LS estimator from the regression of yt on yt−1 . If the null is not imposed, then the estimator is: 2 σ˜ Tm,a = T−1 ∑t=1 zˆ 2t + 2T−1 ∑κ = 1 ∑t = κ + 1 zˆ t zˆ t−κ T

m

T

(6.115)

If the null is imposed, then an estimator is provided by: 2 σ˜ Tm,b = T−1 ∑t=1 z2t + 2T−1 ∑κ = 1 ∑t = κ + 1 zt zt−κ T

m

T

(6.116)

where zt = ∆yt , or in the case of demeaned or detrended data, zt = ∆yˆ˜ t . Phillips (1987a, Theorem 4.2) shows that with a slight strengthening of the moment assumption on zt , then if m → ∞ as T → ∞, such that m = o(T1/4 ), then 2 2 2 under the null; however, the σ˜ Tm,a and σ˜ Tm,b are consistent estimators for σz,lr 2 , which is based on the residuals under the null, is not consistent estimator σ˜ Tm,b under stationary alternatives (see Phillips and Ouliaris, 1990). Also, Perron and 2 introduced below Ng (1996) show that the autoregressive-based estimator of σz,lr 2 . is preferred to σ˜ Tm,a 2 2 and σ˜ Tm,b may be negative if there are large negative The estimators σ˜ Tm,a sample autocovariances. To avoid this, use can be made of kernel estimators which deﬁne a weighting function for the sample autocovariances. The kernel

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T

Dickey-Fuller and Related Tests 241

function ω(κ) deﬁnes the weights ωm,κ and, for example, the revised estimator 2 σ˜ Tm,a (ω) is: 2 σ˜ Tm,a (ω) = T−1 ∑t=1 zˆ 2t + 2T−1 ∑κ = 1 ωm,κ ∑t = κ + 1 zˆ t zˆ t−κ T

m

T

(6.117)

Three of the kernel functions in use in this area are summarised below: Newey-West/Bartlett kernel κ = 1, ..., m;

(6.118)

Parzen kernel ωm,κ = 1 − 6(κ/(m + 1))2 + 6(κ/(m + 1)) κ ≤ (m + 1)/2 ωm,κ = 2(1 − κ/(m + 1))

3

(m + 1)/2 ≤ κ ≤ (m + 1)

(6.119a) (6.119b)

Bohman kernel ωm,κ = (1 − κ/(m + 1)) cos[π(κ/(m + 1)] + sin[π(κ/(m + 1)]/π

(6.120)

The kernel functions give a lower weight to the more distant autocovariances. In the Bartlett kernel, probably the most frequently used, the weights decline linearly with the lag length, κ. For an interesting recent suggestion that develops a nonparametric prewhitened covariance estimator with potential application to PP tests (see Xiao and Linton, 2002). Another issue, analogous to the selection of the maximum lag length in an AR model, is the choice of the truncation parameter, m. The usual methods consider m to be determined by the characteristics of the data; for example, calculating the sample autocovariances and adopting a cut-off rule including only those that are regarded as signiﬁcant. There have been a number of suggestions to automate this process. Although successful in other areas, recent theoretical and simulation results have suggested that an autoregressive estimator of 2 gives better results than kernel-based estimation procedures, but we brieﬂy σz,lr describe the general principle before considering the autoregressive approach. Automatic or plug-in bandwidth estimators specify a data-dependent rule for selecting m (see, for example, Andrews, 1991). In this approach, the ﬁrst step is to minimise the (asymptotic) mean squared error (MSE) associated with estimating a parameter vector or matrix of interest; for example, the variancecovariance matrix of a linear regression model, with possibly heteroscedastic and serially correlated errors. The MSE optimum implies an optimum lag truncation parameter, m∗ , used in modelling the dependence in the errors; m∗ is generally a function of unknown parameters and the kernel function deﬁning the weighting scheme. For example, for the Bartlett kernel, m∗ = 1.1447(ξ T)1/3 ,

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ωm,κ = 1 − κ/(m + 1)

where is ξ a function of unknown parameters in the error process. General expressions for ξ in a number of leading kernel functions and for AR(1), ARMA(1, 1) and MA(q) models have been provided by Andrews (1991, p.835). The last step in obtaining a feasible counterpart for m∗ is to approximate the error structure, here zˆ t , by a parametric scheme and replace the unknown parameters by their estimates. These results are robust to an extension to nonstationary regressors and, hence, cover the unit root case. Since approximation by a parametric scheme is part of this process, an alternative procedure, based on the same general idea, is to approximate the error-generating process by a ‘long’ AR model, a procedure that is familiar from the Said and Dickey (1984) approach to extending the basic DF model. 2 An autoregressive estimator of σz,lr 2 has been used with some success by Perron An autoregressive estimator of σz,lr and Ng (1996) and Ng and Perron (2001). The general approach is familiar from results due to Berk (1974) and Said and Dickey (1984), and the ADF approximation when the errors contain a moving average component. Consider the case where zt is generated by an ARMA process (if the data is demeaned or detrended then, as usual, yˆ˜ t replaces yt ) so that:

(1 − ρL)yt = zt ϕ(L)zt = θ(L)εt ψ(L) = θ(L)−1 ϕ(L)

ψ(L)zt = εt

Then (see, for example, Chapter 3, Equation (3.84)) the model can be written as: ∞

∆yt = γyt−1 + ∑j=1 cj ∆yt−i + εt

(6.121)

where γ = ψ(1)(ρ − 1), and the coefﬁcients cj are from the DF decomposition of A(L) = θ(L)−1 ϕ(L)(1 − ρL) = (1 – αL) – C(L)(1 – L). Thus ARMA errors translate to an inﬁnite ADF representation for ∆yt . The inﬁnite-order ADF regression is truncated to a lag of k–1, as in 2 is: section 6.3.7, and the AR-based estimator of σz,lr 2 σˆ z,lr,AR(k ∗) =

1 ˆ (1 − C(1))

2 2 σˆ ε,k ∗

2 −1 σˆ ε,k ∗ =T ∑t = k εˆ2t,k∗ T

(6.122) (6.123)

where εˆ t,k∗ are the residuals from LS estimation of the truncated version of (6.121). As in section 6.3.6, a sufﬁcient condition governing the expansion of 2 ∗ 1/3 → 0 as T → ∞, and there exist k∗ for the consistency of σˆ z,lr,AR(k ∗ ) is k /T constants, ζ and s, such that ζk∗ > T1/s .

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242 Unit Root Tests in Time Series

Dickey-Fuller and Related Tests 243

Perron and Ng (1996) showed analytically, supported with simulations results, 2 2 2 that σˆ z,lr,AR(k ˜ Tm,a , the latter already being preferred to σ˜ Tm,b ∗ ) is preferred to σ because of its properties under stationary alternatives. Their framework is an extension of the autoregressive root local-to-unity approach to include an MA(1) coefﬁcient local to –1. The local-to-unity asymptotic framework is: y˜ t = ρT y˜ t−1 + zt

zt = εt + θT εt−1 √ θT = − 1 + λ/ T These asymptotics apply to the autoregressive root coefﬁcient at rate T and to the √ MA coefﬁcient local to –1 at rate T, the different rates reﬂecting the different orders of convergence of the least squares estimators. In the limit (1 − L)yt = (1 − L)εt implying yt = εt , but for ﬁnite T, yt is nearly integrated. In this set-up, the LS estimator ρˆ is not consistent and δˆ and τˆ diverge to –∞; the inconsistency property is then passed on to the residuals and func2 2 tions of them, such as σ˜ Tm,a . However, σˆ z,lr,AR(k ∗ ) is based on ∆yt and does not inherit the inconsistency of ρˆ . Moreover, there will also be size distortions as the 2 limiting distributions of δˆ and τˆ using σ˜ Tm,a are a function of λ (see Perron and Ng, 1996, Theorem 3.3, for details). Simulation results for these approaches are considered after the further modiﬁcation suggested by Perron and Ng (1996), which is considered in the next section.

6.8.3 Modiﬁed Z tests Perron and Ng (1996) developed a class of unit root tests suggested by Stock (1999) for a weakly dependent error structure, which can be interpreted as modifying the PP tests; these are referred to as the M tests, for example, MZˆρ and MZˆτ. The motivation for these tests is twofold, both with the common aim of reducing the size distortions present in the original PP tests when the errors are dependent. The modiﬁed PP tests are based on the idea that there are different rates of normalisation under the null and alternative hypotheses; that is, T and √ T, respectively; and the second aspect of the modiﬁed tests is to use an autoregressive estimator of the long-run variance (as in the previous section) rather than a kernel-based estimator, the latter tending to aggravate the problem of size distortion. The ﬁrst two tests are modiﬁcations of Zˆρ and Zˆτ, denoted MZˆρ and MZˆτ, respectively; and the third, MSB, is a new test. These test statistics are as follows

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ρT = 1 − c/T

244 Unit Root Tests in Time Series

for the M0 case: −1 2 2 ˆ 2T−2 MZδ = T yT − σ˜ z,lr

T

∑

−1 y2t−1

(6.124)

t=1

MSB =

−2

T

T

∑

(6.125)

1/2

2 y2t−1 /˜σz,lr

(6.126)

t=1

ˆ The test statistics MZδˆ and MZˆτ are written It can be shown that MSB = Zˆτ/Zδ. as the original PP test plus a modiﬁcation factor, which is zero under the null. Otherwise, the adjustment factor is positive which, given that Zδˆ and Zˆτ are negative for stationary alternatives, will result in a modiﬁed test statistic that is less negative; as the critical values are unchanged, the result will be fewer rejections of the null hypothesis, thus counteracting the oversizing problem. The latter observation implies that the asymptotic distributions of MZδˆ and MZˆτ are the same as for Zδˆ and Zˆτ, respectively. The asymptotic distribution of MSB is positive and MSB tends to zero under HA ; as a result, rejection of the null in favour of the alternative of stationarity occurs with ‘small’ values of MSB (critical values are available in Stock, 1999). These tests are combined with the 2 autoregressive estimator σˆ z,lr,AR(k ∗ ) and generalise to the cases of models M1 and M2 by replacing yt by demeaned or detrended data. 6.8.4 Simulation results Although the intention of the PP tests is to remove the size distortion in the standard DF tests when there are weakly dependent errors, it is not very successful at achieving this in ﬁnite samples, a problem highlighted by Schwert (1989) and reinforced by the simulation results reported in Table 6.6a for MA(1) errors and Table 6.6b for AR(1) errors. The results reported in this section are for 2 ; speciﬁcally, σ 2 two estimators of σz,lr ˜ Tm,a (ω) with a Newey-West kernel and automatic selection of the autocovariance truncation parameter, m, as in Andrews 2 (1991); and σˆ z,lr,AR(k ∗ ) where the truncation lag was set at 6 for T = 200 (and 8 for T = 500, not reported here, but available in Perron and Ng, 1996). The results 2 is used. are also reported when the correct value of σz,lr 6.8.4.i PP tests, Zˆρµ and Zˆτµ , and modiﬁed PP tests, MZˆρµ and MZˆτµ Some of the results in Perron and Ng (1996) are extracted in Tables 6.10a and 6.10b; the former for MA(1) errors, and the later for AR(1) errors. Size distortions 2 2 for Zδˆ µ and Zˆτµ were less severe when σˆ z,lr,AR(k ˜ Tm,a was used, but ∗ ) rather than σ

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T = Zδˆ + (ˆρ − 1)2 2 1/2 ∑Tt=1 y2t−1 MZˆτ = Zˆτ + (1/2) (ˆρ − 1)2 2 σ˜ z,lr

Dickey-Fuller and Related Tests 245

Table 6.10a Effect of MA(1) errors on the size of PP-type tests, 5% nominal size. 2 σˆ z,lr,AR(k ∗)

2 σz,lr

θ1

Zδˆ µ

MZδˆ µ

MSBµ

Zδˆ µ

MZδˆ µ

MSBµ

Zδˆ µ

MZδˆ µ

MSBµ

–0.8 –0.5 –0.2 0 0.2 0.5 0.8

98.3% 41.9% 10.1% 4.4% 3.3% 2.4% 2.6%

98.1% 38.3% 8.8% 3.7% 3.2% 2.2% 2.3%

98.4% 42.5% 11.9% 5.5% 4.2% 3.7% 4.3%

73.6% 11.4% 7.2% 7.9% 7.3% 6.0% 5.9%

8.9% 5.2% 6.2% 7.2% 7.2% 5.7% 5.6%

10.3% 6.3% 7.9% 9.3% 8.6% 7.6% 7.2%

68.7% 8.2% 3.8% 3.9% 4.8% 4.1% 4.5%

0.0% 0.5% 2.7% 3.4% 4.2% 4.0% 4.5%

0.0% 1.8% 3.9% 4.5% 6.1% 5.3% 5.7%

2 σ˜ Tm,a

2 σˆ z,lr,AR(k ∗)

2 σz,lr

θ1

Zˆτµ

MZˆτµ

Zˆτµ

MZˆτµ

Zˆτµ

MZˆτµ

–0.8 –0.5 –0.2 0 0.2 0.5 0.8

98.1% 38.7% 7.0% 3.0% 2.0% 0.9% 1.4%

96.1% 30.2% 4.4% 2.3% 1.4% 0.7% 1.4%

90.3% 16.2% 4.6% 5.8% 5.2% 3.4% 3.2%

6.3% 3.8% 3.7% 4.9% 4.8% 3.2% 3.1%

92.2% 13.9% 2.7% 2.6% 3.0% 2.5% 2.5%

0.0% 0.5% 1.1% 1.8% 2.6% 2.4% 2.4%

Notes: T = 200; 5% nominal size. Source: Extracted from Perron and Ng (1996, Tables 1 and 2).

were still substantial for large negative values of θ1 ; for example, with T = 200 2 and θ = –0.8, actual sizes for Zδˆ µ and Zˆτµ , for a nominal 5% test using σ˜ Tm,a 2 were 98.3% and 98.1%, and 73.6%, and 90.3% using σˆ z,lr,AR(k ∗ ) . It is clear that the original problem of wrong size remains: the PP tests reject too often when θ1 < 0 and do not reject often enough when θ1 > 0. Size distortions, although not quite as substantial, also occur when zt is an AR(1) process (see Table 6.10b). In this case, the tests reject too often for ϕ1 < 0; and when ϕ1 > 0, undersizing 2 2 2 tends to occur with σ˜ Tm,a and oversizing for σˆ z,lr,AR(k ∗ ) and σz,lr . 2 2 Overall, using σˆ z,lr,AR(k ˜ Tm,a (ω), but ∗ ) improves the size ﬁdelity compared to σ the incorrect size problem remains; also, Kim and Schmidt (1990) found that this problem was not resolved by choosing a different kernel function. Note that 2 reduces, but does not remove, the incorrect sizing. using the correct value σz,lr So far, comparing the PP tests with their ADF counterparts (see Table 6.6), they do not improve size retention compared to the best of the latter, which was τˆ µ . 2 The modiﬁed PP tests using the autoregressive estimator σˆ z,lr,AR(k ∗ ) were found 2 2 to be better than the PP tests, with either σ˜ Tm,a or σˆ z,lr,AR(k ∗ ) . For example, for 2 θ1 = –0.8, and using σˆ z,lr,AR(k ∗ ) , the empirical size was reasonably controlled at 2 8.9% for MZδˆ µ and 6.3% for MZˆτµ ; and for ϕ1 = –0.8, using σˆ z,lr,AR(k ∗ ) empirical

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2 σ˜ Tm,a

246 Unit Root Tests in Time Series

Table 6.10b Effect of AR(1) errors on the size of PP-type tests, 5% nominal size. 2 σˆ z,lr,AR(k ∗)

2 σz,lr

ϕ1

Zδˆ µ

MZδˆ µ

MSBµ

Zδˆ µ

MZδˆ µ

MSBµ

Zδˆ µ

MZδˆ µ

MSBµ

–0.8 –0.5 –0.2 0 0.2 0.5 0.8

68.9% 21.1% 7.7% 4.4% 3.5% 1.5% 0.4%

62.9% 16.3% 5.8% 3.7% 1.9% 0.7% 0.3%

66.7% 19.0% 6.9% 5.5% 3.2% 1.3% 0.5%

23.6% 11.0% 7.7% 7.9% 9.2% 9.8% 7.5%

3.4% 6.8% 6.1% 7.2% 7.3% 8.5% 6.5%

4.1% 8.4% 7.6% 9.3% 9.0% 10.7% 8.7%

22.0% 7.1% 5.9% 3.9% 6.1% 6.3% 8.8%

0.8% 2.8% 3.8% 3.4% 4.5% 5.4% 7.4%

0.8% 3.8% 4.6% 4.5% 5.7% 6.5% 9.5%

2 σ˜ Tm,a

2 σˆ z,lr,AR(k ∗)

2 σz,lr

ϕ1

Zˆτµ

MZˆτµ

Zˆτµ

MZˆτµ

Zˆτµ

MZˆτµ

–0.8 –0.5 –0.2 0 0.2 0.5 0.8

66.1% 20.7% 8.3% 3.0% 4.2% 2.3% 1.5%

60.4% 15.9% 6.6% 2.3% 3.7% 2.2% 1.5%

41.8% 13.0% 9.1% 5.8% 10.1% 9.3% 7.4%

4.0% 6.9% 7.3% 4.9% 8.9% 9.2% 7.2%

41.4% 10.4% 7.3% 2.6% 6.6% 6.1% 7.9%

1.7% 3.3% 5.6% 1.8% 5.7% 5.9% 7.9%

Note: T = 200; 5% nominal size. Source: Extracted from Perron and Ng (1996, Tables 1 and 2).

sizes of 3.4% for MZδˆ µ and 4.0% for MZˆτµ were achieved. Of the three modiﬁed 2 ˆ PP tests, and using σˆ z,lr,AR(k ∗ ) , MZδµ and MSB tended to be slightly oversized and MZˆτµ slightly undersized for MA(1) errors; whereas all were slightly oversized for AR(1) errors. The local-to-unity asymptotic framework of section 6.8.2.ii is able to provide a framework to explain the simulation results (see Perron and Ng, 1996). First note that the LS estimator ρˆ is not consistent in this framework and this inconsistency property is passed on to the residuals and functions of them such as the autocovariance function, where each autocovariance is Op (T) and the explosive 2 is estimated by a terms cumulate with the sum of the m lags. Thus, when σz,lr 2 (ω), based on m autocovariances, with kernel estimator of the general form σ˜ Tm,a m → ∞ and (m/T) → 0 as T → ∞, the PP tests diverge to –∞ at a rate of mT, which 2 is known or σ 2 is faster than the rate T which obtains when either σlr ˆ z,lr,AR(k ∗ ) is 2 used. Size distortions are more pronounced for estimators of the form σ˜ Tm,a (ω) 2 compared to σˆ z,lr,AR(k ∗).

2 ∗ ∗ If σˆ z,lr,AR(k ∗ ) is used, with k /T → 0 and k → ∞ as T → ∞, then the dependence of the estimator of the long-run variance on the inconsistent LS estimator ρˆ through its impact on zˆ t is removed. As a result, the distribution of MZδˆ is bounded, with the modiﬁcation factor (T/2)(ˆρ − 1)2 ‘knocking out’ the

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2 σ˜ Tm,a

Dickey-Fuller and Related Tests 247

ˆ Similar arguments carry through to the pseudo-t divergent components of Zδ. versions of the tests, and thus MZˆτ is preferred over Zˆτ. 2 should be An important part of the overall procedure is that estimators of σz,lr

6.9 Power: a comparison of the ADF, WS and MZ tests The local-to-unity framework allows the generation of a family of asymptotic local power functions as T varies for given c. The asymptotic local power functions are the same for a particular DF test, with zt = εt ∼ iid, for example, τˆ µ , and its counterpart PP test, Zˆτµ , with zt possibly serially correlated and heteroscedastic (see Phillips and Perron, 1988). However, this asymptotic equivalence does not carry across to the modiﬁed PP tests because the asymptotic distributions of the latter (under HA ) differ from those of the PP tests, and by implication from the DF tests as well. Hence this opens the possibility that the modiﬁed PP tests may not only hold their size better than the original PP tests, but may be more powerful than the DF and PP tests. The simulation results reported in Perrron and Ng (1996) suggest that there are power gains using the MZ tests compared to the Z tests and the ADF τˆ -type tests. As in the DF case, tests based on δˆ are generally more powerful than those based on τˆ , but in this case without sacriﬁcing size retention. The simulations reported in Table 6.11 have the same general set-up as those underlying Table 6.6. The model is M1 , so that the test statistics are subscripted with µ; concentration on this case is sufﬁcient to indicate the ranking of the test statistics. An automatic lag selection criterion is used (rather than the ﬁxed lag underlying the results in Table 6.6). In this case, AIC (the Akaike information criterion) is used for δˆ µ and τˆ µ , and the selected lag, say AIC(lag), is also used for the PP and the modiﬁed PP tests; lag = AIC(lag) + 2 is used for the WS tests following the suggestion in Pantula et al. (1994) that this is better at protecting the size of those tests. A further examination of the importance of the lag selection criteria is contained in Chapter 9. Power is reported both on an unadjusted and a size-adjusted (SA) basis. When the errors follow an MA(1) process and θ1 = –0.5, there is both undersizing and oversizing of the different test statistics, although τˆ ws µ , alone, is almost perfectly sized, a property that is not shared by δˆ ws . In part this µ may be due to the need for longer lags in the other tests, despite the lag length indicated by AIC. When θ1 = 0.5, size is not as distorted. The semi-parametric tests have similar size, which makes a power comparison

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2 based on ∆yt (or ∆y˜ t more generally), as in σˆ z,lr,AR(k ∗ ) , not on either the residuals zˆ t under the null or the residuals under the alternative. The former are ruled out by conventional asymptotics (that is, coefﬁcients ﬁxed as T → ∞), and the latter by local-to-unity asymptotics.

248 Unit Root Tests in Time Series

Table 6.11 Illustrative size and power comparisons of DF-type tests.

δˆ µ

δˆ ws µ

τˆ µ

τˆ ws µ

Zδˆ µ

MZδˆ µ

Zˆτµ

MZˆτµ

MSBµ

θ1 = –0.5 ρ = 1.0 ρ = 0.95 SA ρ = 0.90 SA

12.4 66.1 52.3 95.4 86.8

10.6 61.9 37.2 93.6 79.3

9.0 49.2 34.5 88.1 77.2

5.0 36.4 36.1 76.2 76.3

9.0 56.4 44.8 94.5 89.8

3.7 27.9 34.2 60.8 67.3

12.3 69.8 45.3 98.4 93.4

2.4 19.7 30.6 52.5 64.8

11.8 62.3 42.1 92.4 81.6

θ1 = 0.5 ρ = 1.0 ρ = 0.95 SA ρ = 0.90 SA

6.7 48.5 40.0 87.4 81.8

10.7 56.9 35.5 88.8 73.0

4.7 30.2 31.6 71.7 72.8

4.8 35.0 35.3 69.1 69.5

9.4 55.3 40.0 89.1 75.9

9.3 54.8 36.8 88.2 74.6

8.0 44.1 33.2 81.4 71.2

7.6 42.3 37.3 78.9 69.9

8.3 54.6 46.4 90.1 78.4

Table 6.11b AR(1) error structure δˆ µ

δˆ ws µ

τˆ µ

τˆ ws µ

Zδˆ µ

MZδˆ µ

Zˆτµ

MZˆτµ

MSBµ

ρ = 1.0 ρ = 0.95 SA ρ = 0.90 SA ϕ1 = 0.3

5.8 48.9 43.8 93.8 92.2

4.7 50.6 52.7 96.1 96.8

5.4 34.8 33.4 84.8 83.5

4.7 52.2 53.7 96.0 96.3

6.2 49.3 43.4 93.5 91.2

5.6 45.9 46.6 91.7 90.1

6.0 36.9 32.3 85.9 82.1

4.9 30.3 30.8 78.9 79.4

6.7 53.1 45.7 95.7 93.4

ρ = 1.0 ρ = 0.95 SA ρ = 0.90 SA ϕ1 = 0.6

5.5 46.2 43.8 90.4 89.2

4.5 49.1 51.9 93.4 94.6

4.5 31.6 33.4 77.6 79.6

4.2 49.6 56.2 93.8 95.5

7.8 55.5 43.1 92.9 87.6

7.7 54.4 43.0 92.3 87.4

6.5 40.8 34.9 84.9 80.4

6.1 38.9 34.5 83.0 79.4

7.0 53.6 44.4 93.2 89.4

ρ = 1.0 ρ = 0.95 SA ρ = 0.90 SA

5.8 44.9 40.1 83.8 80.5

5.0 47.1 47.1 88.0 88.0

5.3 30.6 29.6 69.8 68.4

4.8 47.8 49.0 88.4 89.2

9.2 56.7 39.4 90.4 78.6

9.1 56.6 40.0 90.3 78.7

8.3 42.4 31.1 80.7 69.6

8.1 42.1 31.1 80.3 69.6

7.5 52.2 40.8 88.6 81.3

ϕ1 = 0.0

Note: Emboldened entries indicates good overall performance; SA indicates size-adjusted power.

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Table 6.11a MA(1) error structure

sensible, and MSBµ is the most powerful of these tests. However, τˆ ws µ is the test with the best combination of accurate size and power; although it is not the most powerful on a size-adjusted basis, it is the best test of those that retain their size. When the errors follow an AR(1) process, the weighted symmetric tests are the ˆ ws most powerful, with little to choose between δˆ ws µ and τ µ ; of the semi-parametric ˆ tests, MZδµ and MSB are the more powerful, with power better than the standard DF tests, but not as good as the WS tests. Moreover, size is not as well maintained for the semi-parametric tests compared to the parametric tests. Overall, the characteristics of the various tests depend on the nature of the ˆ ws serial correlation. In the AR case, the WS tests δˆ ws µ and τ µ are dominant; when ws the errors are MA(1), τˆ µ is the better of these two tests and has the best combination of accurate size and good power. Finally, referring back to Table 6.6 and comparing the corresponding results there with those in Table 6.11 serves to highlight the impact of different lag selection rules, the difference being whether the lag length is ﬁxed as in Table 6.6 or is chosen by AIC as in Table 6.11. It is possible to select a lag long enough to deliver an accurate size for the DF τˆ -type tests; for example, τˆ µ and a lag of 8 in both the AR(1) and MA(1) cases, but these are not the most powerful tests (on a size-adjusted basis). Thus either some prior assessment of the likely structure of the errors is important, or a lag selection criteria is required that is long enough to deliver accurate size but short enough to extract the minimum penalty on power. If serially correlated errors are likely, lag selection is an important joint consideration along with which test to use. These issues are considered again in Chapter 9.

6.10 Concluding remarks This chapter has brought together a number of tests of the unit root null hypothˆ esis. The DF δ-type and τˆ -type tests, brieﬂy introduced in the previous chapter, were considered more extensively here, together with a number of developments that are based on these tests. Tests based on both the backward and forward recursions of the difference equation are generally more powerful than the standard DF tests, both when the errors are white noise and when there is dependency in the errors. Whilst the τˆ -type tests are not as powerful as the ˆ δ-type tests, they are more robust to dependent errors and are generally preferred for that reason. The standard DF tests and the WSDF tests work with weakly dependent errors by ‘whitening’ out the regression by including lags of the lagged dependent variable (the ADF regression), whether there is just an AR component and/or an MA component to the errors. An alternative way of approaching the problem of weakly dependent errors is due to Phillips and Perron, who suggested

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Dickey-Fuller and Related Tests 249

making an adjustment to the DF test statistics so that the revised test statistics have the same DF distribution as when the errors are white noise. The adjustment requires estimation of the autocovariance structure of the errors, for which a semi-parametric method was originally suggested; however, an AR-based estimator seems to give PP test statistics with better properties. A different approach to accommodating weakly dependent errors involves estimating the ARMA error structure by some means. ML estimators for this case were considered in Chapter 3, whereas a GLS-type approach is outlined in Chapter 7. Overall, the reader may well be wondering which of the various unit root test statistics is best. There is, however, not yet a single answer to this question and there are still some problems to be addressed. As an indication of the complexity of the answer, note the following. In the case that zt = εt , the most powerful tests so far considered are the (unconditional) ML test and the weighted symmetric versions of the DF tests, with the τˆ ws -type tests being as good as the δˆ ws -type tests ˆ (in contrast to the standard DF case where the δ-type tests are better). However, when the errors are weakly dependent, the weighted symmetric tests are slightly more robust than the ML test; where size is comparable, which is best in terms of power depends on how close ρ is to the unit root. Some of the problems still to be addressed and the chapters where they are considered are as follows:

1. Is yet more power possible? (Chapter 7) 2. Is it possible to control empirical size by bootstrapping the test statistics? (Chapter 8) 3. Is it possible to control deviations from the nominal size by improving the criteria for lag length selection? (Chapter 9) 4. What happens to the properties of the test as a second unit root is approached? (Chapter 10) 5. What tests are possible if the null hypothesis is reversed to become on of stationarity rather than nonstationarity? (Chapter 11) 6. What are the implications of the various test statistics for their inversion to form conﬁdence intervals? (Chapter 12) 7. What are the implications of seasonality for unit root tests? (Chapter 13)

Questions Q6.1 Given the MA(1) process, zt = εt + θ1 εt−1 , obtain the unconditional variance and the long-run variance.

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250 Unit Root Tests in Time Series

Dickey-Fuller and Related Tests 251

A6.1 The unconditional variance is: σz2 ≡ var(zt ) = var(εt + θ1 εt−1 ) = var(εt ) + θ21 var(εt−1 ) + 2cov(εt εt−1 )

where use has been made of var(εt ) = var(εt−1 ) and cov(εt εt−1 ) = 0. To obtain the long-run variance, ﬁrst rewrite the MA(1) process as zt = (1 + θL)εt , then: 2 σz,lr = var(zt |L = 1)

= var[(1 + θL)εt |L = 1] = (1 + θ)2 σε2 The long-run variance uses the ‘trick’, much used in distributed lags, that in a dynamic relationship; for example, zt = εt + θLεt , setting L = 1 gives the long run deﬁned by εt = εt−1 = εt−2 . . . (for an elaboration, see Patterson, 2000, section 9.2). Q6.2 Consider the case where the errors, zt , are generated by an MA(1) process, so that zt = (1 + θ1 L)εt . Why is there likely to be a problem of standard unit root tests becoming oversized as the value of θ1 approaches –1? A6.2 In this case, there is near-cancellation of the roots of the AR and MA polynomial; that is, (1 − L)yt = (1 + θ1 L)εt and as θ1 → –1, yt → εt . Hence yt looks like a stationary time series and the unit root test therefore rejects the null of a unit root; it is ‘fooled’ by the nearstationarity of the data. Q6.3 What is σε2 Σ−1 p in the AR(1) and AR(2) cases? Hence, what is the speciﬁc form of the variance or variance-covariance matrix of φˆ in the asymptotic distribution stated in Equation (6.42)? A6.3 Σp (that is, Σ1 in this example) in the AR(1) case has already been given, but it is a good place to start from for the general case. It is the limit of the 2 as t → ∞, denoted σ2 and given by σ2 (1 − ρ2 )−1 (see Equation variance σy,t y ε 2 2 2 −1 −1 = (1 − ρ2 ), which is the result used in (6.12)). Hence, σε2 Σ−1 1 = σε [σε (1 − ρ ) ] (6.27); note that in AR(1) notation, φ1 = ρ. Σp for p = 2 can be obtained as follows (see also Priestley, 1981, p.128). Start with an AR(2) model:

yt = φ1 yt−1 + φ2 yt−2 + εt

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= (1 + θ21 )σε2

252 Unit Root Tests in Time Series

Then multiply by yt−k and take expectations: E(yt yt−k ) = φ1 E(yt−1 yt−k ) + φ2 E(yt−2 yt−k ) + E(εt yt−k ) Note that E(εt yt−k ) = 0 for k ≥ 1, and if k = 0 then E(εt yt−k ) = E(εt yt ) = σε2 ; and that for a zero mean process, E(yt yt−k ) is the k-th order autocovariance, γ(k), where γ(0) is the variance of yt . Thus for k = 0, 1 and 2:

γ(1) = φ1 γ(0) + φ2 γ(1) γ(2) = φ1 γ(1) + φ2 γ(0) These equations can be solved by substitution, ﬁrst for γ(1), then for γ(2) and, ﬁnally, for γ(0): γ(1) =

φ1 γ(0) (1 − φ2 )

γ(2) =

φ21 + (1 − φ2 )φ2 γ(0) (1 − φ2 )φ2

γ(0) =

(1 − φ2 ) σ2 (1 + φ2 )(1 + φ1 − φ2 )(1 − φ1 − φ2 ) ε

⇒ γ(1) =

φ1 σ2 (1 + φ2 )(1 + φ1 − φ2 )(1 − φ1 − φ2 ) ε

γ(2) =

φ22 + (1 − φ2 )φ2 σ2 φ2 (1 + φ2 )(1 + φ1 − φ2 )(1 − φ1 − φ2 ) ε

Alternatively, and easier to generalise, is to arrange the equations so that they can be obtained as the following solution: 

  γ(0) 1    =  γ(1)   −φ1 γ(2) −φ2

−φ1 1 − φ2 −φ1

 −1  −φ2 σε2    0   0  0 1

As to Σ2 , this has γ(0) on the diagonals and γ(1) on the off-diagonals: γ(0) γ(1) Σ2 = γ(1) γ(0) can be obtained. The variance-covariance matrix of φˆ is from which Σ−1 2 −1 2 2 σε Σ2 and the σε will cancel the σε−2 in Σ−1 2 . In general, let ΣT denote the unconditional variance-covariance, or autocovariance, matrix of yt (see also Chapter 3, section 3.9.1.i), so that assuming for simplicity of notation a sample

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γ(0) = φ1 γ(1) + φ2 γ(2) + σε2

Dickey-Fuller and Related Tests 253

γ(2) γ(1) .. . .. . ...

... ... .. . .. . ...

γ(T − 1) γ(T − 2) .. . .. . γ(0)

         

Then Σp is selected as the p × p block in the top left corner of Σ. Q6.4 Consider obtaining the test statistic δˆ = T(ˆρ − 1) from an ADF(1) model. Suggest two estimators of C(1), one of which imposes the null hypothesis ρ = 1 and the other employs a nonlinear estimator of the structural coefﬁcients. A6.4 The ADF(1) model is: ∆yt = γyt−1 + c1 ∆yt−1 + εt where γ = (1 − ϕ1 )(ρ − 1) and c1 = ϕ1 ρ. The usual approach to use the following test statistic: δˆ = T(ˆρ − 1) =T

γˆ (1 − cˆ 1 )

The rationale for this estimator is that if ρ = 1, then c1 = ϕ1 . An alternative is to impose, rather than assume, ρ = 1 and then estimate the restricted ADF(1): ∆yt = c1 ∆yt−1 + εt with LS estimator of c1 , say c˜ 1 , and then use ρ˜ – 1 deﬁned as: (˜ρ − 1) =

γˆ (1 − c˜ 1 )

Alternatively, use a nonlinear estimator so that the coefﬁcient on yt−1 is (1 − ϕ1 )(ρ − 1) and that on ∆yt−1 is ϕ1 ρ, which necessarily obtains an estimator of ρ that is consistent under the alternative hypothesis |ρ| < 1. These procedures k−1 generalise in the ﬁrst case by extending the restricted ADF to obtain {c˜ j }j=1 , and in the second case by using the relationships between the reduced form coefﬁcients and the structural coefﬁcients outlined in Chapter 3, section 3.8.3, to obtain a nonlinear estimator of ρ. Q6.5 In case of AR(1) errors (1−ϕ1 L)zt = εt , using the results of section 6.7.1.ii, obtain the limit distributions of δˆ and τˆ , and comment on the impact of this structure of errors on these distributions.

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of T observations then:  γ(0) γ(1)  γ(0)  γ(1)  .. ..  ΣT =  . .   .. ..  . .  γ(T − 1) γ(T − 2)

254 Unit Root Tests in Time Series

A6.5 The relevant variances are obtained ﬁrst: E(z2t ) = E(ϕ1 zt−1 + εt )2 = ϕ21 E(z2t−1 ) + E(ε2t ) + 2ϕ12 E(zt−1 εt ) = ϕ2 σz2 + σε2 ⇒ σz2 − ϕ12 σz2 = σε2 ⇒

using E(z2t ) = E(z2t−1 ) = σz2 and E(zt−1 εt ) = 0. 2 , is obtained as follows: The long-run variance, σz,lr

zt = (1 − ϕL)−1 εt ∞

= εt + ∑j=1 ϕj εt−j ⇒ 2 σz,lr = (1 − ϕ)−2 σε2 , using E(εt εs ) = 0 for t = s. 2 in (6.92) or (6.96b) and (6.93) or (6.97b), and Now substitute for σz2 and σz,lr

simplify using 1 − ϕ2 = (1 − ϕ)(1 + ϕ), and note that: σz,lr (1 − ϕ1 )1/2 (1 + ϕ1 )1/2 = σz (1 − ϕ1 ) =

1 2

(1 + ϕ1 )1/2 (1 − ϕ1 )1/2

2 − σ2 σz,lr z

=

2 σz,lr

1 2

(1 − ϕ12 − (1 + ϕ12 − 2ϕ1 ) 1 − ϕ12

2ϕ1 (1 − ϕ1 ) (1 − ϕ1 )(1 + ϕ1 ) ϕ1 = (1 + ϕ1 ) =

1 2

2 − σ2 σz,lr z

σz σz,lr

1 2

ϕ1 (1 + ϕ1 ) ϕ1 (1 + ϕ1 )1/2 = (1 − ϕ1 )1/2 (1 + ϕ1 ) ϕ1 = (1 − ϕ1 )1/2 (1 + ϕ1 )1/2

=

σz,lr σz

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σz2 = (1 − ϕ12 )−1 σε2

Dickey-Fuller and Related Tests 255

ˆ DF + δˆ ⇒D F(δ) τˆ ⇒D

1 + ϕ1 1 − ϕ1

ϕ1 (1 + ϕ1 )

1/2

1 0

F(ˆτ)DF +

1

(6.127)

B(r)2 dr

ϕ1 (1 − ϕ1 )1/2 (1 + ϕ1 )1/2

1

1/2

(6.128)

1 2 0 B(r) dr

The impact of AR(1) dependent errors on the limit distributions depends on the sign of ϕ1 ; the limiting distribution of δˆ is shifted to the right for ϕ1 > 0, and to the left for ϕ1 < 0, given that |ϕ1 | < 1. The former is the more likely case; ˆ DF will result in an undersized test. The hence, using the critical values from F(δ) limiting distribution of τˆ undergoes both a scale change and a location shift; the former from the ﬁrst term in (6.128) and the latter from the second term in (6.128). The location shift takes the sign of ϕ1 . Consider the ﬁrst term in (6.128), and deﬁne Θ = (1 + ϕ1 )/(1 − ϕ1 ) as the inﬂation factor in the variance, then Θ > 1 for ϕ1 > 0 and the variance of F(ˆτ) increases relative to F(ˆτ)DF ; as a second unit root is approached, ϕ1 → 1, Θ → ∞. If ϕ1 < 0, then the variance decreases relative to F(ˆτ)DF and Θ → 0 as ϕ1 → –1.

6.11 Appendix: Response surface function The form of the response functions follows MacKinnon (1991), as extended by Cheung and Lai (1995a, 1995b) and Patterson and Heravi (2003). The general notation is Cj (ˆτ, α, T∗ , k∗ ), which is the estimate of the α percentile of the distribution of test statistic ts for T and k∗ = k – 1. The ‘observations’ are indexed by j = 1, ..., N, and T∗ ≡ T − k adjusts for the actual sample size. The general form of the response function is: Cj (ˆτ, α, T∗ , k∗ ) = κ∞ +

I

J

i=1

j=1

∑ κi /(T∗ )i + ∑ ωj (k∗ /T∗ )j + ξj

(6.129)

Three points should be noted about these response surfaces. First, MacKinnon (1991) and Cheung and Lai (1995a, 1995b) used I = 2 and J = 2, whereas there was a noticeable improvement in ﬁt obtained by allowing both I = 3 and J = 3. These additional terms were included if the absolute value of their t statistic was above 1.9; in nearly all cases this was so. Second, the response functions gave a signiﬁcantly improved ﬁt when speciﬁed with the effective sample size T∗ ≡ T – k rather than the overall sample size T. Third, the more extensive design also gave reductions in the response function equation standard errors for the standard DF test statistics, τˆ , τˆ µ and τˆ β , compared to Cheung and Lai; for example, in the most frequently applied cases, in obtaining the 5th percentile, the reduction in standard error was 12% and 17%, respectively.

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256 Unit Root Tests in Time Series

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A factorial experimental design was used over all different pairings of T and k∗ , where k∗ = 0, 1, ..., 12. The sample size T and the increments (in parentheses) were then as follows: T = 20(1), 51(3), 78(5), 148(10), 258(50), 308(100), 508(200), 908. In all, there were 66 × 13 = 858 sample points from which to determine the response functions for the 1st, 5th and 10th percentiles. The coefﬁcient κ∞ gives an approximate guide as to the critical value for large T. To avoid a dependence on initial conditions, the ﬁrst 100 draws were discarded for each sample size. The response surface coefﬁcients are reported in detail in Table 6.12.

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257

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κ∞ –1.61581 –5.65876 –1.8372 –6.42779

Tests τˆ ρˆ τˆ ws ρˆ ws

Tests τˆ ρˆ τˆ ws ρˆ ws

κˆ 2 –33.3711 –790.263 –116.985 –2571.98

κˆ 3 106.5659 2808.642 –167.066 15523.08 ω ˆ1 1.20448 –18.1035 1.854796 –7.80587

ω ˆ2 –2.39415 9.312984 –4.52767 –98.9153

ω ˆ3 2.499499 –57.3207 11.41488 89.9809

κˆ 2 –30.6711 –494.355 –134.683 –1712.79

κˆ 3 145.4072 1619.858 0 8670.867 ω ˆ1 0.839898 –6.20634 1.622627 –1.3716

ω ˆ2 –1.21031 0.126817 –4.8886 –39.9453

ω ˆ3 0.722283 –21.0683 10.61309 39.15629

κˆ 1 0.026171 50.4526 15.30053 105.6916

κˆ 2 –22.5668 –391.624 –147.068 –1345.63

κˆ 3 108.9953 1287.258 138.8474 6160.095

ω ˆ1 0.696141 –3.18757 1.342096 0.046886

ω ˆ2 –0.94715 –1.0887 –4.00036 –24.8888

ω ˆ3 0.479764 –12.5416 8.993738 26.74716

Model: M0 (no constant and no trend) 10% critical values

κˆ 1 –0.49021 68.47636 14.92144 128.5777

Model: M0 (no constant and no trend) 5% critical values

κˆ 1 –3.03928 117.8137 12.29875 181.6736

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κ∞ –2.5599 –13.5309 –2.655 –14.1318

Tests τˆ ρˆ τˆ ws ρˆ ws

Table 6.12 Response surface coefﬁcients.

Model: M0 (no constant and no trend) 1% critical values

R2 0.921401 0.982363 0.979892 0.959563

R2 0.896397 0.980275 0.978613 0.945676

R2 0.914757 0.984012 0.967415 0.967814

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κ∞ –2.56895 –11.2154 –2.21454 –10.2115

Tests τˆ µ ρˆ µ τˆ ws µ ρˆ ws µ

Tests τˆ µ ρˆ µ τˆ ws µ ρˆ ws µ

κˆ 2 –4.111 1045.566 32.98038 813.305

κˆ 3 –143.64 –9727.2 –458.442 3976.917 ω ˆ1 1.146321 –53.7195 0.519384 –24.7959

ω ˆ2 –2.39972 13.44147 9.469475 –444.926

κˆ 2 –5.97687 278.2346 52.76971 509.0957

κˆ 3 0 –3220.58 –520.291 0 ω ˆ1 0.901746 –27.0831 0.239176 –17.1263

ω ˆ2 –1.46986 11.63782 8.748652 –209.625

κˆ 1 –1.76401 19.99818 –3.93499 –14.6557

κˆ 2 –2.05581 119.392 54.48075 185.9298 0 –1732.33 –507.712 1816.01

κˆ 3

ω ˆ1 0.797342 –17.5616 0.06422 –12.0763

ω ˆ2 –1.23508 4.636452 8.435093 –135.93

Model: M1 (constant and no trend) 10% critical values

κˆ 1 –3.07992 30.69298 –5.10291 –14.5788

Model: M1 (constant and no trend) 5% critical values

κˆ 1 –7.2198 56.06736 –8.00808 3.415239

ω ˆ3 1.320015 –71.537 –8.03787 178.6372

ω ˆ3 1.527349 –105.829 –8.07222 291.1085

ω ˆ3 3.064037 –174.19 –8.12898 551.8304

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κ∞ –3.42982 –20.4576 –3.09127 –19.8306

Tests τˆ µ ρˆ µ τˆ ws µ ρˆ ws µ

Model: M1 (constant and no trend) 1% critical values

R2 0.901355 0.998984 0.963862 0.989377

R2 0.909708 0.998918 0.960205 0.991171

R2 0.971963 0.998422 0.946138 0.987665

259

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κ∞ –3.12798 –18.0093 –2.8998 –17.1815

Tests τˆ β ρˆ β τˆ ws β ρˆ ws β

Tests τˆ β ρˆ β τˆ ws β ρˆ ws β

κˆ 2 28.5592 4921.59 73.49453 1502.652

κˆ 3 –595.064 –38069.8 0 0 ω ˆ1 1.771437 –127.443 2.498481 –122.697

ω ˆ2 –4.5696 79.11798 21.76843 153.6086

κˆ 2 5.05483 2032.085 49.24858 501.134

κˆ 3 –130.831 –15884.4 0 3615.052 ω ˆ1 1.341085 –77.1933 1.107109 –83.0981

ω ˆ2 –2.50943 56.6283 22.86167 148.9066

κˆ 1 –2.97506 25.05023 –7.84296 8.751313

κˆ 2 –4.86709 1439.654 37.78716 347.7821 0 –11572.5 0 2988.873

κˆ 3

ω ˆ1 1.113759 –57.4767 0.408098 –65.8835

ω ˆ2 –1.44179 39.05892 23.01703 123.3953

Model: M2 (constant and trend) 10% critical values

κˆ 1 –5.23793 37.01571 –10.1744 28.28189

Model: M2 (constant and trend) 5% critical values

κˆ 1 –11.1096 36.30394 –16.3209 68.78646

ω ˆ3 1.507824 –323.521 –34.2505 –134.174

ω ˆ3 2.930775 –433.71 –35.2636 –167.123

ω ˆ3 6.323746 –689.384 –35.5365 –152.469

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κ∞ –3.95744 –28.718 –3.74881 –28.4051

Tests τˆ β ρˆ β τˆ ws β ρˆ ws β

Model: M2 (constant and trend) 1% critical values

R2 0.904309 0.999391 0.989690 0.992156

R2 0.930308 0.999269 0.989708 0.990862

R2 0.981384 0.998701 0.986838 0.989576

7

Introduction The primary concern of this chapter is with tests that are designed, at least potentially, to improve the power of the DF-type tests. The ﬁrst topic to be considered is the application of generalised least squares (GLS), which leads to two distinct formulations of the problem of unit root testing. The ﬁrst is perhaps the more natural application that follows from noting that if the errors in a regression model are dependent, a standard solution is to use GLS. The difﬁculty is that whilst it is possible to write the theoretical GLS estimator, in practice a feasible version of this estimator has to be used, and there are a number of ways of doing this. However, at least there is nothing new in principle in this application. The second application of GLS is somewhat different and can be confusing in the sense that it is generally only a partial application of GLS. To approach this it is useful to ﬁrst see the problem not as a problem in unit root testing, but as one in estimation of the ‘trend’ subject to (weakly) dependent errors. The ‘trend’ here includes the case where there is just a constant. One of the most important contributions in this area is due to Grenander and Rosenblatt (1957) who showed that (ordinary) least squares ((O)LS) estimation of the deterministic components in a linear regression model resulted in asymptotically efﬁcient estimators; thus, at least in the limit, the (weakly) dependent errors could be ignored (see also Phillips and Park, 1988). Following this approach results in detrending the data prior to unit root testing by a regression of the ‘raw’ data on the deterministic components in the trend speciﬁcation; the residuals from this detrending process then become the data for subsequent analysis. However, Elliott, Rothenberg and Stock (hereafter ERS) (1996), suggested that as likely alternatives to the null hypothesis of a unit root are processes that are ‘nearly-integrated’, then prior detrending should be

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261

based on these alternatives rather than those far from the nonstationary boundary. This led to local-to-unity detrending, which is a form of quasi-differencing (QD) quite familiar from GLS principles. However, any remaining dependence on the errors is not captured directly by a GLS estimation routine. This can be done either by making an adjustment to the residual sums of squares resulting under the null and a speciﬁc point alternative, the nature of this adjustment being familiar from semi-parametric methods of unit root testing that use an estimator of the ‘long-run’ variance, or by using the quasi-differenced data in otherwise familiar test statistics such as the DF τˆ -type tests. ERS raised the question of how to detrend the data in order to improve the power of the resulting unit root tests. The same goal can also be approached by reducing the bias in the LS estimator, and thereby inducing a rightward shift in the limiting null distribution. If the distributions under stationary alternatives are unaffected, the result will be a more powerful test. This is the effect of using the method of recursive mean or trend estimation, which was outlined in Chapter 4, but is here applied to the problem of obtaining an improvement in the power of a unit root test. Although not an application of GLS, it is included in this chapter as it is related to the more effective estimation of the deterministic components, resulting in an improvement in power. ERS also raised another important issue that has practical implications for unit root testing, which had been raised before (for example, Evans and Savin, 1981; Pantula et al., 1994), but the importance of which had been somewhat neglected. The process generating yt has to start at some point. For example, consider a simple AR(1) model, and suppose t = 0 indexes the pre-sample point and t = 1 indexes the ﬁrst observation, and so on; then one possibility is y0 = u0 = 0, implying y1 = u1 , y2 = ρy1 + u2 , and so on. This does not cause problems under the null, provided that a constant is included in the maintained regression (so, for example, u0 could equally be 10 or 100), but introduces a nonstationarity under the alternative hypothesis because the ﬁrst observation is treated differently from the rest of the observations. A prominent alternative is to let u0 be a draw from the unconditional distribution of yt , with variance σy2 = σu2 = σε2 /(1 − ρ2 ); this ensures stationarity under the alternative. It turns out that the power of unit root tests is generally, perhaps surprisingly, dependent on how far the ﬁrst observation is from trend; if it is a long way off, in units of σu , then a test that is uniformly most powerful (UMP) when u0 = 0, can be less powerful than, for example, a standard DF τˆ test. The chapter concludes with Harvey and Leybourne’s (2005, 2006) suggestion to form a unit root test that is a weighted average of two test statistics, combining one that has good power when the initial deviation is small with one that retains its power when the initial deviation is large (measured in units of σu ). The principles of GLS estimation are outlined in section 7.1, followed in section 7.2 by consideration of a feasible GLS procedure; these methods are

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Improving the Power of Unit Root Tests

illustrated in section 7.3 with a time series from the inﬂuential study by Nelson and Plosser (1982). Section 7.4 is a key section covering much recent material on improving tests for a unit root; it includes the seminal developments by ERS and Elliott (1999) and recent interest in the inﬂuence of the initial condition on the outcome of a unit root test – an empirical example illustrates this sensitivity. The application of GLS and the point optimal approach of ERS is motivated by the aim of increasing the power of unit root tests relative to standard DF tests. An alternative approach, that of recursively demeaning or detrending the data, a topic that was ﬁrst considered in Chapter 4 for bias reduction, is outlined in section 7.5.

7.1 A GLS approach A natural estimation method if the errors in a regression model are dependent is to use a generalised least squares approach. In the case of ARMA errors, this approach is apparent from the error dynamics framework. Consider the following set-up: (1 − ρL)yt = zt

(7.1)

ϕ(L)zt = θ(L)εt

(7.2)

p−1

q

where ϕ(L) = 1− ∑i=1 ϕi Li and θ(L) = 1 + ∑j=1 θj Lj . Thus the errors follow an AR(p – 1 , q) process given by: zt = ∑i=1 ϕi zt−i + εt + ∑j=1 θj εt−j p−1

q

(7.3)

As usual εt ∼ iid(0, σε2 ), and the ARMA process generating zt is assumed to be stationary and invertible, with no roots common to ϕ(z) and θ(z); for simplicity, no mean or trend term is present, but the model is easily amended for these cases. The following notation is carried forward from Chapter 3, section 3.9.1.i: Y1 = (y1 , . . . , yT ) Y1,−1 = (y0 , y1 , . . . , yT−1 ) z = (z1 , . . . , zT ) ε = (ε1 , . . . , εT ) ϕ = (ϕ1 , . . . , ϕp−1 ) Ψ = (ϕ1 , . . . , ϕp−1 , θ1 , . . . , θq )

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262 Unit Root Tests in Time Series

Improving the Power of Unit Root Tests

263

7.1.1 GLS estimation First, consider estimation as a standard problem in a GLS context, then a regression model with sample size T is formulated as: Y1 = ρY1,−1 + z

(7.4)

or by subtracting Y1,−1 from both sides: ∆Y1 = γY1,−1 + z where γ = ρ − 1. The variance matrix of Y1 conditional on Y1,−1 is: var(Y1 |Y1,−1 ) = E(zz )

(7.6a)

= σε2 Γ

(7.6b)

(Note that Γ in (7.6b) was subscripted with T, that is, ΓT , in Chapter 3, Equation (3.114), where it was necessary to emphasise the dimensional distinction between the conditional and unconditional variance matrix, a distinction that is not necessary here.) The GLS solution is obtained by minimising the transformed sum of squares given by: TSS = (Y1 − ρY1,−1 ) Γ−1 (Y1 − ρY1,−1 ) −1

= (∆Y1 − γY1,−1 ) Γ

(7.7)

(∆Y1 − γY1,−1 )

where minimisation is over the estimator analogues of ρ (equivalently γ) and Ψ, which comprise 1 + [(p – 1) + q] = p + q coefﬁcients in all. If the T × T covariance matrix Γ is known, the (conditional) GLS estimator of γ and its variance are: ρˆ GLS = (Y1,−1 Γ−1 Y1,−1 )−1 Y1,−1 Γ−1 Y1

(7.8)

γˆ GLS = (Y1,−1 Γ−1 Y1,−1 )−1 Y1,−1 Γ−1 ∆Y1

(7.9)

2 (Y1,−1 Γ−1 Y1,−1 )−1 σ2 (γˆ GLS ) = σˆ ε,GLS 2 = εˆ εˆ /(T − 1) σˆ ε,GLS

ˆ 1,−1 ) Γ−1 (∆Y1 − γˆ Y1,−1 ) εˆ εˆ = (∆Y1 − γY

(7.10a) (7.10b) (7.10c)

Notice that σ2 (γˆ GLS ) is a scalar, so that a GLS t statistic for the unit root null could be constructed in the usual way; however, such a statistic is as yet infeasible. In general, Γ is not known and a feasible GLS procedure requires a consistent estimator Γˆ of Γ. 7.1.2 Feasible GLS This section is concerned with feasible GLS (FGLS) estimation (see, for example, Galbraith and Zinde-Walsh (hereafter GZW), 1999). Initially, the set-up is

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(7.5)

264 Unit Root Tests in Time Series

a restatement of the infeasible estimator ρˆ GLS , but the purpose is to develop a framework that enables a feasible estimator within a GLS approach. Provided that ϕ(L) is invertible, (7.1) and (7.2) imply: (1 − ρL)yt = zt where Ω(L) = ϕ(L)−1 θ(L)

Let Ω, Φ and Θ be the T × T transformation matrices associated with the polynomials Ω(L), ϕ(L) and θ(L), respectively (examples of these are given below). Thus, in the context of (7.5), z = Ωε = (Φ−1 Θ)ε, and therefore: E[zz ] = E[Ωε(Ωε) ]

(7.11)

= σε2 ΩΩ = σε2 Γ Apart from a scalar, Γ is the variance-covariance matrix of the error process z; evidently Γ = ΩΩ and basic GLS principles require a transformation matrix P such that PΓP = I; the matrix P = Ω−1 satisﬁes this requirement, noting that the transpose of the inverse is the inverse of the transpose and Γ−1 = (ΩΩ )−1 = Ω −1 Ω−1 . Further, from the deﬁnition of Ω(L), note that Ω = Φ−1 Θ, and therefore Ω−1 = Θ−1 Φ. The transformed model that delivers a scalar-diagonal variancecovariance matrix is: PY1 = ρPY1,−1 + PΩε

(7.12a)

= ρΩ−1 Y1,−1 + Ω−1 Ωε −1

= ρΩ

(7.12b)

Y1,−1 + ε

(7.12c)

⇒ P∆Y1 = γ(Ω−1 Y1,−1 ) + ε

(7.12d)

where, as usual, γ = (ρ − 1). A summary of the matrices involved in the GLS framework is provided in Table 7.1.

Table 7.1 Summary of GLS matrices, general and special cases. general case special cases ARMA Ω AR only MA only

PΓP = I

P P = Γ−1

P

Ω

Γ−1 = (ΩΩ )−1

P = Ω−1

Ω = Φ−1 Θ Ω = Φ−1 Ω=Θ

Γ−1 = [(Φ−1 Θ)(Φ−1 Θ) ]−1 = Φ (ΘΘ )−1 Φ Γ−1 = [Φ−1 (Φ−1 )]−1 = Φ Φ Γ−1 = (ΘΘ )−1

P = Θ−1 Φ P=Φ P = Θ−1

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zt = Ω(L)εt

Improving the Power of Unit Root Tests

265

7.1.2.i Simple examples

AR errors First, consider an AR(1) process. The set-up is as follows: (1 − ρL)yt = zt zt = Ω(L)εt

Ω(L) = (1 − ϕ1 L)−1

Then, in vector form, the model is: ∆Y1 = γY1,−1 + z ⇒    y0 ∆y1  ∆y   y 2  1       ∆y3  =  y2     ..   ..  .   . ∆yT yT−1





      γ +       

1 ϕ1 ϕ12 .. . T−1 ϕ1

0 1 ϕ1 .. . ...

0 0 1 .. . ϕ12

... ... ... .. . ϕ1

0 0 0 .. . 1

       

ε1 ε2 ε3 .. . εT

       

(7.13)

The square T × T matrix premultiplying ε is Ω and, in this AR only case, Ω = Φ−1 . Note that the errors z = Φ−1 ε are generated as z1 = ε1 , z2 = ϕ1 z1 + ε2 = ϕ1 ε1 + ε2 , z3 = ϕ1 z2 + ε3 = ϕ12 ε1 + ϕ1 ε2 + ε3 , and so on, whereas Φz = ε implies the following evolution in terms of zt : z1 = ε1 , z2 = ϕ1 z1 + ε2 , z3 = ϕ1 z2 + ε2 , and so on. When there is just an AR component, the transformation matrix is a lower band-diagonal matrix. The matrices Φ and Φ−1 are given below: Φ−1

Φ 

1  −ϕ1   0   .  .  .

0 1 −ϕ1 .. .

0 0 1 .. .

0 0 0 .. . −ϕ1

0 0 0



1    ϕ1  2   ϕ1   .  0   .. ϕ1T−1 1

0... 1... ϕ1 . . . .. .... ϕ1T−2 . . .

0 0 0 .. . ϕ12

... ... ... .. . ϕ1

=   0 1   0  0    0  = 0  ..   . .   .. 1 0

I 0 1 0 .. . 0

0 0 1 .. . ...

... ... ... .. . 0

0 0 0 .. . 1

        

(7.14)

In order to make the procedure feasible, the idea is to ﬁt an AR(1) model to the residuals, zˆ t , from LS estimation ignoring the AR(1) error structure. Next,

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Two special cases of interest that illustrate the general principles are the pure AR case and the pure MA case. In these examples remember that P = Ω−1 ; the ﬁrst case deals with the simpliﬁcation that Ω−1 = Φ, which arises in the AR case, and the second case deals with the simpliﬁcation that Ω−1 = Θ−1 , which arises in the MA case.

266 Unit Root Tests in Time Series

ˆ −1 of Ω−1 using noting that P = Ω−1 = Φ in this case, construct an estimator Ω the estimated value of ϕ1 , from which the following FGLS estimators are then obtained: γˆ FGLS = (Y1,−1 Γˆ −1 Y1,−1 )−1 Y1,−1 Γˆ −1 ∆Y1

(7.15)

2 (Y1,−1 Γˆ −1 Y1,−1 )−1 σˆ (γˆ FGLS ) = σˆ ε,FGLS

ˆΩ ˆ =Φ ˆ −1 Φ ˆ −1 and σˆ 2 ˆ FGLS ; where Γˆ = Ω ε,FGLS is deﬁned from the residuals using γ ˆ ˆ FGLS Y1,−1 ). Note that the data vectors in the transformed LS that is, εˆ = P(∆Y 1 −γ model are as follows: PY1 = (y1 , y2 − ϕ1 y1 , . . . , yT−1 − ϕ1 yT−2 , yT − ϕ1 yT−1 ) PY1,−1 = (y0 , y1 − ϕ1 y0 , . . . , yT−2 − ϕ1 yT−1 , yT−1 − ϕ1 yT−2 ) The extension to an AR(p) process is straightforward, requiring ﬁtting an ˆ −1 is AR(p) model to zˆ t , although, of course, the second step of obtaining Φ computationally more complex. Practically, choosing p implies a selection criterion, for which leading candidates are the set of information-based criteria or a marginal-t selection; such criteria are considered in greater detail in Chapter 9. MA errors Consider the simple MA(1) example, then the GLS set-up is: (1 − ρL)yt = zt zt = Ω(L)εt

Ω(L) = θ(L) = (1 + θ1 L) 

     

∆y1 ∆y2 .. . ∆yT





    =    

y0 y1 .. . yT−1

       γ+          

1 θ1 0 .. . 0 0 0

0 1 θ1 .. . 0 0 0

0 0 1 .. . 0 0 0

... ... ... .. . ... ... ...

0 0 0 .. . 1 θ1 0

0 0 0 .. . 0 1 θ1

0 0 0 .. . 0 0 1



ε1 ε2 ε3 .. .

           εT−2    εT−1 εT

            

(7.17)

Notice that Ω(L) = θ(L), and hence Ω = Θ, which is a lower band-diagonal matrix, with θ1 below the unit diagonal. In this case, taking the route of transforming the data vectors by P, note that P = Θ−1 , and for detail on the calculation of Θ−1 see Box and Jenkins (1970), Zinde-Walsh (1988, 1990) and Zinde-Walsh and Galbraith (1991). The resulting estimator has the same general GLS structure;

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(7.16)

Improving the Power of Unit Root Tests

267

that is:

2 (Y1,−1 Γˆ −1 Y1,−1 )−1 σˆ (γˆ FGLS ) = σˆ ε,FGLS

(7.18) (7.19)

ˆΘ ˆ and σˆ 2 ˆ FGLS . where Γˆ = Θ ε,FGLS is deﬁned from the residuals using γ As in the AR(1) case, one possible feasible GLS estimator is as follows. In the ﬁrst stage, estimate ∆yt = γyt−1 + zt by LS ignoring the possible serial correlation structure. Next ﬁt an MA(1) process to the LS residuals zˆ t and obtain an estimate ˆ of Θ; then Θ ˆ −1 is an estimator of Θ−1 . Finally, use Θ ˆ −1 ( = Ω ˆ −1 in this case) Θ to obtain the (feasible) GLS estimator. The extension to an MA(q) process just implies ﬁtting an MA(q) model to the LS residuals zˆ t . In a variation of this procedure, the residuals under the null hypothesis could be used; that is, γY1,−1 is omitted from the regression (γ = 0 as under the null). As in the construction of the PP tests, using the residuals under the null does not give consistency under stationary alternatives, but using the residuals under the alternative implies that any problems in estimating γ, for example, ﬁnite sample bias, are transmitted to the residuals. 7.1.3 Approximating an MA process by an AR model An alternative to the (direct) procedure of estimating an MA(q) process, is to approximate it by an autoregression, which is a principle that also underlies the 2 2 autoregressive estimator, σˆ z,lr,AR(k ∗ ) , of σz,lr , through the ADF approximation of MA errors. There are some variants of this method, and three are described here. 7.1.3.i Using the AR version of FGLS In the ﬁrst case, the MA errors are approximated by a ‘long’ autoregression, and the AR coefﬁcients from this approximating autoregression are used in the FGLS procedure; thus the problem is treated as due to AR not MA errors and so has a 2 direct analogy with estimation of σˆ z,lr,AR(k ∗ ) and the ADF regression. To illustrate the options, consider the simplest version of the error dynamics model, with the errors generated by an MA(q) process. The error model is ﬁrst written in MA form and then inverted to an inﬁnite AR representation; the AR representation is then truncated and, with the LS residuals, zˆ t , replacing the unknown errors, the empirical model for the errors is estimated. In summary, the elements are: ∆yt = γyt−1 + zt

(7.20a)

zt = θ(L)εt ϕ(L)zt = εt

(7.20b) ϕ(L) = θ(L)−1

(7.20c)

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γˆ FGLS = (Y1,−1 Γˆ −1 Y1,−1 )−1 Y1,−1 Γˆ −1 ∆Y1

∞

zt = ∑i=1 ϕi zt−i + εt = ∑i=1 ϕi zt−i + εt,k + 1 k

∞ ϕ z + εt i = k + 1 i t−i

εt,k + 1 = ∑

zˆ t = ∑i=1 ϕˆ i zˆ t−i + εˆ t,k + 1 k

inﬁnite AR representation

(7.21a)

truncated AR representation

(7.21b)

approximation error

(7.21c)

empirical model for errors

(7.21d)

The LS errors, zˆ t , shown here do not impose the null (as noted above, an alternative is to impose ρ = 1, that is, γ = 0, in which case the errors are just ∆yt ). Let Φk be the transformation matrix constructed from the truncated AR(k) representation. Whilst the exact GLS transformation in the case of MA errors requires premultiplication by P = Ω−1 = Θ−1 in order to ensure white-noise residuals, the use of Φk means that the transformed model is: Φk Y1 = ρΦk Y1,−1 + Φk Θε

(7.22)

In this case, as k increases (for given sample size), Φk Θ should get ‘closer’ to the identity matrix (if Φk = Θ−1 , then Φk Θ is exactly the identity matrix), but the transformation will not, in general, be exact for ﬁnite k in the sense of leading to the identity matrix (see Zinde-Walsh, 1988; GZW, 1997, for a discussion of these issues). The estimated coefﬁcients ϕˆ i are used to form the transformation matrix ˆ k . The end result is a feasible GLS estimator based on the (approximate) AR Φ transformation: −1 ˆ −1 γˆ FGLS,k = (Y1,−1 Γˆ −1 k Y1,−1 ) Y1,−1 Γk ∆Y1 2 −1 (Y1,−1 Γˆ −1 σˆ 2 (γˆ FGLS,k ) = σˆ ε,FGLS,k k Y1,−1 )

(7.23) (7.24)

ˆ −1 Φ ˆ −1 and σˆ 2 ˆ FGLS,k . where Γˆ k = Φ k ε,FGLS,k is deﬁned from the residuals using γ k

7.1.3.ii Using the MA version of FGLS via an AR approximation The second case comprises two sub-cases: again, an approximating autoregression is estimated, but the relationships between the AR and MA coefﬁcients are used to obtain an estimator of the MA coefﬁcients, which are then used in an MA version of the FGLS procedure. The two sub-cases just depend on whether the number of estimated AR coefﬁcients, k, is greater than or equal to the number of MA coefﬁcients, q. The resulting estimation procedure is non-iterative, as in the previous case.

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269

The relationship between the AR and MA lag coefﬁcients if ϕ(L) = θ(L)−1 is: ψ1 = θ1 ψ2 = − θ1 ϕ1 + θ2 ϕi = − θ1 ϕi−1 − θ2 ϕi−2 . . . − θi−1 ϕ1 + θi

i
.. . i=q

.. . ϕk = − θ1 ϕk−1 − θ2 ϕk−2 . . . − θk−1 ϕ1

k>q

(7.25)

For example, the MA(1) case is familiar: ϕ1 = θ1 , ϕ2 = −θ21 , ϕ3 = θ31 , . . ., ϕk = (−1)k−1 θk1 . These recursions can conveniently represented as a system of k × q equations:     1 0 0 ... ... 0 0 ϕ1   ϕ   −ϕ1 1 0 ... ... 0 0    2   θ1     .   .  ϕ3   −ϕ  −ϕ1 1 ... ... . 0 θ2  2       ..      . . . . .  θ3   . = . . . . .      . . . . . 1 0  .   ϕ    .   q   −ϕq−1 −ϕq−2 . . . . . . . . . −ϕ 1 .   1  .     .   . . . . . . .  θ q  .   .. .. .. .. .. .. ..  ϕk −ϕ . . . . . . . . . −ϕ −ϕ −ϕ k−1

k−2

k−(q−1)

k−q

(7.26) Say: Φk = ΠΞ where Φk is the k × 1 vector of ϕi coefﬁcients, Π is the k × q matrix in square brackets and Ξ = (θ1 , . . . , θq ) is the (column) vector of MA coefﬁcients. There are two estimation methods based on these relationships due to Durbin (1959) and GZW (1994), differing in whether k > q or k = q. These estimators are consistent given k → ∞ as T → ∞ such that k/T → 0. In both cases, the ˆ k and Π, ˆ and Ξ is the q × 1 estimated AR parameters ϕˆ j replace ϕj , resulting in Φ vector of unknowns. In the case that k > q, the estimator Ξˆ is a least squares solution to the set of equations, and when k = q, the equations are inverted to ˜ as follows (respectively): obtain the solution, Ξ, ˆ Π ˆ Φ ˆ −1 )Π ˆk k>q Ξˆ = (Π

(7.27a)

ˆk ˆ −1 Φ Ξ˜ = Π

(7.27b)

k=q

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ϕq = − θ1 ϕq−1 − θ2 ϕq−2 . . . − θq−1 ϕ1 + θq

270 Unit Root Tests in Time Series

Having obtained an estimator of the MA structure, next form an estimator of ˆ Ξ) ˜ Ξ), ˆ and Θ( ˜ respectively, where (.) indicates the funcΘ; for example, say, Θ( ˜ respectively, tional dependence on the estimator in parentheses with Ξˆ and Ξ, and then obtain the FGLS estimator based on this estimator of the covariance matrix, designated FAGLS (for feasible approximate GLS):

2 ˆ Ξ) ˆ −1 Y1,−1 )−1 (Y1,−1 Γ( σˆ 2 (γˆ FAGLS ) = σˆ ε,FAGLS

˜ Ξ) ˜ −1 Y1,−1 )−1 Y1,−1 Γ( ˜ Ξ) ˜ −1 ∆Y1 γ˜ FAGLS = (Y1,−1 Γ( 2 ˜ Ξ) ˜ −1 Y1,−1 )−1 (Y1,−1 Γ( σ˜ 2 (γ˜ FAGLS ) = σ˜ ε,FAGLS

(7.28) (7.29) (7.30) (7.31)

2 ˆ Ξ) ˆ Ξ) ˜ Ξ) ˜ Ξ) ˆ Ξ) ˆ = Θ( ˆ Θ( ˆ and Γ( ˜ Ξ) ˜ = Θ( ˜ Θ( ˜ ; and σˆ 2 where Γ( ˜ ε,FAGLS are ε,FAGLS and σ deﬁned from the residuals based respectively on γˆ FAGLS and γ˜ FAGLS . GZW (1994) examined the ﬁnite sample performance of the Durbin (k > q) estimator, their own (k = q) MA estimator and the ML estimator in a small simulation study using MA(1) and MA(2) errors. In terms of choosing k, they found that the rule that k should grow at the rate O(T1/3 ) (see Berk, 1974), combined with k = 8 for T = 100 as a reference point, provided a reasonable rule of thumb. No one estimator dominated the others in terms of bias and mean squared errors, despite the theoretical advantage of ML over both non-iterative methods and the apparent advantage of Durbin’s method over GZW’s method in terms of using more information. They found that ML could result in non-invertible estimates when the root(s) were close to unity and Durbin’s method could result in larger bias.

7.1.3.iii ARMA errors The obvious extension of the separate AR and MA models is the combination of these two elements into an ARMA model. In principle, the GLS estimator ˆ −1 . The framework requires Ω−1 = Θ−1 Φ and a feasible GLS estimator requires Ω in the previous sections can be extended to the case of ARMA errors; that is, obtain estimates of the AR and MA coefﬁcients from the (inﬁnite) AR representation of an ARMA process, provided that the MA component is invertible. Whilst this can be done (see, for example, Fuller, 1976, 1996; GZW, 1997), the latter authors suggest that, in contrast to the pure MA model, the performance of the resulting estimator is poor in ﬁnite samples even though it is consistent. Three-stage estimation for ARMA models An alternative procedure is to combine the ADF and FGLS approaches, which is illustrated below with an ARMA(1, 1) model of the error dynamics. This is a procedure with three steps: 1. ﬁrst, estimate the AR structure (only) through an ‘ADF’ regression; 2. second, the residuals from the step 1 are used to estimate an MA(q) model;

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ˆ Ξ) ˆ −1 Y1,−1 )−1 Y1,−1 Γ( ˆ Ξ) ˆ −1 ∆Y1 γˆ FAGLS = (Y1,−1 Γ(

Improving the Power of Unit Root Tests

271

3.

ﬁnally, a FGLS estimator is obtained using a transformation matrix based on the second step. The method is illustrated with ARMA(1, 1) error dynamics. The model is: (1 − ρL)yt = zt (1 − ϕ1 L)zt = (1 + θ1 L)εt

⇒ ARMA(1, 1) error structure

Then combine the two components:

(1 − ϕ1 L)(1 − ρL)yt = vt

vt = (1 + θ1 L)εt

A(L)yt = vt

A(L) = (1 − ϕ1 L)(1 − ρL)

This leads to an ‘ADF’ regression, but with an MA(1) error: ∆yt = γyt−1 + c1 ∆yt−1 + vt

γ + vt = (yt−1 , ∆yt−1 ) c1 As the AR structure is easy to estimate, the idea is to incorporate estimation of the AR components into a ﬁrst step, which leaves (possible) MA components to be accounted for separately. Thus, unlike the usual ADF regression, on which it is based, the last equation differs because vt = (1 + θ1 L)εt = εt . However, all that is required is to obtain the LS residuals, vˆ t , which follow from ignoring the MA structure; and then, as in the pure MA case, estimate an MA(1) model using vˆ t . ˆ of Θ, and The resulting estimator of the transformation matrix is, as before, Θ ˆ −1 is an estimator of Θ−1 , which is then used to obtain the FGLS estimator so Θ with (yt−1 , ∆yt−1 ) as the regressors. In matrix form, the model and estimator are:

γ +v ∆Y1 = Y1,−1 ∆Y1,−1 c1 = Wϑ + v

(7.32)

ϑˆ FGLS = (W Γˆ −1 W)−1 W Γˆ −1 ∆Y1

(7.33)

2 σˆ 2 (ϑˆ FGLS ) = σˆ ε,FGLS (W Γˆ −1 W)−1

(7.34)

ˆΘ ˆ ; γˆ FGLS is now the ﬁrst element of ϑˆ FCGLS and its estimated where Γˆ = Θ standard error is the ﬁrst element of σˆ 2 (ϑˆ FGLS ). This procedure easily generalises. If the error dynamics are ARMA(p, q), then the ﬁrst stage is to estimate an ADF(p – 1) regression and then ﬁt an MA(q) model ˆ to obtain the FGLS estimator in a regression to the LS residuals; ﬁnally, use Θ with regressors (yt−1 , ∆yt−1 , . . . , ∆yt−(p−1) ).

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(1 − ϕ1 L)(1 − ρL)yt = (1 + θ1 L)εt

272 Unit Root Tests in Time Series

7.2 FGLS unit root tests If the estimation purpose was solely to parameterise the error dynamics, then that task has been achieved; however, the error dynamic coefﬁcients are intermediate to the eventual purpose of carrying out a unit root test, a natural route being to extend the DF-type tests, but with estimation by FGLS.

Any of the FGLS estimators of the previous sections will serve as examples. For example, consider (7.15) and (7.16), restated here for convenience: γˆ FGLS = (Y1,−1 Γˆ −1 Y1,−1 )−1 Y1,−1 Γˆ −1 ∆Y1 2 (Y1,−1 Γˆ −1 Y1,−1 )−1 σˆ (γˆ FGLS ) = σˆ ε,FGLS

(7.35) (7.36)

ˆ Then δ-type and τˆ -type statistics can be formed as: δˆ FGLS = Tγˆ FGLS τˆ FGLS =

γˆ FGLS σˆ (γˆ FGLS )

(7.37) (7.38)

Test statistics formed in this way, using a consistent estimator of Γ, have DF limiting distributions (see, for example, GZW, 1999). If γˆ FGLS is based on an ADF regression, then δˆ FGLS is modiﬁed accordingly (see Chapter 3, section 3.8.3; Chapter 6, section 6.3.6).

7.2.2 GZW unit root test (ARMA variation) GZW (1999) suggest a variation on the procedure of section 7.1.3.iii that reverses the order of modelling the AR and MA components, but uses the same motivation. It is assumed that there is good reason – for example, through model identiﬁcation procedures – to consider that an MA component may be present in the DGP. Then the estimation and testing procedure is as follows: 1.

impose the unit root null and ﬁt a ‘low order’ MA process to ∆yt (if drift is allowed under the null, ∆yt is regressed on a constant and the residuals are used); let Σˆ denote the covariance matrix of the estimated MA process, and hence obtain Σˆ −1 ; for example, by direct inversion or by the methods considered in Zinde-Walsh and Galbraith (1991);

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7.2.1 DF-type tests based on FGLS

Improving the Power of Unit Root Tests

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273

estimate the ‘standard’ ADF(k – 1) regression by FGLS using Σˆ −1 ; that is, the regression model and estimator are:   γ    c1   (7.39) ∆Y1 = Y1,−1 ∆Y1,−1 . . . ∆Y1,−(k−1)  .  .  +v  .  ck−1

ϑˆ FGLS,GZW = (W Σˆ −1 W)−1 W Σˆ −1 ∆Y1 σˆ 2 (ϑˆ FGLS,GZW ) = σˆ v2 (W Σˆ −1 W)−1

(7.40) (7.41)

vˆ = ∆Y1 − Wϑˆ FGLS,GZW σˆ v2 =

vˆ vˆ T − (k + 1)

(7.42)

ˆ The δ-type and τˆ -type tests are calculated in the usual way using the estimated value of γ, which is the ﬁrst element of ϑˆ FGLS,GZW ; for later reference, the resulting test statistics are denoted δˆ FGLS,GZW and τˆ FGLS,GZW . In a sense, the suggested procedure ‘neutralises’ the presence of MA error terms provided that the estimated order of the MA component is not less than the true order. If this is the case, then the asymptotic distribution of the FGLSbased pseudo t test is the same as the corresponding ‘standard’ DF asymptotic distribution. This is also the case if a constant or constant and trend is included in the regression, in which case the ADF includes the relevant deterministic terms (or they are removed by prior detrending). The ‘neutralisation’ should, however, achieve a better ﬁnite sample performance for the FGLS test statistic over (i) its unaugmented counterpart and (ii) a counterpart where a ‘long’ autoregression takes the place of the MA component, thus avoiding the approximation of the MA component by a ‘high’ order truncated AR component. Moreover, the lag length in the ADF does not need to increase with the sample size to ensure consistency; this is because the MA component has been accounted for in the ﬁrst step. Thus, by removing the need of the ADF to approximate the MA component by increasing the lag length so that the included lags are there just for any AR component, it should be possible to both preserve size and increase power (which suffers when long lags are included in the ADF). The simulation evidence presented in GZW (1999) suggests that size is better maintained using the FLGS procedure compared to a standard ADF test with a ﬁxed lag. However, there is no explicit power comparison of the FGLS with the standard ADF tests because of the need to control size in the latter case.

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= Wϑ + v

However, size can be controlled for the standard DF tests if the lag is allowed to increase. To examine this issue further a small simulation study was undertaken. The set-up was as that underlying Table 6.6; that is, the DGP was (1 − ρL)yt = zt , with the errors modelled either as AR(1), (1 − ϕ1 L)zt = εt , εt ∼ niid(0, 1), or MA(1), zt = (1 + θ1 L)εt , with ϕ1 = 0.0, 0.3, 0.6 and θ1 = –0.5, 0.5; when ρ < 1, the initial observation was drawn from N(0, (1 − ρ2 )−1 ). For comparability with other methods, the lag length for the ADF was chosen by AIC. A constant was included in the maintained regression, so the focus is on test statistics with a µ subscript. The simulation assessment mimics a practical situation in which it is not known in the ﬁrst stage that there is an MA component, so an initial assessment is introduced using AIC to select the order of the MA model, with q = 0, 1, 2. If the order is greater than zero, then the second stage ADF is estimated by FGLS, otherwise LS is used. The order has then to be chosen in the ADF, but as this does not now need to approximate the MA component it can be short, and AIC is used in this stage. A direct alternative to the FGLS procedure is ML estimation of an ARMA model, as described in Chapter 3, and the Shin-Fuller (SF) ML method is included for comparison, the relevant test statistic being LRUC,µ . Again, to make the simulation more realistic, the order of the ARMA model for the errors is not assumed known, and AIC was used to select the order, with maximum lag of 2 on each of the AR and MA components for zt . Introducing a search procedure imparts a further element of random variation in the ﬁnite sample null distribution of the test statistics, which should be taken into account, generally implying that the relevant critical values are increased in absolute value. In fact the implied changes to the methods based on ADF estimation were slight; however, the changes to the critical values for the ML test statistic were signiﬁcant and need to be noted. The critical values reported in Chapter 3, Table 3.1 (from Shin and Fuller, 1998) are from simulations of the ﬁrst-order AR model (that is, they assume no ARMA structure for the errors). For example, the percentiles for 90%, 95% and 99%, with T = 200, are approximately 1.07, 1.75 and 3.43; however, simulating the search and estimate procedure gives corresponding percentiles of approximately 1.56, 2.54 and 4.80, which are used here and differ from the ‘no search’ case. Rather than report all of the previous test statistics, some are ruled out as they suffer size retention problems, especially with MA errors. Of the PP semi2 parametric estimators, MSBµ is included and uses the AR estimator σˆ z,lr,AR(k ∗), ∗ with k = 8. Of the DF tests, τˆ µ is included on the basis that, whilst it is not the most powerful with iid errors, it is best at size retention with AR or MA errors; and, on the same basis, τˆ ws µ is included from the WS tests. The set of test statistics ˆ FGLS,GZW and LRUC,µ , and the results considered is: τˆ µ , τˆ ws , MSB µ , δˆ FGLS,GZW , τ µ are summarised in Table 7.2.

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Table 7.2 Size and power comparisons: CGLS and ML test statistics.

τˆ µ

MSBµ

δˆ FGLS,GZW

τˆ FGLS,GZW

LRUC,µ

ϕ1 = 0.0 ρ = 1.0 ρ = 0.95 SA ρ = 0.90 SA

5.0 24.4 26.1 55.7 58.3

4.4 35.8 39.7 70.0 74.4

13.9 63.4 35.7 90.5 67.8

6.5 48.4 41.1 93.2 89.4

5.3 33.0 33.6 83.4 81.9

5.0 37.7 37.9 85.2 85.5

ϕ1 = 0.3

τˆ µ

τˆ ws µ

MSBµ

δˆ FGLS,GZW

τˆ FGLS,GZW

LRUC,µ

ρ = 1.0 ρ = 0.95 SA ρ = 0.90 SA

4.9 23.9 24.2 53.7 54.2

4.1 34.9 39.4 67.3 72.1

13.3 64.1 37.1 89.6 69.7

6.1 48.3 41.4 89.9 86.9

5.4 30.6 28.4 76.8 74.0

3.1 30.5 45.1 79.3 89.8

ϕ1 = 0.6

τˆ µ

τˆ ws µ

MSBµ

δˆ FGLS,GZW

τˆ FGLS,GZW

LRUC,µ

ρ = 1.0 ρ = 0.95 SA ρ = 0.90 SA

5.1 22.4 21.3 48.3 46.7

4.4 32.7 36.3 63.1 66.4

14.0 61.5 33.2 87.2 62.6

7.2 45.5 35.1 86.3 77.7

6.0 29.7 26.4 70.0 65.7

3.2 29.4 39.8 73.8 84.1

τˆ µ

τˆ ws µ

MSBµ

δˆ FGLS,GZW

τˆ FGLS,GZW

LRUC,µ

θ1 = –0.5 ρ = 1.0 ρ = 0.95 SA ρ = 0.90 SA

5.3 25.7 25.3 64.3 63.8

5.0 36.4 36.1 76.2 76.3

11.1 58.0 31.4 86.0 64.5

12.3 67.4 41.6 96.6 85.7

8.2 48.9 34.9 87.6 78.0

4.2 33.9 42.0 73.2 79.3

θ1 = 0.5

τˆ µ

τˆ ws µ

MSBµ

δˆ FGLS,GZW

τˆ FGLS,GZW

LRUC,µ

ρ = 1.0 ρ = 0.95 SA ρ = 0.90 SA

5.6 25.2 24.3 55.1 53.2

4.8 35.0 35.3 69.1 69.5

5.9 32.9 29.1 72.0 67.1

8.1 50.8 37.9 89.0 78.3

5.7 30.9 27.9 69.9 66.3

2.9 34.6 44.7 83.3 91.1

MA(1) error structure

Note: Best or good performances shown in bold; SA indicates size-adjusted power.

First, consider size in the case of iid errors: τˆ µ , τˆ µ and LRUC,µ have close to the correct size, whilst δˆ FGLS,GZW is slightly oversized. As to power, τˆ ws µ loses out quite considerably in terms of power by using the lag(AIC) + 2 rule rather than the AIC(lag) rule (see Chapter 6, Table 6.8). Whilst δˆ FGLS,GZW is the most powerful (on a size-adjusted basis) of the tests with more accurate size, the ranking

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AR(1) error structure τˆ ws µ

ˆ is not uniform, with τˆ ws µ and δFGLS,GZW competitive; LRUC,µ is best, although undersized, as ρ departs more from the unit root. In the case of MA(1) errors, when θ1 = –0.5, τˆ µ , τˆ ws µ and LRUC,µ maintain their size reasonably well, whereas the GLS tests are oversized. Of the reasonably sized tests, LRUC,µ is superior in terms of power. When θ1 = 0.5, δˆ FGLS,GZW is still oversized whilst LRUC,µ becomes undersized. A comparison in terms of power, ˆ FGLS,GZW depends in how close ρ is to unity; no one test of τˆ ws µ , MSBµ and τ dominates, although LRUC,µ is best on a size-adjusted basis. Overall, it is clear that no one test statistic is best in all the situations considered; both the FGLS and ML test statistics have something to add depending on the nature of the process generating the errors.

7.3 Illustration This section applies the unit root tests to a time series chosen from the inﬂuential set collated by Nelson and Plosser (1982), as extended by Schotman and Van Dijk (1991). In this case, the data are for the US unemployment rate, 1890 to 1988. The intention is to illustrate the various test statistics rather than seek an example with test conﬂict, although that is an important issue when multiple tests are applied. 7.3.1 US Unemployment rate The data are graphed in Figure 7.1. A priori it could be argued that a bounded positive variable, such as the unemployment rate, cannot conceptually be a random walk; there is a positive probability that a random walk will cross the zero axis and will not have a ﬁnite upper or lower limiting bound, and neither of which can be satisﬁed for the unemployment rate. A priori, there is a strong anticipation that the unit root tests will lead to rejection of the unit root null hypothesis. Nevertheless, the series has been analysed extensively both in its original form and by taking the logarithms (see, for example, Xiao and Phillips, 1998). The log series is used here, with the test statistics for the original series reported in a question. 7.3.1.i Standard ADF estimation As to prior issues, there is no evidence of the need for a linear time trend under the stationary alternative and, hence, to include one would reduce the power of the test. The usual ADF regression, in this case ADF(3), was selected by the AIC, BIC (Bayesian information criterion) and marginal-t and the results are summarised in Table 7.3; the LS estimate of ρ is ρˆ = 0.519 and the test statistics are δˆ µ = 96(ˆρ − 1) = –46.19 (the sample size T is sometimes taken to include the

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276 Unit Root Tests in Time Series

Improving the Power of Unit Root Tests

277

LS ∆yˆ˜ t ‘t’ ∆yˆ˜ t ‘t’ FGLS ∆yˆ˜ t ‘t’

yˆ˜ t−1

∆yˆ˜ t−1

∆yˆ˜ t−2

∆yˆ˜ t−3

–0.223 –3.50 –0.286 –3.97

–

–

–

0.378 3.88

–0.206 –2.23

0.23 2.52

–0.203 –3.32

0.182 1.78

– –

– –

Note: Effective sample period: 1894–1988 = 95 observations.

lags, in which case T = 96 + 4), and τˆ µ = –3.97; both indicate rejection of the null. This forms the baseline for comparison. 7.3.1.ii PP tests The PP tests and their modiﬁcations derive from the basic DF regression, which is summarized in the ﬁrst row of Table 7.3. The values of δˆ µ and τˆ µ are –21.47 and –3.50, respectively. These are then modiﬁed to obtain the PP versions of the tests. The relevant variances on which the adjustment is based are σz2 and 2 , which are estimated by σ 2 and σ 2 σz,lr ˜ z,a ˆ z,lr,AR(k ∗ ) , respectively (see Chapter 6, 2 = 0.159 and σ 2 section 6.8.2.ii). The values obtained were: σ˜ z,a ˆ z,lr,AR(3) = 0.364,

25

20

15 % p.a. 10

5

0 1890 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 Figure 7.1 US unemployment rate (Nelson and Plosser series).

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Table 7.3 Estimation results for US unemployment, Nelson and Plosser data.

278 Unit Root Tests in Time Series

Table 7.4 Summary of unit root test values for US unemployment data.

5% cv

δˆ ws µ

Zδˆ µ

MZδˆ µ

τˆ µ

τˆ ws µ

–46.19 –14.54

–42.73 –14.32

–45.29 –14.76

–42.91 –14.34

–3.97 –2.87

–3.23 –2.58

Zˆτµ

MZˆτµ

MSBµ

δˆ FGLS,GZW

τˆ FGLS,GZW LRUC,µ

–4.90 –2.96

–4.64 –2.87

0.11 0.17

–23.82 –16.20

–3.32 –2.94

8.53 2.50

Notes: Lag selection details: ADF, lag(AIC) = 3; WS, lag(AIC) + 2 = 4; FGLS, lag(AIC) = MA(2), ADF(1); UCML, lag(AIC) ⇒ ARMA(0, 1) for zt ⇒ ARMA(1, 1) for yt .

where the lag length was selected by minimising AIC. The difference implies that there will be a noticeable adjustment to the test statistics δˆ µ and τˆ µ , from the serial correlation correction, which become –45.29 and –4.90, respectively. The MZ versions of these test statistics make an additive adjustment, which in the case of MZδˆ µ is (T/2)(ˆρ − 1)2 = (95/2)(−0.2236)2 = 2.38, and, therefore, MZδˆ µ = –45.29 + 2.38 = –42.91. The adjustment to Zˆτµ is (1/2ˆσz,lr,AR(k∗) )ST (ˆρ − 1)2 , where ST = (∑Tt=1 y˜ 2t−1 )1/2 and y˜ t refers to the √ demeaned data; numerically the adjustment is (1/2 0.364)38.82(−0.2236)2 = 0.258, so that MZˆτµ = –4.90 + 0.258 = –4.64. Finally, MSBµ is obtained as Zˆτµ /Zδˆ µ = 0.11. The critical values for δˆ µ , MZδˆ µ , Zˆτµ and MZˆτµ are the same as their parametric counterparts, and so unequivocally imply rejection. The 5% critical value for MSBµ is approximately 0.17, with rejection of the null for values smaller than this, so rejection also follows.

7.3.1.iii FGLS (GZW version) The preferred model in the ﬁrst stage, with the null imposed, is MA(2), the results of which are then used in FGLS estimation of an ADF(k∗ ) model; in this case, k∗ = 1 was selected by AIC and BIC. The estimated MA(2) model was: ∆yˆ˜ t = (1 + 0.178L − 0.357L2 )˜εt , and FGLS estimation of the ADF(1) model is reported in Table 7.3. The test statistics are δˆ FGLS,GZW = –23.82 and τˆ FGLS,GZW = – 3.32 (see Table 7.4); both indicate rejection of the null hypothesis at conventional signiﬁcance levels. Maximum likelihood estimation of the unconditional ARMA model resulted in (1 − 0.582)yˆ˜ t = zt , zt = (1 + 0.578)˜εt , so that ρ˜ = 0.582 and θ1 = 0.578, with a ‘t’ statistic of 5.61; also σ˜ (˜ρ) = 0.0982, so that the LR-based τˆ -type test statistic is τ˜ µ = [(0.582 – 1)/0.0982] = –4.27. The restricted model is (1 − ρµ )y˜ t = zt , zt = (1 + 0.400)˜εt , where ρµ = (1 – 4/95) = 0.958. The resulting value of LRUC,µ was 8.53, which is clearly signiﬁcant whether the 95% quantile is taken as 1.75

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5% cv

δˆ µ

Improving the Power of Unit Root Tests

279

(Shin and Fuller, 1998; see Chapter 3) or 2.5, allowing for the search procedure carried out here.

The previous sections have suggested a number of ways of approaching the problem of testing the unit root null hypothesis in the context of errors that are weakly dependent, speciﬁcally that they are generated by an ARMA process. The tests make the conventional assumption that under the alternative hypothesis of stationarity, the deterministic components are best removed by demeaning/detrending, so that the resulting residuals are used in estimation and testing (or a procedure that is equivalent). However, it is usually the case that some persistence is exhibited under the alternative; indeed, the alternative is likely to be quite local to the unit root hypothesis, which suggests undertaking the demeaning/detrending in a local-to-unity framework. This approach has been termed GLS even though it relates to the special case where the errors are generated by an AR(1) process and the resulting GLS transformation matrix can be interpreted as quasi-differencing the data, given an estimator of the AR(1) coefﬁcient. Hence, a better description of the process would be quasi-differencing, but the method is widely referred to as GLS, or sometimes efﬁcient detrending. With that caveat in mind, this section considers such developments, initially due to ERS (1996) and Elliott (1999). In many situations a UMP test does not exist; this is the case even in the simple case of testing a hypothesis about the mean µ0 against the two-sided alternative, HA : µA = µ0 . It is also the case for unit root tests that no UMP test exists if the size-α likelihood-ratio critical region depends on the alternative hypothesis. To examine this situation further, consider the set of alternative hypotheses given by H0 : ρ = 1 and HA : ρ = ρ¯ < 1, where ρ¯ is one of a set of alternative values for ρ. In general, Cox and Hinkley (1974) note three possibilities when there is no UMP test, all of which have relevance to unit root tests: 1. use the most powerful test for a typical or representative ρ¯ under HA ; 2. maximise power for an alternative very local to ρ = 1; 3. maximise the weighted power for a range of local alternatives. Option (3) is attractive and avoids the dependence on a particular value of ρ under HA , but requires the speciﬁcation of a weighting function. The development in ERS addresses point (2) above, that of paying more attention to demeaning/detrending the data under the local alternative. (See also Burridge and Taylor, 2000, for an alternative interpretation of the gain in power for ˆ ERS procedure.) The standard DF δ-type and τˆ -type tests lack power when the (dominant) root is close to, but not actually, a unit root; a situation described

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7.4 Tests that are optimal for particular alternatives (point-optimal tests)

280 Unit Root Tests in Time Series

as near-integrated rather than integrated. Whilst the weighted symmetric and (unconditional) maximum likelihood versions of these tests generally have better power properties, the question still arises as to whether other tests, explicitly designed to exploit the weakness of the standard tests in the sense of focusing on the near-unit root under the alternative, can do even better.

There are two novel aspects to the suggestions by ERS. The ﬁrst is to focus on the local-to-unity framework in specifying the alternative hypothesis; thus the alternative is HA : ρ = ρ¯ = 1 + c¯ /T < 1 ⇒ c¯ < 0. The second is to suggest test statistics that do not directly involve ML or GLS estimation, although the initial development is in terms of the likelihood function and likelihood ratio tests. 7.4.1.i Known deterministic components This section is an introduction to the conceptual basis of the unit root tests suggested by ERS (1996). The simpliﬁcation adopted here is that the coefﬁcients on the deterministic components are assumed known; this simpliﬁcation is relaxed in section 7.4.1.iii. Consider the following simple set-up, which is written to emphasise that the data-generating process is a trend function subject to serially correlated errors, for which the common factor framework is most appropriate: yt = µt + ut (1 − ρL)ut = zt

t = 1, ..., T

(7.43)

t = 2, ..., T

(7.44)

The trend function in (7.43) will typically be either µt = µ or µt = β0 + β1 t, say λ xt where λ = µ and xt = 1 or λ = (β0 , β1 ) and xt = (1, t) , respectively. Also let σu2 = σz2 /(1 − ρ2 ), which is the unconditional variance of ut , where σz2 is the variance of zt . Suppose, initially, that µt is known; hence ut = (yt − µt ), which is the demeaned/detrended yt without estimation error, is also known and all that is required is to focus on: ∆ut = γut−1 + zt

(7.45)

where γ = (ρ − 1). The following initial condition is assumed to apply: u0 = 0 ⇒ u 1 = z1

initial condition

(7.46)

t=1

(7.47)

It turns out that the condition (7.46), known as the initial condition assumption, is material to the analysis. Note that (7.46) implies y0 = µ0 , so that the initial observation is ‘on trend’. An alternative initial condition and its implications are considered in section 7.4.3.

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7.4.1 Likelihood ratio test based on nearly-integrated alternatives

Improving the Power of Unit Root Tests

281

Let z = (z1 , z2 , . . . , zT ) , E(z) = 0 and E(zz ) = Σz = σε2 Γ (the notation Σz is more convenient here), then the conditional log-likelihood function, given γ¯ = ρ¯ − 1, is: T 1 1 ¯ u−1 )] ln 2π − ln |Σz | − [(∆u − γ¯ u−1 ) Σ−1 z (∆u − γ 2 2 2

(7.48)

where ∆u = (u1 , u2 − u1 , . . . , uT − uT−1 ) , u−1 = (0, u1 , u2 , . . . , uT−1 ) . Note that the ﬁrst term in ∆u is u1 and the ﬁrst term in u−1 is zero: these terms arise from the initial condition. Under the null hypothesis γ¯ = 0, so that the log-likelihood function simpliﬁes to: CLL(0, u) = −

1 T 1 ln 2π − ln |Σz | − [∆u Σ−1 z ∆u] 2 2 2

(7.49)

The difference in the log-likelihood functions under the alternative and null hypotheses is given by: ¯ = CLL(γ, ¯ u) − CLL(0, u) DLL(γ)

(7.50)

1 ¯ u−1 ) − ∆u Σ−1 = − [(∆u − γ¯ u−1 ) Σ−1 z (∆u − γ z ∆u] 2 1 ¯ u Σ−1 = − [γ¯ 2 u−1 Σ−1 z u−1 − 2γ z ∆u−1 ] 2 1 ¯ T−1 u−1 Σ−1 = − [c¯ 2 T−2 u−1 Σ−1 z u−1 − 2c z ∆u] 2 The last line uses γ¯ = ρ¯ − 1 = c¯ /T under the local alternative. Then −2 × DLL(¯ρ) gives the likelihood ratio test statistic, LLR(¯ρ), as follows: ¯ T−1 u−1 Σ−1 ¯ 0) = c¯ 2 T−2 u−1 Σ−1 LLR(γ, z u−1 − 2c z ∆u

(7.51)

¯ 0) lead to Notice that LLR(γ¯ , 0) is equivalently LLR(¯ρ, 1). Small values of LLR(γ, rejection of H0 against c = c¯ . By an application of the Neyman-Pearson lemma, the critical region based on the likelihood ratio is best (see, for example, Harvey, 1990, section 5.3) and the LR test statistic is most powerful for the particular ¯ 0) simpliﬁes to: (local) alternative ρ¯ = 1 + c¯ /T. In the event that Σz = σε2 I, LLR(γ, ¯ 0) = LLR(γ, =

1 2 ¯ ∆u−1 ) (γ¯ u−1 u−1 − 2γu σε2

(7.52a)

1 2 −2 [c¯ T u−1 u−1 − 2c¯ T−1 u−1 ∆u] σε2

(7.52b)

In this case, assuming σε2 is known, there is nothing to estimate as γ¯ = ρ¯ − 1 = c¯ /T is speciﬁed by the local alternative and u−1 and ∆u are known. Notice that the ‘error’ term in (7.44) is zt not εt , as previously, so there may be further dependence in the error structure; for example, zt may be generated by an ARMA process and the framework is easily extended to more general error processes. Suppose that {zt }Tt=1 is a dependent sequence, an obvious case

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¯ u) = − CLL(γ,

282 Unit Root Tests in Time Series

being where an ARMA process generates the errors. To illustrate, suppose that zt = (1 + θ1 L)εt then: yt = µt + ut

(7.53a)

(1 − ρL)ut = zt

(7.53b)

zt = (1 + θ1 L)εt

(7.53c)

u1 = z1

Bearing in mind the initial condition, the T × T variance-covariance matrix for z = (ε1 , ε2 + θ1 ε1 , . . . , εT + θ1 εT−1 ) is E(Θεε Θ ) = σε2 ΘΘ = Σz , where Θ is a lower band-diagonal matrix, with the unit element on the diagonal and θ1 on the ﬁrst off-diagonal, see Equation (7.17). The likelihood framework is then given by (7.48) to (7.51). ERS (1996, Theorem 1) show that the desirable properties of the LR test are not affected by (unknown) slowly evolving deterministic components; for example, µt = ln(t) and µt = tδ for δ < 1/2. The case where µt = µ or µt = β0 + β1 t with unknown coefﬁcients is dealt with explicitly below. 7.4.1.ii Unknown deterministic components The important development in this section is the ﬁrst step to relax the assumption that the coefﬁcients on the deterministic components are known. By a simple rearrangement of Equations (7.43) and (7.44), and setting ρ = ρ¯ , the model can be written as: y1 = µ1 + u1 (1 − ρ¯ L)yt = (1 − ρ¯ L)µt + zt

t=1

(7.54)

t = 2, ..., T

= (µt − ρ¯ µt−1 ) + zt

(7.55)

and u1 = z1 . It is then convenient to collect all T observations into a matrix representation. Y1 is deﬁned as before, as is µt ; speciﬁcally, let µt = λ xt , where λ = µ and xt = 1 or λ = (β0 , β1 ) and xt = (1, t) . Then: Yρ¯ ≡ Pρ¯ Y1 = (y1 , y2 − ρ¯ y1 , y3 − ρ¯ y2 , . . . , yT − ρ¯ yT−1 )   1 0 0 0 0  −¯ρ 1 0 0 0      0 −¯ρ 1 0 0  Pρ¯ ≡  T×T   .. .. .. ..     . . . . 0 0 . . . 0 −¯ρ 1

(7.56)

T×1

Xρ¯ ≡ Pρ¯ X

(7.57)

(7.58)

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(7.53d)

Improving the Power of Unit Root Tests

283

Where Xρ¯ is T × 1 if xt = 1 or T × 2 if xt = (1, t) , then, respectively, X is as follows: X = (1, . . . , 1)

1 1 ... X= 1 2 ...

1 t

... ...

1 T

Yρ¯ = Xρ¯ λ + ω

(7.59)

This approach has been described as ‘GLS’ detrending as λ = (β0 , β1 ) is the vector of trend coefﬁcients; thus, if, for example, λ˜ ρ¯ is a consistent LS estimator of λ from (7.59), then y˜ ρ¯ ,t ≡ yt – λ˜ ρ¯ xt is the ‘GLS’ detrended data for ρ = ρ¯ . When just a constant is estimated, the data are usually demeaned by a regression of Y1 on X = (1, . . . , 1) , which is the special case of (7.59) with ρ¯ = 0; however, under the local-to-unity alternative, ρ¯ is close to unity rather than close to or equal to zero, and taking this into account involves multiplying the equation through by Pρ¯ , which is akin to a GLS-type procedure. However, this transformation does not necessarily result in a scalar diagonal variance-covariance matrix in the estimating equation; hence ‘full’ GLS has not been applied and as a result some have preferred to describe this approach as quasi-differencing the data prior to regression. For example, apart from the ﬁrst observation, the typical element of Yρ¯ is yt − ρ¯ yt−1 , which is a ‘quasi-difference’ of yt using ρ¯ , rather than unity. A more extensive analysis of this approach is undertaken in the section 7.4.5.

7.4.1.iii A likelihood-based test statistic, with estimated trend coefﬁcients Given ρ¯ , the data can be constructed as in (7.56) and (7.58), so that the loglikelihood function analogous to (7.48), but without the assumption that the coefﬁcients on the deterministic components are known, is: CLL(λ, Σz |¯ρ) = −

T 1 1 ln 2π − ln |Σz | − [(Yρ¯ − Xρ¯ λ) Σ−1 z (Yρ¯ − Xρ¯ λ)] 2 2 2

(7.60)

where maximisation is over λ and Σz . The conditioning on ρ¯ = 1 + c¯ /T is equivalently a conditioning on c¯ and so a family of CLL functions is deﬁned indexed by c¯ . The resulting ML estimators, conditional on ρ¯ , are denoted λ˜ ρ¯ and Σ˜ z,¯ρ , with residual vector ω ˜ ρ¯ = Yρ¯ – Xρ¯ λ˜ ρ¯ .

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In this notation, the equation of interest, to estimate the trend coefﬁcients, λ conditional on ρ¯ is:

284 Unit Root Tests in Time Series

The unit root hypothesis is ρ = 1, with the consequence that Yρ¯ and Xρ¯ reduce accordingly to Y(1) and X(1) as follows: Y(1) ≡ P(1) Y1 = (y1 , y2 − y1 , y3 − y2 , . . . , yT − yT−1 ) X(1) ≡ P(1) X

T×1 T × 1 or T × 2

(7.61) (7.62)

CLL(λ, Σz |1) = −

1 T 1 ln 2π − ln |Σz | − [(Y(1) − X(1) λ) Σ−1 z (Y(1) − X(1) λ)] 2 2 2

(7.63)

with resulting estimators λ˜ (1) and Σ˜ z,(1) , and residual vector ω ˜ (1) = Y(1) – X(1) λ˜ (1) . Minus twice the difference in log-likelihoods is: ρ) − LL(λ˜ (1) , Σ˜ −1 |1) LLR(¯ρ, 1) = − 2 LL(λ˜ ρ¯ , Σ˜ −1 z,¯ρ |¯ z,(1) ω ˜ −ω ˜ ρ¯ Σ˜ −1 ˜ ρ¯ + ln |Σ˜ −1 | − ln |Σ˜ −1 =ω ˜ (1) Σ˜ −1 z,¯ρ ω z,¯ρ | z,(1) (1) z,(1)

(7.64)

where H0 is rejected for small values of LLR(¯ρ, 1). It may be helpful to consider the simpler case with Σz = σε2 I, where σε2 is unknown, then the log-likelihood functions simplify as follows: CLL(λ, σε2 |¯ρ) = −

T T 1 ln 2π − ln σε2 − 2 [(Yρ¯ − Xρ¯ λ) (Yρ¯ − Xρ¯ λ)] 2 2 2σε

(7.65)

with ML estimators given by: λ˜ ρ¯ = (Xρ¯ Xρ¯ )−1 Xρ¯ Yρ¯

(7.66)

2 −1 ˜ ρ¯ ω ˜ ρ¯ σ˜ ε,¯ ρ =T ω

(7.67)

2 only differs The ML estimator λ˜ ρ¯ is also the LS estimator applied to (7.59) and σ˜ ε,¯ ρ from the LS estimator in not making a correction for degrees of freedom; but note that λ˜ ρ¯ implicitly uses y1 (see Equation (7.56)) so it is not the conditional LS estimator. 2 into CLL(λ, σ2 |¯ Substituting λ˜ ρ¯ and σ˜ ε,¯ ε ρ) results in: ρ 2 CLL(λ˜ ρ¯ , σ˜ ε,¯ ρ) = − ρ |¯

=−

T T 1 2 ln 2π − ln σ˜ ε,¯ ω ˜ ρ¯ ω ˜ ρ¯ ρ− 2 2 2 2˜σε,¯ ρ

(7.68)

T T T 2 ln 2π − ln σ˜ ε,¯ ρ− 2 2 2

(7.69)

2 . Similarly: where the second line uses ω ˜ ρ¯ ω ˜ ρ¯ = T˜σε,¯ ρ 2 ˜ =− CLL(λ˜ ρ¯ , σ˜ ε,¯ ρ = 1) = CLL(λ|1) ρ |¯

T T T 2 ln 2π − ln σ˜ ε,(1) − 2 2 2

(7.70)

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The resulting log-likelihood function is:

Improving the Power of Unit Root Tests

285

Hence, minus twice the difference in log-likelihoods is: 2 ˜ σ˜ ε2 |¯ρ) = T(ln σ˜ 2 − ln σ˜ ε,¯ LLR(λ, ρ) ε,(1)

Taking the exponential of LLR results in:

2 T σ˜ ε,(1) 2 ˜ LR(λ, σ˜ ε |¯ρ) = 2 σ˜ ε,¯ ρ

(7.71)

(7.72)

ST =

2 −σ 2 T(˜σε,¯ ˜ ε,(1) ) ρ

=T

2 σ˜ ε,(1) 2 σ˜ ε,¯ ρ 2 σ˜ ε,(1)

(7.73)

−1

7.4.1.iv The power envelope In the local-to-unity framework, with γ¯ = (¯ρ − 1) = c¯ /T, there is a test statistic for each value of c¯ , with the critical values and power depending on c¯ . In such a case there is not a single UMP test but a ‘family’ of Neyman-Pearson tests indexed by c˜ ; thus there is a power function for each value of c˜ . The notation for this function is π(c, c¯ ). Since each test is UMP at the point c˜ = c, the outermost limit of power is given by the function π(c, c¯ = c), say π(c, c); this function ‘picks’ off the points at which each test is most powerful and deﬁnes the power envelope. Thus π(c, c¯ ) ≤ π(c, c) with equality holding at c¯ = c; graphically, this implies that the power function π(c, c¯ ) is tangent to the envelope power function, π(c, c¯ = c). The power envelope is a general concept in the sense that it can be constructed for any unit root test that has a dependence on a point alternative indexed by c¯ . It is also the case that for some tests π(c, c¯ ), with c˜ = c, is close to π(c, c¯ = c) for a reasonably wide range of values of c¯ , which is a desirable property of a test designed to have power for alternatives close to the unit root; it means that the test constructed for c¯ can be used for c¯ = c, and the practical problem is to choose a value for c¯ . The choice of the particular value of c¯ , denoted c˜ below, is taken up in the next section. ERS (1996) show that the asymptotic (size α) power function for the family of LR tests indexed by c¯ is given by: 1 π(c, c¯ ) = P c¯ 2 B2c (r)dr−c¯ B2c (1) < b(c¯ ) (7.74) 0

where Bc (r) = ot ec(t−r) dB(r),Bc (0) = 0, B(r) is standard Brownian motion and 1

P c¯ 2

0

B20 (r)dr−c¯ B20 (1) < b(c¯ ) = α

(7.75)

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An alternative form of the test statistic is:

286 Unit Root Tests in Time Series

7.4.2 A family of simple-to-compute test statistics The second contribution of ERS, noted earlier, was to suggest a family of test statistics that are simpler to compute than the LR tests. In the case of weakly dependent errors, assuming that the errors were generated by an ARMA process, the approach taken in Chapter 3 was to maximise the log-likelihood function directly or, as in this chapter, obtain the GLS estimator – in both cases taking into account the error structure. However, it is possible to construct a feasible test that is asymptotically equivalent to the LR test. This is referred to as the PT test, constructed as follows: PT (c¯ ) =

S(¯ρ) − ρ¯ S(1) 2 σ˜ z,lr

(7.76)

where the respective residual sums of squares are: S(¯ρ) = ∑Tt=1 ω ˜ 2t,¯ρ and T 2 S(1) = ∑t=1 ω ˜ t,(1) . The former residuals are deﬁned from LS estimation of (7.59) and the latter by using Y(1) and X(1) in an analogous regression. The denomi2 is a consistent estimator of the long-run variance of z (see nator quantity σ˜ z,lr t section 7.4.2.ii). Rejection occurs for small values of PT (c¯ ); that is, it is a lefttailed test, with critical values provided below (see Table 7.5, p. 295). When it is necessary to distinguish the deterministic terms, the test statistics will be denoted PT,µ (c¯ ) and PT,β (c¯ ). The asymptotic theory for PT (c¯ ) is valid under weaker assumptions than so far assumed; speciﬁcally (ERS, 1996, Condition C), it is just required that u0 has a bounded second moment (that is, it does not have to equal zero) for ρ in the neighbourhood of unity; {zt } is a stationary, ergodic sequence with [rT] (summable) ﬁnite autocovariances and T−1/2 ∑t=1 zt /σz,lr ⇒ B(r). The test statis2 in addition to the two residual tic PT is simple to compute, only requiring σ˜ z,lr 2 have already been discussed (see, sums of squares; several possibilities for σ˜ z,lr for example, Chapter 6, section 6.8.2). The limiting distributions of PT (c¯ ) for the zero mean, constant mean and linear trend cases are given by ERS (1996, Theorem 2):

PT (c¯ ), PT,µ (c¯ ) ⇒D c¯ 2 PT,β (c¯ ) ⇒D c¯ 2

1 0

1 0

B2c (r)dr − c¯ B2c (1)

(7.77)

V2c (r, c¯ )dr − (1 − c¯ )V2c (1, c¯ )

(7.78)

r Vc (r) = Bc (r) − r ςBc (1) + 3(1 − ς) sBc (s)ds 0

ς = [(1 − c¯ )/(1 − c¯ + c¯ 2 /3)]

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7.4.2.i The PT tests

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287

Choosing c˜ , the representative value of c¯ Note that there is a test statistic for each value of ρ¯ or, effectively, c¯ , so that there is a family of tests indexed by c¯ . The question arose earlier as to choosing c¯ for the purposes of a single test statistic rather than a multiplicity of such tests (that is, one for every possible value of c¯ ). ERS suggest choosing a representative value of c¯ , denoted here as c˜ , where the power function π(c, c¯ = c˜ ) takes a value of 0.5; this is usually a good choice in the sense that there is not too much curvature in the power function for c˜ = c at this value. The corresponding value of ρ is denoted ρ˜ = 1 + c˜ /T. The suggested values of c˜ are c˜ = 7.0 for the demeaned case, xt = 1, and c˜ = –13.5 for the detrended case, xt = (1, t) . This dependence will be explicitly acknowledged in the notation PT (c˜ ) for the test statistic; other notation will also reﬂect this choice; for example, Equation (7.59) becomes Yρ˜ = Xρ˜ λρ˜ + ωρ˜ .

2 7.4.2.ii Estimating σz,lr 2 . For example, in the simple ERS (1996) use an autoregressive estimator of σz,lr case of Equations (7.43) and (7.44), where by assumption zt = εt , then there is no additional structure in zt and the residuals can be obtained from a LS regression of the usual form; that is:

y˜ t = ρˆ y˜ t−1 + zˆ t

(7.79a)

˜t y˜ t = yt − µ

(7.79b)

σˆ z2 = (T − 1)−1 ∑

T zˆ 2 t=2 t

2 = σˆ z2 σz,lr

(7.79c) (7.79d)

˜ t is a consistent estimator of the deterministic component and ρˆ is the where µ ˜t = µ ˆ t (as the deterusual LS estimator. In effect ERS and Elliott (1999) use µ ministic components are included in the regression directly); that is, the LS estimator of the trend components. However, this breaks the link with the GLS detrended (QD) data. Ng and Perron (2001) use the QD data, so that detrended yt is obtained as: y˜ ρ˜ ,t = yt − λˆ ρ˜ xt

(7.80)

where λˆ ρ˜ is the LS estimator of the trend coefﬁcients from the regression of Yρ˜ on Xρ˜ ; the y˜ ρ˜ ,t are then used instead of y˜ t in (7.79a).

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The limiting distributions are the same for the no mean and constant mean cases – see (7.77).

288 Unit Root Tests in Time Series

2 is obtained from the ADF extension Generally, zt = εt and an estimator of σz,lr of (7.79a), that is:

∆y˜ t = γˆ y˜ t−1 + ∑i=1 cˆ i ∆y˜ t−i + zˆ t,k k−1

(7.81a) (7.81b)

2 ˆ (1 − C(1))

σˆ z2 = (T − k)−1 ∑t = k + 1 zˆ 2t,k T

(7.81c)

If the QD data are used, then y˜ ρ˜ ,t replaces y˜ t . Ng and Perron (2001) found that using QD data tended to improve the properties of the test statistics.

7.4.3 GLS detrending and conditional and unconditional tests ERS (1996) note that their development relates to the case with a ﬁxed initial condition u0 = 0 (or bounded second moment for u0 ), which implies u1 = z1 , under both the null and alternative hypotheses. A different assumption is that under HA , u1 is drawn from its unconditional distribution. Given |ρ| < 1, the variance of the unconditional distribution of ut is σu2 = σz2 /(1 − ρ2 ), where σz2 is the variance of zt ; this violates Condition C of ERS (1996) because σu2 → ∞ as ρ → 1. In this case the LR-based tests of ERS are no longer point-optimal and the power of the tests is affected. Elliott (1999, Assumption A) considers the alternative assumption as follows: u0 = 0 ⇒ u1 = z1 when ρ = 1, but u1 ∼ (0, σu2 ) when ρ < 1; this will be referred to as the unconditional initial assumption. This is usually the preferred assumption (see, for example, Pantula et al. 1994), as it implies that under HA , the initial observation is part of the same (stationary) process generating the remainder of the observations and, in particular, it has the same unconditional variance. This problem is familiar in a GLS context when an AR(1) process is generating the errors (see, for example, Greene, 2006), and the solution is to deﬁne a transformation matrix that restores the regression model to one with a scalar-diagonal variance-covariance matrix, taking the initial observation into account. First, consider the case with the ﬁxed initial condition assumption, and, for simplicity, initially assume that zt = εt and µt = µ. Then the complete generating process is: u1 = z1

t=1

(7.82a)

yt = µ + u t

t = 1, . . . , T

(7.82b)

ut = ρut−1 + zt t = 2, . . . , T

(7.82c)

z t = εt

(7.82d)

t = 1, . . . , T

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2 σˆ z,lr =

σˆ z2

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289

Collecting observations, the error process is:       



u1 u2 u3 .. . uT



1 ρ ρ2 .. .

      =      

ρT−1

0 1 ρ .. . ...

0 0 1 .. . ρ2

... ... ... .. . ρ

0 0 0 .. . 1

       

z1 z2 z3 .. . zT

       

(7.83)

The structure of Equation (7.83) is just as the earlier (full) GLS case in the AR(1) example in section 7.1.2.i (see Equation (7.13)), with ρ here replacing ϕ1 . Thus, to parallel the previous notation, write Equation (7.83) as follows: u = Φ−1 z = Φ−1 ε

assuming that z = ε

(7.84)

so that Φ−1 is the matrix premultiplying (z1 , . . . , zT ) in Equation (7.84). The transformation matrix associated with Φ−1 , such that ΦΦ−1 = I, is:     Φ=    

1 −ρ 0 .. . 0

0 1 −ρ .. . ...

0 0 1 .. . 0

0 0 0 .. . −ρ

0 0 0



      0  1

On reference to (7.59) it is evident that the matrix Pρ¯ that serves to transform the data, that is, Yρ¯ ≡ Pρ¯ Y1 and Xρ¯ ≡ Pρ¯ X, is just Φ evaluated at ρ = ρ¯ . Now suppose that instead of u1 = z1 = ε1 , which implies var(u1 ) = σε2 , u1 is a draw from (0, σε2 /(1 − ρ2 )). The transformation matrix that is now required will be denoted ΦU to distinguish it from Φ, the subscript indicating the draw of u1 from the unconditional distribution. The ﬁrst element of ΦU is (1 − ρ2 )1/2 ; that is:       ΦU =     

(1 − ρ2 )1/2 −ρ 0 .. . 0 0

0 1 −ρ .. . 0 0

0 0 1 .. . 0 0

0 0 0 .. . ... ...

0 0 0 .. . 1 −ρ

0 0 0 .. . 0 1

          

In this case, the transformation matrix – that is the analogue of Pρ˜ – will be denoted PU,˜ρ and is ΦU evaluated at ρ = ρ˜ . The resulting transformed data

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

290 Unit Root Tests in Time Series

matrices, analogous to (7.56) and (7.58), are: YU,¯ρ ≡ PU,¯ρ Y1

(7.85)

= ((1 − ρ¯ 2 )1/2 y1 , y2 − ρ¯ y1 , y3 − ρ¯ y2 , . . . , yT − ρ¯ yT−1 ) XU,¯ρ ≡ PU,¯ρ X

(7.86)

Only the ﬁrst element in each case differs from Yρ¯ and Xρ¯ , respectively. If ρ˜ = 1 + c˜ /T replaces ρ¯ (as it does in practice), then this is reﬂected in the notation; for example, YU,˜ρ for (7.85) and XU,˜ρ for (7.86). The ERS method, whichever assumption about u1 is chosen, has been referred to as GLS detrending. However, the analogy is only perfect if zt = εt ; in the event that zt = εt , then is still some structure in the error process that has not been removed, which is why the detrending has also been referred to as quasidifferencing (QD), since the transformation matrix only removes part of the serial correlation structure. 7.4.4 Unconditional case: the QT family of test statistics The test statistic suggested by Elliott (1999) for the ‘unconditional’ case, which parallels the PT statistic in the conditional case, is referred to as the QT statistic. The procedure is exactly as in the case of the PT test. In summary, regress YU,¯ρ on XU,¯ρ to obtain the residual sum of squares, such that:

SU (¯ρ) = ω ˜ U,¯ρ ω ˜ U,¯ρ = ∑t=1 ω ˜ 2U,t,¯ρ T

ω ˜ U,˜ρ = YU,¯ρ − XU,¯ρ λ˜ U,¯ρ λ˜ U,¯ρ = (XU,¯ρ XU,¯ρ )−1 XU,¯ρ YU,¯ρ

Analogous quantities are deﬁned for the case restricted by the null hypothesis, ρ = 1, but note that the residual sum of squares under the null is unchanged (see the comment following Equation (7.76)). Thus the QT test statistic is: QT (c¯ ) =

SU (¯ρ) − ρ¯ S(1) 2 σ˜ z,lr

(7.87)

Where, as before, ρ¯ = 1 + c¯ /T. As in the case of PT (c¯ ) tests, the notation QT,µ (c¯ ) and QT,β (c¯ ) distinguishes the value of c¯ used in QT (.) and the different trend functions. There is in principle one test statistic of the form (7.87) for each value of c¯ ; but, in practice, a ‘representative’ value of c¯ denoted c˜ is chosen and it is this value that is used in computing the QD data, so that the test statistic is referred to as QT (c˜ ). Note also that ρ˜ = 1 + c˜ /T is the parameter value for the

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= ((1 − ρ¯ 2 )1/2 x1 , x2 − ρ¯ x1 , x3 − ρ¯ x2 , . . . , xT − ρ¯ xT−1 )

Improving the Power of Unit Root Tests

291

7.4.5 Exploiting the quasi-differencing interpretation An interesting interpretation of the procedures leading to the PT and QT tests suggests variations in other unit root tests. For example, the set of DF unit root tests depends on the deterministic components included in the maintained regression or, alternatively, the data are ﬁrst regressed on the deterministic components and the residuals from this prior regression are then used in the DF regression. By viewing the ERS detrending procedure as quasi-differencing the data, the DF tests are easily modiﬁed. The properties of the estimator of the mean or trend are not generally the focus of unit root tests; however, not only is the good design of unit root tests important, there should also be good design for trend estimation, which can be viewed as subject to the ‘nuisance’ of possibly serially correlated errors. The prior detrending (or just demeaning) can be justiﬁed by reference to the Grenander-Rosenblatt (GR) (1957) theorem, which shows that OLS detrending is asymptotically efﬁcient. In this context it can then be seen that the ERS procedure approaches the problem of detrending from a different perspective. Example Consider the case where the alternative to the unit root null is the following trend stationary process: (yt − µt ) = ut (1 − ρL)ut = zt zt = εt

|ρ| < 1

t = 1, ..., T

(7.88a)

t = 2, ..., T

(7.88b)

t = 1, . . . , T

(7.88c)

u1 = z1

(7.88d)

⇒ ut = (1 − ρL)−1 εt

(7.88e)

In a standard context, the data can be detrended by a prior regression of yt on the components of µt , for example, a constant or a constant and a time trend, to obtain the detrended series (residuals), y˜ t ; then the DF regression with ∆y˜ t as the dependent variable is estimated, but without a constant or a time trend.

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QD data. As in the case of the PT (.) tests, rejection occurs for small values of the sample test statistic, QT (.) (see Table 7.6 for some critical values). Elliott (1999) shows that c˜ = –10 is a good choice in the sense that the power function is tangential to the power envelope at 50% power for both the QT,µ (c˜ ) and QT,β (c˜ ) versions of the test. Thus QT,µ (−10) and QT,β (−10) are the notations for the QT -type test statistic of Equation (7.87) calculated using c˜ = –10, and hence ρ˜ = 1 − 10/T. The limiting distributions of QT,µ and QT,β are somewhat more complex than their PT counterparts, and the reader is referred to Elliott (1999, Theorems 1 and 2 and Lemmas 1 and 2) for the detail.

Evidently, as shown by (7.88b) and (7.88e), the errors can be viewed as being generated by an AR(1) process or by an inﬁnite MA process if |ρ| < 1. The error structure does not therefore satisfy the ‘standard’ assumptions of least squares theory, and an alternative estimation method is GLS, which is asymptotically efﬁcient but infeasible because ρ is not known. What justiﬁes the use of OLS rather than GLS is the GR theorem that OLS and GLS estimators of the trend parameters are asymptotically equivalent. It is also possible to construct several feasible GLS (FGLS) estimators that share this asymptotic equivalence, at least in the I(0) case. The precise construction of a FGLS estimator depends on the initial condition assumption, which for simplicity is assumed to be the ﬁxed initial condition. Consider the following set-up: y1 = µ1 + u1

t=1

(1 − ρ˜ L)yt = (1 − ρ˜ L)µt + zt

(7.89a)

t = 2, ..., T

= (µt − ρ˜ µt−1 ) + zt

(7.89b)

where µt = β0 + β1 t. Then the data for the QD detrending regression model are: Yρ˜ = (y1 , y2 − ρ˜ y1 , y3 − ρ˜ y2 , . . . , yT − ρ˜ yT−1 )

(7.90a)

Xρ˜ ,1 = (1, 1 − ρ˜ , 1 − ρ˜ , . . . , 1 − ρ˜ )

(7.90b)

Xρ˜ ,2 = (1, 2 − ρ˜ , . . . , t − ρ˜ (t − 1), . . . , T − ρ˜ (T − 1)) Xρ˜ = Xρ˜ ,1 Xρ˜ ,2

(7.90c) (7.90d)

Apart from the ﬁrst observation, the data are quasi-differenced: the typical observation on the dependent variable is yt − ρ˜ yt−1 ; the typical observation on the ﬁrst explanatory variable is (1 − ρ˜ ); and the typical observation on the second explanatory variable is t − ρ˜ (t − 1). Thus the detrending regression model Yρ˜ = Xρ˜ λ + ω is, apart from the ﬁrst observation, a regression using QD data; and the estimator of λ = (β0 , β1 ) obtained from (7.59), is the LS estimator, λ˜ ρ˜ , of the trend coefﬁcients in this model, where λ˜ ρ˜ = (β˜ 0,˜ρ , β˜ 1,˜ρ ) , say. The trend coefﬁcients, λ˜ ρ˜ , are then used to deﬁne the ‘efﬁciently’ detrended – that is, QD detrended – data, y˜ ρ˜ ,t , for DF or ADF regressions of the standard form, where: y˜ ρ˜ ,t = yt − (β˜ 0,˜ρ + β˜ 1,˜ρ t)

t = 1, . . . , T

(7.91)

The QD detrended data is conditional on ρ = ρ˜ and it is used in place of the LS detrended data y˜ t in the usual DF/ADF regression model and associated test statistics. Setting ρ˜ = 0 gives the LS detrended data. In the case that the assumption about the initial condition is that it is a draw from the unconditional distribution, then the ﬁrst observation in the data vector for the dependent variable is replaced by (1 − ρ˜ 2 )1/2 y1 and the ﬁrst observation in each of the data

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Improving the Power of Unit Root Tests

293

vectors for the explanatory variables is replaced by (1 − ρ˜ 2 )1/2 (see Equations (7.85) and (7.86), and see Q7.6 for a summary of the procedure). In the event that zt = εt , one can apply the usual procedures to ‘whiten’ the errors (equally, one of the GLS options described in this chapter could be used), with the same caveats about choosing an appropriate rate of expansion for the ADF lag length. The ADF regression using QD data is: k−1

(7.92)

Note that, as with the usual detrending, the regression model (7.92) does not include a constant or a trend. The resulting test statistics are often referred as GLS versions of the DF test statistics, although, as noted, this is not an entirely accurate description. A superscript glsc or glsu for the conditional and unconditional initial assumptions, respectively, will serve to distinguish the glsc glsu two versions; for example, τˆ β and τˆ β are the conditional and unconditional counterparts to τˆ β . Some critical values for different values of c˜ are provided in ˆ Tables 7.5 and 7.6. The analogous δ-type tests are considered in the next section. 7.4.6 ADF tests using QD data Xiao and Phillips (1998) have obtained the asymptotic null distributions for the ˆ ADF δ-type tests applied to QD data and, by extension, the τˆ -type test. Xiao and Phillips show that:

1¯ Bc (r)dB(r) σ˜ z 0 Tγˆ ⇒D (7.93) 1 σ˜ z,lr B¯ c (r)2 dr 0

where γˆ is obtained from LS estimation of (7.92). B¯ c (r) is given by:

1 − c¯ 01 Xc (r)B(r)dr ¯Bc (r) = B(r) − X(r) 0 Xc (r)dB(r) 1 0 Xc (r)Xc (r)dr

(7.94)

where X(r) = (1, r) , Xc (r) = (−c¯ , 1 − c¯ r) and c¯ = 0 if the trend function includes a constant (see Xiao and Phillips, 1998, section 3). (Note that Xiao and Phillips use the notation W(r), not B(r), for standard BM.) σ˜ z,lr and σ˜ z are consistent estimators of σz,lr and σz , respectively, which can be obtained from (7.81b) and (7.81c), (and see also Chapter 6, section 6.3.6); that is, using data that is not quasi-differenced or, analogously, using data that is quasi-differenced from LS estimation of (7.92). As in section 6.3.6, multiplying (7.93) through by the inverse sigma ratio results in:

1¯ σ˜ 0 Bc (r)dB(r) ˆδglsc ≡ Tγˆ z,lr ⇒D ˆ DF i = µ, β = F(δ) (7.95) 1 i 2 ¯ σ˜ z 0 Bc (r) dr

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∆y˜ ρ˜ ,t = γy˜ ρ˜ ,t−1 + ∑j=1 cj ∆y˜ ρ˜ ,t−j + εt,k

294 Unit Root Tests in Time Series

ˆ is the estimated standard error of γˆ from (7.92). where σ˜ (γ) The versions of these tests based on the unconditional distribution assumpglsu glsu tion are δˆ i and τˆ i . These have limiting distributions of the same form as (7.95) and (7.96), differing only in the underlying projections, since the detrended or demeaned data on which they are based is deﬁned differently for the ﬁrst observation (see Equations (7.85) and (7.86)). For more on the limiting distributions of these and related test statistics, see Elliott (1999, p.773) and M¨ uller and Elliott (2003, especially Table I). The limiting distributions depend on the speciﬁcation of the trend function and the value of c¯ ; the former introduces no new issues, and the second is an issue already encountered. In fact, as noted above, the size-adjusted power is not very sensitive to c˜ , the (single) selected value of c¯ ; for example, Xiao and Phillips (1998) illustrate this point for c˜ between –2.5 and –7.5 for the constant only case and for c˜ between –10 and –15 for the trended alternative. 7.4.7 Critical values This section comprises two tables of critical values for the test statistics outlined earlier. Table 7.5 is relevant for the ﬁxed initial condition (or bounded second moment), and Table 7.6 for the case that the initial condition is a draw from the unconditional distribution (with unbounded second moment as ρ → 1). 7.4.8 Illustration of a power envelope and tangential power functions 7.4.8.i Graphical illustrations This section gives some graphical illustrations of the concepts introduced in the glsc previous sections. Figure 7.2 shows the 5% critical values PT,µ , τˆ µ and τˆ µ , for T = 100, as c varies. Note that there is a substantial variation in the critical values glsc for PT,µ and τˆ µ ; however, the standard DF test, τˆ µ , is virtually invariant to c. Figure 7.3 shows the power envelope, π(c, c), for PT,µ (c), for the case with T = 100; also shown are two of the tangential power functions π(c, c˜ ) from which the power envelope is constructed; these are for c˜ = –7 in the left-hand panel and c˜ = –10 in the right-hand panel, which correspond to ρ˜ = 0.93 and 0.90, respectively. Note that the power functions π(c, −7) and π(c, −10) are throughout very close to the power envelope. Also note that in the former case, the function π(c, −7) is tangential to π(c, c) at power of 50%, which illustrates the choice of c˜ = –7 for construction of tests based on the ERS principle, although

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It can be inferred from these results that the τˆ -type test based on (7.92) has the following limiting distribution:   1 ¯ Bc (r)dB(r)  γˆ  glsc τˆ i ≡ ⇒D  0 (7.96) 1/2  = F(ˆτ)DF i = µ, β 1 ˆ σ˜ (γ) ¯ c (r)2 dr B 0

Improving the Power of Unit Root Tests

295

PT,µ (−7) 100 200 ∞ glsc τˆ µ (−7) 100 200 ∞ PT,β (−13.5) 100 200 ∞ glsc τˆ β (−13.5) 100 200 ∞

1%

5%

10%

1.95 1.91 1.99

3.11 3.17 3.26

4.17 4.33 4.48

–2.77 –2.69 –2.55

–2.14 –2.08 –1.95

–1.86 –1.76 –1.63

4.26 4.05 3.96

5.64 5.66 5.62

6.79 6.86 6.89

–3.58 –3.46 –3.48

–3.03 –2.93 –2.89

–2.74 –2.64 –2.57

Note: Reject H0 : ρ = 1 if the sample value lies to the left of the 100α% critical value.

Table 7.6 Critical values (initial condition: drawn from unconditional distribution).

QT,µ (−10) 100 200 ∞ glsu τˆ µ (−10) 100 200 ∞ QT,β (−10) 100 200 ∞ glsu τˆ β (−10) 100 200 ∞

1%

5%

10%

3.25 3.11 3.06

4.88 4.66 4.65

6.17 5.97 5.93

–3.59 –3.44 –3.26

–2.78 –2.79 –2.73

–2.47 –2.47 –2.46

2.21 2.14 2.11

2.91 2.91 2.86

3.50 3.47 3.46

–3.77 –3.78 –3.69

–3.23 –3.20 -3.17

–2.97 –2.94 –2.91

Note: Reject H0 : ρ = 1 if the sample value lies to the left of the 100α% critical value. Sources: ERS (1996, Table I), Elliott (1999, Table 1) and own calculations based on 100,000 replications.

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Table 7.5 Critical values (ﬁxed initial condition).

296 Unit Root Tests in Time Series

–1.6

20 18

–1.8 16 –2

14

τˆ glsc µ

PT,µ(c)

12

–2.2 5% cv –2.4

8 6

–2.6

τˆµ

4 –2.8 2 0 0

5

10 –c

15

20

–3

0

5

10 –c

15

20

Figure 7.2 Variation in 5% critical values for test statistics.

1

1 power envelope

power envelope

0.9

0.9

0.8

0.8

0.7

power function close to power envelope

0.7

50% power

0.6

0.6

power 0.5

power 0.5

0.4

0.4 point of tangency

0.3

0.3

0.2

~ c = c = –10

0.2 ~ c = c = –7

0.1 0

0.1 0

5

10 –c

15

20

0

0

5

10 –c

15

20

Figure 7.3 Illustration of power envelope for ρT,µ (c).

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5% cv 10

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297

1 0.9 τˆ glsc µ (–7)

0.8 0.7

PT,µ(–7)

power envelopes virtually the same

0.6

0.4

τˆµ

0.3 0.2 0.1 0

0

2

4

6

8

10 –c

12

14

16

18

20

Figure 7.4 Comparison of power envelopes.

the tangency suggests that this value should not be too critical. Figure 7.4 conglsc tinues the comparison and shows that the power envelopes for PT,µ and τˆ µ are virtually indistinguishable, and well above that for τˆ µ . 7.4.8.ii Power gains? The question of what are the power gains from using QD techniques can be addressed on two levels. First, as far as the conditional tests are concerned, ERS glsc show that the asymptotic power functions for PT,µ and τˆ i , i = µ, β are indistinguishable, and very much better than those for their standard DF counterparts – a point illustrated in Figure 7.4. On this basis, the latter would not be used. In practical situations, the lag length has to be determined, and the PT tests 2 . In these circumstances, there is a slight advanrequire an estimator of σz,lr glsc

tage to the τˆ i versions of the test statistics, which are better at preserving size (otherwise power comparisons are generally not very helpful). As is usually the case, error structures that exhibit some weak dependence, especially MA processes with a near unit root, can cause problems with size preservation, but these problems can largely be overcome if the modiﬁed AIC of Ng and Perron (2001) (MAIC), is used as the lag selection criterion, rather than BIC, AIC or a ﬁxed lag length; see Chapter 9 for an elaboration of lag selection criteria. As far as the unconditional tests are concerned, in terms of large sample power, there is little to choose between the QT tests compared to the weighted symmetric tests that use either QD data or standard OLS detrending.

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power 0.5

298 Unit Root Tests in Time Series

The second level of practical assessment links the issue of the importance of the initial observation to the power of the test; this is a sensitivity that was highlighted by, for example, DeJong et al. (1992), but which has also been the subject of recent research (see, for example, M¨ uller and Elliott, 2003; Elliott and M¨ uller, 2006; Harvey and Leybourne, 2006). This research shows that this is an important practical issue that has a bearing on the selection of a test statistic.

The question thus arises of which of the two initial conditions (ﬁxed or a draw from the unconditional distribution) is more appropriate or if there is another way of dealing with the problem. The ﬁrst point to assess is whether the initial assumption matters. If it does, then it does matter how the transformation matrix deals with the ﬁrst observation. The large sample (T = 1,000) power, reported by Elliott (1999, Table 2) shows that the QT test dominates the PT test when the unconditional distribution is used. However, the simulation evidence for ﬁnite samples does not provide a clear decision in this respect; the PT tests tend to be somewhat more powerful for –10 < c < 0, whereas the QT tests tend to be more powerful for –20 < c < –10. The comparison is complicated by slight differences in size (the PT tests tend to be slightly oversized relative to the QT tests). There is, though, no clear overall dominance. To examine this problem further, it is embedded in a more general framework that nests several possibilities. Let u1 = y1 − µ1 ≡ ξ be a draw from a distribution that is iid(0, (κσu )2 ), with κ ≥ 0 and varying; thus, κ = 0 corresponds to the zero initial condition and κ = 1 to u1 being a draw from the unconditional distribution, with variance σu2 . The idea is to assess the sensitivity of unit root tests to variations in ξ controlled by variations in κ in this set-up. An alternative set-up is to treat ξ as a ﬁxed quantity, rather than a random draw, so that κ can be positive or negative (see, for example, M¨ uller and Elliott, 2003; Harvey and Leybourne, 2005, 2006). The ﬁrst point to note is that, at least under the null, the initial condition acts as a nuisance parameter so that different values of ξ result in simple mean shifts in the data. This implies that the differences can be accommodated by including a constant in the regression model, or prior demeaning of the data, which renders the distribution of a unit root test statistic under the null invariant to the unknown initial condition. (For an analysis of the case when there is no constant is included in the maintained regression and the initial condition varies, see Perron and Vodounou, 2001.) Under the alternative hypothesis, a change in the value of ξ adds the term ∆ξ ρt to yt , which affects the distribution under the alternative and thus the power of a test. To illustrate the sensitivity of some of the unit root tests to the initial condition, Figure 7.5 plots the simulated (size-adjusted) power for PT,µ (−7), τˆ µ and glsc

τˆ µ (−7), when u1 is a draw from N(0, (κσu )2 ), with κ varying, and ρ = 0.95;

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7.4.9 Sensitivity to the initial condition

Improving the Power of Unit Root Tests

299

0.35 with constant; ρ = 0.95 0.3

τˆ glsc µ (–7)

0.25 power

τˆµ

0.15

0.1

0

0.5

1

1.5

2

2.5

κ Figure 7.5 Power as scale of initial observation varies, π(κ|ρ); constant in estimated model.

the case illustrated is for T = 100, so that c = –5; qualitatively similar results were obtained for other sample sizes and for other values of ρ (see Harvey and Leybourne, 2005). The plotted function is generically referred to as π(κ|ρ); that is, power as a function of κ given ρ. A more extensive set of test statistics is considered below. What is very clearly evident from Figure 7.5 is that the initial condition does glsc indeed matter. Note that τˆ µ is slightly more powerful than PT,µ , but they are close to being indistinguishable (conﬁrming ERS’s ﬁndings that their power functions are almost identical), being most powerful when κ = 0 (for which they were designed), with power declining to zero as κ increases. The case when a trend is included is shown in Figure 7.6, for the test statistics τˆ β , PT,β (−13.5) glsc

and τˆ β (−13.5); the same general message is apparent, but, as is now familiar, the inclusion of the trend (which is superﬂuous under the null) reduces power. Similar results were also obtained for the unconditional versions of these tests. There is one constructive point to note from these ﬁgures, which is that the power of the simple DF test does not decrease as κ increases; for example, in Figure 7.5, note that the function π(κ|ρ) for τˆ µ crosses those for PT,µ and τˆ GLSC at µ about κ = 1.5; that is, where the initial observation is drawn from N(0, (1.5σu )2 ). The scale of the implied possible deviations from the long-run mean is quite feasible in practice (as is shown below with an empirical example), and hence the question of which test statistic to use is indeed a problematic issue.

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PT,µ(–7)

0.2

300 Unit Root Tests in Time Series

0.115 with trend; ρ = 0.95

0.11 0.105

τˆglsc β

0.1 PT,β(–13.5)

0.095 power 0.09

τˆβ

0.08 0.075 0.07 0.065

0

0.5

1

1.5

2

2.5

κ Figure 7.6 Power as initial observation varies, π(κ|ρ), trend in estimated model.

7.4.10 A test statistic almost invariant to the initial condition It is evident that the size of the initial condition is critical to the characteristics of a unit root test. The usual case to consider in empirical research is one where the beginning of the available data does not signal the start of the process, but it is simply a practical matter of when records were started. There are some exceptions – for example – when new ﬁnancial instruments are introduced or new organisations are formed; however, generally, a priori, there is likely to be little that is known about the initial condition. Of course, after the (perhaps forced) decision of the sample starting date, the deterministic components can be estimated and an ‘informal’ estimate of the initial condition (for example, the deviation from constant or trend) can be formed; however, despite intuition, basing the choice of a test statistic on the observed u1 is not quite as straightforward as it might seem. The point is that it is quite possible to ‘design’ the outcome of a unit root test by combining a sample point with a test with good or bad power, depending on the desired outcome, for the resulting initial condition. For example, it is known that for small-scale initial conditions, ERS tests have high power, but standard DF τˆ -type tests have low power; hence choosing the former is more likely to lead to rejection. The situation is reversed for large-scale initial conditions, with the DF more likely to lead to rejection. Equally, to maximise the chance of not rejecting the null, a researcher could choose a DF τˆ -type test in the ‘small’ initial condition case, or choose an ERS test in the ‘large’ initial condition case. Also, it is desirable to have tests that are not susceptible to markedly different results

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0.085

301

just because the sample on which they are based happens to start in one period rather than another. As a practical aside, if a process generated under the alternative hypothesis has been in existence for some time before the sample start then it may be quite reasonable to assume that the initial condition is a draw from a distribution much like the unconditional distribution, in which case QT-type tests would be appropriate. At this stage, a brief reminder about the interpretation of the alternative hypothesis is worthwhile: HA always speciﬁes |ρ| < 1, corresponding to an interpretation of mean reversion; if, additionally, u1 is a draw from a distribution with variance (κσu )2 , with κ = 1, then mean reversion is extended to second-order stationarity; so this is the HA appropriate for QT-type tests, whereas PT-type tests just refer to mean reversion. If there is no particular knowledge about the initial condition, two possible solutions to this problem have been suggested. The one considered in the next section is to combine two test statistics with different power characteristics as a function of the initial condition; that is, combine a test that has high power for a ‘small’ initial condition with one that has high power for a ‘large’ initial condition. Of necessity this cannot dominate either test statistic over the range of the initial condition; but, if something was known about the initial condition, then that information, subject to qualiﬁcations, could guide the choice of test statistic. Second, a solution is to maximise a weighted average power function, where the weights reﬂect the possible range of the initial condition; this idea is due to Elliott and M¨ uller (hereafter EM) (2006) and also involves a form of ‘averaging’ (see also M¨ uller, 2009). The idea is to obtain a test that does not sacriﬁce (too much) power for small |ξ |, but does not lose power as |ξ | increases or, better still, is invariant to |ξ |. Unfortunately, a test that is invariant to ξ cannot be obtained for all values of ρ. EM (2006) suggest a test that is invariant to ξ for the point ρ = r, but still depends on ξ for other values of ρ. Assuming Gaussianity for {zt }, the optimal invariant test for the ‘point’ HA : ρ = r, is an infeasible GLS-based test (denoted Pinv in EM, 2006). An alternative is therefore necessary, and the starting point is a family of tests where power varies for members of the family as ξ varies. As the power problem exists for near-unit root alternatives, the local-to-unity framework, in which HA : ρ = 1 + c/T, c < 0, is adopted. It is convenient in this section (especially to enable cross-reference to EM, 2006) to use the notation HA : ρ = 1 − g/T, g > 0, so that g ≡ –c. Then the criterion function is: π(g, ξ ) =

∞ −∞

p(test rejects|ρ = 1 − g/T, ξ = x)dF(x)

This is a weighted power function; the part under the integral is just the power of a test at the point ρ = 1 − g/T, which in turn is a function of g. The integral is then over values of ξ = x, as x varies, so mapping out the range of possible

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Improving the Power of Unit Root Tests

values of ξ . The speciﬁcation of F(x), the cumulative distribution function of x, depends on ξ , which is distributed as (0, (κσu )2 ). In the Gaussian case, F(x) is taken to be the cumulative normal distribution function. Denote the test statistic that maximises π(g, ξ ) as Q (i) (g, f), where i = µ, β, for the mean and trend cases, respectively, and f is a parameter to be speciﬁed. EM (2006, p. 296) show that the power of Q (i) (g, f) converges to zero or unity for a large enough initial condition (strictly, they use an appropriately scaled version of ξ ). The parameter f governs the inﬂuence of ξ on the power of the test, so the (i) idea is to set f = f∗ , say, such that asymptotic power under the local alternative speciﬁed by g does not converge to zero or unity for large initial conditions. EM (i) (2006) show that the appropriate values of f∗ are 3.8 for i = µ and g = 10, and 3.968 for i = β and g = 15; and the values of g relate to those chosen by Elliott (1999) corresponding to the tangent of the power function at 50% power. The feasible test statistic suggested by EM (2006) for i = µ, β, is: (i)

(i)

(i) (i)

(i) (i) ˆ (i) (g, f) = q(i) + q(i) (T−1/2 Q y 1 )2 + q2 (T−1/2 y T )2 + q3 T−1 y 1 y T 0 1

(i)

+ q4

(i)

∑t=2 ( y t−1 )2 T

(7.97)

(i) −1 ˜ (i) ˜ (i) where: y t = σˆ z,lr yt ; yt is the demeaned data for i = µ and the detrended data (µ)

(µ)

(i)

(β)

ˆ where µ ˆ = y¯ = ∑Tt=1 yt /T, and y˜ t is the for i = β, respectively, y˜ t = yt − µ, 2 is a residual from a regression of a constant and a linear time trend; and σˆ z,lr 2 , the long-run variance. The parameters in (7.97) are consistent estimator of σz,lr given by: (µ)

q0 = − g (µ)

q1 = − g(1 + g)(f − 2)/(2 + gf) (µ)

q2 = (2g − gf + g2 f)/(2 + gf) (µ)

q3 = 2g(f − 2)/(2 + gf) (µ)

q 4 = g2 (β)

q0 = − g (β)

q1 = (8g2 + g3 (8 − 3f) − g4 (f − 2))/(24 + 24g + 8g2 + g3 f) (β)

q2 = (8g2 + g3 (8 − 3f) − g4 k)/(24 + 24g + 8g2 + g3 f) (β)

q3 = (8g2 + 2g3 (4 − 3f))/(24 + 24g + 8g2 + g3 f) (β)

q 4 = g2

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Improving the Power of Unit Root Tests

303

ˆ (i) (g, λ). Table 7.7 Critical values for Q ˆ (µ) (10, 3.8) Q 100 200 ∞

1% –7.29 –7.48 –7.70

5% –6.04 –6.20 –6.40

10% –5.03 –5.16 –5.37

ˆ (β) (15, 3.968) Q 1% –10.32 –10.79 –11.24

5% –8.87 –9.33 –9.77

10% –7.83 –8.26 –8.70

f = f(µ) =

4g − 2 + 2e−2g (1 − e−2g )g

(µ)

f∗ = 3.8 for g = 10

f = f(β) =

2 (eg − 1)

−

2(2 + g)2 48 + 24g − 8g2 − 8g3 + 4g4 + (eg (g − 2)2 + g2 − 4) g3 (g − 2)

(β)

f∗ = 3.968 for g = 15 (i)

ˆ (i) (g, f∗ ) are given in Table 7.7. Some critical values for Q EM (2006) note that the local asymptotic power is relatively ﬂat as a function of scaled ξ , especially for alternatives where the power is approximately 50%; this is similarly a feature of the small sample Monte Carlo results, but the compromise nature built into the test construction means that it is dominated by other tests or variants of the same test over different parts of the range of initial conditions. For example, for very large initial conditions, it is better to ˆ (µ) (10, ∞), or τˆ µ , rather than Q ˆ (µ) (10, f∗(µ) ), whereas for small initial condiuse Q (µ) ˆ (10, 1). Some graphical illustrations of the trade-off tions it is better to use Q are given in the following section after considering two weighted test statistics also designed to offer some protection in the case that the scale of initial condition is not known a priori. 7.4.11 Weighted test statistics to achieve robustness to the initial condition Harvey and Leybourne (HL) (2005) construct an estimator that brings together the beneﬁcial characteristics of the QD-based test statistics and the robustness of the simple DF τˆ -type test statistics; and on the same principles, HL (2006) ˆ (i) (g, f) test statistic and the DF τˆ -type consider a weighted average of the EM Q test statistic. What is going to be critical to such an aim is the scale of the initial condition, so it is ﬁrst helpful scale ξ to be in units of σu , so that ξc ≡ (y1 − µ1 )/σu ≡ ξ /σu is the scaled initial condition. It is possible that the data can tell us something

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Sources: Asymptotic critical values from EM (2006, Table 1); other values own calculations, based on 50,000 simulations.

304 Unit Root Tests in Time Series

7.4.11.i Combining test statistics ˆ as a function of |ξˆc |, which The quantity ξˆc can be used to form a weight α, is used to combine two component test statistics with different but desirable characteristics into a linear combination. The design of the weighting function should ensure that as |ξˆc | → ∞, then αˆ → 0, and as |ξˆc | → 0, then αˆ → 1; the weight is therefore in the interval such that αˆ ∈ [0, 1]. HL (2005, 2006) have suggested two versions of a weighted average test statistic. These are: glsc

ˆ τi + αˆ ˆ τi τˆ DFG = (1 − α)ˆ i

i = µ, β

(7.98)

ˆ (i) (g, f)/ω(i) ˆ τi + αˆ Q τˆ DFQ = (1 − α)ˆ i

i = µ, β

(7.99)

where ω(i) is a scale factor to ensure that component statistics have similar variances; HL (2006) use ω(µ) = 2.0 and ω(β) = 2.7, for the mean and trend cases, respectively, which they found to give good overall results in terms of power. As to the weights, the general form is: αˆ = exp(−υ |ξˆc |)

(7.100)

The weighting function has the characteristic that as |ξˆc | → 0, αˆ → 1, so that glsc ˆ (i) (g, f)/ω(i) ; and as |ξˆc | → ∞, αˆ → 0 so that τˆ DFG → τˆ DFG → τˆ i and τˆ DFQ → Q i i i DFQ → τˆ i . On the basis of simulation experiments and obtaining good τˆ i and τˆ i power for small and large values of |ξˆc |, HL (2005, 2006) suggest ν = 0.4 for τˆ DFG i and ν = 0.3 for τˆ DFQ . The asymptotic null distributions of the weighted statistics i are given in HL (2005, 2006). Some critical values are provided in Table 7.8. 7.4.11.ii Power and the initial condition The sensitivity of power to the initial condition is a feature shared by (nearly all of) the more powerful tests considered in this and previous chapters. Figure 7.7 brings together the following nine statistics, split into two groups (as distinguishing more than ﬁve lines on a single graph is difﬁcult). (The simulation ˆ ws details were as in section 7.4.9.) The ﬁrst group comprises: τˆ max µ , τ µ , LRUC,µ rma (see the left-hand panel of Figure 7.7); the test τˆ µ is considered in section 7.5 glsc

and will be referred to therein. The second group comprises: PT,µ (−7), τˆ µ (−7),

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ˆ 1 )/ˆσu , where µ ˆ 1 is the estimated value of about ξc from the estimator ξˆc = (y1 − µ the trend for t = 1, and σˆ u is an estimator of σu , both estimators being based on observations post-dating t = 1. HL (2005) suggest σˆ u2 = (T − 1)−1 ∑Tt=2 uˆ 2t , where {uˆ t }T2 are the LS residuals from the regression of yt on µt , t = 2, . . . , T; other estimators of σˆ u2 are possible, and one that conditions on a structure for the {ut } process is illustrated below. The estimator ξˆc is consistent under the general alternative ρ < 1, but it is not consistent under the local alternative; however, HL (2005, Theorem 1), show that there is a monotonic relationship between |ξˆc | and |ξc |, so that the former is informative about the latter.

Improving the Power of Unit Root Tests

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, τˆ DFQ . Table 7.8 Critical values for τˆ DFG i i i=β

1%

5%

10%

1%

5%

10%

τˆ DFG i 100 200 ∞

–2.99 –2.86 –2.79

–2.43 –2.33 –2.23

–2.14 –2.06 –1.96

–3.77 –3.66 –3.58

–3.22 –3.13 –3.05

–3.77 –3.66 –3.58

τˆ DFQ i 100 200 ∞

–3.58 –3.62 –3.67

–2.99 –3.04 –3.10

–2.55 –2.59 –2.65

–3.93 –4.00 –4.09

–3.37 –3.47 –3.56

–3.93 –4.00 –4.09

Sources: From HL (2005, 2006) and own calculations based on 100,000 simulations.

0.4

0.3 τˆmax µ

τˆ glsc µ (–7)

0.35 0.25

Q T,µ(–10) 0.3

τˆ rma µ

PT,µ(–7)

0.2

0.25

power

τˆ glsu µ (–10)

power 0.2

0.15 LRUC,µ

0.15 Q(µ)(10, 3.8)

0.1 0.1

τˆ ws µ 0.05

0

0.5

1

1.5

2

2.5

0.05 0

0.5

κ

1

1.5 κ

2

2.5

Figure 7.7 Power comparision, ρ = 0.95.

glsu

ˆ (µ) (10, 3.8), (see the right-hand panel of Figure 7.7); QT,µ (−10), τˆ µ (−10) and Q throughout, ρ = 0.95. Of the ﬁrst group, τˆ max µ is the most powerful and then there is little to choose glsu

between the other three tests. In the second group, initially taking τˆ µ (−10) apart in the comparison, these tests dominate those in the ﬁrst group except as glsu κ > 1.4; τˆ µ (−10) is exceptional in its pattern, in that power does not decline monotonically with κ, but stays within a fairly narrow band whatever the value

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i=µ

306 Unit Root Tests in Time Series

0.4 0.35 glsc

τˆ µ (–7) 0.3 τˆDFG µ

0.25

τˆµ

power

0.15 glsu

τˆ µ (–10)

0.1 0.05

0

0.5

Q(µ)(10,3.8) 1

1.5

τˆ DFQ µ 2

2.5

κ Figure 7.8 Power comparison: the weighted tests.

of κ, and is most powerful in the set of test statistics in Figure 7.7 when κ exceeds about 1.3. The HL weighted tests τˆ DFG and τˆ DFQ are brought into the comparison in µ µ glsc

Figure 7.8, which also shows the component tests τˆ µ and τˆ µ (−7) in the forˆ (µ) (10, 3.8) in the latter case. This graph shows that the mer case, and τˆ µ and Q power performance depends critically on κ. Recall that it is not intended that glsc the weighted average test statistics are competitors to τˆ µ (−7) if the initial condition is known to be small or to τˆ µ if the initial condition is known to be large; rather that, in the absence of such knowledge, they offer a way of ‘hedging’ one’s bets. Note that for κ ≈ 1.2, there is little to choose amongst all the tests. An illustration of the interpretation of some of these test statistics is considered in the next section.

7.4.12 Illustration: US industrial production To illustrate the various tests of this section and draw out some of the practical implications, the time series for the logarithm of US industrial production (monthly, s.a.) is used. The complete time period is 1919m1 to 2005m12, comprising 1,044 observations; however, for purposes of illustration, different starting dates are chosen to show the importance of the ‘ﬁrst’ observation even in a large sample.

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0.2

Improving the Power of Unit Root Tests

307

5 4.5 4 3.5 logs

3

2 1.5 1 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 Figure 7.9 US Industrial production (logs, p.a., s.a.).

The data are shown in Figure 7.9, and there is clear evidence of a trend. OLS estimation of a linear trend using all the data gave the following results: (yt − [1.6139 + 0.00315t]) = uˆ t

t = 1, ..., T

uˆ t = 0.989uˆ t−1 + zˆ t

t = 2, ..., T

σ˜ z2 = 0.007 σˆ u2 =

0.007 (1 − 0.9892 )

= 0.0343 √ σˆ u = 0.0343 = 0.185 The ﬁtted trend is superimposed on the data in Figure 7.9, where it is evident that some deviations from trend, particularly for the period before 1950, are quite substantial. By this method, the estimate of the unconditional variance of ut is σˆ u2 = 0.0343. An alternative method, is to estimate the trend over t = 2, ..., T (which is unchanged to the decimal places shown), and use σ˜ u2 = (T − 1)−1 ∑Tt=2 uˆ 2t = 0.0334, which is close to the derived estimate of 0.0343. The latter is used for illustrative purposes. The estimate of the scaled deviation from trend in t = 1 (that is, 1919m1) ˆ 1 )/ˆσu = (1.6874 − [1.6139 + 0.00315])/0.185 = 0.38. This estimate is ξˆc = (y1 − µ suggests that the tests based on the ﬁxed initial condition are likely to be more powerful than those based on the unconditional initial condition. However, starting the sample elsewhere in the overall period may change this conclusion.

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2.5

308 Unit Root Tests in Time Series

2 0

ˆξc

–2 –4

scaled deviation from trend of first observation

–6 1918 1920 1922 1924 1926 1928 1930 1932 1934 1936 1938 1940

1 0.8

weight increases as scaled deviation increases

α ˆ 0.6 0.4 0.2 1918 1920 1922 1924 1926 1928 1930 1932 1934 1936 1938 1940 ˆ Figure 7.10b Estimated weights, α.

0 glsc

τˆ β

–1 –2 –3 –4

glsc τˆβ 5% cv for τˆβ –5 5% cv for τˆ β –6 1918 1920 1922 1924 1926 1928 1930 1932 1934 1936 1938 1940

Figure 7.11a DF and GLSC unit root tests.

0 –1 –2 –3 –4 –5

τˆDFG β

5% cv for τˆ DFG β

rejection of null below the line

–6 1918 1920 1922 1924 1926 1928 1930 1932 1934 1936 1938 1940

Figure 7.11b Weighted unit root test.

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Figure 7.10a Estimated scaled residuals, ξˆc .

Improving the Power of Unit Root Tests

309

less than unity in absolute value, there is little difference between the two test statistics, with values of both within the non-rejection region, and this is the case until about 1932. However, thereafter the initial observation is a long way from trend (see Figure 7.10a) and the values of the test statistics start to diverge glsc as ξˆc increases in absolute value; for example, at one point τˆ β = –0.67, but τˆ β = –5.15; the former suggesting ﬁrm non-rejection of the null, the latter just as ﬁrmly rejecting the null hypothesis. This could, quite understandably, be rather confusing! However, it is at this point that |ξˆc | = 5.14, which, as it is in units of σˆ u , is a substantial deviation from trend. The weights as given in Equation (7.106) applied to this series are shown in Figure 7.10b, and illustrate that as glsc |ξˆc | increases, the weight assigned to τˆ β decreases, implying that the weight assigned to τˆ β increases (towards unity). The resulting weighted test statistic, τˆ DFG , is shown in Figure 7.11b. β

7.5 Detrending (or demeaning) procedure by recursive mean adjustment The ﬁnal method to be considered in this chapter, with a view to a potential improvement in the power of a unit root test, is the recursive mean adjustment (RMA) procedure suggested by So and Shin (1999) and Shin and So (2002), which was described in Chapter 4, and also has a counterpart in providing RMA versions of the DF unit root test statistics. RMA unit root tests are included in this chapter as they relate to an alternative to ‘efﬁcient’ detrending to adjust the data for deterministic components. In principle, the RMA tests are just as their corresponding DF test statistics, but they use the recursively demeaned or detrended data. As a brief reminder, ﬁrst consider the case where a mean is removed from the data either using the ‘global’ mean y¯ G = ∑Tt=1 yt /T over all T observations, or y¯ = ∑Tt=2 yt /(T − 1) over T – 1 observations. The data are then demeaned by subtracting y¯ G or y¯ from yt , ¯ respectively, are used in any subsequent regressions. and then yt − y¯ G or yt − y, In contrast, the recursive mean only uses observations prior to period t in the calculation of the mean.

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To illustrate this point, the start of the sample period is moved forward one month at a time for 20 years and the ξˆc is re-estimated each time; the resulting ξˆc ˆ 1 )/ˆσu varies from are graphed in Figure 7.10a. This ﬁgure shows that ξˆc = (y1 − µ 1.05 to –5.14, and hence this is a range that is likely to affect the characteristics of the test statistics. Almost any conclusion can be reached depending on choice of starting point. glsc The values of τˆ β and τˆ β as the sample moves forward one month at a time are graphed in Figure 7.11a. The graph shows that when ξˆc is small, practically

310 Unit Root Tests in Time Series

The RMA procedure gives a sequence of estimators of the mean, say {y¯ rt }Tt=1 : y¯ r1 = y1 ; y¯ r2 = 2−1 ∑i=1 yi ; y¯ r3 = 3−1 ∑i=1 yi ; ...; y¯ rt = t−1 ∑i=1 yi ; ...; y¯ rT = T−1 ∑i=1 yi 2

3

t

T

Note that y¯ rT = y¯ G , so that in only one case do the two means coincide. The RMA means are related by the recursion (see also Chapter 4, Equation (4.43)): y¯ rt = y¯ rt−1 (t − 1)/t + yt /t

The recursive means can differ quite markedly from the global mean, especially in the early part of the sample and for an integrated process; of course, y¯ rt → y¯ G as t → T. ¯ the recursive mean, y¯ rt−1 , at Now, rather than demean the data using y¯ G or y, time t – 1, is used. For example, in the simplest case, the regression model (see Chapter 4, Equation (4.44)), is: yt − y¯ rt−1 = ρ(yt−1 − y¯ rt−1 ) + ζt

(7.102)

where ζt = (1 − ρ)(µ − y¯ rt−1 ) + εt . The null hypothesis is, as usual, H0 : ρ = 1. (Note that here ρ rather than φ1 is the preferred notation.) Counterparts to the DF tests are straightforward, being: δˆ rma ρrma − 1) µ = T(ˆ

(7.103)

τˆ rma ρrma − 1)/ˆσ(ˆρrma ) µ = (ˆ

(7.104)

where ρˆ rma is the LS estimator from (7.102) and σˆ (ˆρrma ) is its estimated standard error. A similar procedure can be applied if the deterministic components comprise a constant and a linear trend; this method was outlined in Chapter 4 (see Equations (4.46) and (4.47)). In the simplest case, the resulting DF regression is: ˆ rt−1 ) = ρ(yt−1 − µ ˆ rt−1 ) + ζt (yt − µ

(7.105)

ˆ rt = βˆ 0,t + βˆ 1,t t, and βˆ 0,t and βˆ 1,t are the LS estimators based on a regression where µ ˆ rt is then lagged of yt on a constant and t, including observations to period t; µ ˆ rt−1 is just the once to avoid correlation with the error term. The term yt−1 − µ recursive residual from the preliminary detrending regression at t – 1. As noted in Chapter 4, the preliminary regression of yt on a constant and t requires three observations to generate nonzero residuals. The RMA counterparts to the DF test are denoted δˆ rma and τˆ rma and are as in (7.103) and (7.104), respectively, β β but with ρˆ rma obtained from LS estimation of (7.102). The corresponding ADF version of the maintained regression with recursive mean adjustment is (as Chapter 4 of the form of Equation (4.53)), but with some slight changes in notation to reﬂect developments after Chapter 4): ˆ rt−1 ) = ρ(yt−1 − µ ˆ rt−1 ) + ∑j=1 cj ∆yt−j + εt,k (yt − µ k−1

(7.106)

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(7.101)

Improving the Power of Unit Root Tests

311

100 200 500

100 200 500

1%

δˆ rma µ 5%

1%

δˆ rma β 5%

10%

10%

–15.36 –15.84 –15.73

–9.64 –9.79 –9.74

–7.03 –7.09 –7.14

–18.14 –18.31 –18.22

–11.71 –11.83 –11.46

–8.68 –8.77 –8.53

1%

τˆ rma µ 5%

10%

1%

τˆ rma β 5%

10%

–2.53 –2.53 –2.52

–1.88 –1.89 –1.88

–1.53 –1.54 –1.55

–2.47 –2.50 –2.47

–1.81 –1.85 –1.82

–1.46 –1.49 –1.47

Source: Own calculations based on 100,000 replications. See also Shin and So (2002) for δˆ rma and τˆ rma µ µ .

The RMA procedure reduces the bias of the LS estimator. This was noted in Chapter 4 (see especially Table 4.4) where RMALS was effective in the neighbourhood of the unit root, relatively more so in the with-trend case. This bias reduction leads to a rightward shift in the limiting null distribution, with critical values less negative than in the standard DF case, as can be noted from Table 7.9. Indeed, as the bias in the LS estimator increases with the order of the polynomial trend in the deterministic component, the scope for the rightward shift is greater in the with-trend case, as is evidenced by the greater shift in the null distribution, compared to the constant mean case. However, the limiting distributions of the estimators for stationary alternatives are unaffected by the √ RMA procedure, for which T(ˆρrma − ρ) ⇒D N(0, 1 − ρ2 ) as in the standard LS case. Hence with the revised critical values, there will be more rejections for a given stationary alternative, implying that the RMA versions of the DF tests will be more powerful than their standard counterparts (see So and Shin, 1999; Shin and So, 2002). In some Monte Carlo simulations for sample sizes T = 50, 100 and 250, Shin and So (2002) report that the RMA test τˆ rma outperforms its standard DF µ counterpart and is nearly as powerful as the unconditional maximum likelihood test, LRUC,µ , and the weighted symmetric test, τˆ ws µ . The power improvement is also maintained for weakly dependent errors arising from AR(2) and ARMA(1, 1) data-generating processes. See also the left panel of Figure 7.7 for a comparison of τˆ rma with several other unit root tests. µ Critical values under the unit root null for the usual signiﬁcance levels are given in Table 7.9; note that there is almost no variation in the critical values as function of T.

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Table 7.9 Critical values for τˆ rma and τˆ rma µ β .

312 Unit Root Tests in Time Series

The focus of this chapter has been on GLS or GLS-type methods in some form, although a brief excursion into recursive mean/trend estimation was warranted when viewed as an alternative way of estimating the trend component. Whilst the beneﬁts of GLS at a theoretical level are well known, the key question is whether they can be translated into practical beneﬁts once GLS is made feasible. The answer to this question is equivocal. Not all of the possible feasible methods of estimation were used to obtain a unit root test, although that might be a worthwhile study. The variant that was used performed well enough on the usual criteria of size retention and power, but was not exceptional, and of the test statistics considered in Table 7.1, the weighted symmetric versions of the DF τˆ -type tests were generally superior overall. The second area in which a GLS-type technique was used was in ‘efﬁcient’ detrending. The idea here is to exploit the fact that many economic time series conform to the stylised view that if the process generating them does not contain a unit root then there is substantial persistence; that is, the stationary alternative is nearly (but not quite) integrated rather than clearly stationary. In that case, the detrending (or simply demeaning) is better done by assuming an alternative close to the unit root; in an established terminology each possible alternative is deﬁned by a ‘point’ ρ = 1 + c/T for c < 0. By way of a related digression, demeaning or detrending by way of recursive adjustment was also outlined. Under certain assumptions on the initial condition (a bounded second moment for all ρ in the neighbourhood of unity), feasible point-optimal tests exist and are simple to compute for each value of c˜ , where c˜ locates the localto-unity alternative. By choosing c˜ in such a way that the power function is tangential to the power envelope when power is at 50%, a single test statistic is obtained (referred to as a PT-type test). In the case of data that are demeaned, then c˜ = –7, whereas if a linear trend is used, then c˜ = –13.5. However, the initial condition on which these results are based is somewhat awkward in the sense that it does not deﬁne a strictly stationary alternative. In the case that the initial value is drawn from the unconditional process for the errors, the implied quasi-differencing changes for the ﬁrst observation, with resultant changes in the test statistics (now referred to as QT-type tests). Nevertheless, the same set of general principles applies in the form of the test statistics, but with c˜ = –10 for both the mean and trend cases. Applying QT-type tests when PT-type tests would be, in principle, preferable, and vice versa, whilst costly, in terms of the asymptotic power envelope, is not so costly in terms of power loss in ﬁnite samples as far as the τˆ -type versions of these tests was concerned. The developments arising from this work highlighted the potential importance of the initial condition, which distinguishes

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7.6 Concluding remarks

313

the PT and QT tests. The distance, appropriately scaled, of the initial observation from the trend under the alternative, can have a substantial effect on the power of a unit root test and should inform practical use of the various tests. Given the general dependence of the power of a unit root test on the initial condition, it is natural to seek a test that is relatively invariant to the scale of the initial condition. Two possible solutions to this problem have been suggested; Elliott and M¨ uller (2006) have suggested a new test, whereas Harvey and Leybourne (2005, 2006) have suggested weighting together two tests with different characteristics. The obvious question, ‘which test is best?’, now has to be supplemented either in the question or in the answer with the qualiﬁcation that it depends on the initial condition!

Questions Q7.1 Consider the model given by: (1 − ρL)yt = zt zt = εt + θ1 εt−1 + θ2 εt−2 Explain how to obtain an FLGS estimator. A7.1 In vector terms the model in terms of observable variables is:     ε     1 0 0 0 0  1  y0 ∆y1  ε2   θ  0 0   1 1 0    ∆y   y  ε3        2 1   0 0   .  =  .  γ +  θ2 θ1 1    ..   .   .   .    .   .   .   0 θ2 θ1 .. 1 0   ∆yT yT−1  εT−1  0 0 0 θ2 θ1 1 εT The square matrix on the right-hand side is Θ, which is the T × T transformation matrix associated with the lag polynomial θ(L); as there is no AR lag polynomial, Ω = Θ. Writing the model as: ∆Y1 = γY1,−1 + z

(7.107)

then z = Ωε = Θε and E(zz ) = σε2 Γ with Γ = ΘΘ . The infeasible GLS estimator and standard error are: γˆ GLS = (Y1,−1 Γ−1 Y1,−1 )−1 Y1,−1 Γ−1 ∆Y1 2 σ(γˆ GLS ) = σˆ ε,GLS (Y1,−1 Γ−1 Y1,−1 )−1

A feasible estimator is obtained by the following procedure: ﬁrst, estimate (7.107) by LS to obtain the residual vector zˆ ; second, estimate an MA(2) model

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Improving the Power of Unit Root Tests

314 Unit Root Tests in Time Series

results

yˆ˜ t−1

∆yˆ˜ t−1

∆yˆ˜ t−2

∆yˆ˜ t−3

–0.133 –2.62 –0.197 –3.71

–

–

–

0.427 4.62

–0.167 –1.78

0.236 2.53

–0.224 –3.48

0.168 1.67

– –

– –

LS ∆yˆ˜ t ‘t’ ∆yˆ˜ t ‘t’ FGLS ∆yˆ˜ t ‘t’

for

US

Notes: Levels data; effective sample period: 1894–1988 = 95 observations.

Table 7.11 Summary of unit root test values for US unemployment data.

5%

5%

δˆ µ

δˆ ws µ

τˆ µ

τˆ ws µ

Zδˆ µ

MZδˆ µ

–37.48 –14.54

–37.89 –14.32

–3.71 –14.76

–3.48 –14.34

–37.08 –2.87

–36.24 –2.58

Zˆτµ

MZˆτµ

MSBµ

δˆ FGLS,GZW

τˆ FGLS,GZW

LRUC,µ

–4.36 –2.96

–4.26 –2.87

0.12 0.17

–25.88 –16.20

–3.48 –2.94

2.83 2.50

Notes: Levels data; lag selection details: ADF, lag(AIC) = 3; WS, lag(AIC) + 2 = 4; FGLS, lag(AIC) = MA(1), ADF(1); UCML, lag(AIC) ⇒ ARMA(0, 1) for zt ⇒ ARMA(1, 1) for yt .

ˆ and hence Γˆ from the estimates of θ1 and θ2 ; using zˆ ; third, construct Θ, fourth, form the FGLS estimator that replaces Γ−1 by Γˆ −1 . (See, for example, section 7.1.3 for a method of estimating the MA model.) Q7.2 Consider the summary in Tables 7.10 and 7.11 of the unit root statistics applied to the level of the US unemployment series used in section 7.3.1, where logarithms were used. Note also that unconditional ML estimation resulted in (1 − 0.801)∆yˆ˜ t = (1 + 0.456)˜εt , where ∆yˆ˜ t now refers to the demeaned levels. Reassess the ﬁndings relative to those in that section. A7.2 The effect of using the original series is to move the estimate of ρ closer to unity; the estimate of ρ from the ADF regression is 1 + (–0.197/(1 – (0.427 – 0.167 + 0.236)) = 0.61, and the unconditional ML estimate is 0.80, compared to 0.52 and 0.58, respectively, for the log of the series, but this change is not sufﬁcient to change the conclusion that the unit root null hypothesis can be rejected. Otherwise, the only marked change is in the value of LRUC,µ which falls to 2.83, but is still signiﬁcant. In sum, both the test using the levels and

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Table 7.10 Estimation unemployment.

Improving the Power of Unit Root Tests

315

logs of the data result in rejection, which is a consistent outcome (that is, both series can be stationary). However, quite often the problem is the reverse one in the sense that unit root tests on the level and the log of a series both lead to non-rejection, which is a problem taken up in Units Roots, Volume 2.

X = (1, . . . , 1)

1 1 ... X= 1 2 ...

1 t

1 T

... ...

A7.3 This is a straightforward application of the principles in section 7.4.1.ii. In the ﬁrst case, they are:     Pρ˜ X =    

1 −˜ρ 0 .. . 0

=

0 1 −˜ρ .. . ...

0 0 1 .. . 0

0 0 0 .. . −˜ρ

1

1 − ρ˜

1 − ρ˜

1

0

...

P(1) X =

0

0

0 0 0



      0  1

...

1 − ρ˜

 1 1    1   ..  .  1

In the second case, they are:     Pρ˜ X =     = P(1) X =

1 −˜ρ 0 .. . 0

0 1 −˜ρ .. . ...

0 0 1 .. . 0

0 0 0 .. . −˜ρ

1 1

1 − ρ˜ 2 − ρ˜

1 − ρ˜ 3 − 2˜ρ

1 1

0 1

... ...

0 1

0 1

0 0 0



      0  1

... ...

1 1 1 .. . 1

1 2 3 .. . T

       

1 − ρ˜ T − (T − 1)˜ρ

Q7.4 Consider the GLS transformation matrix, denoted P in general, but now under the assumption that the intialisation is replaced by

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Q7.3 What do the matrices Xρ˜ ≡ Pρ˜ X and X(1) ≡ P(1) X look like for the following X, which are the typical cases in practice:

316 Unit Root Tests in Time Series

u1 ∼ iid(0, σε2 /(1 − ρ2 )). Show that the required transformation matrix PU,˜ρ is:

     PU,˜ρ =     

(1 − ρ˜ 2 )1/2 −˜ρ 0 .. . 0 0

0 1 −˜ρ .. . 0 0

0 0 1 .. . 0 0

0 0 0 .. . ... ...

0 0 0 .. . 1 −˜ρ

0 0 0 .. . 0 1

          

A7.4 A transformation is sought such that for the ﬁrst observation P1 u = ε1 , where P1 is the ﬁrst row of the desired transformation matrix, P. By assumption, 2 2 u1 is not serially correlated and has variance σε /(1−ρ ). Hence choosing the ﬁrst 2 element of P1 , say P11 , as P11 = (1 − ρ ), with all other row elements equal to zero, will do the trick since var(P1 u) = P211 σε2 /(1 − ρ2 ) = (1 − ρ2 )[σε2 /(1 − ρ2 )] = σε2 ; evaluating this element at ρ = ρ˜ , with all elements in the transformation matrix as in Pρ˜ , results in PU,˜ρ . The practical effect of the unconditional draw assumption for the ﬁrst error is that quasi-differencing results in ﬁrst ‘observations’ of (1 − ρ˜ 2 )1/2 y1 and (1 − ρ˜ 2 )1/2 x1 for the dependent and explanatory variables, rather than y1 and x1 , respectively. Q7.5.i Consider the MA(2) process zt = εt + θ1 εt−1 + θ2 εt−2 . Assuming that this is invertible, obtain the corresponding inﬁnite AR representation. Q7.5.ii Verify the relations between the MA and AR coefﬁcients given by the equations of (7.25) for this example. A7.5.i Invertibility implies the existence of (1 + θ1 L + θ2 L2 )−1 zt = εt . Note that the simple inﬁnite sum (1 + θL + (θL)2 . . .) = (1 − θL)−1 ; by a change of variable, set θ = (θ1 + θ2 L), so that (1 + θ1 L + θ2 L2 )−1 = (1 + (θ1 + θ2 L)L)−1 . Thus the righthand side of the following relationship gives the inﬁnite AR representation:

(1 + [−(θ1 + θ2 L)]L + [−(θ1 + θ2 L)]2 L2 . . .) = (1 − [−(θ1 + θ2 L)]L)−1

Expand the left-hand side for, say, the ﬁrst three terms and collect coefﬁcients on like powers of Lj to establish the pattern:

(1 − θ1 L + (θ21 − θ2 )L2 + (2θ1 θ2 − θ31 )L3 . . .)

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

Improving the Power of Unit Root Tests

317

j Thus the ﬁrst three coefﬁcients in the expansion of ϕ(L)zt = (1 − ∑∞ j=1 ϕj L )zt = εt are:

ϕ 1 = θ1 ϕ2 = − θ21 + θ2 = − θ1 ϕ1 + θ2 ϕ3 = θ31 − 2θ1 θ2 = − θ1 ϕ2 − θ2 ϕ1 .. .

.. .

.. .

.. .

ϕj = − θ1 ϕj−1 − θ2 ϕj−2 Alternatively, factor θ(L) such that: θ(L) = (1 + θ1 L + θ2 L2 ) = (1 − λ1 L)(1 − λ2 L) where θ1 = − (λ1 + λ2 ) and θ2 = λ1 λ2 ; and, taking the inverse: θ(L)−1 = (1 − λ1 L)−1 (1 − λ2 L)−1 = (1 + λ1 L + (λ1 L)2 . . .)(1 + λ2 L + (λ2 L)2 . . .) = (1 + (λ1 + λ2 )L + [λ1 (λ1 + λ2 ) + λ22 ]L2 + [λ1 (λ1 (λ1 + λ2 ) + λ22 ) + λ32 ]L3 + . . .). Noting that the expansion is valid by the assumptions |λ1 |<1 and |λ2 |<1, and using the equalities for θ1 and θ2 in terms of λ1 and λ2 , the result follows. A7.5.ii The question also asks for veriﬁcation against the general case, which follows. The appropriate simpliﬁcation of the general case is: ϕ1 = θ1 ϕ2 = − θ1 ϕ1 + θ2 = − θ21 + θ2 ϕ3 = − θ1 ϕ2 − θ2 ϕ1 = θ31 − 2θ1 θ2 .. .

.. .

ϕi = − θ1 ϕi−1 − θ2 ϕi−2 Q7.6.i Summarise the ERS procedure for ‘efﬁcient’ detrending and explain how it can be viewed as quasi-differencing. Q7.6.ii What difference in the procedure is implied by treating the initial condition as a draw from the unconditional distribution? A7.6.i. The steps in ERS test are as follows: Demean or detrend the data according to the following scheme: 1.(i)

Deﬁne the quasi-differencing (QD) parameter, ρ˜ = 1 + c˜ /T.

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.. .

318 Unit Root Tests in Time Series

(ii) 2.(i)

Depending on whether the data is being demeaned or detrended, use c˜ = −7 for the former and c˜ = −13.5 for the latter. Dependent variable: The ﬁrst observation is y1 , the remainder are yt − ρ˜ yt−1 , t = 2, ... , T; thus construct:

(ii)

The ‘explanatory’ variable is just a constant in this case. The ﬁrst ‘observation’ is 1, the remainder are 1 − ρ˜ , t = 2, ... , T; thus construct: Xρ˜ = (1, 1 − ρ˜ , 1 − ρ˜ , . . . , 1 − ρ˜ )

3.

˜ , from: Use the QD data in an LS regression to estimate a constant, µ Yρ˜ = Xρ˜ λ + v

4.

˜ ρ˜ is the LS estimator of λ from this regression. where λ˜ ρ˜ = µ The QD detrended data is then (see Equation (7.91)): ˜ ρ˜ y˜ ρ˜ ,t = yt − µ

5.

t = 1, . . . , T

Use y˜ ρ˜ ,t in a DF or ADF regression model of the standard form (see Equation (7.92)).

If detrending the data, then proceed as above, except that the data vectors in step (2)(ii) for the ‘explanatory’ variables are: Xρ˜ ,1 = (1, 1 − ρ˜ , 1 − ρ˜ , . . . , 1 − ρ˜ ) Xρ˜ ,2 = (1, 2 − ρ˜ , . . . , t − ρ˜ (t − 1), . . . , T − ρ˜ (T − 1)) The detrending regression model in step (3) is: Yρ˜ = Xρ˜ λ + v ˜ ρ˜ , β˜ ρ˜ ) is the LS estimator of λ from this regression. where λ˜ ρ˜ = (µ The QD detrended data is now: ˜ ρ˜ − β˜ ρ˜ t) y˜ ρ˜ ,t = yt − (µ

t = 1, . . . , T

A7.6.ii In the case of the unconditional initial condition, replace y1 in Yρ˜ by (1 − ρ˜ )1/2 y1 and the 1 in the ﬁrst element of Xρ˜ ,1 and Xρ˜ ,2 by (1 − ρ˜ )1/2 ; then proceed as above, but note that λ˜ U,˜ρ = (β˜ 0,U,˜ρ , β˜ 1,U,˜ρ ) and the QD detrended data, y˜ U,˜ρ,t , is: y˜ U,˜ρ,t = yt − (β˜ 0,U,˜ρ − β˜ 1,U,˜ρ t)

t = 1, . . . , T

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Yρ˜ = (y1 , y2 − ρ˜ y1 , y3 − ρ˜ y2 , . . . , yT − ρ˜ yT−1 )

8

Introduction A bootstrap procedure was described in Chapter 4 as a means of reducing the bias in LS estimators, and in Chapter 5 the grid bootstrap procedure, due to Hansen (1999), was outlined as a means of constructing a conﬁdence interval in a situation where the underlying quantiles are not constant. This chapter picks up the bootstrap theme as it applies speciﬁcally to unit root testing. Perhaps the ﬁrst question to ask is: why bootstrap a unit root test? There are two possible answers to this question. First, as far as the second moment is concerned, the asymptotic properties of the DF-type tests only require a ﬁnite variance. However, simulation of the ﬁnite sample quantiles involves an assumption about the generating distribution of the errors, and this is usually taken to be normal. If, in fact, the distribution of the errors satisﬁes the variance condition but is other than normal, inference should, in principle, be based on simulation using the correct ﬁnite sample distribution. The bootstrap procedure naturally allows for this problem by basing the bootstrap replications on draws from the empirical distribution of the residuals from an initial regression. A second possible answer to the ‘why bootstrap’ question lies in the ‘inﬁdelity’ of nominal and actual size when the errors are generated by an MA process. In these situations we know (see Schwert’s 1987 simulation study) that the size distortions in realistic cases can be so substantial as to completely prejudice accurate inference. For example, suppose that the errors are generated by an MA(1) process, with an MA coefﬁcient of –0.8, then with, say, T = 100, and using a conventional DF critical value, the actual size of the ADF τˆ µ test is about 50%, and the actual size of its PP equivalent approaches 100%, both for tests at a nominal 5% level. Consider a test at the 5% nominal level but which has an actual size of 100%, this means that the test always rejects when the null is true, rather than rejecting just 5% of the time; the rejections are said to be spurious. Of course, this also 319

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Bootstrap Unit Root Tests

means that such a test has spurious power, since one way of maximising power is to always reject the null. Not only does such over-rejection imply misleading decisions in individual cases, it implies misleading comparisons in simulation analysis, since an assessment of the power of different tests requires that they have the same size, otherwise the difference in power may just be due to the difference in size. A power comparison can neutralise the size discordancy by generating the required critical value or values from the simulated distribution, known as size-adjusted power, so that actual (or simulated) size equals nominal size by construction, and hence enables different tests to be compared. The procedure of size-adjusting rather naturally offers the means of the solution to the size inﬁdelity problem. The essence of the latter is that there is not a single distribution, for all values of a nuisance parameter or parameters, from which it is valid to draw the particular quantile(s), say the 5% left-sided quantile. Given this statement of the problem, the general form of the solution is apparent: simulate the correct quantile; then, at least subject to the accuracy of the simulation design, the nominal and actual size will be the same. Though simply stated, there is more than one way to achieve this aim. This chapter describes two bootstrapping schemes designed to overcome the size inﬁdelity problem. Consider a situation in which the errors are serially correlated and, in particular, are generated by an MA process; one solution is to proceed along the lines of Said and Dickey (1984) of approximating the MA process by a truncated AR process, where the truncation lag is selected to increase with the sample size, but at a slower rate. This results in an ADF regression, which is estimated as the ‘initial’ regression in the bootstrap scheme with and without the restriction of a unit root. The ﬁrst of these regressions provides standard test statistics and the second serves to deﬁne an empirical distribution function of residual errors from which resampling takes place to generate the random input to the bootstrap data; the latter is generated by the hypothesised DGP, which, in this context, must impose a unit root. Having generated, say, B bootstrap samples, the data are used to estimate, or mimic, the initial regression; the required test statistics are generated for the B samples and then ordered (sorted) to obtain the bootstrap distribution from which the bootstrapped quantile(s) is(are) obtained. Rejection or non-rejection follows in the standard way by comparing the test statistic(s) from the initial regression with the bootstrap quantile. This, in essence, is the scheme suggested by Chang and Park (2003). A slightly different approach is suggested by Psaradakis (2001). The initial regression is a simple AR(1) model, as in the PP case, with no extension for serially correlated errors; however, as in Chang and Park (2003), the Said and Dickey approach of estimating a truncated autoregression is used for the ﬁrst difference of the data (which may be adjusted for a nonzero mean), and the residuals from this regression serve to deﬁne the empirical distribution function from which the random input to the bootstrap data is generated. The bootstrap

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321

regressions take the pattern of the initial regression, which has no ADF augmentation; otherwise, the principle of obtaining the bootstrap distribution of the test statistic(s) and using the bootstrap quantile(s) is as in the Chang and Park scheme. There are some differences of detail between these two schemes; for example, Chang and Park prefer the AIC to select the lag length, whereas Psaradakis prefers a two-sided t test at a 10% signiﬁcance level. What is important in the choice of criterion is to ensure that the lag length is not truncated too short to provide an adequate approximation to an MA process. Also, the reported simulation results in the former case relate to the demeaned case, whereas in the latter they refer to the detrended case, so the precise details are not comparable. A small simulation study is therefore also reported in this chapter to enable a consistent comparison. Section 8.1 outlines the basic principles of setting up the bootstrap routine for the case of a unit root; this section covers issues such as how to deal with the deterministic terms and the possibility of non-iid errors. Two bootstrap frameworks, those due to Psaradikis (2001) and Chang and Park (2003), are given in detail. The results of a simulation study to compare these approaches are given in section 8.2 and an empirical illustration is reported in section 8.3. Whilst the focus in this chapter is on bootstrapping the standard ADF regression, the bootstrap can be applied to any of the unit roots tests; for example, Chang and Park (2003) brieﬂy report that the ERS procedure results in more powerful tests than their DF counterparts, as one would expect, but is not any different in correcting for size.

8.1 Bootstrap schemes with an exact unit root The grid bootstrap procedure suggested by Hansen overcomes the problem identiﬁed by Basawa et al. (1991), which concerns the generation of the bootstrap replications in the case of a unit root. When the data are generated by ∆yt = zt with zt = εt ∼ iid, and the ﬁtted AR(1) model does not include a constant, Basawa et al. show that if there is a unit root, then constructing the (conventional) bootstrap sample from the recursion ybt = ρˆ ybt−1 + eˆ t , where the eˆ t are draws with replacement from the LS residuals εˆ t and ρˆ is the LS estimate, results in an asymptotic distribution for the bootstrap t statistic that is random. Thus, unlike the stationary case, the limit distributions of the t statistic from LS estimation and from the bootstrap are not the same, and the bootstrap procedure is invalid. The grid bootstrap overcomes this problem because the bootstrap replications are generated from a recursion that imposes the unit root at the point in the grid that ρr = 1 (see step (6) of Chapter 5, section 5.5.1.ii; and where the notation φr was used, which coincides with ρr in this simple case).

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Bootstrap Unit Root Tests

If the DGP is an AR(p) model, p > 1, then the problems associated with the AR(1) case when there is a unit root, carry across to the bootstrap estimator of p φ = ∑i=1 φi and its associated t statistic. However, the bootstrap estimators, φˆ bi , of √ the individual AR coefﬁcients, φi , are still valid in the sense of T-consistency and convergence in distribution of T1/2 (φˆ bi − φi ) to a normal distribution, as in the case of the LS estimators. The conventional bootstrap is also valid for linear combinations of φi not proportional to φ and smooth nonlinear functions of φi and φ such as the long-run multiplier (see Inoue and Kilian, 2002). However, the central problem of the invalidity of the conventional bootstrap for φ remains. There are several bootstrap replication schemes that avoid this problem; as noted, Hansen’s grid-bootstrap procedure, considered in Chapter 5, section 5.5.2, is one of them. We ﬁrst consider how the random inputs to the bootstrap replications may be generated. 8.1.1 The random inputs to the bootstrap replications An aspect common to the different schemes is the need to impose the unit root in the bootstrap recursion, thus if the goal is to construct a unit root test, then the bootstrap generation process must impose the unit root; in the simplest case, this is ybt = ybt−1 + eˆ t . However, there is some variation in how the sequence of bootstrap errors {eˆ t }Tt=2 is obtained. (We assume here that the underlying model is AR(1); in the more general AR(p) case, the lower summation or sequence limit is p + 1.) Two variants are as follows. S1. Sample eˆ t from the centred LS residuals, εˆ t −ε¯ˆ, from the initial regression, where εˆ t = yt − ρˆ yt−1 and ε¯ˆ is the sample mean of {ˆεt }T2 (see, for example, Ferreti and Romo, 1996). S2. Impose the unit root in initial LS estimation to give the restricted residuals ¯ t , where ∆y ¯ t is ε˜ t = ∆yt , and ensure that these are centred so that ˜εt − ε¯˜ = ∆yt − ∆y T the sample mean of {∆yt }2 ; this is sampling from the centred differences of yt (see, for example, Swensen, 2003a). There is no difference asymptotically between these two schemes either under the null (see Ferretti and Roma, 1996; Heimann and Kreiss, 1996) or in terms of asymptotic power (Swensen, 2003a, 2003b). These schemes are appropriate for a bootstrap unit root test, which we will describe in detail in a moment. In practice, the initial LS regression will include a constant and, possibly, a time trend. First, we deal with the case of an estimated AR(1) model with constant. Now, whilst keeping the same bootstrap recursion, in addition to sampling from the centred residuals or the centred differences, there is the further sampling scheme given by: S3. {eˆ t }Tt=2 can be obtained as draws from εˆ t = yt − (µ ˆ ∗ + ρˆ yt−1 ); there is no need ˆ to centre et , as the sum of residuals is zero by construction.

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Bootstrap Unit Root Tests

323

S4. When a trend is included in the initial regression, the LS residuals are εˆ t = yt – (βˆ ∗ + βˆ ∗ t + ρˆ yt−1 ), and this is a fourth candidate for sampling in the bootstrap 0

1

recursions. Swensen (2003a) shows that the implications of the four resampling schemes do not differ asymptotically for the null distribution and the power function, but they may differ in ﬁnite samples. 8.1.2 A bootstrap approach to unit root tests The construction of a conﬁdence interval, or an assessment of power for unit root tests, requires consideration of the bootstrap away from the unit root. The bootstrap, with recursion ybt = ρˆ ybt−1 + eˆ t , is known to be valid both in the stationary and explosive directions; that is, for |ρ| < 1 and ρ > 1 (see Bose, 1988; Basawa et al., 1989, respectively, and also Ferreti and Romo, 1996), but not for ρ = 1. Thus Hansen’s grid-bootstrap scheme, by generating the bootstrap recursion with ρr ∈ Gρ , gets the right bootstrap recursion both for ρr = 1 and elsewhere in the parameter space. The discussion of how to construct the bootstrap recursions leads naturally into obtaining a direct bootstrap test of the unit root. The next section deals with the AR(1) case, and the section following considers the extension to a general AR(p) model. 8.1.3 The AR(1) case Consider the case of data generated by ∆yt = zt with zt = εt ∼ iid, with ﬁnite variance (that is, a driftless random walk with uncorrelated errors). We assume that an AR(1) model is ﬁtted to this data (the maintained regression), with three possible speciﬁcations of the deterministic components of the model corresponding to: (i) no constant, no trend; (ii) constant, no trend; and (iii) constant and trend. Trend here refers to a deterministic linear trend. As a simple point of reference we start with the no constant, no trend speciﬁcation, although it is practically the least likely of the models to be ﬁtted. Variations can then be considered relative to this base point. A point on notation: throughout this chapter, the bootstrap t statistic is generically denoted τˆ b , with subscripts as appropriate to denote which version is being used; a (realised) sample τˆ -type statistic is denoted τˆ s , again with additional subscripts as usual to reﬂect deterministic components.

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Swensen (2003a) shows that the schemes S1, S2 and S3 do not differ asymptotically in their implications for the null distribution, or the asymptotic power functions of the bias and t-type unit root tests, and these are the same as for the standard unit root tests (which ensures the validity of the bootstrap). In an economic context, many time series exhibit a marked trend, thus it often makes sense to include a time trend in the ﬁtted regression (or detrend) since the alternative to a unit root, with or without drift, is stationary deviations around a deterministic trend. This gives rise to another sampling scheme, S4.

324 Unit Root Tests in Time Series

In this case, the regression model is yt = ρyt−1 + zt , with zt = εt . (This could equally be written in AR notation as yt = φ1 yt−1 + εt , but the preference here is for the error dynamics version of the model.) The residual sequence from LS estimation is {ˆεt }T2 , where the ‘t’ statistic for the unit root hypothesis is τˆ s = (ˆρ −1)/ˆσ(ˆρ); and σˆ (ˆρ) = σˆ ε (∑Tt=2 y2t−1 )−1/2 where σˆ ε2 = (T − 2)−1 ∑Tt=2 εˆ 2t . The aim of the exercise is to construct a bootstrap (BS) test statistic to test the null hypothesis of a unit root against the alternative hypothesis ρ < 1 (equivalently φ1 < 1). The steps can be brieﬂy described given that they correspond to a special case of the grid bootstrap described in Chapter 5, section 5.5.2. The bootstrap replications must follow the recursion ybt = ybt−1 + eˆ t , where the eˆ t sequence is drawn with replacement from one of the sampling schemes S1 or S2. That is, as a constant is not included in the estimated model, centre the residuals from a regression without a constant or use the centred differences of yt . Next, estimate the bootstrap regression for each replication, resulting in the bootstrap estimate yˆ bt = ρˆ b ybt−1 . The bootstrap LS estimator, bootstrap t statistic and bootstrap standard error are, respectively: ρˆ b = ∑t=2 (ybt ybt−1 )/ ∑t=2 (ybt−1 )2 T

τˆ b =

T

(ˆρb − 1) σˆ (ˆρb )

σˆ (ˆρb ) = σˆ ε (∑t=2 (ybt−1 )2 )1/2 T

(8.1a) (8.1b) (8.1c)

Note that σˆ ε is the estimate from the initial LS regression. The bootstrap equivalent of σˆ ε could also be used (see Psaradakis, 2001); however, Swensen (2003a, ˆ Remark 4) ﬁnds slightly better power when σˆ ε is used. The δ-type test statistic b b ˆ based on the bootstrap LS estimator is δ = T(ˆρ − 1). The B bootstrap replications are used to obtain (an approximation to) the sampling distribution of the unit root test statistic under the null. The decision rule is based on a comparison of the sample value of the test statistic from the initial LS estimation with the α-quantile from the bootstrap sample distribution, where the bootstrap recursion imposes the unit root. This contrasts with the standard procedure of comparing the sample value of the test statistic with the α-quantile from the DF distribution. The sample value of a test statistic is given an s subscript. First, consider the t-type test. Let τˆ s = (ˆρ − 1)/ˆσ(ˆρ) be the sample value of the t statistic from the initial LS regression, and let τˆ bs α be the α-quantile from the bootstrap distribution of the t statistic; if τˆ s < τˆ bs α , then unit root null is rejected at the α% signiﬁcance level. The procedure is analogous to the usual comparison, the exception being that the bootstrap distribution, not the DF distribution, is used to obtain the α% critical value.

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8.1.3.i The AR(1) case: no constant

325

The bootstrap p-value of the test statistic τˆ s can be calculated by counting the number of times that τˆ b ≤ τˆ s , then dividing by the number of replications B in order to obtain a probability estimate. That is, obtain (say) κbs = #[ˆτb ≤ τˆ s ]/B, where #[.] is the counting function incrementing by 1 if the statement is true, and 0 otherwise. Then at the α signiﬁcance level reject H0 : ρ = 1 against HA : ρ < 1 if κbs < α, otherwise do not reject. Why does this work? From the deﬁnition of an α-quantile, τˆ bs τb ≤ τˆ bs τb ≤ τˆ s ) = κbs in effect α is such that P(ˆ α |H0 ) = α; thus, P(ˆ bs obtains the κ -quantile under the null. If the estimated p-value, κbs of τˆ s , is less than α, then the sample evidence, in the form of τˆ s , is judged as inconsistent with the null hypothesis. A similar procedure applies to δˆ b .

8.1.3.ii Variations due to different speciﬁcations of the deterministic terms So far it has been assumed that the estimated model does not contain a constant or a trend; and it has also been assumed that the DGP is a driftless random walk. This section is concerned with relaxing both sets of assumptions. As to the ﬁrst of these issues, one possibility is to demean or detrend the data as appropriate and use the residuals from this prior part of the process. An alternative procedure works directly with the data as observed. In the ﬁrst case, a constant is included in the initial LS regression model, resulting in the LS estimated model yt = µ∗ + ρyt−1 + εˆ t , where µ∗ = (1 − ρ)µ, with sequence of residuals {ˆεt }T2 . The sample t statistic and related quantities for the unit root hypothesis are: τˆ s = (ˆρ − 1)/ˆσ(ˆρ) σˆ (ˆρ) = σˆ ε (∑t=2 (yt−1 − y¯ (−1) )2 )−1/2 T

y¯ (−1) = ∑t=2 yt−1 /(T − 1) T

Data are generated by the bootstrap recursions as before; that is, from ybt = ybt−1 + eˆ t , where the eˆ t are obtained by resampling with replacement from one of the following schemes: S1 (centred LS residuals), S2 (centred differences) or S3 (LS residuals from the model with a constant). The estimated bootstrap ˆ ∗b + ρˆ b ybt−1 , with relevant bootstrap quantities given by: regression is yˆ bt = µ ρˆ b =

∑Tt=2 (ybt − y¯ b )(ybt−1 − y¯ b(−1) ) ∑Tt=2 (ybt−1 − y¯ b(−1) )2

ˆ ∗b = y¯ b − ρˆ b y¯ b(−1) µ τˆ b =

(ˆρb − 1) σˆ (ˆρb )

(8.2a) (8.2b) (8.2c)

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T

y¯ b = ∑t=2 ybt /(T − 1) T

y¯ b(−1) = ∑

T yb /(T − 1) t=2 t−1

(8.2d) (8.2e) (8.2f) (8.2g)

If a linear trend is included in the initial LS regression, then the estimated model is: yt = βˆ ∗0 + βˆ ∗1 t + ρˆ yt−1 + εˆ t . Within the bootstrap, the recursions are as before (because drift is not assumed in the unit root process), with the sequence of eˆ t obtained by resampling with replacement from one of the schemes S1, S2, S3 ˆ ∗b ˆ b ybt−1 and, again, or S4. The estimated bootstrap regression is yˆ bt = βˆ ∗b 0 + β1 t + ρ other calculations are as before. If the DGP is also thought to include a drift, a more substantive change has to be made. In this case, the underlying DGP is ∆yt = β1 + εt . To accommodate this in the bootstrap recursions, an estimate of β1 is required. One possibility (see Psaradakis, 2001), is the average change in ∆yt over the sample, say d¯ = ∑Tt=2 ∆yt /(T − 1); the bootstrap recursion is then ybt = d¯ + ybt−1 + eˆ t . It is likely that the presumption of drift under the null would be matched with least squares estimation of a model including a trend, and so the remainder of the bootstrap scheme is as in the with-trend case.

8.1.4 Extension to higher-order ADF/AR models It is often the case that an AR(1) model is inadequate to capture the dynamics for economic time series and we need to consider the bootstrap extension to the situation when the error zt is serially correlated. For example, suppose that an AR(2) model is considered appropriate then, as in the grid-bootstrap example of section Chapter 5, section 5.5.1.ii., one way of dealing with this is to reformulate the maintained regression as an ADF(1) model. A bootstrap unit root test follows, with the bootstrap recursion as in step 5 of that section, but now the bootstrap recursion is ybt = ybt−1 + cˆ 1 ∆ybt−1 + eˆ t . Notice that the unit root restriction is imposed in the recursions and two values of ybt ; that is, yb1 and yb2 , will be required to initialise the process. By extension, higher-order ADF processes can be accommodated, as in the standard ADF case, by amending the bootstrap recursion to p−1 ybt = ybt−1 + ∑j=1 cˆ j ∆ybt−j + eˆ t ; the decision rule to select p is discussed further below. Higher-order AR/ADF models can be viewed as being generated by an AR(1) model plus an AR process for the generation of the errors as in the error dynamics framework. A more general framework allows the possibility that the errors are generated either by an AR or an MA process, or a combination of both. The ‘sieve’ bootstrap, which is described next, addresses this more general case.

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σˆ (ˆρb ) = σˆ ε (∑t=2 (ybt−1 − y¯ b(−1) )2 )1/2 T σˆ ε2 = (T − 3)−1 ∑t=2 εˆ 2t

327

For the bootstrap procedure to be valid, the bootstrap distributions have to converge to the respective limiting distributions of the non-bootstrap statistics. This issue is addressed by Psaradakis (2001, especially Remark 2), Chang and Park (2003, Theorem 2) and Swensen (2003a, Theorem 1), among others. Although the precise detail of the assumptions on the generating process differ, these authors show that the bootstrap is valid provided that the bootstrap recursions are generated under the unit root null, with an appropriate resampling scheme for {eˆ t } and, in the case of weakly dependent errors, that there is an appropriate rate of expansion of the mechanism that is included in the bootstrap algorithm to account for such weak dependency. For example, Chang and Park (2003, Theorem 2) show that their bootstrap test statistics based on ˆ DF/ADF extensions of the δ-type and τˆ -type statistics have the same limiting distributions as their standard counterparts. They show that as T → ∞, then: pbs (δˆ b ≤ δˆ α ) − p(δˆ ≤ δˆ α ) →p 0 where pbs (.) is the probability taken with respect to the distribution of {ybt }Tt=1 ˆ DF . The difference in the conditional on {yt }Tt=1 and δˆ α is the α% quantile of F(δ) asymptotic size of the bootstrap procedure for δˆ using the α% quantile relative to the asymptotic size from the limiting (non-bootstrap) null distribution of the standard procedure tends to 0 in probability. A similar result holds for τˆ b and for extensions to include deterministic components. 8.1.5 Sieve bootstrap unit root test The sieve bootstrap has been suggested in the context of a unit root test by Psaradakis (2001) (see also Park, 2003; Chang and Park, 2003). Typically, stationary ARMA processes are chosen to model the serial correlation in zt , but the procedure will work for quite general linear processes. The idea of a ‘sieve’ is based on an approach due to Said and Dickey (1984) in the context of a standard extension of the DF test to the case of ARIMA(p, 1, q) models, where p and q are unknown and have to be selected according to a criterion. Consider an ARMA process for zt with a non-degenerate MA component then, assuming invertibility, an inﬁnite AR representation exists. The sieve method works with this representation. However, since an inﬁnite-order process cannot be estimated, the order of the autoregression needs to be truncated in a way that depends upon the number of observations to ensure consistency of the LS estimators. This idea, suggested by Said and Dickey (1984), can also be used in the bootstrap context and is valid for linear processes more general than ARMA. Truncation criteria that have been used include AIC, BIC, FPE (ﬁnal prediction error), the Ng and Perron (2001) criterion, modiﬁed AIC (MAIC) and t or F tests on the marginal lag in the ADF. Whilst such criteria are explored more fully in the next chapter, some discussion of an appropriate procedure is included here to make this chapter self-contained.

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The set up is conveniently viewed in an error dynamics framework which, in summary, is the basic AR(1) model for yt supplemented by an ARMA model for the errors, which results in a potentially inﬁnite ADF model: an AR(1) model for yt

ϕ(L)zt = θ(L)εt ⇒

(8.3a)

ARMA model for the errors

ψ(L)zt = εt ⇒

invert MA polynomial, ψ(L) = θ(L)

A(L)yt = εt ⇒

A(L) = θ(L)−1 ϕ(L)(1 − ρL)

(8.3b) −1

ϕ(L)

(8.3c) (8.3d)

using the DF decomposition on A(L) ⇒ ∞

∆yt = γyt−1 + ∑i=1 ci ∆yt−i + εt

(8.3e)

Of particular relevance for the following bootstrap sequence is the case where ρ = 1; then in (8.3a) ∆yt = zt and in (8.3c) ψ(L)∆yt = εt . Truncating the ψ(L) polynomial, we have: zt = ∑j=1 ψj Lj zt + ε∗t,k k−1

(8.4)

j where ε∗t,k = ∑∞ j = k ψj L zt + εt . The interpretation of this step is that the errors, which in this case are just the ﬁrst differences of yt under the null hypothesis, are approximated by an AR(k – 1) process.

8.1.5.i Chang and Park sieve bootstrap The bootstrap set-up described here in essence follows that of Chang and Park (2003); the set-up in Psaradakis (2001) is slightly different, and the variations are described in the section following this one. For simplicity of notation the bootstrap is outlined using yt as the typical data ˆ t , where µ ˆ t is observation. If the data are demeaned or detrended, then yˆ˜ t = yt – µ the estimator for µt , replaces yt and, in effect, the test statistics and the bootstrap then relate to the residuals, yˆ˜ t , from the prior demeaning/detrending. Although the test statistics are the same in principle, whichever of the three schemes of no adjustment/demean/detrend is appropriate, the null distributions of the test statistics are not the same. This will be picked up automatically in the bootstrap procedure. The bootstrap steps are now described. 1.

Initial estimation 1(a). The initial regression is the (unrestricted) truncated ADF regression: ∆yt = γyt−1 + ∑j=1 cj ∆yt−j + εt,k k−1

t = k + 1, . . . , T

(8.5)

Estimated by LS, this is: ˆ t−1 + ∑j=1 cˆ j ∆yt−j + εˆ t,k ∆yt = γy k−1

(8.6)

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(1 − ρL)yt = zt ⇒

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where {ˆεt,k }Tt= k + 1 are the LS residuals. The selection of k will generally represent a practical decision on the lag length rather than prior knowledge, and this is discussed below. From this (single) sample, calculate the unit root statistics of interest, here the normalised bias coefﬁcient, δˆ s , and the τˆ -type statistic, τˆ s :

τˆ s =

Tγˆ

(8.7)

ˆ (1 − C(1)) γˆ ˆ σˆ (γ)

(8.8)

k−1 ˆ ˆ ˆ is the estimated standard error of γˆ ; C(1) where σˆ (γ) = ∑j=1 cj ; and the subscript s on a test statistic serves to emphasise that these values are from the sample. The notation τˆ s is used here for the t statistic on γˆ ; this is then taken generically to refer to τˆ , τˆ µ or τˆ β as the case is appropriate.

1(b). Estimate the ADF regression restricted by the imposition of a unit root, so that γ = 0, resulting in: ∆yt = ∑j=1 c˜ j ∆yt−j + ε˜ t,k k−1

(8.9)

˜ j ; and the estimates c˜ j Note that under the null ρ = 1, in which case c˜ j = ψ and residuals ε˜ t,k correspond to those restricted under the unit root null. Swensen (2003a) suggests using the Yule-Walker procedure rather than LS k−1 ˜ ˜ to estimate this regression. Also, for later reference, deﬁne C(1) = ∑j=1 cj . 1(c). Centre the restricted residuals to deﬁne ε˜ ct,k = ε˜ t,k − ε˜¯k where ¯ε˜k = ∑T k + 1 ε˜ t,k /(T − k); these centred, restricted residuals serve to deﬁne the empirical distribution function, E, from which draws are taken in the bootstrap sequence. (The centring is a relatively minor adjustment as prior (global) demeaning or detrending will ensure that the mean of yt is zero over the sample used in the prior regression.) For future reference, deﬁne the residual variance estimator based on the centred, restricted residuals: σ˜ ε2 = (T − k)−1 ∑t = k + 1 (˜εct,k )2 T

2.

(8.10)

Construct the bootstrap equivalent of zt = ∆yt by the recursion: zbt = ∑j=1 c˜ j zbt−j + e˜ bt k−1

t = k + 1, . . . , T; b = 1, . . . , B

(8.11)

Sampling of the errors e˜ bt is from the sequence {˜εct,k }Tk + 1 , where each e˜ bt has an equal probability of occurring in the sample. How to start the recursion is discussed below; however, once started, zbk + 1 is generated, which given {c˜ j }1k−1 and {e˜ k + 2 }, enables zbk + 2 to be generated, and so on.

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δˆ s =

330 Unit Root Tests in Time Series

3.

Given the sequence {zbt }, the next step is to generate the bootstrap data ybt for each replication (that is, each bootstrap sample). This is done imposing the unit root, ρ = 1, so that the recursion is: ybt = ybt−1 + zbt

(8.12)

ybt = yb0 + ∑i=1 zi + ∑i = k + 1 zbi k

t

(8.13)

The choice of initialisation, yb0 , is discussed below. 4.

Next, use the bootstrapped observations, ybt of step 3, to estimate the bootstrap ADF regression for each bootstrap sample, resulting in: (k −1) b cˆ j ∆ybt−j + eˆ bt

b ∆ybt = γˆ b ybt−1 + ∑j=1

(8.14)

where t = kb + 1, . . . , T; b = 1, . . . , B. The bootstrap residuals are {eˆ bt }Tk + 1 ; b the bootstrap ADF lag length is (kb − 1); the estimated LS coefﬁcients from one bootstrapped sample are γˆ b and cˆ bj , j = 1, . . . ,(kb − 1); and the bootstrap standard error of γˆ b is denoted σˆ (γˆ b ). As the notation indicates, the lag length in the bootstrap estimation is not necessarily the same as in the initial regression, and not necessarily the same in different bootstrap samples. (For consistency of notation, the bootstrap residuals should also have a kb subscript, but clarity is not endangered by omitting this.) Also deﬁne: T (ˆσεb )2 = (T − kb )−1 ∑k + 1 (eˆ bt − e¯ˆ bt )2 b

(8.15)

Thus (ˆσεb )2 is just as σ˜ ε2 of (8.10) in construction, but uses the bootstrapped residuals. 5.

There is just one estimate of γˆ b per bootstrap sample, but an alternative to cˆ bj is the initial restricted estimate c˜ j from step (1)(b). The cj coefﬁcients are ˆ required in order to calculate an estimate of C(1). The δ-type test statistic can be deﬁned using either cˆ bj or c˜ j in the denominator to give the following choice: δˆ bCP =

Tγˆ b ˆ b (1)) (1 − C

(8.16a)

δ˜ bCP =

Tγˆ b ˜ (1 − C(1))

(8.16b)

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An initialisation, yb0 , is needed, as is clear from back-substitution of the recursion, which gives the partial sum representation of the process:

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τˆ bCP =

γˆ b σˆ (γˆ b )

(8.17a)

τ˜ bCP =

γˆ b σ˜ (γˆ b )

(8.17b)

In (8.17a), σˆ (γˆ b ) is the coefﬁcient standard error from the bootstrap regression; whereas σ˜ (γˆ b ) in (8.17b) uses the same bootstrap quantities in its construction, except for σ˜ ε2 of (8.10) in place of (ˆσεb )2 . The simulation results in Chang and Park (2003) suggest that δˆ bCP and τˆ bCP have slightly better properties than δ˜ bCP and τ˜ bCP . 6.

Repeat the bootstrap sampling procedure B times, thus b = 1, . . . , B. Sort the B bootstrap values of δˆ bCP and τˆ bCP , and hence obtain their (bootstrap) cumulative distributions F(δˆ bCP ) and F(ˆτbCP ), respectively. Quantiles of interest can thus be obtained from these distributions; for example, the ˆb ˆ bs 5% quantile of F(δˆ bCP ), say δˆ bs 0.05 , satisﬁes #[δCP ≤ δCP,0.05 ]/(B + 1) = 0.05, where #[.] is the counting function.

7.

Decision rule: at the α signiﬁcance level (asymptotically) reject the null ˆ s< hypothesis of a unit root if δˆ s < δˆ bs CP,α or, where the t statistic is used, if τ τˆ bs CP,α . Different deterministic speciﬁcations are indicated in the usual way by subscripts, thus the usual set of test statistics comprises: δˆ bCP , δˆ bCP,µ and and τˆ b , τˆ b and τˆ b . δˆ b CP,β

CP

CP,µ

CP,β

There are a number of details that have to be decided in a practical application, discussion of these has so far been deferred. 1.

Lag length The dependence of the lag lengths k (and kb ) on some choice criteria was suppressed in the notation of the initial LS regression and bootstrap regression steps. In the case of an inﬁnite autoregression, the rate of expansion of k is usually a function of T, say k = k(T). A minimum implication for consistency is that k → ∞ as T → ∞, such that k/T→ 0; that is, k increases as T increases, but not as fast as T, implying that any rule should have ∆k(T)/∆T < 1. For example, k∗ /T1/3 → 0 as T → ∞, and there exist constants, ζ > 0 and s, such that ζk∗ > T1/s (see Said and Dickey, 1984). Chang and Park (2003) only require k(T) = op ([T/ ln T]1/2 ); whereas Psaradakis (2001) speciﬁes k(T) = op ([T/ ln T]1/4 ). These assumptions are, however, for proofs of consistency and are not designed for use in a single sample for which a level rather than a rate of

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There is also a choice of using either using (ˆσεb )2 or σ˜ ε2 in the bootstrap t statistic:

332 Unit Root Tests in Time Series

2.

Initialisation Initialisation is required at two points in the bootstrap sequence. The ﬁrst of these relates to the pre-sample differences, zbt for t = 0, –1, . . . , –(k – 2), which are required if the bootstrap data are to be generated starting at t = 1. For example, suppose an AR(1) process generates the errors, then the recursions start with zb1 = c˜ 1 zb0 + e˜ 1 , but zb0 is not available and an initialisation is required. One option is to set the pre-sample values to zero. Alternatively, the bootstrap recursion for the errors could be started at the ﬁrst time sample data is available, which is the scheme outlined in step 2. For example, in the restricted ADF(1), suppose that y1 and y2 are available, as is c˜ 1 from (8.9), then the recursion can start at t = 3; thus: zb3 = c˜ 1 z2 + e˜ 3 ;

zb4 = c˜ 1 zb3 + e˜ 3 ; . . . ;

zbT = c˜ 1 zbT + e˜ T ;

for b = 1, . . . , B

where z2 = ∆y2 = y2 −y1 is actual (conditioning) data. Although the asymptotic effect of the starting values is negligible, in ﬁnite samples this procedure induces a dependency of subsequent zbt on the initial conditions. An alternative is to randomise the starting values; for example, draw the starting values at random from {∆yt }T2 , or in a random block of length (k − 1). The second required initialisation is that which starts the ybt sequence, with initialised value denoted yb0 ; the choice is not material asymptotically (provided yb0 is stochastically bounded). This choice could also be randomised; however, typically yb0 is set to zero. This choice is sensible if yb0 refers to demeaned or detrended data, since the residuals from a prior regression have a zero mean by construction; a unit root regression without a constant should not be used unless the speciﬁcation of the alternative hypothesis implies a zero mean. The recursion speciﬁed in step 3 above assumes that yb0 is the value requiring initialisation, then actual data in the form of ∑kj=1 zj is used until the recursion for zbt can take over. 3.

Deterministic components An alternative strategy to demeaning or detrending the data is followed by Psaradakis (2001) who, in the case of an included time trend, adjusts

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expansion is required. In practice there are three frequent choices. In their simulation work, Chang and Park (2003) prefer AIC for the more likely situation where the autoregressions in (8.9), (8.11) and (8.14) are not ﬁnite; they found that it had better properties than BIC and MAIC (see Chapter 9 for further details of the Ng and Perron criterion). Psaradakis (2001) notes that in an MA(1) model a 10% two-tailed t test on the signiﬁcance of the longest lag is preferred in his simulations to AIC, BIC and FPE, especially where θ1 , the ﬁrst-order MA coefﬁcient, is large negative, for example, θ1 = –0.8.

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zt and the bootstrap recursions for yt , and includes the deterministic function µt directly in the bootstrap regressions (and in the initial regression – see step (1) above); the variations are considered in the next section.

The bootstrap scheme suggested by Psaradakis (2001) is slightly different from Chang and Park’s scheme. Instead of estimating an ADF regression both in the initial LS estimation and in the bootstrap loop, step 1 is to estimate a simple DF regression (the analogy here is with Phillips and Perron’s approach; see Chapter 6). Then, as in step 2, estimate a truncated AR regression using zt = ∆yt ¯ where y¯ for the cases of µt = 0 and µt = µ; and for µt = β0 + β1 t use zt = ∆(yt − y), is the (global) mean, y¯ = T−1 ∑Tt=1 yt . The deterministic terms are included in the regressions, so that demeaning/detrending is not done prior to the regressions. The Psaradakis scheme is referred to below as the P-scheme; analogously, Chang and Park’s scheme will be referred to as the CP-scheme. In the revised set-up for the Psaradakis approach the steps are referred to as P1 etc., to distinguish them from the CP-scheme. These are as follows, starting with the initial regression: P1(a).

ˆ t−1 + zˆ t ∆yt = µ ˆ ∗t + γy

the initial regression

(8.18)

The sample test statistics are the same in principle; that is, the normalised bias and the pseudo t statistic, but differ in practice because they apply to (8.18) rather than (8.6); these are denoted δˆ P = Tγˆ and τˆ P = γˆ /ˆσ(γˆ ). A subscript will be added to the test statistic as necessary to indicate the deterministic speciﬁcation. In contrast to the CP bootstrap, there is no choice of lag length in this step; thus there is no attempt to ‘whiten’ the errors, zt , and in principle these are not replaced by iid errors εt . Also, the ‘raw’ data are used rather than the demeaned or detrended data. P1(b). Estimate the ADF regression restricted by the unit root: this is as in the CPscheme, with the regression restricted by the imposition of a unit root, but note that the speciﬁcation of the dependent variable differs from CP in the with-trend case: ∆zt = ∑j=1 c˜ j ∆zt−j + ε˜ t,k

zt = ∆yt for µt = 0 or µt = µ

(8.19a)

∆zt = ∑j=1 c˜ j ∆zt−j + ε˜ t,k

¯ for µt = β0 + β1 t zt = ∆(yt − y)

(8.19b)

k−1 k−1

P1(c). Centre the residuals ε˜ t,k (again, as in the CP-scheme) to obtain ε˜ ct,k . P2. Construct the bootstrap equivalent of zt , as in the CP-scheme step 2, zbt , by the recursion: zbt = ∑j=1 c˜ j zbt−j + e˜ bt k−1

b = 1, . . . , B

(8.20)

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8.1.5.ii Psaradakis bootstrap scheme

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Psaradakis (2001) suggests starting this recursion from t = 1, with pre-sample values of zbt set to zero. The errors e˜ bt are sampled from the centred residuals obtained in step P1(c).

ybt = ybt−1 + zbt

if µt = 0 or µt = µ

(8.21)

ybt = y¯ + ybt−1 + zbt

if µt = β0 + β1 t

(8.22)

P4. In the CP-scheme, step 4 is to estimate the bootstrap equivalent of the ADF regression; however, as this is not the starting point in the P-scheme, in the latter the bootstrap estimation mimics the unaugmented regression in P1(a), resulting in: ˆ b ybt−1 + zˆ bt ∆ybt = µ ˆ ∗b t +γ

b = 1, . . . , B

(8.23)

where ˆ indicates the result from estimation with the bootstrap sample; for example, zˆ bt is the bootstrap residual. P5. The test statistics to be bootstrapped; that is, δˆ P and τˆ P , are obtained from P4 where the bootstrap regression makes no adjustment for the dependency in the errors (as in the Phillips-Perron (PP) approach to unit root testing). The bootstrap test statistics are: δˆ bP = Tγˆ b τˆ bP =

(8.24)

γˆ b σˆ (γˆ b )

(8.25)

where γˆ b and σˆ (γˆ b ) are obtained from (8.23) in step P4. The test statistics do not have the same asymptotic null distributions as the ADF versions of the tests. Otherwise, and as is usual, the bootstrap sampling procedure is repeated B times and the quantiles of the bootstrap distributions are obtained, with decision rules, in principle, as in the CP-scheme. As also for the CP-scheme, the inclusion of deterministic terms will be indicated by a subscript on the test statistic; for example, δˆ bP,µ and τˆ bP,µ for the tests with a constant in the maintained regression. The Psaradakis scheme is initially motivated as in the PP approach, in which an unaugmented DF regression is ﬁrst estimated, so that even though the errors are serially correlated, there is no attempt at this stage to make any adjustment. However, in the PP approach the test statistics are adjusted to give them the same distributions as their DF counterparts, with the adjustment factor based on the difference between the variance of zt and the variance of εt . However,

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P3. As in the CP-scheme, the next step is to generate the bootstrap data ybt for each replication, which must be done imposing the unit root, ρ = 1. The recursions need to distinguish whether there is a trend in the maintained (initial) regression as the data are not detrended prior to use in the scheme:

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335

2 to produce asymptotically pivotal test The PP approach is to estimate σz2 and σz,lr statistics, but that procedure is not followed here. Although in other contexts basing the bootstrap on pivotal quantities has been found to be advantageous, Swensen (2003a) suggests that this may not produce an improvement here, but the proposition remains to be tested. Given that the bootstrap is more demanding that the standard approach based on the DF statistics or some variant thereof, the question arises, what are the beneﬁts of bootstrapping? There are two key areas that the bootstrap can address. First, one problem with standard tests is that they do not deal well with certain kinds of weakly dependent error processes: a case in point being as simple as an MA(1) process with a large negative coefﬁcient; for example, zt = εt − 0.8εt−1 . In two inﬂuential studies, Schwert (1987, 1989) showed that actual sizes of DF and PP tests were severely distorted, and a number of contributions since that article have sought to address the problem of the inﬁdelity of the actual size to the nominal size. Second, although normality is not necessary for the validity of the DF tests, an assumption of normality is used in simulations to generate the standard tables of ﬁnite sample critical values for these tests. The bootstrap takes draws from the empirical distribution function, thus any non-normality represented in the residuals from the initial LS regression is ‘captured’ in the bootstrap procedure.

8.2 Simulation studies of the bootstrap unit root tests Having outlined two bootstrap procedures it is of interest to see how well they perform. As noted above, the CP-scheme directly bootstraps test statistics from the ADF regression, and the bootstrap versions of these tests have the same limiting distributions as their conventional DF counterparts. In contrast, in the P-scheme a simple unaugmented DF regression is ﬁrst estimated, as in the PP approach, but with the asymptotic distribution depending upon the particular structure of the departure of the distribution of zt from being iid. The bootstrap distribution of a particular test statistic automatically captures such departures. The implied comparisons are twofold. One level of comparison is for each of the bootstrap schemes against the conventional approach; and, second, whether one bootstrap scheme is better than the other. The simulation results in Chang and Park (2003) are not directly comparable with those in Psaradakis

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in the Psaradakis bootstrap approach this is not necessary as bootstrapping in a sense automatically generates the correct distribution for the particular model characteristics, so it is the distribution rather than the test statistic that changes from one application to another. In the presence of weakly dependent errors, the limiting null distributions of the standard and the bootstrapped test statistics 2 (see Chapter 6, section 6.7). depend on the unknown parameters σz2 and σz,lr

336 Unit Root Tests in Time Series

8.2.1 Chang and Park In the simulation setup of Chang and Park (2003), the error dynamics are generated by the MA(1) process zt = εt + θ1 εt−1 , with θ1 = (–0.8, –0.4, 0, 0.4, 0.8); AIC is preferred by CP to both BIC and the Ng and Perron (2001) procedure in steps (1) and (4) to choose the order of the truncated autoregression; there were 5,000 simulations with B = 5,000. Results were reported for T = 50 and T = 100, and we concentrate on the latter, with a summary of the results reported in Table 8.1 for θ1 = 0 and θ1 = –0.8. The reported results indicate the bootstrap tests are generally better than their Dickey-Fuller counterparts at maintaining empirical size close to the nominal size. (In the comparative study reported in section 8.2.3, the DF tests did maintain their size for θ = 0; see that section for a possible explanation of the difference.) In the case of θ1 = 0, the sizes of the bootstrap tests were 4.5% for δˆ bCP,µ and 6.2% for τˆ bCP,µ. The bootstrap tests also did reasonably well in the more difhad ﬁcult cases; for example, with θ1 = –0.8 and a nominal size of 5%, δˆ b CP,µ

an empirical size of 10.1% compared to 47.4% for the ADF version; and τˆ bCP,µ had an empirical size of about 11% compared to 36.2% for the standard ADF τˆ -type test. In the case of ρ = 0.9, the simulations relate to power, but a comparison is only sensible in this context where the nominal and actual size of the tests is close; but because of the considerable size distortions of the conventional DF tests when θ1 = 0, this means that a comparison of power cannot generally be undertaken. Overall, as is evident from the results reported in Table 8.1, the size distortions are substantial for the conventional tests, with marked improvements for the bootstrapped tests. 8.2.2 Psaradakis In a simulation study of his bootstrap procedure, Psaradakis (2001) considers a number of popular unit root tests; for example, the ADF τˆ -type test statistic, the Z-type semi-parametric tests of Phillips and Perron, and the modiﬁed versions of these (see Chapter 6, section 6.8). Also, as well as the usual case of normal errors, errors are drawn from a double exponential distribution and a χ2 distribution with 6 degrees of freedom. The DGP is yt = ρyt−1 + εt – 0.8εt−1 , and ρ = (1, 0.99, 0.95, 0.90, 0.85); the maintained regression includes a trend (whereas the

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(2001) to be described below. In the former, the deterministics in the maintained regression are speciﬁed as µt = µ rather than µt = β0 + β1 t as in the latter; and to select the truncation lag, Chang and Park use the AIC with the maximum lag set at 20 for T = 100, whereas Psaradakis prefers the marginal t test in a general-to-speciﬁc strategy. We ﬁrst summarise the key ﬁndings of each study and then undertake a comparison of the two approaches by way of a small simulation study. (See also Palm et al., 2008, for a comparative study of bootstrap procedures in the case of a unit root.)

Bootstrap Unit Root Tests

337

ˆ δ-type tests Bootstrap: δˆ bCP,µ

ADF: δˆ µ

τˆ -type tests Bootstrap: τˆ bCP,µ

ADF: τˆ µ

4.5% 13.3% 24.1%

12.9% 30.0% 49.1%

6.2% 11.6% 21.1%

8.2% 18.2% 34.2%

ρ=1 ρ = 0.95 ρ = 0.9 θ1 = 0.8

10.1% 20.2% 26.7%

47.4% 73.5% 82.4%

11.0% 21.1% 30.1%

36.2% 61.9% 77.5%

ρ=1 ρ = 0.95 ρ = 0.9

4.8% 10.7% 16.3%

20.4% 37.7% 50.2%

6.1% 11.7% 17.2%

9.1% 18.9% 28.3%

θ1 = 0 ρ=1 ρ = 0.95 ρ = 0.9 θ1 = –0.8

Notes: The DGP is yt = ρyt−1 + εt + θ1 εt−1 , where εt is niid(0, 1). The maintained ˆ˜ t = γ yˆ˜ t−1 + zt where yˆ˜ t = yt − y¯ and y¯ = ∑Tt=1 yt /T. Power is not regression is ∆ y size adjusted and the truncation lag is decided by AIC. Source: Chang and Park (2003, Tables III and IV).

reported results in CP (2003) relate to the demeaned case). The preferred selection criterion for the lag length was a two-sided 10% t test on the signiﬁcance of the coefﬁcient on the longest lag. 8.2.2.i Summary of results Psaradakis (2001) ﬁnds that his bootstrap test statistics δˆ bP,β and τˆ bP,β are the only ones to maintain the correct size across the range of experimental designs (see Table 8.2 for a selected summary of the results). Recall that these results are not directly comparable to the CP results, because of the differences in the DGP and the lag selection criterion. For example, for a 5% nominal size, the bootstrap tests δˆ bP,β and τˆ bP,β had an empirical size of 5.4% compared to close to 100% for the PP versions of the tests; that is, these tests nearly always reject, and about 50% for the modiﬁed Z-type δˆ test and about 50% for the ADF and modiﬁed Z-type τˆ test. There is not much variation in the results across different error distributions. The simulation results suggest that when the errors are generated by an MA(1) process, with θ1 = –0.8, there is a substantial advantage to the bootstrap tests δˆ b P,β

and τˆ bP,β over the non-bootstrapped versions, and the nominal size is accurate for these tests. Although in a comparison using size-adjusted power, δˆ bP,β and τˆ bP,β tend not to be more powerful than the alternative tests considered, especially the ADF τˆ -type test, this is to rather miss the advantage of the bootstrapping: nominal size is so unreliable in this situation that the nominal size of the nonbootstrapped tests is misleading. In practice, size-adjusted power is not available

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Table 8.1 Simulated size and power of the Chang and Park ADF tests (5% nominal size, T = 100).

338 Unit Root Tests in Time Series

Table 8.2 Simulated size (ρ = 1) and power (ρ = 0.9) of Psaradakis bootstrap tests for 5% nominal size, T = 100; θ1 = –0.8 throughout. τˆ -type tests

Error distribution

Bootstrap: δˆ bP,β

Zδˆ β

Normal ρ=1 ρ = 0.9

5.4% 11.7%

96.6% 48.8% 10.6% 11.2%

5.4% 12.0%

53.7% 100% 17.5% 8.6%

4.6% 9.5%

95.9% 51.0% 12.3% 10.8%

5.3% 9.8%

53.8% 99.8% 49.6% 17.0% 10.8% 10.8%

5.9% 9.6%

95.9% 52.2% 10.7% 10.3%

6.2% 9.6%

55.3% 95.8% 51.0% 12.3% 8.7% 10.3%

Double-exp ρ=1 ρ = 0.9 χ2 (6) ρ=1 ρ = 0.9

MZδˆ β

Bootstrap: τˆ bP,β

ADF: τˆ β

Zˆτβ

MZˆτβ 47.3% 11.1%

Notes: The DGP is yt = ρyt−1 + εt − 0.8εt−1 and the maintained regression included a constant and a linear time trend as well as yt−1 ; power is size adjusted. The truncation lag is determined by two-sided marginal t test at the 10% signiﬁcance level. Source: Extracted from Psaradakis (2001, Table 1).

if the critical values that are used (in conventional tests) do not deliver a close correspondence between nominal and actual size – and they do not in the cases considered with MA errors. (A secondary but important point on the low power of the tests reported here relates to the inclusion of a constant and trend in the deterministics of the maintained regression: these are redundant under both the null and the alternative and considerably reduce the power of the tests.) 8.2.2.ii Lag truncation criteria Psaradakis (2001) ﬁnds that the lag truncation criteria matter: for both bootstrap test statistics, if the AIC is used then the empirical size for a 5% nominal size test, with T = 100, is 19.2%, whilst it is about 35% if BIC is used, but close to the nominal 5% level if the two-sided marginal-t criterion is used. Using a 10% signiﬁcance level for the two-sided t test ensures that the lag length is long enough (mean lag of 5.4) compared to AIC (mean lag 4.0) and BIC (mean lag 3.3) to provide a satisfactory approximation to the inﬁnite AR inversion of the MA(1) process. If the 10% t test selection criterion is used, then this also suggests that the Chang and Park versions of the tests may have even better size conformity for θ1 = –0.8. This and other issues are considered in the next section, whereas the importance of the lag selection criteria is considered in Chapter 9. 8.2.3 A brief comparison of the methods of Chang and Park and Psaradikis The simulation results reported in sections 8.2.1 and 8.2.2 leave some practical issues unresolved. In particular, is there anything to choose between the different bootstrap procedures using the same selection criterion? In order to provide

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ˆ δ-type tests

339

some guidance on these issues, the Chang and Park and Psaradakis versions of the bootstrap tests were simulated using the marginal t statistic lag selection criterion and AIC for choosing the lag length. The DGP was yt = ρyt−1 + εt + θ1 εt−1 and the maintained regression included a constant, with T = 200. To assess size, ρ = 1 and θ1 = 0, –0.4, –0.8; and to assess power, ρ = 0.95 and ρ = 0.9; there were 1,000 simulations and 999 bootstrap replications. In addition to the bootstrap tests, the standard ADF versions of δˆ µ and τˆ µ were also included; the critical values for these tests are obtained from response surfaces that use the adjustment for lag length suggested by Cheung and Lai (1995a, 1995b), as updated by Patterson and Heravi (2003) (see Table 6.12 of Chapter 6). 8.2.3.i AIC versus marginal-t as lag selection criteria The ﬁrst set of simulations are for ρ = 1 and θ1 = 0, therefore, assess size in the simplest situation (see Table 8.3a). When the lag is selected by AIC, the empirical size is reasonably close to 5% in most cases, but slightly undersized for δˆ bP,µ in both versions. The actual sizes of the ADF δˆ µ and τˆ µ tests were not found to be as inaccurate as Chang and Park report (12.9% and 8.2%, respectively, whereas we ﬁnd 4.6% and 5.4%); this may be due to a difference in critical values since those used here adjust for sample size and lag length, although not for the search procedure. The AIC versions of the tests tend to perform better than their marginal-t counterparts; as expected, the former choose a shorter lag length and so power is not as penalised relative to the zero lag case. As to power, referring here to the AIC selected tests, when ρ = 0.95 the Chang and Park versions of the test are more powerful than the Psaradakis versions; however, when ρ = 0.90 this conclusion is reversed, suggesting slightly different gradients to the power curves. The δˆ versions of the tests are more powerful than the τˆ -type tests when the lag is selected by AIC, but this is not uniformly so when marginal-t is used. There is some deterioration in size for some of the tests when moderate MA(1) serial correlation is introduced into the error process (see Table 8.3b for the case θ1 = –0.4). The conventional tests are oversized at about 10% or 12% for a nominal 5% size. Overall, the Psaradakis versions of the tests are best in this respect, and these versions of the tests are also best when it comes to assessing power. Note that all versions of the test are always less powerful when the lag truncation criterion is the marginal-t test compared to AIC, and the relatively small differences in size do not account for these differences. Thus, given that size in this case is well-maintained, using AIC (the average lag using AIC is about 1.9 whereas it is about 3.5 with the t test), there is no advantage to selecting a longer lag and power suffers quite markedly if this is the case. Also, on a pairwise comparison for each method, the δˆ versions of the tests are more powerful than the τˆ tests; this ﬁnding conﬁrms the ranking with the standard versions of

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Bootstrap Unit Root Tests

340 Unit Root Tests in Time Series

ADF: δˆ µ

CP: δˆ bCP,µ

P: δˆ bP,µ

ADF: τˆ µ

CP: τˆ bCP,µ

P: τˆ bP,µ

ρ=1

4.6%

4.6%

3.6%

5.4%

6.4%

4.6%

ρ = 0.95 ρ = 0.90

48.0% 94.4%

47.2% 94.4%

42.4% 96.6%

34.8% 82.4%

37.2% 85.4%

33.2% 89.4%

ˆ δ-type; lag selected by 10% t-test (two-sided) ρ=1 ρ = 0.95 ρ = 0.90

ADF: δˆ µ 7.6% 51.0% 88.6%

CP: δˆ bCP,µ 5.8% 30.2% 67.4%

P: δˆ bP,µ 3.0% 23.4% 81.8%

τˆ -type tests; lag selected by 10% t test (two-sided) ADF: τˆ µ 5.8% 33.2% 69.0%

CP: τˆ bCP,µ 5.8% 39.8% 75.2%

P: τˆ bP,µ 4.6% 20.6% 78.2%

Table 8.3b Simulated size: θ1 = –0.4 and ρ = 1; simulated power: θ1 = –0.4 and ρ = 0.95, θ1 = –0.4 and ρ = 0.9 ˆ δ-type; lag selected by AIC τˆ -type tests; lag selected by AIC

ρ=1 ρ = 0.95 ρ = 0.90

ADF: δˆ µ

CP: δˆ bCP,µ

P: δˆ bP,µ

ADF: τˆ µ

CP: τˆ bCP,µ

P: τˆ bP,µ

12.2%

10.2%

5.6%

9.4%

9.8%

5.6%

61.0% 95.0%

49.4% 91.2%

72.8% 100%

45.6% 85.4%

42.8% 83.8%

70.4% 99.8%

ˆ δ-type; lag selected by 10% t test (two-sided) ρ=1 ρ = 0.95 ρ = 0.90

ADF: δˆ µ 11.8% 57.6% 89.4%

CP: δˆ bCP,µ 8.2% 47.2% 84.4%

P: δˆ bP,µ 5.2% 65.0% 99.8%

τˆ -type tests; lag selected by 10% t test (two-sided) ADF: τˆ µ 8.4% 35.0% 74.0%

CP: τˆ bCP,µ 8.4% 36.2% 74.6%

P: τˆ bP,µ 5.6% 63.0% 99.8%

these tests. However, in the case of the standard tests, the advantage in power is counteracted by the deterioration in size ﬁdelity when there is strong MA serial correlation. The next set of simulations addresses the question of whether this is also the case for the bootstrapped versions of the tests when θ1 = –0.8. As shown in Table 8.3c, the lack of size retention is evident when θ1 = –0.8 across all tests. The various tests are oversized, and more so than in the case of θ1 = –0.4; this means that rather than reject at the nominal rate (size) of 5% when the null hypothesis is correct, the tests reject more often than is justiﬁed (> 5%); there is a spurious over-rejection of the null hypothesis of nonstationarity in favour of stationarity. Failing use of the correct (ﬁnite sample) null distribution,

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Table 8.3 Comparison of bootstrapping procedures. Table 8.3a Simulated size: θ1 = 0 and ρ = 1; simulated power: θ1 = 0 and ρ = 0.95, θ1 = 0 and ρ = 0.9 ˆ δ-type tests; lag selected by AIC τˆ -type tests; lag selected by AIC

Bootstrap Unit Root Tests

341

Table 8.3c Simulated size: θ = –0.8 and ρ = 1; simulated power: θ1 = –0.8 and ρ = 0.95, θ1 = –0.8 and ρ = 0.9 ˆ δ-type; lag selected by AIC τˆ -type tests; lag selected by AIC CP: δˆ bCP,µ

P: δˆ bP,µ

ADF: τˆ µ

CP: τˆ bCP,µ

P: τˆ bP,µ

39.1% 96.5% 100%

27.6% 65.6% 95.4%

14.0% 30.0% 82.0%

25.6% 85.4% 100%

17.6% 64.8% 94.8%

14.2% 29.6% 82.8%

ˆ δ-type; lag selected by 10% t test (two-sided) ρ=1 ρ = 0.95 ρ = 0.90

ADF: δˆ µ 45.3% 98.6% 100%

CP: δˆ bCP,µ 20.4% 55.0% 85.6%

P: δˆ bP,µ 8.8% 38.8% 90.0%

τˆ -type tests; lag selected by 10% t test (two-sided) ADF: τˆ µ 32.4% 92.1% 100%

CP: τˆ bCP,µ 17.6% 48.6% 80.8%

P: τˆ bP,µ 9.6% 38.6% 89.8%

Note: Entries in bold indicate good or best performance.

the critical values need to be larger negative to achieve the correct rate of rejection. The practical issue is therefore whether the bootstrap procedure can lead to size being more accurately maintained. To some extent, size suffers with all versions of the bootstrap test, but the Psaradakis versions of the tests do quite well at holding back the spurious increase in size, and slightly more so when the lag is selected by the marginal-t criterion. For example, when the lag was selected by AIC, the empirical sizes of δˆ bCP,µ and δˆ bP,µ were 27.6% and 14%, respectively, but these were 20.4% and 8.8% when the lag was selected by marginal-t; there was no marked difference in empirical size or empirical power between δˆ bP,µ and τˆ bP,µ . The lag could be increased further to improve size retention; for example, Swensen (2003a) found that a lag length of 12 maintained an accurate size when θ1 = –0.8; however, the cost is a loss of power. 8.2.3.ii Summary Overall there is no single conclusion: much depends on the extent of the serial correlation. In the absence of serial correlation, δˆ bCP,µ , with the lag selected by AIC, is best because the known power advantage of this form of a unit root test can be exploited, whilst choosing a relatively parsimonious lag. If the MA serial correlation is moderate, the simulation results suggest that the advantage, both in terms of maintaining size and obtaining the best power, lies with the Psaradakis versions of the tests; and power is better with the AIC selection criterion. When there is near-cancellation in the unit root, as in the case θ1 = − 0.8, the Psaradakis tests maintain their advantage, but do not deliver perfect actual sizes. The advantage is now with the marginal-t procedure to select the lag length because it chooses a longer lag than the AIC.

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ρ=1 ρ = 0.95 ρ = 0.90

ADF: δˆ µ

342 Unit Root Tests in Time Series

Table 8.4 ‘Aysmptotic’ size: θ1 = –0.8 and ρ = 1 (T = 2,000). ˆ δ-type; lag selected by 10% t test (two-sided) CP: δˆ bCP,µ 6.00%

P: δˆ bP,µ 5.4%

ADF: τˆ µ 5.4%

ˆ δ-type; lag selected by 10% t test (two-sided) ρ=1

ADF: δˆ µ 4.6%

CP: δˆ bCP,µ 5.8%

P: δˆ bP,µ 5.0%

CP: τˆ bCP,µ 5.6%

P: τˆ bP,µ 5.2%

τˆ -type tests; lag selected by 10% t test (two-sided) ADF: τˆ µ 4.4%

CP: τˆ bCP,µ 5.0%

P: τˆ bP,µ 5.4%

8.2.4 ‘Asymptotic’ simulations Finally, some simulations were undertaken with T = 2,000 and θ = –0.8, to indicate two related aspects of the properties of the test statistics: ﬁrst, that the problem with over sizing is a ﬁnite sample problem; and second, that, at least asymptotically, the test statistics are correctly sized (a minimum requirement that the bootstrap procedure is valid). The number of simulations and bootstrap replications were set quite low at 500 and 199, respectively; this was partly for practical reasons, but also, as it happens, to show that the bootstrap delivered acceptable results. The results are reported in Table 8.4, and show that despite the relatively small number of bootstrap replications, the tests are quite uniformly well sized at 5% or close thereto, whichever selection criterion is used. (The results for power are not reported separately, as simulated power is 100% for ρ = 0.95 and ρ = 0.90, uniformly for all tests.)

8.3 Illustration: US unemployment rate The essence of the bootstrap procedure is to provide a better approximation to the null distribution of the test statistic than is provided by the ‘conventional’ null distribution, which does not explicitly allow for any features present in the serial correlation structure of the input noise, εt . Thus, where εt lacks, or has relatively weak, serial correlation, the bootstrap is not expected to yield an advantage over standard procedures. If the bootstrap is applied in such a case, the bootstrap null distribution and the conventional null distribution will show relatively minor differences. Otherwise, if the bootstrap is to correct for oversizing, then, for a test at the 5% level, ceteris paribus, the left-sided 5% quantile of the bootstrap distribution must be to the left of the corresponding quantile for the conventional distribution. In other words, if oversizing is to be corrected, then, for a given sample value of the ADF test statistic, the p-value

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ρ=1

ADF: δˆ µ 5.6%

τˆ -type tests; lag selected by 10% t test (two-sided)

Bootstrap Unit Root Tests

343

11 10 9 8 7 %p.a.

5 4 3 2

1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Figure 8.1 US unemployment rate (monthly data).

Table 8.5 ML estimation of ARIMA(2, 0, 2) for US unemployment rate. yt − y¯

¯ −1 (yt − y)

¯ −2 (yt − y)

εˆ t−1

εˆ t−2

Estimated ‘t’

1.823 (42.78)

–0.831 (–19.74)

–0.850 (–16.03)

0.246 (6.15)

from the bootstrap distribution will exceed the p-value from the conventional distribution. (In fact, this does not always occur, as in the illustration below.) This illustration uses the US unemployment rate, with monthly (seasonally adjusted) data for the period 1949m1 to 2004m12, a total of 672 observations. The data are graphed in Figure 8.1. As in Chapter 7, there is a question to be considered as to whether the unemployment rate is a legitimate series to be modelled as an unbounded random walk, but there are many reported examples of such a series being used. The lag lengths chosen for the conventional ADF tests are 12 by AIC and 15 by marginal-t. These long lags suggest that serial correlation might well be of an MA form and an ARMA model would provide a better representation of the DGP. This was conﬁrmed by estimation of ARMA(p, d, q) models with d = 0; that is, the null was not imposed at this stage, and p ≤ 4, q ≤ 4. The estimated ARIMA(2, 0, 2) model is reported in Table 8.5 and is representative of the general structure, with asymptotic ‘t’ statistics in parentheses.

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6

344 Unit Root Tests in Time Series

No lag Lag(AIC) p-value Lag(t) p-value

ADF: δˆ µ

CP: δˆ bCP,µ

P: δˆ bP,µ

ADF: τˆ µ

CP: τˆ bCP,µ

P: τˆ bP,µ

–16.426 3.4% –19.504 2.3%

–16.426 3.9% –19.504 2.4%

–5.698 n.r. 7.6% n.r. 8.2%

–2.695 8.0% –2.842 5.8%

–2.695 6.2% –2.842 4.5%

–1.730 n.r. 18.4% n.r. 20.0%

Notes: Lag(AIC) and Lag(t) are the unit root test statistics with lags chosen by AIC and marginal-t, respectively. The test statistics δˆ bP,µ and τˆ bP,µ are from regressions without lags; ‘n.r.’ = ‘not relevant’ as there are no lags in the initial regression for this version of the test statistic.

The AR coefﬁcients clearly suggest the possibility of a unit root, with a sum of 0.992. Whilst a test for a unit root could be based on this ARMA model, here it serves a different purpose of indicating that the signiﬁcant MA components suggest serial correlation, which could be a problem for conventional tests. The sum of the MA coefﬁcients is about –0.6, suggesting that using critical values for the conventional ADF tests may lead to an oversized test. The various test statistics and p-values are reported in Table 8.6. The conventional ADF test statistics are δˆ µ = –16.426 and τˆ µ = –2.695 for the AIC selected lag of 12 and δˆ = –19.504 and τˆ µ = –2.842 for the marginal-t selected lag of 15. The next issue is to obtain the p-values of these test statistics. In the case of the non-bootstrapped procedures, there are (at least) three possibilities. The ﬁrst is to use the null distribution as, for example, tabulated by Fuller (1976, 1996) and extended to response surfaces by MacKinnon (1991). The second is to use a null distribution that corrects for the lag length in ﬁnite sample, as in Cheung and Lai (1995a, 1995b) or Patterson and Heravi (2003); the former limits the lag length to 8, and the latter to a lag length of 12; nevertheless, some adjustment in ﬁnite samples is likely to better than none. However, in this case the adjustment will be negligible given the size of the sample. The third, and preferable, case is to simulate the null distribution taking into account the search procedure over the lag length. In this case the p-values are 3.4% and 2.3% for δˆ µ by AIC and marginal-t, respectively, and 8.0% and 5.8% for τˆ µ by AIC and marginal-t, respectively. At a conventional 5% signiﬁcance level, δˆ µ leads to rejection of the null, whereas in the case of τˆ µ the decision is for non-rejection of the null, at least marginally. Next, we consider the bootstrap p-values, which were based on 9,999 bootstrap replications. In the CP bootstrap scheme the sample value of the test statistic is as in the ADF case, but it is the (simulated) null distribution that changes. The CP versions deliver broadly the same result as the conventional ADF p-values, whether the lag is chosen by AIC or marginal-t, giving bootstrap p-values of 3.9% and 2.4% for the sample value of δˆ µ , and 6.2% and 4.5% for

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Table 8.6 Test statistics for US unemployment rate.

345

the sample value of τˆ µ , respectively. The situation is somewhat different for the Psaradakis bootstrap distributions and hence bootstrap p-values. In this case the test statistics are those relating to the unaugmented DF regressions, with sample values of δˆ bP,µ and τˆ bP,µ equal to –5.698 and –1.730, respectively. The bootstrap pvalues corresponding to lag selection by AIC and marginal-t are 7.6% and 8.2%, respectively, for δˆ bP,µ , and 18.4% and 20%, respectively, for τˆ bP,µ. These values suggest non-rejection of the null hypothesis. Whilst conﬂicting in their outcomes, the results of the CP and Psaradakis versions of the bootstrap test are at least consistent with the simulation results reported earlier. Recall that the problem of oversizing that; is too many rejections of the null of a unit root, increases with the extent of MA induced serial correlation, and in this case, the simulation results reported in Tables 8.3b and 8.3c suggested that the Psaradakis versions of the test make a correction for size, resulting in fewer rejections of a true null hypothesis. Nevertheless, the question of conﬂicting outcomes (that is ‘test conﬂict’,) is something that cannot be overlooked when multiple tests are applied and is taken up as a topic in Chapter 9.

8.4 Concluding remarks This chapter has considered some variants on bootstrap schemes for unit root tests. Although the focus has been on bootstrapping the normalised bias and τˆ -type tests in DF, ADF and PP type regressions, the principle is quite general and may be applied to different regression forms and test statistics; for example, the GLS-type detrending procedure of ERS could be bootstrapped using the same general scheme as for bootstrapping the conventional ADF test, although in this case Chang and Park (2003) report no marked improvement. The correction of size distortions is a vital aspect of the practical application of unit root tests, and bootstrap tests offer one means of achieving substantial, if not complete, size corrections. However, from the various simulation studies it is clear that unless long enough lag lengths are selected in the initial and bootstrap regressions, the size correction is not achieved. For example, Swensen (2003a) found that it was necessary to choose the lag length to be about 4 for θ1 = –0.4, but 12 for θ1 = –0.8. Psaradakis (2001) preferred the twosided marginal-t procedure to both the AIC and the BIC, although Chang and Park (2003) used the AIC. We found that the AIC was satisfactory when there was moderate MA(1) serial correlation, θ1 = –0.4, and slightly more powerful than when the marginal-t procedure was used. However, with substantial MA(1) serial correlation, for example, θ1 = –0.8, the advantage lay more clearly with marginal-t. Unless the strength of the serial correlation is known beforehand,

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Bootstrap Unit Root Tests

one seeks a procedure that will protect against unknown serial correlation without being too demanding on power; this issue is taken up in greater detail in the next chapter. The issue of choosing an appropriately long lag length raises the question of whether the advantage in the bootstrapped unit root tests is simply because proper attention is being paid to selecting ‘long’ enough lags. That is, what is the marginal contribution of the bootstrapping procedure? That there is value added from bootstrapping is indicated from the simulations in which the same, in principle, bootstrap and conventional unit root test statistic is used with the same selection criterion. Generally, as the serial correlation becomes more severe, the greater the size distortion of the conventional tests; both bootstrap versions of the test statistics tend to correct the size ‘inﬁdelity’, with an advantage, as indicated from the small simulation study reported in section 8.2.3, to the Psaradakis versions of the test statistics, despite their non-pivotal nature. A more extensive study of the comparative properties of different bootstrap tests would seem to be a useful line of research. For further research on the bootstrap in related contexts, see Kim and Hwang (2004), Poskitt (2008) and Murphy and Izzeldin (2009).

Questions Q8.1 Brieﬂy describe the essential differences between the Chang and Park and the Psaradakis bootstrap schemes (the CP-scheme and the P-scheme, respectively). A8.1.i The initial regressions differ: in the CP-scheme, the test statistics are based on an augmented DF regression, whereas in the P-scheme no adjustment is made for serially correlated errors. Hence the distributions of the different versions of the ‘same’ test statistics are not the same, asymptotically or otherwise, in the presence of serial correlation. A8.1.ii In the CP-scheme the data is demeaned or detrended prior to estimation, whereas in the P-scheme the deterministic function is directly included in the regression model; this is a relatively minor difference. A8.1.iii The different schemes require slightly different bootstrap set-ups. In particular, in the CP-scheme, step (4), is to estimate the bootstrap equivalent of the ADF regression, but in the P-scheme the bootstrap estimation mimics the unaugmented regression not an ADF regression. Q8.2 What are the asymptotic distributions of the t-type tests suggested by: (i) Chang and Park, and (ii) Psaradakis? A8.2.i The asymptotic distribution of τˆ bs CP is just the conventional DF distribution, where the precise form of the Brownian motion depends on the

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346 Unit Root Tests in Time Series

Bootstrap Unit Root Tests

347

deterministic terms speciﬁed in µt (see Chang and Park, 2003, Theorem 1). In the case of no prior adjustment of the data, or µt = 0, then τˆ bs CP weakly converges to its standard DF counterpart; that is: 1

B(r)dr ⇒D 0 1/2 ≡ F(ˆτ)DF 1 2 0 B(r) dr

(8.26)

where B(r) is a standard Brownian motion process deﬁned on r ∈ [0, 1]. (For the technical details of background of statements regarding weak convergence ⇒D and stochastic order in the bootstrap case, see Theorem 2 and its development in Chang and Park, 2003.) The notational conventions on the different forms of Brownian motion follow Psaradakis (2001). For the asymptotic distributions in the demeaned and detrended cases see section 6.3.3. A8.2.ii In this case, we just need to refer to the asymptotic distributions of the PP versions of the DF tests (see Equations (6.106) and (6.109), and Psaradakis, 2001, Lemma 1). Q8.3 The simulation results in Psaradakis (2001, Table I) show that sizecorrected power for the ADF t-type test is greater, often substantially so, than the bootstrapped test τˆ bP,β . For example, with T = 100, θ1 = –0.8, ρ = 0.9 and normal errors, the simulated power of the ADF test, τˆ β , is 17.5% compared to 12.0% for the bootstrapped test. Does this imply that it would be better to use the ADF test? A8.3 No: recall that whilst size-adjusted power offers a helpful comparison of different tests, its practical use is limited since one of the design aspects of an inference procedure is to control the size of the test. Thus in the ADF example the simulated size for a 5% nominal size, using conventional DF critical values, was 53.7% (see Table 8.2); this indicates that the correct critical value to use is quite substantially more negative than the conventional value and depends on θ1 .

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τˆ bCP

9

Introduction The chapter is concerned with two related issues. The ﬁrst concerns the need to select a lag or truncation parameter when there is serial dependence in the errors. Unit root tests can become severely mis-sized in the presence of serially correlated errors, especially if there is an MA component to the errors; however, increasing the lag length in an ADF regression, or increasing the bandwidth in a semi-parametric estimator of the long-run variance, may have costs in terms of a loss of power if the some of the included lags are superﬂuous. It is important, therefore, to have an assessment of different methods of selecting the ‘truncation’ parameter. The emphasis in this chapter is on parametric models, but similar considerations apply to semi-parametric methods. To ﬁx ideas, section 9.1 deals explicitly with the case where the errors are weakly dependent and generated by a stationary ARMA process. However, the approach generalises to cases where the errors are generated by a linear process not restricted to the ARMA class (see, for example, Berk, 1974; Wang et al., 2002; Chang and Park, 2002). The primary methods for lag selection are based on information criteria, such as AIC and BIC, or on hypothesis testing applied to the marginal coefﬁcients. The latter could be in the framework of a simple-to-general or general-to-simple approach, where the former works out from a minimum lag to see if it needs to be lengthened and the latter works in from a maximum lag to see if it can be shortened. We subject only the latter to a detailed examination as a simple-to-general approach is not generally recommended as a lag selection criterion because of the potential inconsistency of the procedure. The second issue relates to the common occurrence in econometrics of using multiple tests for a single characteristic, this includes the possibility of combining the same test statistic with different lag selection criteria. The single characteristic of interest is whether a unit root is present in the DGP. Given 348

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Lag Selection and Multiple Tests

the multiplicity of such tests, only some of which have been presented in previous chapters, it is not surprising that some practitioners report several tests, although it is not clear that such an approach has necessarily taken into account the impact on the overall type I error. As noted, even using just one test statistic will imply a multiple testing procedure if different lag selection criteria are used; thus a ‘test’ should be understood to comprise a test statistic; for example, τˆ µ , combined with a lag selection method. The framework outlined below in section 9.2 has a quite general application to other areas of econometric testing; for example, multiple tests for serial dependence are commonplace in the speciﬁcation of empirical equations. The section shows that when multiple tests are involved, then there will generally be conﬂict as well as agreement in the test outcomes. Finally, some of the issues raised in the chapter are considered in the context of an empirical illustration in section 9.3.

9.1 Selection criteria In the case of errors generated by an ARMA process, two cases can be distinguished. First, ML and GLS methods of estimation explicitly model the ARMA components, as in Chapters 6 and 7, and the problem is that of identifying the orders of the AR and MA components; these might be regarded as direct methods of estimation. Second, methods based on the ADF regression extend the lag length in order to ‘whiten’ the residuals, and the problem is to determine how long the lag needs to be in order to deliver desirable properties to the unit root test; by contrast, this is generally an indirect method of dealing with the serial correlation in the errors. In the ﬁrst case, the problem of order determination has been studied extensively elsewhere and is not the primary focus of this chapter. The usual solution to this problem is to base order selection on information criteria, which remains a valid consideration when there is a unit root. Rather, the focus in this chapter is on the second case, where there may not be a true order of the ADF regression, although there are implications for selecting orders of the AR and MA components for the direct estimation methods. Within the ADF-type approach, there are two cases. In the ﬁrst case, there is no MA component, so the errors are generated by a ﬁnite-order AR process. In this case there is a true order of the AR process. In the second case, there is an MA component either with or without an AR component; then a method that approximates the MA component by extending the AR lag length order does not seek a true order, since that is inﬁnite, but an approximating order that delivers desirable properties to the test statistic. For example, in the context of an AR approximation to an MA process, the issue of how good the approximation is relates to choosing the lag length that maintains the size of the test and ensures that the power properties of the test are retained.

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Lag Selection and Multiple Tests 349

The simulations reported in Chapters 6 and 7 have indicated that incorrect selection of the lag length will have an effect on the size of the test. Further, even if the correct size can be achieved, the power of the test will be affected by including lags that are unnecessary to that aim. Also note two points concerning the ﬁnite sample distributions of unit root test with nonzero lags. First, even if the lag length is known, the ﬁnite sample distribution is not generally the same as the zero lag case. Second, a typical practical procedure is to make the lag selection depend on the data, implying a search procedure that determines the lag length; in effect the chosen lag length becomes a random rather than a ﬁxed variable, which also implies a change to the ﬁnite sample distribution of a test statistic. 9.1.1 ADF set-up: reminder In this section we consider the case where zt is generated by a stationary ARMA process. Assuming invertibility of the MA component and no roots in common in the AR and MA polynomials, the error dynamics model is familiar but is restated below for convenience: (1 − ρL)yt = zt

(9.1a)

ϕ(L)zt = θ(L)εt A(L)yt = εt ∆yt = γyt−1 +

(9.1b) A(L) = θ(L)

−1

ϕ(L)(1 − ρL)

(9.1c)

∞

∑ cj ∆yt−j + εt

(9.2)

j=1

where the last step is simply an application of the ADF decomposition, j A(L) = (1 − αL) − C(L)(1 − L) where C(L) = ∑∞ j=1 cj L (see Chapter 3). Deterministic terms have not been included explicitly in (9.1a), but can be accommodated by reinterpreting yt as the demeaned or detrended series. Two cases are distinguished. If there is an MA component then the ADF lag polynomial A(L) is of inﬁnite order and the problems are, ﬁrst, to determine an appropriate rate of expansion as T increases; and second, to determine an appropriate lag order for a ﬁxed sample size. The ﬁrst aspect is relevant to simulation settings; for example assessing the asymptotic properties of a test statistic, and the second to empirical applications where just one sample of size T is available. First, consider the case where there is an MA component, then the ADF lag polynomial is of inﬁnite order and the problem is to decide on the truncation parameter. That is, to decide on the parameter k in the following: ∆yt = γyt−1 + ∑j=1 cj ∆yt−j + εt,k ∞ ∞ εt,k = ∑j=k+1 aj yt−1 + ∑j = k cj ∆yt−j + εt k−1

(9.3) (9.4)

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350 Unit Root Tests in Time Series

Lag Selection and Multiple Tests 351

Note that truncating the ADF(∞) model to ADF(k∗ ), where k∗ = k – 1, introduces the ‘remainder’ term εt,k , and the problem can be viewed as choosing k∗ such that εt,k is negligible in some sense. In the case that there is just an AR polynomial of order p, then, as there is a correct order, the problem is slightly different. The set-up is: k−1

(9.5) (9.6)

where I(p > k) is the indicator function, such that I(p > k) = 1 if p > k and 0 otherwise. ˆ The implications for the δ-type tests of estimating an ADF regression as an approximation to ARMA errors are slightly different from those for the τˆ -type tests; the essential point is that the coefﬁcient, γ, on yt−1 is no longer (ρ − 1), ˆ This point and the necessary hence the δˆ test statistic is no longer formed as Tγ. developments were covered in Chapters 5 and 6. A brief recap follows, with the notation somewhat more explicit to recognise the truncation involved in ˆ forming the feasible version of δ. If the errors zt are generated by an ARMA process, then γ = ψ(1)(ρ − 1) where ψ(L) = θ(L)−1 ϕ(L) and ψ(1) = θ(1)−1 ϕ(1); hence: γ (ρ − 1) = ψ(1) =

θ(1) γ ϕ(1)

(9.7)

As a preliminary next step, assume that ψ(1) is known, but γ is estimated, then ˆ the limiting null distribution of (δ|ψ(1)) is: 1 Tγˆ 0 B(r)dB(r) ˆ (δ|ψ(1)) = (9.8) ⇒D 1 2 ψ(1) 0 B(r) dr 1 B(1)2 − 1 ˆ DF ≡ F(δ) = 1 2 2 0 B(r) dr ˆ DF is the standard DF asymptotic distribution (see Said and Dickey, where F(δ) 1984, p. 606; Xiao and Phillips, 1998, p.30). What is necessary to maintain this distributional equality is a consistent estiˆ mator of ψ(1) say ψ(1). Under the null hypothesis, ψ(1) can be replaced by 1 − C(1) = 1 − ∑∞ c but, as noted, this is not practical because of the inﬁnite j=1 j upper limit on the summation. Assume that a truncation lag kˆ ∗ is determined ˆ k∗ (1) = 1 − ∑kˆ ∗ cj , then the corresponding δˆ test statistic is: and let C j=1

ˆ kˆ ∗ ) = δ(

Tγˆ ˆ k∗ (1) 1−C

(9.9)

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∆yt = γyt−1 + ∑j=1 cj ∆yt−j + εt,k p ∞ εt,k = I(p > k) ∑j = k+1 aj yt−1 + ∑j=k cj ∆yt−j + εt

352 Unit Root Tests in Time Series

Note that in contrast to the τˆ test, the δˆ test also involves the coefﬁcients on the stationary variables, c = (c1 , . . . , ck−1 ) . Also let τˆ (kˆ ∗ ) be the corresponding τˆ ˆ kˆ ∗ ) test statistic, then the questions of interest are under what conditions will δ( ∗ ˆ DF and τˆ (kˆ ) ⇒D F(ˆτ)DF . The conditions relate to the expansion rate of ⇒D F(δ) kˆ ∗ , which are considered in section 9.1.3; prior to that consideration, the next section illustrates the sensitivity of size and power to lag length selection.

To illustrate the nature of the problem, dependent errors were generated by MA(1) and AR(1) processes, respectively, and the size and power of the (A)DF tests δˆ µ and τˆ µ were examined for lag lengths k∗ = 0, 1, . . . , 10; the nominal size for the simulations is 5% and T = 200. For the power calculations ρ = 0.95. The results for MA(1) errors are considered ﬁrst (see Figures 9.1–9.3). As to size, the ﬁrst point to note is the benchmark on the graph that the tests are correctly sized when the empirical size cuts the horizontal line at the value 0.05. Consider Figure 9.1 for δˆ µ and Figure 9.2 for τˆ µ , in which empirical size is graphed for θ1 = 0.0, –0.3 and –0.6, the negative values being more likely for economic data. In the case of θ1 = 0.0, note that τˆ µ is much better behaved than δˆ µ , since the latter shows oversizing when superﬂuous lags are included, whereas the former maintains its nominal size to within 1% point. Both tests are oversized for θ1 = –0.3, but whereas δˆ µ does not achieve the correct size even for

1 θ1 = –0.6

0.8

Empirical 0.6 size 0.4 θ = –0.3 1 0.2 0

0

1

size does not settle to 5% as the lag increases

θ1 = 0

2

3

4

5

6

7

8

9

5

6

7

8

9

lag 1

θ1 = –0.6

Empirical 0.8 power 0.6 0.4

θ1 = –0.3 θ1 = 0 0

1

2

3

4 lag

Figure 9.1 MA(1) errors: size and power δˆ µ .

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9.1.2 Illustration of the sensitivity of size and power to lag length selection

Lag Selection and Multiple Tests 353

1 0.8

θ1 = –0.6

Empirical 0.6 size 0.4

θ1 = –0.3 θ1 = 0

0.2 0

0

1

2

3

4

5

6

7

8

9

5

6

7

8

9

1.5

θ1 = –0.6

θ1 = –0.3

1 Empirical power

0.5 0

θ1 = 0 0

1

2

3

4 lag

Figure 9.2 MA(1) errors: size and power τˆ µ .

1 θ1 = 0 0.5 0

0

1

2

3

4

5

6

7

8

9

10

1 θ1 = –0.3

minimum lag necessary to remove size distortions

0.5 0

0

1

2

3

4

5

6

7

8

9

10

0.4 minimum lag necessary to remove size distortions

θ1 = –0.6 0.2 0

0

1

2

3

4

5 lag

6

7

8

9

10

Figure 9.3 MA(1) errors: relative frequency of lags chosen by AIC.

k∗ = 10, τˆ µ is within 1% point of the nominal size with k∗ = 3, and is perfectly sized at k∗ = 5; moreover, this accurate sizing is maintained with lags that are superﬂuous to the aim of removing size distortions. Note, however, that these

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lag

superﬂuous lags have a cost in terms of a loss in power, as shown in the lower graph to Figure 9.2, where power declines beyond the minimum lag necessary to remove size distortion; for example, there is power of 36% at k∗ = 3, but 25% at k∗ = 8. The case with θ1 = –0.6 tells the same overall story, but shows that k∗ = 7 is now needed to remove size distortions for τˆ µ , although size distortions are not removed for δˆ µ . Although the cost to being one lag or so out is not too great, the decline in power is sharper as θ1 becomes more negative. As a precursor to the next section, the lag length was also selected by AIC, with the distribution of lag lengths for the MA(1) case shown in Figure 9.3. The modal lags were: 0 for θ1 = 0.0; 1 for θ1 = − 0.3; and 3 for θ1 = − 0.6. The percentage of lags chosen at or above (and hence superﬂuous to) those required to remove size distortions for τˆ µ was 22% for θ1 = –0.3 and only 9% for θ1 = –0.6. Selecting the lag by AIC, tends to underestimate the lag length required to correct for size distortions, a situation that worsens as θ1 becomes more negative. For example, when θ1 = − 0.3, a lag length of 1, the modal AIC lag, results in an actual size about twice the nominal size, whereas when θ1 = − 0.6 using the modal AIC lag of 3 results in an actual size nearly three times the nominal size. As θ1 becomes more negative, the lag required to remove size distortions departs more from the modal lag. Overall, it is evident that using AIC as the lag selection criterion when the errors follow an MA(1) process, results in a lag length that is generally too short to remove the size distortion and gives a misleading impression of power. A fortiori, these concerns will also apply to lag selection by BIC, which, when it differs from AIC, (generally) selects a shorter lag length. The discussion of the results for the AR(1) errors is briefer than for MA(1) errors as the key points have already been outlined. Also, in this case there is a correct lag length of one for the ADF; hence, what the simulations indicate is whether size and power suffer if superﬂuous lags are included. The relevant ﬁgures are Figures 9.4–9.6. The graphs are shown for positive values of ϕ1 = ψ1 (the equality holds because there is no MA component), which are illustrative of wider set, not reported here; positive ϕ1 are likely to be more relevant, and note that as ϕ1 → 1, then a second unit root is approached. Figures 9.4 and 9.5 show the size and power of δˆ µ and τˆ µ , respectively, for ϕ1 = 0, 0.3 and 0.9. Note that whilst δˆ µ has (nearly) matching nominal and empirical size at the correct lag of one, adding superﬂuous lags increases the empirical size, which does not stabilise as lags are added; as a result the empirical size is double the nominal size at a lag of 8. Note from the lower panel of Figure 9.4 that power increases as the lag length increases; however, this is misleading in the sense that the increase in power is not adjusted for the increase in size shown in the upper panel. The behaviour of τˆ µ contrasts with that of δˆ µ (see Figure 9.5). τˆ µ is better behaved, although not as powerful as δˆ µ if size can be controlled. The empirical

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354 Unit Root Tests in Time Series

Lag Selection and Multiple Tests 355

0.2

ϕ1 = 0 ϕ1 = –0.3 ϕ1 = –0.6

0.15 Empirical size

0.1 size does not stabilise with superfluous lags

0.05 0

0

1

2

3

4

5

6

7

8

9

lag

0.6 Empirical power

0.4

0

ϕ1 = 0 ϕ1 = –0.3 ϕ1 = –0.6

power increases because size is not controlled

0.2 0

1

2

3

4

5

6

7

8

9

lag Figure 9.4 AR(1) errors: size and power δˆ µ .

0.2 ϕ1 = 0.9

0.15

ϕ1 = 0

Empirical 0.1 size

size stabilises

0.05 0

ϕ1 = 0.3 0

1

2

3

4

5

6

7

8

9

8

9

lag 0.4 0.3

0.1 0

ϕ1 = 0.3

ϕ1 = 0

Empirical 0.2 power

power declines with superfluous lags

ϕ1 = 0.9 0

1

2

3

4

5

6

7

lag Figure 9.5 AR(1) errors: size and power τˆ µ .

size is very stable provided that a lag of at least one is chosen. As expected power declines as superﬂuous lags are added, but the rate of decline is fairly slow; the costs to a loss of power are when the selected lag is quite a long way from the correct lag length of one, for example for ψ1 = 0.3, power declines from 32% at a

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0.8

356 Unit Root Tests in Time Series

1 ϕ1 = 0

0.5 0

1

2

3

4

5

6

7

8

9

10

7

8

9

10

11

12

7

8

9

10

11

12

1

0

0

1

2

3

4

5

6

1 ϕ1 = 0.9

0.5 0

0

1

2

3

4

5

6 lag

Figure 9.6 AR(1) errors: relative frequency of lag chosen by AIC.

lag of one, to 23% for a lag of eight. Figure 9.6 is a histogram of the lags selected by AIC; the correct lag is chosen 82%, 71% and 70% of the time for ϕ1 = 0, 0.3 and 0.9, respectively. What is clear from this preliminary analysis is that whilst much can be done to formulate test statistics that are more powerful than the family of DF test statistics, an equally important issue is to have a practical method of determining the lag length that removes size distortions at the least cost to power. Selecting the lag length by AIC does not fulﬁl these criteria, neither does selection by BIC as this method results in even shorter lag lengths. The next section considers the lag selection issue in greater detail.

9.1.3 Consistency and simulation lag selection criteria The importance of choosing the lag can be approached from two aspects: ﬁrst, from the viewpoint of establishing the asymptotic properties of estimators and illustrating the ﬁnite-sample sensitivity using simulations with different sample sizes; second, the question of selecting a lag in a single sample, which is the usual practical situation. The following two conditions on the expansion rate for the lag length, due to Berk (1974) and Said and Dickey (1984), for a stationary and invertible ARMA error processes, have been a starting point for assessing lag selection rules and which have been referred to in previous chapters (see, for example, Chapter 6).

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ϕ1 = 0.3

0.5

Lag Selection and Multiple Tests 357

A2: There exist positive constants, ζ and s, such that k∗ > T1/s /ζ. This states that k∗ is bounded from below by a positive multiple of T1/s to ensure that k∗ is not too small. This is a lower bound condition that restricts k∗ to be at least a polynomial rate in T and rules out values of k∗ that are proportional to log T. This condition is relevant for the properties of cˆ = (cˆ 1 , . . . , cˆ k−1 ) . ˆ of π = (γ, c1 , . . . , ck−1 ) under Consistency of the (vector) least squares estimator π the null, is proved by Said and Dickey (1984, Theorem 5.1)), with both A1 and A2 used in their proof, and γˆ is consistent at rate T, known as ‘super-consistency’, √ whilst cˆ is consistent at rate T. A proof of the consistency of cˆ under the stationary alternative is provided by Lopez (1997). It is helpful to distinguish the different assumptions that are required for the ˆ distributional results to hold for the τˆ -type and δ-type tests. The differences arise ˆ because the τˆ -type tests, unlike the δ-type tests, do not require an estimate of c = (c1 , . . . , ck−1 ) ; that is, the coefﬁcients on the stationary variables ∆yt−j , j = 1, . . . , k – 1. We start with the results for the τˆ -type tests. ADF τˆ -type tests Ng and Perron (1995, Lemma 2.1) show that γˆ is consistent at rate T with A1 alone; that is, k∗ = o(T1/3 ). This implies that if k∗ is chosen, such that just A1 is satisﬁed, then: τˆ (k∗ ) ⇒D F(ˆτ)DF

τˆ µ (k∗ ) ⇒D F(ˆτµ )DF

τˆ β (k∗ ) ⇒D F(ˆτβ )DF

(9.10)

This result also holds if k∗ is selected by an information criterion (see section 9.1.4.iii), a result also due to Ng and Perron (1995, Theorem 4.3). Chang and Park (2002) considered the case where zt is a general linear process, i such that zt = ∑∞ i=0 πi L εt , where {εt } is a martingale difference sequence (MDS) r |π | < ∞ for some r ≥ 1 (a condition satisﬁed by ARMA processes). with ∑∞ |i| i i=0 Note that an MDS may exhibit conditional heteroscedasticity, for example, as in second-order stationary ARCH (autoregressive conditional heteroscedasticity) and GARCH (generalised autogressive conditional heteroscedasticity) type processes. They show that, as far as τˆ tests are concerned, the result (9.10) still holds for the faster rate k∗ = o(T1/2 ) compared to the slower rate k∗ = o(T1/3 ) of A2. ˆ ADF δ-type tests √ First consider the consistency of cˆ . The condition A2 is sufﬁcient to ensure T√ consistency of cˆ . Also, T-consistency is maintained if A2 is replaced by: ∞

A2a : T1/2 ∑j = k |cj | → 0 as k → ∞ and T → ∞

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A1: k∗ /T1/3 → 0 as T → ∞ and k∗ → ∞; this states that k∗ should expand with T, but at a slower rate than T1/3 , thus k∗ = o(T1/3 ) or, if the rule is data-dependent, op (T1/3 ). This is an upper-bound condition that prevents k∗ from growing too quickly relative to T.

The condition A2a is more obviously focused on controlling the error arising from truncating the inﬁnite regression since it relates to the approximation arising from truncating the lag. Both A2 and A2a rule out values of k∗ proportional to log T, a rate which is important in practice because it follows from information-based lag selection criteria (see section 9.1.4.iii). If A2 or A2a are not satisﬁed, cˆ will still be consistent but at a slower rate √ than T, so this is a relevant condition for the approximation of a model with an MA component. Faster convergence is expected to improve ﬁnite sample performance of the selection criterion. ˆ Xiao and Phillips (1998) prove a result equivalent to (9.10) for ADF δ-type tests using assumptions A1 and A2 (their assumption 4), so that: ˆ ∗ ) ⇒D F(δ) ˆ DF δ(k

δˆ µ (k∗ ) ⇒D F(δˆ µ )DF

δˆ β (k∗ ) ⇒D F(δˆ β )DF

(9.11)

Since A2a is basically equivalent to A2, it could replace A2 in this result. Chang and Park’s (2002) results for the slightly more general (MDS) set-up noted above (which includes ARMA error processes) are that (9.11) holds if: k∗ = o(T/ log T)1/2 ) ∗

k = o(T

1/3

)

for a homogeneous MDS for a possibly heterogeneous MDS

Finally, note that when there is no MA component in the error process, the term given by (9.6) must at some point become zero by assumption A2a; in fact, for consistency of cˆ , and for (9.11) to hold, k∗ does not have to grow with T, provided it is chosen to be at least equal to the true order. 9.1.4 Lag selection criteria The conditions A1 and A2 (or A2a) provide appropriate lag selection rules for situations involving a sequence of samples; for example, proofs of asymptotic properties, rather than the rules appropriate to a single sample of size T. Hence some means of making a practical choice is required in a single sample, which is consistent with the assumptions were it to be applied to samples of increasing sizes. However, it is useful to start from the point where a ﬁxed lag is used, and then explore rules that set k as a function of T or are data-dependent. 9.1.4.i Fixed lag An example of a ﬁxed k∗ occurs when the lag length is set to coincide with the frequency of the data; for example, 4 for quarterly data, 12 for monthly data. However, the ﬁxed lag rule takes no account of the characteristics of the data and suffers from problems with size and/or power. As noted in section 9.1.2 (see Figures 9.1 and 9.2) it was observed that the empirical size did not settle to the nominal size for δˆ µ for all values of θ1 = 0 considered, although correct sizing could be obtained for τˆ µ with a sufﬁciently large lag length. Ng and Perron

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358 Unit Root Tests in Time Series

Lag Selection and Multiple Tests 359

9.1.4.ii Choose k∗ as a function of T An alternative to the ﬁxed lag rule is one based on a deterministic function of the sample size. An example of a deterministic rule is one of the following form: k∗ (j, s) = int[(T/100)1/s j]

(9.12)

where int[.] is the integer part of the expression (see, for example, Schwert, 1989). Typical applications have chosen j = 4 (12) and s = 4. Although k∗ (j, s) does increase with T, the rule is fairly insensitive; for example, in the case of j = 4 and s = 4, k∗ (j, s) = 4 for T between 100 and 244, and then k∗ (j, s) = 5 for T between 245 and 506. Note that this rule implies k∗ (j, s) = O(T1/s ), so that (T1/s /T1/3 )→ 0 as T → ∞ provided that s > 3, which justiﬁes the typical choice of s = 4. Schwert’s (1989) simulation rules have sometimes been seen as model selection rules, but a problem with this practice is that their performance in the latter role depends upon the AR and MA parameters, so that the characteristics of the sample data should affect the choice of lag length. Schwert’s simulations conﬁrm that in a DGP with an MA(1) component, then as θ1 → –1, size is better maintained with j = 12 rather than j = 4. It seems sensible, therefore, to build data-dependency into a selection rule. 9.1.4.iii Data-dependent rules Information criteria-based selection rules Information criteria rules are popular means of selecting empirical models. The idea is to use a criterion that has a trade-off between ﬁt, as measured by the residual variance, and ‘complexity’, as indicated by the number of regressors. The particular objective function is of the general form: 2 IC(n) = log[˜σε,k (n)] + n × C(T, n)/T

(9.13)

where n is the number of regressors including any deterministic terms (which may have been removed by a prior detrending procedure). The penalty function is C(T, n) > 0, with the property that C(T, n)/T → 0 as T → ∞, so that asymptotically the lag length is determined by the ﬁrst term. Let K be the number of deterministic terms, then in an ADF(k – 1) = AR(k) regression, n = K + (k − 1)

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(1995) also report simulation results that show that the difﬁculty of achieving correct size with weakly dependent errors. On this basis, it might be thought that one strategy is to set the ﬁxed lag large enough to cope with any anticipated serial correlation in the errors; for example, perhaps a lag of 12 in a sample of T = 200 observations. However, there are two problems with this strategy: ﬁrst, it ignores the possibility that lag will need to vary with the sample size; and second, superﬂuous lags generally affect the power of the test.

360 Unit Root Tests in Time Series

+ 1 = K + k; note that if there are no deterministic terms, then n = k. The effective sample comprises T# = T – k observations, and this number, rather than T, is sometimes used in deﬁning IC(n). Also, the information criterion is alternatively written multiplying IC(n) in equation (9.11) by T (or T# ), but this does alter the selected value of n. The residual variance estimator σ˜ 2 (n) does not make an adjustment for degrees of freedom, but does use the actual number of observations in the sample, thus: T

(9.14)

and εˆ t,k is the t-th residual from LS estimation of (9.3). The objective function ∗ . Note that IC(n) is to be minimised by choosing k∗ from a range 0 ≤ k∗ ≤ kmax ∗ for ﬁxed K, the information criterion, IC(n), only varies with k . Let k˜ ∗ be the notation for the k∗ that minimises IC(n), where the comparison is made on a ∗ ; if the selected lag common sample based on an a priori maximum lag, kmax ∗ order is less than kmax then the model can be re-estimated with the additional observations. The penalty, or complexity, function for increasing the number of lags is C(T, n), with different penalty functions distinguishing the particular form of the IC; for example: Akaike Information Criterion : AIC(n) = ln[˜σ2 (n)] + n × 2/T Schwarz Information Criterion : BIC(n) = ln[˜σ2 (n)] + n × ln T/T Hannan-Quinn Information Criterion : HQIC(n) = ln[˜σ2 (n)] + n × ln(ln T)/T so that C(T, n) = 2, C(T, n) = ln T/T, C(T, n) = ln(ln T)/T, respectively. The BIC (for Bayesian information criterion) is sometimes referred to as SIC, although BIC is used here. Ng and Perron (1995) show that information rules lead asymptotically to log T proportionality in the selection of k∗ (see also Shibata, 1980; Hannan and Diestler, 1988). The key result is lim[k˜ ∗ (T)/b] log T = 1 as T → ∞, where b is a constant and k˜ ∗ (T) is the selected lag for a sample of size T, so that the use of an information criterion leads to truncation lags proportional to log T. As noted above in section 9.1.3, if k∗ is selected by an information criterion, then the limiting distributions of the DF τˆ -type tests are as in the simple DF case (see Equation (9.10)), but this convergence rate is not sufﬁciently fast for ˆ the same result to apply to the δ-type tests with errors driven by a heterogeneous MDS. The log T proportionality rule does not satisfy A2, or equivalent formu√ ˆ lations, which is important for the δ-type tests; hence the T-consistency of cˆ j fails to hold. Modiﬁed AIC: MAIC Consider the ADF model of equation (9.3) where the problem is to determine the appropriate truncation to the ADF lag, k∗ = k – 1. Ng and Perron (2001)

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2 σ˜ ε,k (n) = (T − k)−1 ∑t = k + 1 εˆ 2t,k

Lag Selection and Multiple Tests 361

have suggested modiﬁed versions of the information criteria, which have been found to lead a better performance for DF tests. They differ from the standard information criteria by including a measure of the departure from the value of γ = 0 under the null. The modiﬁed form of the Akaike information criterion, designated MAIC, is given by: 2[κ(k∗ ) + k∗ ] ∗ ) (T − kmax

(9.15)

where: κ(k∗ ) = (1/˜σk2∗ )γˆ 2 ∑t = k∗ + 1 y2t−1 max T

∗ )−1 ∑t = k∗ + 1 εˆ 2t,k σ˜ k2∗ = (T − kmax max T

(9.16) (9.17)

As usual, if deterministic components are included in the speciﬁcation of µt , then an estimator of µt is used and y˜ t replaces yt throughout. The form of the adjustment in (9.15) suggests the following general way of modifying the information criteria: MIC(k∗ ) = ln[˜σ2 (k∗ )] +

[κ(k∗ ) + k∗ ] C(T, n) ∗ ) (T − kmax

(9.18)

∗ ); and MHQIC MAIC results from C(T, n) = 2; MBIC from C(T, n) = ln(T − kmax ∗ from C(T, n) = ln[ln(T − kmax )]. The key difference between the standard and modiﬁed versions of the information criteria is due to the inclusion of a term in γˆ 2 due to κ(k∗ ) in (9.16), which is an indication of the departure from the null hypothesis; ceteris paribus, at the margin the selected lag length will be that which favours the null hypothesis. Ng and Perron (2001, especially Figure 1) show that if the errors are generated by an MA(1) process, then κ(k∗ ) varies little with k∗ for θ1 > 0, but is sensitive to θ1 < 0. In the latter case there can be a substantial discrepancy in approximating an MA(1) process by an AR process, unless a long lag is used.

General-to-speciﬁc sequential selection tests Another popular selection criterion is based on a general-to-speciﬁc (G-t-S) search on the lag length. The G-t-S principle starts from a maximum lag length to be considered, and then assesses, by means of a simple algorithm, whether the longest lag, regarded as marginal in this context, is actually needed. A stopping rule is required for the G-t-S principle to be feasible. The G-t-S principle is particular attractive where, as in this case, there is a natural order from a higher lag order to a lower lag order, which reduces the number of combinations to be evaluated. This selection criterion requires the ∗ , to be considered, and a signiﬁcance level prior choice of a maximum lag, kmax for each of the tests on the marginal coefﬁcient(s); then an algorithm is based

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MAIC(k∗ ) = ln[˜σ2 (k∗ )] +

on a deletion criterion for the last g lags. The most often used version of this ∗ ) be the t statistic for the hypothesis that test is with g = 1; for example, let t(kmax ∗ ) is zero; if |t(k∗ )| is signiﬁcant then the coefﬁcient on the last lag in ADF(kmax max ∗ ∗ ∗ ∗ − 1); decide that k = kmax , otherwise estimate ADF(kmax –1) and calculate t(kmax ∗ ∗ − 1)| is signiﬁcant and consimilarly, decide in favour of ADF(kmax –1) if |t(kmax tinue with the sequence if the t statistic is not signiﬁcant. The last possible test in the sequence is the ADF(1) which, if not rejected, leads to a simple DF model. The selected lag from such a procedure is denoted kˆ ∗ . There are variations on this procedure; for example, use a joint test, such as an F or χ2 (g) test, on the last g ≥ 1 lags. Such a lag selection rule will be denoted G-t-S(g), where g ≥ 1. ∗ A second variation on a G-t-S principle starts with an F-test (or χ2 ) on all kmax ∗ lags then, if not rejected, tests all (kmax – 1) lags, and so on, so that only the last test is a one degree of freedom t test. Whichever version is used, there is a stopping rule and the overall signiﬁcance level of the procedure will generally exceed the size of an individual test. ∗ For consistency of the procedure, kmax should not be ﬁxed as T increases. Ng ∗ and Perron (1995) show that if kmax satisﬁes A1 and A2a, and a G-t-S(g) rule is ∗ . If the latter increases at a rate used, then kˆ ∗ increases at the same rate as kmax faster than log(T), then so will the former, and as a result τˆ (kˆ ∗ ) ⇒D F(ˆτ)DF . An important practical question is the signiﬁcance level of each test, and, for ∗ samples of increasing size, the rate at which kmax increases. Typically, a fairly generous signiﬁcance level has been used at each stage; for example, 10%, 15% or even 20%; the latter seemingly required to ensure that a sufﬁciently long lag is chosen when the errors follow and MA(1) process. However, it is as well to be aware that multiple testing implies that the overall type I error is generally greater than the type I error at any particular stage. Such issues are taken up below in section 9.3. 9.1.5 Simulation results To illustrate some of the key issues, we ﬁrst concentrate on a comparison of size and power of the DF test τˆ µ ; δˆ µ is not included in this comparison, as the results already reported show that, although more powerful than τˆ µ when the errors are not dependent, it has an actual size that, in ﬁnite samples, does not settle to the nominal level as the lag length increases. Note also that whilst the comparison here uses the DF test τˆ µ , Ng and Perron (2001) report the results of an extensive simulation experiment with ten test statistics or their variants, and ﬁnd that the MAIC works well in controlling size with quasi-differenced data (that is, GLS detrending) for the PT tests and their DF variants (of Chapter 7, sections 7.4.2 and 7.4.6) and for the QD variations of the modiﬁed Z statistics (of Chapter 6, section 6.8.3).

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362 Unit Root Tests in Time Series

Lag Selection and Multiple Tests 363

Size: ρ = 1.0

τˆ µ

τˆ µ : G-t-S

τˆ µ : AIC

τˆ µ : MAIC

θ1 = –0.6 Average lag θ1 = –0.3 Average lag θ1 = 0.0 Average lag θ1 = 0.3 Average lag θ1 = 0.6 Average lag

80.2 0 24.8 0 5.0 0 2.3 0 2.5 0

7.7 4.7 5.6 4.1 5.2 3.8 5.4 4.2 5.3 4.9

7.3 3.5 6.3 1.9 5.5 0.8 5.8 2.0 5.1 3.8

4.6 4.2 3.8 2.3 3.9 1.0 3.9 2.3 3.9 3.8

Power: ρ = 0.95 θ1 = –0.6 Average lag θ1 = –0.3 Average lag θ1 = 0.0 Average lag θ1 = 0.3 Average lag θ1 = 0.6 Average lag

100 0 92.0 0 33.5 0.0 6.3 0 3.7 0

43.9 4.6 35.2 4.1 32.8 3.8 34.7 4.2 39.5 4.9

40.7 3.1 40.0 1.8 34.8 0.67 35.7 2.0 37.7 3.8

28.0 4.8 23.6 2.7 22.9 0.9 23.4 2.4 26.1 3.7

Note: Bold entries indicate best or good performance.

The cases reported in Tables 9.1 and 9.2 are for MA(1) errors with θ1 = –0.6, –0.3, 0, 0.3, 0.6, and for AR(1) errors with ϕ1 = –0.6, –0.3, 0, 0.3, 0.6, and ρ = 1.0, 0.95, where T = 200 (a wider range of parameters was considered, but not reported as the conclusions are not altered). When ρ < 1, the initial conditions are drawn from the unconditional distribution for yt . The lag selection criteria included in this comparison are: AIC, MAIC and G-t-S using the marginal-t criterion – that is, G-t-S(1) – with a two-sided 20% signiﬁcance level; smaller signiﬁcance levels were also tried, but 20% was necessary to give the G-t-S criterion good size properties as the absolute value of θ1 increased. In the case of a pure AR error, the question resolves to selecting the true order rather than the best approximation as in the MA(1) case. 9.1.5.i MA(1) errors As to size, when θ1 < 0 and ρ = 1.0, G-t-S and MAIC are slightly better than AIC; if the serial correlation is moderate then G-t-S is best, but MAIC is best as θ1 → –1. Note that these results are not explained just by a long lag, since

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Table 9.1 MA(1) errors: empirical size and power of τˆ µ using different lag selection criteria.

364 Unit Root Tests in Time Series

Size: ρ = 1.0

τˆ µ

τˆ µ : G-t-S

τˆ µ : AIC

τˆ µ : MAIC

ϕ1 ϕ1 = –0.6 Average lag ϕ1 = –0.3 Average lag ϕ1 = 0.0 Average lag ϕ1 = 0.3 Average lag ϕ1 = 0.6 Average lag

50.9 0 17.8 0 5.0 0 2.6 0 4.4 0

5.9 4.0 5.7 3.9 6.2 3.8 5.9 6.4 6.4 4.0

6.1 1.7 6.3 1.6 5.8 0.7 6.1 6.2 6.2 1.6

4.2 1.9 4.2 1.8 4.3 0.9 4.2 3.8 3.8 1.8

Power: ρ = 0.95 ϕ1 = –0.6 Average lag ϕ1 = –0.3 Average lag ϕ1 = 0.0 Average lag ϕ1 = 0.3 Average lag ϕ1 = 0.6 Average lag

92.8 0 53.4 0 12.9 0 1.5 0 3.0 0

16.8 4.0 17.2 3.9 17.3 3.7 16.5 4.0 15.7 4.0

14.7 1.7 15.3 1.5 14.4 0.7 14.9 1.6 14.5 1.6

9.4 2.0 9.3 1.8 9.7 0.9 9.8 1.5 9.4 1.8

Note: Bold entries indicate best or good performance.

using G-t-S results in a longer average lag than MAIC. When θ1 = 0 and ρ = 1.0, size is reasonably well maintained, despite the additional variation induced by the search process. All methods do quite well when θ1 > 0 and ρ = 1.0; G-t-S and AIC result in slight oversizing, whereas MAIC results in slight undersizing. Throughout, G-t-S results in the selection of the longest average lag. When ρ = 0.95, the simulations relate to power. There is a loss in power using MAIC relative to G-t-S and AIC, (where the comparison is sensible), which is not entirely explained by the slight undersizing. Overall, MAIC is superior for the case with a large negative MA(1) coefﬁcient, but it does not have a particular advantage when the negative MA(1) coefﬁcient is not ‘too’ large and when there is a positive MA(1) component. 9.1.5.ii AR(1) errors In the case of AR(1) errors, the problems are known to be less severe than with MA(1) errors, as the problem of near-cancellation of the unit root is not present. It is replaced by a different problem, namely that a second unit root is

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Table 9.2 AR(1) errors: size and power of τˆ µ , different lag selection criteria.

approached as ϕ1 → 1. Also, there is now a correct lag of one for the ADF. When ρ = 1.0, there is some slight undersizing with MAIC, which tends to increase as ϕ1 increases, but none of the lag selection methods are associated with severe problems. As expected, not augmenting the DF regression leads to a size inﬁdelity, with oversizing for ϕ1 < 0 and undersizing for ϕ1 > 0. Selection by G-t-S leads to the longest average lag, which exceeds the true value. As to power, when the lag length is selected by by G-t-S or AIC, power is similar and better than when MAIC is used (even when power is size adjusted); although G-t-S does imply a penalty for choosing superﬂuous lags (see Figure 9.5, lower panel), power as a function of lag length shows a fairly slow decline, which keeps the G-t-S procedure competitive. 9.1.5.iii Lag lengths Also of interest is the question of how multiple tests are related and we touch on this brieﬂy here and more extensively in section 9.3. Note that, faced with the same regression information, the different criteria may select different lags, and the degree of agreement can be assessed by considering the probability of (A) each method choosing the same lag. Thus let L(A) = {Lr }Rr=1 be the sequence of lags chosen by method A over the replications r = 1, . . . , R; and let L(A) , L(B) and L(D) be the sets of lags resulting from selection by G-t-S, AIC and MAIC, respectively. (A) (B) Then, for example, pˆ AB = ∑Rr=1 I(Lr = Lr )/R, where I(.) is the indicator function, estimates the probability of choosing the same lag length on replication r, with methods A and B, that is, G-t-S and AIC. The estimates pˆ ij are reported in Tables 9.3 and 9.4 for the three possible combinations of pairs. MA(1) errors In the case of MA(1) errors, see Table 9.3. Generally, G-t-S has least in common with MAIC, despite the similarly long average lag when θ1 = 0, and AIC has more in common with MAIC than with G-t-S. When θ1 = 0, G-t-S has about 37% of lag lengths in common with AIC and 28% with MAIC; this occurs because the marginal signiﬁcance level has been set at a value that is generous when the errors are actually iid, hence G-t-S tends to then over-parameterise the lag length. AR(1) errors In the case of AR(1) errors, see Table 9.4, the set with least in common is that which compares the lags chosen by G-t-S and MAIC, with broadly only 30% of the cases resulting in the same lag length. This is, perhaps, surprising given that these criteria are similar in tending to choose a longer lag than AIC, but note that this also occurred with MA(1) errors. The set with most in common is

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Lag Selection and Multiple Tests 365

366 Unit Root Tests in Time Series

ρ = 1.0

pˆ AB

pˆ AD

pˆ BD

θ1 = –0.6 θ1 = –0.3 θ1 = 0 θ1 = 0.3 θ1 = 0.6

59.8 45.9 36.7 45.5 62.3

47.4 35.6 28.2 32.6 39.6

54.0 61.3 64.7 60.8 55.0

ρ = 0.95 θ1 = –0.6 θ1 = –0.3 θ1 = 0 θ1 = 0.3 θ1 = 0.6

55.8 44.0 36.7 45.2 61.9

41.2 32.1 25.7 28.4 31.0

31.8 45.4 55.1 46.7 40.4

Notes: A is G-t-S, B is AIC and D is MAIC, respectively; the estimated joint probabilities are pˆ AB , pˆ AD and pˆ BD .

Table 9.4 Intersection of sets using different lag selection criteria, % same choice, AR(1). ρ = 1.0

pˆ AB

pˆ AD

pˆ BD

ϕ1 = –0.6 ϕ1 = –0.3 ϕ1 = 0 ϕ1 = 0.3 ϕ1 = 0.6

45.2 45.7 38.4 44.5 43.7

30.1 31.0 26.7 28.6 32.0

61.3 60.7 64.9 57.3 64.7

ρ = 0.95 ϕ1 = –0.6 ϕ1 = –0.3 ϕ1 = 0 ϕ1 = 0.3 ϕ1 = 0.6

45.0 44.5 39.6 44.1 44.2

29.1 29.2 26.2 24.6 29.8

56.6 56.4 56.4 46.9 60.9

Note: See notes to Table 9.3.

that which compares AIC and MAIC, with approximately 55–65% in common depending on the value of ϕ1 . The remaining (paired) combination is G-t-S and AIC, which results in the same lag in about 40–45% of cases. Overall, Tables 9.3 and 9.4 show that it makes a difference which lag selection criterion is used; in a sense this is obvious if the criteria are to have an impact

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Table 9.3 Intersection of sets using different lag selection criteria, % same choice, MA(1).

Lag Selection and Multiple Tests 367

in controlling size in the face of weakly dependent errors. However, it is not just a case of choosing a longer lag than say would result from AIC, since G-t-S and MAIC tend to choose longer lags, but in different samples. This question of what difference this makes is considered further in section 9.2.5.ii.

The application of multiple tests is a common practice in unit root testing. It may, for example, occur in testing for the lag order for a single test and/or in calculating several tests for a unit root The latter practice is not surprising given the number of test statistics that are available, but the implications for the overall type I error and the probability of conﬂict are not usually formalised. One might ﬁnd, for example, that two tests are reported and if there is disagreement in the outcomes, one is preferred to the other. This disagreement is referred to here as test conﬂict, which does not, in general, have a probability of zero. Typically, what is usually reported when a new test statistic is introduced is a comparison of test power, whereas other characteristics, such as test dependency are relevant. This section considers some of the issues that arise when multiple tests for a unit root are applied to the same data set. Some examples illustrate the applicability of the principles to cases of practical interest. 9.2.1 Test characteristics Consider the simplest case of two tests for a unit root, then the test outcomes may agree: either both reject or both do not reject H0 ; or the test outcomes may disagree, one leads to rejection, but the other does not. If both tests either reject or not reject, the practitioner may think no more of the situation; however, if one test leads to rejection, but the other does not, a conﬂict arises. A possible assumption here is that the former test is more powerful than the latter test and the overall conclusion is to reject the null hypothesis. However, an interesting question is what is the probability of such conﬂict? Even two tests with identical power will generally conﬂict: the key is whether they are not rejecting in the same cases. This is a relevant issue as shown in the previous section on lag selection see especially Tables 9.3 and 9.4. The answer to the question of how probable agreement and conﬂict are lies in the extent of test dependence, which also determines the actual overall type I error of the multiple testing procedure; that is, the probability that at least one of the tests falsely rejects the null hypothesis. An analysis of the power of different procedures is insufﬁcient to answer these questions, since only exceptionally is the probability of test conﬂict equal to the difference in power. In using two tests, it is possible that a less powerful test rejects when the more powerful test does not; a case in point would be where, say, using τˆ µ leads to rejection, but

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9.2 Multiple tests

368 Unit Root Tests in Time Series

using τˆ ws µ does not. One might think that because the latter is more powerful than the former this possibility is disallowed; however, this is not correct. The key issues are illustrated with two tests, with the test statistics generically denoted tA and tB ; the generalisation to n ≥ 3 tests is straightforward. The extent of test agreement and test conﬂict can be considered (i) conditional on the null hypothesis being correct and (ii) conditional on the alternative hypothesis being correct and the following sections deal with both cases.

The test statistics tA and tB are assumed to be deﬁned on the real line, with (marginal) probability density functions under the null, given by fA (tA |H0 ) and fB (tB |H0 ), respectively. In the unit root context H0 is ρ = 1. For simplicity assume that each test statistic has a left-sided rejection region, as is the case with the DF δˆ and τˆ test statistics; for example, for test statistic tA , H0 is rejected if the sample cv value of tA is less than the α% quantile cvA,α , where −∞A fA (tA |H0 )dxA = α. Consider simulating the null distribution of tA with R replications and mapping the outcomes {tA,r }Rr=1 into the sample space ΩA with the r-th element deﬁned as 1 if tA,r < cvA,α and 0 otherwise; for example, the derived outcomes ΩA = (1, 0, 0, 0, 1) indicate rejection on replications 1 and 5 and non-rejections on replications 2, 3 and 4. Next deﬁne A as the event that tA leads to rejection, which is the set of elements of ΩA equal to unity, and hence the event that tA leads to non-rejection is Ac , which is the set of zero elements. Deﬁne similar quantities for test statistic tB . Next consider the meaning of A ⊂ B, which is critical for the deﬁnition of test dominance. This means that if tB leads to rejection then so does tA , but there are some cases when tB rejects but tA does not; A = B means that tests A and B are rejecting in exactly the same cases. Given the sets A and B, the following events of interest can be deﬁned, distinguishing between whether the tests agree or disagree. Agreement occurs if each test leads to rejection or each test leads to non-rejection. The ﬁrst case occurs if ΩA,r = 1 and ΩB,r = 1. An estimate of the probability that both tests reject, that is p(A ∩ B), is provided by: p(A ∩ B) = lim

R→∞

= lim

R→∞

∑r=1 I(tA,r < cvA,α R

∑r=1 I(ΩA,r = 1 R

and tB,r < cvB,α )/R

and ΩB,r = 1)/R

(9.19)

Next, both tests lead to non-rejection when ΩA,r = 0 and ΩB,r = 0, which corresponds to entries in the sets Ac and Bc . An estimate of p(Ac ∩ Bc ) is provided by: p(Ac ∩ Bc ) = lim

R→∞

= lim

R→∞

∑r=1 I(tA,r ≥ cvA,α R

∑r=1 I(ΩA,r = 0 R

and tB,r ≥ cvB,α )/R

and ΩB,r = 0)/R

(9.20)

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9.2.2 Deﬁning the indicator sets

Lag Selection and Multiple Tests 369

Table 9.5 Joint probabilities for two test statistics. A

Ac

B Bc

p(A ∩ B) p(A ∩ Bc ) p(A)

p(Ac ∩ B) p(Ac ∩ Bc ) p(Ac )

p(B) p(Bc ) 1

The remaining two cases are of test disagreement (conﬂict). First, on the r-th replication test A leads to rejection and test B leads to non-rejection, so that ΩA,r = 1 and ΩB,r = 0 and, second, test A leads to non-rejection and test B leads to rejection, so that ΩA,r = 0 and ΩB,r = 1. The relevant probabilities can be estimated, respectively, by: p(A ∩ Bc ) = lim

∑r=1 I(ΩA,r = 1

and ΩB,r = 0)/R

(9.21)

p(Ac ∩ B) = lim

∑r=1 I(ΩA,r = 0

and ΩB,r = 1)/R

(9.22)

R→∞ R→∞

R R

To summarise, carrying out both tests leads either to agreement or conﬂict; thus, taking pairs from the outcome sets (A, Ac ) and (B, Bc ), gives four possibilities, which are presented in a simple 2 × 2 event table, as in Table 9.5. The diagonal elements are the probabilities that the outcomes agree, either both rejecting or both not rejecting, whereas the off-diagonal elements are the outcomes that disagree, when one test rejects and the other does not reject. The probabilities of agreement, p(G) and conﬂict, p(C), are, therefore: Agreement: p(G) = p(A ∩ B) + p(Ac ∩ Bc ) Conﬂict: p(C) = p(A ∩ Bc ) + p(Ac ∩ B) Exhaustive: p(G) + p(C) = 1 So far it has been assumed that H0 is correct; however, the situation where HA is correct is also of interest and can be set up in the same way. For example, suppose that ρ = 0.95, then we are still interested in whether tA,r < cvA,α , which corresponds to rejection of a false H0 at the α% signiﬁcance level and, alternatively, tA,r ≥ cvA,α , leading to non-rejection, which implies an incorrect decision since H0 is false. The power of a single test, say tA , can be estimated from R replications by: ˆ p(A|H A ) = lim

R→∞

∑r=1 I(tA,r < cvA,α )/R R

where A is the set of rejections and tA,r < cvA,α leads to rejection. Of greater interest in the present context is estimation of the probability of joint rejection

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Outcome

370 Unit Root Tests in Time Series

Table 9.6 Probabilities under independence of tests given H0 .

B Bc

A

Ac

p(A ∩ B) = α2 p(A ∩ Bc ) = α(1 − α) p(A) = α2 + α(1 − α) = α

p(Ac ∩ B) = (1 − α)α p(Ac ∩ Bc ) = (1 − α)2 p(Ac ) = (1 − α)α + (1 − α)2 = (1 − α)

p(B) = α p(Bc ) = (1 − α) 1

ˆ ∩ B|HA ) = lim p(A

R→∞

= lim

R→∞

∑r=1 I(tA,r < cvA,α R

∑r=1 I(ΩA,r = 1 R

and tB,r < cvB,α )/R

and ΩB,r = 1)/R

(9.23)

Other probabilities and their estimates are obtained as in (9.20), (9.21) and (9.22), but conditional on HA . Even if two tests have the same power for a given size, there will generally be outcomes in the conﬂict sets. That is, the tests are not rejecting or rejecting in different cases, hence giving rise to conﬂict. Unless the tests are perfectly dependent, they will reject or not reject in different cases. The simplest case to analyse is independent tests, so we start with that case. 9.2.3 Overall type I error Suppose the two test statistics tA and tB are used, with rejection of H0 following if either or both tests reject. Let A and B denote the rejection sets conditional on H0 , then the overall type I error is αΣ ≡ p(A ∪ B) = p(A) + p(B) − p(A ∩ B). For two independent tests carried out at the same size α then αΣ = 1 − (1 − α)2 ; for n tests αΣ = 1 − (1 − α)n ; this limit has been used for multiple tests of different characteristics, where independence may be a good working hypothesis. Given a value for αΣ it is possible to solve for the size of each individual test, that is, α = 1 − (1 − αΣ )1/n – for example if n = 4 and αΣ = 0.1, then α = 0.026. The probabilities of interest for two independent tests, each undertaken at size α, are summarised in Table 9.6. The probabilities of agreement and conﬂict, respectively, are therefore p(G) = α2 + (1 − α)2 and p(C) = 2α(1 − α). To illustrate take α = 0.05, then p(A ∪ B) = 0.0975, that is an overall type I error of 9.75%, with p(G) = 0.905 and p(C) = 0.095. Thus, for independent tests each carried out at a 5% signiﬁcance level, the probability of conﬂicting test outcomes is 9.5%. However, in tests for the same characteristic, such as a unit root, there is more likely to be dependence between the tests, which reduces the overall type I error and the probability of conﬂict. First, consider the case A ⊂ B, that is all the elements in the rejection set A are also in rejection set B, then p(B|A) = 1,

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obtained by:

Lag Selection and Multiple Tests 371

0.25 0.2 0.15 5 P(A ∩ B)

P(Ac ∩ B)

0.05 0 5

5 P(A ∩ Bc)

0 tA

0 –5 –5

tB

P(Ac ∩ Bc)

Figure 9.7a Bivariate normal with η = 0.75.

and therefore p(A ∪ B) = p(B). (This follows as p(B|A) = p(A ∩ B)/p(A) = 1 ⇒ p(A ∩ B) = p(A), so p(A ∪ B) = p(A) + p(B) − p(A ∩ B) = p(B).) If tests A and B are undertaken at the same size α, then perfect dependence is A = B, there is no information in set A that is not in set B, and vice versa, implying p(A) = p(B) = α, and therefore p(A ∪ B) = α. Thus, for two tests the lower and upper limits of the cumulative type I error are αΣ = α and αΣ = 1 − (1 − α)2 , corresponding to perfect dependence and perfect independence, respectively. The implications for p(G) and p(C) are intuitive as in the case of perfect dependence the tests always agree, therefore p(G) = 1 and p(C) = 0. Generally, for two tests carried out each at size α, if H0 is correct, then 0 ≤ p(C) ≤ 2α(1 − α), where the lower limit corresponds to perfect dependence and the upper limit to perfect independence. As an example of the impact of dependence on the overall type I error and the probability of conﬂict, consider the situation where the joint pdf of the test statistics tA and tB is bivariate normal with correlation coefﬁcient η. Then the probabilities can be obtained analytically. This is illustrated in Table 9.7 for η = 0, 0.5, 0.75, 0.95 and 1 and in Figures 9.7a and 9.7b for η ∈ [0, 1]. For example, if η = 0.5 and α = 0.05, then p(A ∪ B) = 0.05 + 0.05 – 0.012 = 0.088 and p(C) = 2(0.038) = 0.076, whereas if η = 0.95 then p(A ∪ B) = 0.063 and p(C) = 2(0.013) = 0.026. The various probabilities are illustrated graphically by shaded areas in Figure 9.7a for the bivariate case with η = 0.75; for example, the probability of agreement is the sum of the two diagonal boxes from North-West to South-East. Then in Figure 9.7b the probability of conﬂict, p(C), is graphed as a function of η; notice from this ﬁgure that p(C) is a nonlinear function of η, with the value

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0.1

372 Unit Root Tests in Time Series

0.1 evaluated conditional on H0

0.09 independence

0.08 0.07 0.06 Probability 0.05 of conflict

0.03 0.02 0.01 0

perfect dependence 0

0.1

0.2

0.3

0.4

0.5 η

0.6

0.7

0.8

0.9

1

Figure 9.7b Probability of conﬂict, two tests, bivariate normal. Table 9.7 Illustrative probabilities if tests are distributed as joint normal, α = 0.05, with dependency indicated by η. η=0

η = 0.5 Ac

A B 0.0025 Bc 0.0475 p(C) 0.095

A

0.0475 0.012 0.9250 0.038 0.076

η = 0.75 Ac 0.038 0.912

η = 0.95

A

Ac

0.022 0.028 0.056

0.028 0.922

η = 1.00

A

Ac

A

Ac

0.037 0.013 0.026

0.013 0.937

0.05 0.00 0.000

0.00 0.95

Note: Table entries are probabilities of the joint events given by a row element and a column element; for example, p(A ∩ B) = 0.0025 for η = 0.

for η = 0 corresponding to independent test and the value for η = 1 corresponding to perfectly dependent tests. There is a gradual decline in p(C) until η ≈ 0.8 and a sharp decline thereafter. The more usual situation is that the dependency between tests is not known; however, the probabilities of interest can be obtained by simulation. For example, p(A ∩ Bc ) and p(Ac ∩ B) can be estimated from (9.21) and (9.22) for ﬁnite R, and these are illustrated in section 9.2.4 for two situations of interest in unit root testing. 9.2.4 Test power and test dominance The case where, for a given size, test B is more powerful than test A and where if test A leads to rejection then so does test B, is a rather special situation, since then not only does test B dominate test A in power, the rejection set B, that is using test tB , includes all the rejections using test A. What conditions deliver this

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0.04

Lag Selection and Multiple Tests 373

9.2.4.i Test dependency function Test dependency is the extent to which the two tests lead to the same decision. It corresponds, in an absolute sense, to the probabilities p(A ∩ B) and p(Ac ∩ Bc ), so that in each case the decisions are the same. In a relative (or conditional) sense the relevant probabilities are p(A|B) and p(B|A), and p(Ac |Bc ) and p(Bc |Ac ). A conditional test dependency function relates p(A|B) or p(B|A) to the parameter in the hypothesis test; that is: p(A|B) = dA|B (ρ) p(B|A) = dB|A (ρ) where the functions dA|B (ρ) and dB|A (ρ) indicate the dependence of the conditional probabilities on ρ. Generally, as ρ moves away from its value under the null, ρ0 , both dA|B (ρ) and dB|A (ρ) will tend to 1. The functions dA|B (ρ) and dB|A (ρ) plotted as a function of ρ are referred to as the test dependency curves.

9.2.4.ii Test conﬂict function The test conﬂict curve is a plot of a conﬂict probability, either p(A ∩ Bc ) or p(Ac ∩ B), as a function of ρ. As in the case of test dependency, test conﬂict can also be deﬁned in a conditional sense, with the probabilities p(Ac |B) = 1 – p(A|B) and p(Bc |A) = 1 – p(B|A), both considered as a function of ρ. Notice that these probabilities can be read off from the corresponding test dependency curve by measuring from 1 rather than 0. The probability of test conﬂict is only exceptionally just the difference in power, that is, p(Ac ∩ B) = p(B) – p(A) ≥ 0, iff A ⊆ B, implying p(B|A) = 1. Now test statistic tB cannot be less powerful than test statistic tA since A ⊆ B implies p(A) ≤ p(B). This addresses the common (mis)conception that just knowing the difference in power is all that is needed to know about two test statistics. Generally, this is not the case: the test conﬂict function is only exceptionally the difference in the power functions.

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result? The ﬁrst is simply p(B) > p(A); the second is A ⊂ B; that is, all rejections from test A are also in test B’s rejection set. But A ⊂ B implies p(B|A) = 1, so that test A adds nothing to test B; it is perfectly dependent on test B. These conditions may be satisﬁed at a point in the parameter space or throughout the parameter space; if the latter is the case, then from the perspective of power, test A is redundant. This property of dominance is over and above that of a test being more powerful.

374 Unit Root Tests in Time Series

1 0.9 0.8 0.7

τˆws µ

τˆµ

0.6

0.4

ˆτ ws µ (test B) is more

0.3

powerful than ˆτµ (test A)

0.2 0.1 0 0.7

0.75

0.8

0.85 ρ

0.9

0.95

1

Figure 9.8 Power functions of τˆ µ and τˆ ws µ .

9.2.5 Illustrations: different test statistics and different lag selection criteria 9.2.5.i Illustration 1: a comparison of τˆ µ and τˆ ws µ To illustrate power, test dependencies and test conﬂict, simulated data were generated from yt = ρyt−1 + εt , where εt ∼ niid(0, 1) and 0 ≤ ρ ≤ 1; and, as usual, H0 was taken as ρ = 1. There were 25,000 replications, with T = 100 and a size of 5% (α = 0.05) for each of two tests tA and tB . The test statistics tA and tB c c are here τˆ µ and τˆ ws µ , with outcome sets (A, A ) and (B, B ), respectively, where A and B are the sets of rejections. Whether these rejections are correct depends on whether the null or alternative hypothesis is true. A comparison of power is shown in Figure 9.8 where it is evident that τˆ ws µ is more powerful than τˆ µ for 0 ≤ ρ ≤ 1. It might therefore be thought that ˆ µ , but this is incorrect; what is of interest is τˆ ws µ would completely dominate τ the dependencies between the test statistics. Figure 9.9 shows the conditional test dependencies and, by reading down from 1 on the right-hand scale, it also shows the conditional test conﬂicts; for example, 1 – p(A|B) = p(Ac ∩ B)/p(B). Conditional test dependency at the null is 48.4%, implying conditional test conﬂict of 51.6%, which can be read down on the right-hand scale of the graph. ˆ µ only rejects in The interpretation of this result is that at H0 when τˆ ws µ rejects, τ about one-half of such cases and so conﬂicts about one-half the time. As ρ moves away from its value under H0 , conditional dependency increases and so conﬂict declines; that is when one test rejects the other also rejects, but

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power 0.5

Lag Selection and Multiple Tests 375

1

0

0.9

0.1

dependency at ρ = 0.9 for tB = τˆw s is 0.917

0.7

0.6

0.5

0.2

µ

0.3

dA|B(ρ)

dB|A(ρ) 0.4

dependency at ρ = 0.9

conflict

for tA = τˆµ is 0.594

0.4 0.7

dependency at the null is 0.48 0.75

0.8

0.85 ρ

0.9

0.95

1

Figure 9.9 Conditional test dependency functions.

note that there is still a noticeable difference between the decisions. Away from the null, as expected, p(B|A) > p(A|B) and at ρ = 0.9 the conditional test dependencies have increased to 91.7% and 59.4% for p(B|A) and p(A|B), implying that conditional test conﬂicts have declined to 8.3% and 40.6%, respectively. Unconditional test dependence at the null, p(A ∩ B), is 2.4%, so the overall size (type I error) of applying both tests at the 5% level is p(A) + p(B) – p(A ∩ B) = 7.6%, about midway between the limits of 5% and 9.75%. Figure 9.10 shows the two unconditional test conﬂict curves, which relate to p(A ∩ Bc ) and p(Ac ∩ B), and the overall conﬂict curve relating to p(C), each as a function of ρ; the ﬁrst refers to rejection with τˆ µ but non-rejection with τˆ ws µ , and the second to non-rejection with τˆ µ but rejection with τˆ ws . The probability of µ test conﬂict at the null for each type of conﬂict is 2.6%, and hence 5.2% overall at the null in the application of both tests. Conﬂicts of the kind (A ∩ Bc ) do occur, with a maximum probability of about 3% at ρ = 0.9125, these correspond to the less powerful test, τˆ µ , rejecting when c the more powerful test does not, τˆ ws µ . Conﬂicts of the kind (A ∩ B) occur in a substantial number of cases, with a maximum of about 23% for ρ = 0.875; note that p(Ac ∩ B) > p(B) – p(A) except when ρ is well into the stationary region. The overall probability of conﬂict is the sum of the probabilities for the two conﬂicts, with a maximum of 25% at ρ = 0.875.

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0.8 agreement

376 Unit Root Tests in Time Series

0.3 tA = τˆµ

P(Conflict)

tB = τˆws µ

0.25 P(Ac ∩ B)

0.2

probability 0.15

0.1

0.05

P(A ∩ Bc)

0 0.7

0.75

0.8

0.85 ρ

0.9

0.95

1

Figure 9.10 Unconditional test conﬂict.

9.2.5.ii Illustration 2: using different lag selection methods The second illustration considers applying the same generic test, but with different methods to select the lag length. Provided both methods meet at least assumptions A1 and A2, their use will lead to consistent estimators, so that asymptotically with T there should be the following: no difference in power; perfect test dependency, and hence no conﬂict. However, what is of practical interest is how far the ﬁnite sample counterparts reﬂect these asymptotic properties. Consider Table 9.3 in section 9.1.5.iii, which reports the intersection of the sets of lag lengths chosen by different methods. It is evident that three widely used methods of lag selection result in different lag lengths being selected on quite a large number of occasions. The selection of different lags for the same generic test statistic is a necessary condition for test conﬂict, but it is not sufﬁcient as the test result at a given signiﬁcance level may be not be affected by the different choice of lags. Hence, the concern of this example is to see if it does make a difference. To illustrate the concepts, selection is alternatively by G-t-S and MAIC for the lags in the ADF version of τˆ µ . These two are chosen because they tend to result in similar mean lags, especially when the errors are weakly dependent and follow an MA(1) process, the situation which is know to cause problems for unit root testing. For present purposes, tA is τˆ µ with G-t-S and tB is τˆ µ with MAIC; the G-t-S rule uses a two-sided signiﬁcance level of 20%. The probabilities are ﬁrst illustrated

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P(B) – P(A)

Lag Selection and Multiple Tests 377

1 G-t-S

0.9

ρ = 1, θ1 = 0

0.8

MAIC

0.7 0.6

tA is τˆµ with G-t-S

power 0.5 0.4 0.3 0.2 0.1 0 0.5

0.55

0.6

0.65

0.7

0.75 ρ

0.8

0.85

0.9

0.95

1

Figure 9.11 Power of τˆ µ using G-t-S and MAIC.

for θ1 = 0 and ρ = [0.5, 1], with a nominal 5% size for each test and T = 200. There are two ways for a researcher to proceed when part of the procedure involves a data-dependent lag search; one is to use the critical values assuming no search procedure, which appears to be the dominant empirical practice, perhaps justiﬁed on the basis that asymptotically the searching makes no difference; second, the critical values can be made search-dependent by simulating the ﬁnite sample critical values when combined with the lag search. Although the latter procedure is generally preferable, the former is probably the dominant practice and we adopt to show what happens in practice. Note from Figure 9.11, that tA is more powerful than tB for all ρ. There is a slight difference in size at the null, with test A and test B having sizes of 4.8% and 4.3%, respectively. Although relatively minor, these differences imply that p(A|B) = p(B|A) at the null, so the reader should not be surprised that the two conditional test-dependency curves shown in Figure 9.12 do not pass through the same point at the null, ρ = 1. The probability that tA will reject given that tB has rejected is much larger than the other way round; for example at ρ = 0.9, p(A|B) is about 95% whereas p(B|A) is 62%. The probabilities of conditional test conﬂict can be obtained by reading down from 1 on the right-hand scale; for example, at ρ = 0.9, p(Ac |B) = 5% and p(Bc |A) = 38%. These show that given that tB leads to rejection, then using tA only leads to non-rejection in 5% of these cases, whereas given that tA rejects, using tB leads to non-rejection in 38% of these cases.

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tB is τˆµ with MAIC

378 Unit Root Tests in Time Series

0

1 dA|B(ρ)

0.9

0.7

ρ = 1, θ1 = 0

0.2 dB|A (ρ)

tA is τˆ µ with G-t-S

0.3

tB is τˆµ with MAIC

conflict

0.6

0.5

0.4 0.5

0.55

0.6

0.65

0.7

0.75 ρ

0.8

0.85

0.9

0.95

1

Figure 9.12 Conditional test dependencies, G-t-S and MAIC.

The unconditional probabilities of test conﬂict are shown in Figure 9.13. First note that p(Ac ∩ B) is fairly small, although it increases as ρ → 0.95, whereas p(A ∩ Bc ) is much larger, for example, over 30% at ρ = 0.9. Generally, the overall probability of conﬂict largely reﬂects the importance of p(A ∩ Bc ), but as ρ → 1, p(Ac ∩ B) is also important. For example, at ρ = 0.95, there is about a 21% probability of conﬂict, comprising 5% of cases in which tA leads to non-rejection, whilst tB leads to rejection and 16% of cases where tA leads to rejection, whilst tB leads to non-rejection. Whilst the results for θ1 = 0 indicate the general picture, there are some differences of emphasis when θ1 < 0 and, to illustrate, the case with θ1 = –0.6 is also considered in Figures 9.14–9.16. Now, as the value of θ1 is away from the null, the difference in size between the two search procedures is more notable, with test A having a size of 6.8% and test B maintaining its size better at 4.6%; although even the former would be regarded as ‘reasonably’ close in general terms. Nevertheless, this means that p(A|B) = p(B|A)at the null. Test tA is still more powerful than test tB (see Figure 9.14) (although power is not size adjusted), and is also still more inclusive of the test outcomes conditional on test B, using tB , than the other way around (see Figure 9.15). Finally, see Figure 9.16, the overall probability of conﬂict is less than when θ1 = 0, but still noticeable, in parts of the parameter space, peaking at 21% for ρ = 0.925. Moreover, the probability of conﬂict is more than the difference in power for ρ > 0.8; for example, at ρ = 0.925, the difference in power is 16% whereas the probability of conﬂict p(C) is 21%.

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0.8 agreement

0.1

Lag Selection and Multiple Tests 379

0.35 c P(A ∩ B )

P(Conflict) 0.3

ρ = 1, θ1 = 0

0.25

tA is τˆµ with G-t-S

0.2

tB is τˆ µ with MAIC

P(A) - P(B)

0.1 0.05 0 0.5

c P(A ∩ B)

0.55 0.6

0.65 0.7

0.75 0.8 ρ

0.85 0.9

0.95

1

Figure 9.13 Unconditional test conﬂict, τˆ µ with G-t-S and MAIC.

1 G-t-S

0.9 ρ = 1, θ1 = –0.6

0.8 0.7 0.6 0.5

MAIC

tA is τˆ µ with G-t-S

tB is τˆµ with MAIC

0.4 0.3 0.2 0.1 0 0.5

0.55

0.6

0.65

0.7

0.75 ρ

0.8

0.85

0.9

0.95

1

Figure 9.14 Power of τˆ µ using G-t-S and MAIC.

Overall, these probabilities indicate that the implications of using different methods of lag selection are not benign. Conﬂicts can be expected to arise. Indeed, the very nature of using different lag selection methods to control

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0.15

380 Unit Root Tests in Time Series

1

0 dA|B(ρ)

0.9

0.8 agreement 0.7

ρ = 1, θ1 = –0.6

0.1

dB|A (ρ)

0.2

test A is τˆµ with G-t-S test B is τˆµ with MAIC

0.3

0.5

0.4 0.5

0.55

0.6

0.65

0.7

0.75 ρ

0.8

0.85

0.9

0.95

1

Figure 9.15 Conditional test dependencies, τˆ µ with G-t-S and MAIC.

0.25 P(Conflict)

ρ = 1, θ1 = –0.6 0.2 c P(A ∩ B )

0.15 P(A) – P(B) 0.1

test A is τˆµ with G-t-S test B is τˆµ with MAIC

0.05

0 0.5

P(Ac ∩ B)

0.55

0.6

0.65 0.7

0.75 0.8 ρ

0.85 0.9

0.95

1

Figure 9.16 Unconditional test conﬂict, τˆ µ with G-t-S and MAIC.

size when the errors are weakly dependent implies conﬂict. Of the methods compared in the illustration, and based on the concepts of test dependence and dominance, selection by G-t-S dominates MAIC, although that conclusion depends upon the marginal signiﬁcance level used in the G-t-S tests.

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conflict 0.6

Lag Selection and Multiple Tests 381

This illustration uses time series data on US wheat production (harvested) for the period 1866 to 2008, a total of 143 observations. The data are clearly trended, as shown in Figure 9.17, and so the unit root tests are based on removing a linear trend. To illustrate some of the issues arising from lag selection, the unit root test statistic is τˆ β combined with one of three selection criteria for the ADF lag length; that is, AIC, the MAIC adaptation of the AIC and G-t-S using the marginal-t statistic. The sensitivity of the lag length to the selection criteria is illustrated in Figures 9.18a–9.18c, where the values of three selection criteria are plotted against the lag length for the ADF. The value of the test statistic τˆ β for each lag length is shown in Figure 9.18d, with the 5% critical value also shown (variations due to the changing lag length are marginal). The resulting selected lag lengths are 1 by AIC, but 4 by MAIC, see sub-ﬁgures 9.18a and 9.18b. The G-t-S lag length depends on the marginal signiﬁcance level. The absolute critical values (which are constant to two decimal places) for a 10% and 20% (two-sided) signiﬁcance level are also shown on sub-ﬁgure 9.18c and these values imply a lag of 3 for a 20% two-sided test, but a lag of 0 for a 10% two-sided test. Thus, a choice of 0, 1, 3 or 4 could be justiﬁed by selectively using one of these criteria! The issue of lag length is also critical to the result of the unit root test, as is evident from sub-ﬁgure 9.18d; for example, using a 5%

15

14.5

14

13.5

13

12.5 12 1860

1880

1900

1920

1940

1960

1980

2000

Figure 9.17 US (log) wheat production.

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9.3 Empirical illustration

382 Unit Root Tests in Time Series

(a) –508

(b) –450

–510

–460

–512 MAIC –470

AIC –514

MAIC lag = 4

–480

–516 AIC lag =1

(c)

–490 0

1

2

3

4

5

6

7

2

0

1

2

3

4

5

6

7

lag

(d) –2 10% cv

1.5

–3 20% cv

marginal |t| 1 0.5

5% cv

τ^ β –4 –5

G-t-S lag = 3

reject null do not reject null 0

0

1

2

3

4

5

6

7

–6

0

1

2

3

4

5

6

7 lag

Figure 9.18 The impact of different lag lengths.

critical value, the null hypothesis would be rejected for lags 0, 1 and 2, but not rejected for lags 3 through to 7. The lag choices might be distinguished by the additional criterion; for example, that the residuals pass a standard test for the absence of serial dependence. To this effect, an LM-type test for ﬁrst-order serial correlation gives p-values (lag length) of 13% (0), 30% (1), 14% (2), 54% (3) and 94% (4), suggesting that a lag of 3 or 4 may be preferable to shorter lags, which in turn implies non-rejection of the null hypothesis. This example illustrates agreement and conﬂict in the test outcomes. There are in effect three tests when each test is combined with a different lag selection procedure. Referring to these as tests A (G-t-S), B (AIC) and D (MAIC), with outcome sets deﬁned by rejection of the null hypothesis, then note that the test outcomes comprise two outcomes that are conﬂicts; that is Ac ∪ B and B ∪ Ac , and one outcome of agreement, Ac ∪ Dc .

9.4 Concluding remarks It is almost inevitably the case that the lag length, or some other truncation parameter, has to be chosen in a practical implementation of a unit root test and this procedure may well be a critical part of the analysis. The leading criteria are those based on an information criteria or a modiﬁcation thereof, or as part of a hypothesis testing strategy seeking to restrict the lag length to be equal to or less than some speciﬁed maximum lag. In principle, where a search strategy is

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–518

built into the testing procedure, the simulated ﬁnite sample distributions should reﬂect this aspect so that the critical values are likely to differ from the case when it is assumed that the lag length is known (usually assumed to be zero). In the presence of serial dependence due to an MA(1) error, there is a tendency for AIC to under-parameterise the lag length, resulting in over-sized tests. In the case of AR(1) errors, the problems were less severe, but over-parameterisation, and hence superﬂuous lags, could lead to a decline in power. Overall, in terms of selecting lags by a single criterion without additional knowledge of the DGP of the errors, the G-t-S principle, with a fairly generous marginal signiﬁcance level (msl), performed reasonably well. A case could be made for making the msl a declining function of the sample size, but that possibility was not pursued here. In empirical studies one sometimes (perhaps usually?) ﬁnds either or both multiple tests for a unit root and the use of multiple lag selection criteria. Multiple testing implies that, exceptional circumstances aside, there will be occasions when the conclusion will be sensitive to which test or which lag selection procedure has been used. A framework was provided in section 9.2 to enable the interpretation of test conﬂict agreement and conﬂicts. Generally some conﬂict is to be expected and it will not always be of the form that the more powerful test will lead to rejection, whereas the less powerful test does not. The opposite is possible; the key to understanding this outcome is to look at what is happening in each case (for example, in a simulation). It is only exceptionally the case that the set of rejected outcomes of the more powerful test includes all of the rejected outcomes of the less powerful tests. The possibility of agreement and conﬂict in test outcomes was illustrated by the application of the DF unit root test τˆ β to US wheat production. The test results were sensitive to the selected lag length, and so by a judicious choice of criterion different conclusions could be reached. The reader should be aware of this possibility when interpreting published results.

Questions Q9.1 A researcher reports the sample values of the two unit root test statistics ˆ µ but not rejected τˆ µ and τˆ ws µ , ﬁnding that the unit root null is rejected using τ ws ws using τˆ µ . He is puzzled because τˆ µ is the more powerful test statistic. Explain the puzzle. ˆ µ , the feature of interest is A9.1 Although τˆ ws µ is generally more powerful than τ whether the tests lead to the same decisions given a speciﬁc case or cases; a test statistic that is both more powerful and rejects whenever the less powerful test ˆ µ , so that rejects is said to be dominant. In this sense τˆ ws µ does not dominate τ examples may occur in practice where using τˆ µ leads to rejection but using τˆ ws µ does not. Consider the simulation example, with results graphed in Figure 9.9.

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Lag Selection and Multiple Tests 383

384 Unit Root Tests in Time Series

This showed conditional test dependency at H0 : ρ = 1, was 48.4%, implying conditional test conﬂict of 51.6%.

A9.2 First, the cumulative type I error is αΣ ≡ p(A ∪ B ∪ D) = 1 − (1 − α)3 , where D is the set of rejections using the third test statistic. If α = 0.05, then αΣ = 0.1426; that is, 14.26%. First consider the case of independent tests, then the joint probability that all three tests reject is α3 , whereas the joint probability that all three tests do not reject is (1 − α)3 ; all other cases involve an element of disagreement. The probability of agreement is, therefore, p(G) = α3 + (1 − α)3 and the probability of conﬂict is p(C) = 1 – [α3 + (1 − α)3 ]; for example, if α = 0.05, then p(G) = 0.8575 and p(C) = 0.1425. Notice that for ‘small’ α, p(C) ≈ αΣ because α3 ≈ 0.

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Q9.2 Consider the case where three tests statistics are calculated to test a particular null hypothesis, with each being applied at the α signiﬁcance level, rejecting the null hypothesis if at least one of the tests leads to rejection. What are the limits to the cumulative type I error in such a procedure? Illustrate your answer for α = 0.05.

10

Introduction This chapter is concerned with new issues that arise in testing for two unit roots. The techniques described here are easy to generalise to the less likely case of more than two unit roots. Series that are generated with two unit roots involve the double summation of shocks, a process that tends to produce a smooth time series; in contrast, a series generated by a single unit root, which involves a single summation of shocks, is less smooth, although generally smoother than an I(0) series; the limiting case of the latter being the ‘spiky’ pattern produced by a white noise series. Do such series arise in economics? Whilst not as prevalent as candidates for I(1) behaviour, there is evidence, visual and more formal, that some economic series may have been generated by I(2) processes. For example, Sen and Dickey (hereafter SD) (1987) suggest that the time series of US population is an I(2) candidate variable; Haldrup (1998) suggests that stock series and series in nominal prices are similarly I(2) candidates; and Juselius (2009) considers some nominal price indices as candidate I(2) variables in the context of the PPP (purchasing power parity) puzzle. One straightforward extension to test for the I(2)-ness of a series is to impose a single unit root and then test for a second unit root; thus the data analysis is in terms of ∆yt rather than yt and there is nothing new in principle, with the panoply of test statistics available for the unit root case also available for the two unit root case. This procedure is easily generalised since, if a variable is a candidate for I(3) behaviour, then the sequence starts with a test using ∆2 yt , which imposes two unit roots, and then test for a further unit root; if the further, in this case, third, unit root is rejected, drop down to using ∆yt ; and if the second unit root is rejected, drop down to the standard case using yt . This sequence was suggested by Dickey and Pantula (hereafter DP) (1987).

385

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Testing for Two (or More) Unit Roots

An alternative is to formulate the restrictions that two (or more) unit roots impose on the coefﬁcients of an AR(p) model, p ≥ 2, and then test these restrictions jointly by an F-test or other generalised testing principle; for example, an LM principle looks attractive in the general case as the null model is easy to specify. An F-type test of the joint restrictions was suggested by Hasza and Fuller (hereafter HF) (1979), who also showed how to reparameterise the AR model so that the test statistic was based on setting two coefﬁcients to zero by way of an extension of the DF decomposition of a lag polynomial. A version of this test using the backward and forward difference equations leading to symmetric estimation (as in the weighted symmetric unit root test described in Chapter 6), was suggested by SD. Rodrigues and Taylor (hereafter RT) (2004a) drew out the importance of the initial conditions in the I(2) case for a number of widely used parametric tests. A number of other approaches have also been suggested. For example, semi-parametric methods have been suggested by Haldrup (1998) and Shin and Kim (1999). Section 10.1 starts this chapter by laying out the preliminaries associated with allowing for two unit roots, including the characteristics of data generated by a process with two unit roots. This is followed in section 10.2 by the HF decomposition of a lag polynomial, which is analogous to the DF decomposition of Chapter 3, but in the context of more than one unit root. The use of the DF test when the DGP contains two unit roots rather than one is the subject of section 10.3, which is then followed by two sections, 10.3 and 10.4, outlining the HF and DP tests, and variants thereof, for multiple unit roots. Power is considered in section 10.6, and two empirical applications illustrate the tests in action in section 10.7. A summary of the initials widely used in this chapter follows: DF, Dickey-Fuller; ADF, augmented DF; HF, Hasza-Fuller; AHF, augmented HF; SD, Sen-Dickey; ASD, augmented SD; DP, Dickey-Pantula; ADP, augmented DP; RT, Rodrigues-Taylor.

10.1 Preliminaries A case that is of interest in some empirical situations is when there is a second unit root in the process generating yt . We start with the simplest case that will allow this possibility. Consider the following, and by now familiar, simple error dynamics model: (1 − ρL)yt = zt

(10.1a)

(1 − ϕ1 L)zt = εt

(10.1b)

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386 Unit Root Tests in Time Series

Testing for Two (or More) Unit Roots 387

⇒ (1 − ρL)(1 − ϕ1 L)yt = εt

This model is used in a number of places in this chapter and is referred to as the starting model. For the moment, the models in this section are presented without the complication of deterministic terms. In the case of I(2) tests, it turns out that how these are dealt with, and the importance of the initial conditions, is more complex than in the case of tests for a single unit root; hence these issues are considered in detail below. Returning to the model of (10.1), the case with ρ = 1 is familiar enough, con(1) stituting the usual H0 , now referred to as H0 to distinguish it from the case of (2)

(2)

two unit roots, which will be referred to as H0 ; for example, in this case H0 is ρ = 1 and ϕ1 = 1. The corresponding alternative hypotheses will also now be (1) (2) (1) distinguished as HA and HA , respectively. The former is, as before, HA : |ρ| < 1. The alternative hypothesis for the two unit root null is, however, now more (2) complex. We could just state that HA : ρ = 1 and/or ϕ1 = 1, which covers the following three possibilities, distinguished by adding a subscript: (2)

(1)

(i)

yt is I(1), H1,A = H0 : ρ = 1, |ϕ1 | < 1

(ii)

yt is I(0), H2,A : |ρ| < 1 and |ϕ1 | < 1

(iii)

yt is explosive, H3,A : at least one of ρ and ϕ1 is such that |ρ| > 1, |ϕ1 | > 1.

(2)

(2)

(2)

Under the null hypothesis H0 , the data-generating process is: ∆2 yt = εt

(10.2) (1)

The DGP of (10.2) is in contrast to the null under H0 , which in the simplest case is ∆yt = εt . The issue of concern in this chapter is to test for the presence of two unit roots in the DGP. However, before looking at the detail of such test, it may be helpful to consider some typical characteristics of I(2) series and some of the empirical series that have been modelled as potentially I(2). 10.1.1 I(2) characteristics The ﬁrst point to note is the smoothness of an I(2) series. This can best be seen by starting with an I(0) series, for example, the simplest case is a white noise series, such as εt . This will give rise to a ‘spiky’ time series graph with, by deﬁnition, no serial correlation structure. As serial correlation is introduced it tends to smooth out the spiky nature of the series, for example, consider yt = zt with zt = ρzt−1 + εt , which becomes smoother as ρ → 1; in the limit this is a unit root process, which is as smooth as this process can get before becoming explosive.

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(10.1c)

388 Unit Root Tests in Time Series

Now consider an invertible ﬁrst-order model and its solution in terms of an initial value and weighted innovations: (1 − ρL)yt = εt

(10.3)

yt = (1 − ρL)−1 εt = ρt y0 + ∑j=0 ρj Lj εt t−1

The inverse operator (1 − ρL)−1 sums the weighted innovations which, together with the ‘starting’ value y0 weighted by ρt , gives the level of the series yt . When ρ = 1, the operator becomes ∆−1 ≡ (1 − L)−1 , which is the summation operator analogous to the integration operator for continuous series; in this case, yt is integrated of order one, being the sum of white noise components. The series of εt realisations is very spiky, but the summation process smoothes out the individual irregularities in εt to produce an integrated series that is much smoother than its individual inputs. This also works with inputs that are not white noise, but have some time-series structure, whilst still being I(0). Just as in the I(1) case, much can be learned from the simple random walk process yt = yt−1 + εt , which is an AR(1) model with a single unit root. At this stage we use the analogous I(2) process with two unit roots; that is, ∆2 yt = εt or, explicitly in terms of lagged values, yt = 2yt−1 – yt−2 + εt . Let y−1 and y0 denote the starting values of the process, such that, for example, y1 = 2y0 – y−1 + ε1 ; then, by continuing this sequence, the following are obtained: y2 = 2y1 − y0 + ε2 = 2(2y0 − y−1 + ε1 ) − y0 + ε2 = 4y0 − 2y−1 + 2ε1 − y0 + ε2 = 3y0 − 2y−1 + 2ε1 + ε2

y3 = 2y2 − y1 + ε3 = 6y0 − 4y−1 + 4ε1 + 2ε2 − (2y0 − y−1 + ε1 ) + ε3 = 4y0 − 3y−1 + 3ε1 + 2ε2 + ε3 .. .

yt = t(y0 − y−1 ) + y0 + ∑j=1 (t − j + 1)εj t

= t(y0 − y−1 ) + y0 + ∑i=1 ∑j=1 εj t

i

(10.5)

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(10.4)

The ﬁrst two terms in (10.5) show the inﬂuence of the starting values; otherwise the series is the double summation of the shocks to the series, and it is this that accounts for one of the key visual features of an I(2) series – that is, its smoothness – especially relative to an I(1) series. However, note a point that will turn out to be signiﬁcant in achieving invariance of test statistics for two unit roots. Unlike the solution to a random walk without drift, (10.5) includes a trend term if y0 = y−1 , so we anticipate that including a trend in the demeaning process will be necessary if invariance is to be achieved, even though there is no drift in the underlying process. Apart from the starting values, equation (10.5) could have been derived by multiplying ∆2 yt = εt through by ∆−2 , the double summation operator, ‘solving’ yt as yt = ∆−2 εt ; moreover, multiplying through by ∆−1 , implies that ∆yt = ∆−1 εt , so that the ﬁrst difference of an I(2) series is an I(1) series, which is characterised by the (single) summation of the shocks. 10.1.2 Illustrative series Figure 10.1 shows simulated I(d) series for d = 0, 1, 2, 3. The series evidently become smoother as d increases. The interesting practical question is whether this smoothness is characteristic of any economic time series. Candidate series

25

4 I(0)

20

2

I(1)

15 yt

yt

0

10 –2 –4

5 0

100

200

300

400

0

500

15

6,000

4,000

0

100

300

400

500

400

500

× 105

10

I(2)

I(3)

yt

yt

5

2,000

0

200

0

100

200 300 time

400

500

0

0

100

200 300 time

Figure 10.1 Simulated I(d) series, d = 0, 1, 2, 3.

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Testing for Two (or More) Unit Roots 389

390 Unit Root Tests in Time Series

3

0.3 earnings inflation rate (annualised)

2.5

0.2

2 log $

0.1 1.5 hourly earnings

0

1

–0.1 1970

1980

1990

2000

1970

1980

1990

2000

Figure 10.2a US average hourly earnings (logs).

9

0.3

8

0.2

7

0.1

6

0

5

–0.1

annual inflation rate

4 1900

1950

2000

–0.2 1900

1950

2000

Figure 10.2b Denmark CPI (logs).

include those in nominal terms, such as wages and price levels, and stocks that may be cumulating I(1) variables. Some series that are I(2) candidates are shown in Figures 10.2a, 10.2b, 10.3a and 10.3b. The ﬁrst of these is the average hourly earnings of production workers in the US, seasonally adjusted, monthly data from 1964m1 to 2007m12, $ per hour, which is graphed in logs in Figure 10.2a (left panel), with the ﬁrst difference of this series shown in the right panel of the ﬁgure (graphed as the annualised change, so that this is the per annum earnings inﬂation rate). The smoothness in the series is evident, as is the ‘spiky’ nature of the ﬁrst differenced series. The second series is the log CPI for Denmark over the period 1900 to 2007 (1900 = 100 in levels) (see Figure 10.2b, with the left panel for the series and the right panel for the ﬁrst difference of the series. The data are annual and illustrate that smoothness is not by itself a sufﬁcient criterion for the I(2) property; smoothness can be achieved by temporal aggregation alone, whereas the ﬁrst difference of an I(2) series must also display random-walk type behaviour. The third and fourth series are stock series (see Figures 10.3a and 10.3b). The former is US M1, seasonally adjusted, monthly data from 1959m1 to 2007m12,

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0.5

Testing for Two (or More) Unit Roots 391

7.5

0.06

7

0.04

6.5

0.02

6 0

stock of M1

–0.02

5 4.5 1960

1970

1980

1990

2000

monthly M1 growth rate

–0.04 1960

1970

1980

1990

2000

Figure 10.3a US M1 (logs).

16

0.08 0.06

monthly credit growth rate

14 0.04 12

0.02 0

10 –0.02

outstanding personal credit 8 1943

1963

1983

2003

–0.04

1950 1960 1970 1980 1990 2000

Figure 10.3b US consumer credit (logs).

$billions, which is shown in logs in Figure 10.3a (left panel), with the ﬁrst difference shown in the right panel. The series is visually ‘smooth’, but not quite as smooth as the average earnings series (its sample standard deviation is about 40% larger). The spike in the monthly growth rate of M1 toward the end of the sample, is associated with ‘9/11’, and may reﬂect the cautionary withdrawal of ‘cash’ balances. Finally, the fourth series is outstanding consumer credit in the US (see Figure 10.3b); the data is monthly for the period 1943m1 to 2007m12. The levels series exhibits the smoothness expected of an I(2) series and there is some serial correlation in the ﬁrst difference of the series that might be suggestive of a random-walk element. 10.1.3 Alternatives to I(2) It is important in the case of a series that might be I(1) to form the alternative hypothesis in such a way that it can provide an explanation of the key data characteristics. For example, in the case of a series that has sustained movements in one direction, it will be important to include a trend in an otherwise stationary (1) alternative; to omit a trend component, when HA is true, will lead to a DF-type

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5.5

392 Unit Root Tests in Time Series

Table 10.1 A taxonomy of possibilities in the two root case.

|δ1 | > 1

Unit root

δ1 = 1

Explosive

|δ1 | < 1

Stationary |δ2 | > 1 |δ1 | > 1, |δ2 | > 1 with quadratic trend

Unit root δ2 = 1 |δ1 | > 1, δ2 = 1 with linear trend

Explosive |δ2 | < 1 |δ1 | > 1, |δ2 | < 1

δ1 = 1, δ2 = 1 two unit root null

δ1 = 1, |δ2 | < 1 |δ1 | < 1, |δ2 | < 1

unit root test that will not reject the null – this should not be surprising as there is no explanation of the data trend under the alternative! The question then arises as to what forms the alternative hypothesis for a test of the I(2)-ness of a series. Consider the starting model, (1 − ρL)(1 − ϕ1 L)yt = εt , which is a DGP with two roots. Let δ1 = ρ−1 denote the root of (1 − ρL) and δ2 = ϕ1−1 denote the root of (1−ϕ1 L), then the region for each root can be divided into three parts, corresponding to a stationary root, a unit root and an explosive root. This gives a 3×3 table of possibilities, one of which refers to the null hypothesis and the others to alternatives (see Table 10.1). Because of the symmetry of the roots, not all the possibilities are distinct; for example, |δ1 | > 1, |δ2 | < 1 is indistinguishable from |δ1 | < 1, |δ2 | > 1, hence there are six distinct pairings and the table of possibilities is triangular. Also, not all of these possibilities are those for which the tests outlined here were designed. For example, if the left-tail critical values are used, as in the DP test, which is a simple extension of the DF procedure, the alternative hypothesis is of stationarity rather than an explosive root; however, by using the right tail the test becomes that for a second root. The table also needs to be constructed so that the alternative hypothesis can match the variational characteristics of yt when there are two unit roots; for example, comparing two unit roots with two stationary roots make little sense as the latter cannot generate the growth in yt ; for example, what would be needed is a quadratic trend. If the series of interest is analysed in logarithms, which is often the case for economic time series, and consider the case where yt is a price level, then ∆yt is the inﬂation rate and ∆2 yt is the ‘acceleration’ rate. Under the null, the price level is I(2) and the inﬂation rate is I(1), so that stationarity only occurs with the acceleration rate. Three candidates are likely to be of interest as alternatives to the I(2) price level: (i) stationarity of both roots combined with a quadratic trend; (ii) ∆yt is stationary about a linear trend; that is, one unit root and one stationary root, combined with a linear trend; (iii) one root is unity and one root is explosive, which implies that ∆yt has an explosive root.

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δ2 → δ1 ↓ Stationary

Testing for Two (or More) Unit Roots 393

This section is preliminary in the sense of developing the simple starting model of Equation (10.1) to explore the implications of a second unit root. The restrictions associated with the presence of two unit roots can be formulated directly on the AR representation of the model; however, it would be convenient if, as in the case of the ADF representation of an AR model, there was a way of writing the AR model so that a test for one or two unit roots could be simply focused on one or two coefﬁcients, rather than imposing a set of restrictions that will vary with the AR order. This can be done by way of the HF decomposition of a lag polynomial, a topic that is introduced by an example in this section and dealt with more fully in the next section. The model of (10.1), with ρ = 1 has the following ARIMA(1, 1, 0) representation: ∆yt = ϕ1 ∆yt−1 + εt

(10.6)

In the limiting case, ϕ1 = 1 and, therefore, ∆2 yt = εt . From the deﬁnition ∆2 ≡ (1 − L)(1 − L) = (1 − 2L + L2 ), it is evident that there are two unit roots. In the integration order notation, yt ∼ I(2), so that ∆yt ∼ I(1) and ∆2 yt ∼ I(0); if |ϕ1 | < 1, then ∆yt ∼ I(0) implying yt ∼ I(1). In terms of the AR(2) notation of Chapter 3, the model of (10.1) can be written as: φ(L)yt = εt

(10.7) 2

φ(L) = (1 − φ1 L − φ2 L )

(10.8)

On comparison of (10.1) and (10.7), (1 − φ1 L − φ2 L2 ) = (1 − ρL)(1 − ϕ1 L), and therefore, φ1 = ρ + ϕ1 and φ2 = −ρϕ1 ; thus, in the case of two unit roots, that is, ρ = 1 and ϕ1 = 1, then φ1 = 2 and φ2 = –1. (In passing, note that the notation A(L) is used in place of φ(L) if, potentially, the AR representation has been obtained by inverting an MA polynomial.) The ADF(1) representation of (10.7) is: ∆yt = (φ1 + φ2 − 1)yt−1 + (−φ2 )∆yt−1 + εt = (ρ + ϕ1 − ρϕ1 − 1)yt−1 + ρϕ1 ∆yt−1 + εt

(10.9a) (10.9b)

There are two points to note about (10.9) that are useful when testing for two unit roots. First, if either one or two unit roots are present, the coefﬁcient on yt−1 is equal to zero; this follows because φ(1) = 0 if one or more unit roots is present; second, if two unit roots are present, then the coefﬁcient on ∆yt−1 is equal to 1, otherwise |ρ| ≤ 1 and |ϕ1 | < 1 ⇒ |ρϕ1 | < 1. Thus, subtracting ∆yt−1 from both sides of (10.9b), we have: ∆2 yt = γ1 yt−1 + γ2 ∆yt−1 + εt

(10.10)

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10.1.4 Some factorisations and restrictions

394 Unit Root Tests in Time Series

(1 − ρL)yt = zt

(10.11a)

(1 − ϕ1 L − ϕ2 L)zt = εt

(10.11b)

It is then convenient to factorise the lag polynomial ϕ(L) as follows: ϕ(L) = (1 − ϕ1 L − ϕ2 L) = (1 − λ1 L)(1 − λ2 L)

(10.12a) (10.12b)

where ϕ1 = (λ1 + λ2 ) and ϕ2 = −λ1 λ2 . Two unit roots corresponds to ρ = 1 and one of λi = 1, say λ1 = 1, with |λ2 | < 1. Then with these restrictions, the resulting model is: (1 − L)(1 − λ2 L)∆yt = εt

(10.13a)

(1 − λ2 L)∆2 yt = εt

(10.13b)

⇒ ∆2 yt = λ2 ∆2 yt−1 + εt

(10.14)

Now the AR representation of the model of (10.11) is AR(3): yt = φ1 yt−1 + φ2 yt−2 + φ3 yt−3 + εt

(10.15)

The imposition of two unit roots results in: φ1 = 2 + φ3

(10.16)

φ2 = − (1 + 2φ3 )

(10.17)

φ3 = λ 2

(10.18)

These are linear restrictions and, hence, easy to impose. For example, let λ2 = 0.5, then φ1 = 2.5, φ2 = –2 and φ3 = 0.5; and the polynomial φ(L) = (1−φ1 L− φ2 L2 − φ3 L3 ) = (1 − 2.5L + 2L2 − 0.5L3 ) has roots 1, 1, 2; also note that φ(1) = 0, indicating that there is at least one unit root in this lag polynomial. Whilst it is helpful to see how the AR coefﬁcients are related by the restriction of two unit roots, and hence in principle see how such restrictions could be tested, it is

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where γ1 = (ρ + ϕ1 − ρϕ1 − 1) = (φ1 + φ2 − 1) and γ2 = (ρϕ1 − 1) = −(φ2 + 1). This convenient representation is due to Hasza and Fuller (1979) and regressions of the form of (10.10), will be referred to as the HF regression (analogous to the DF regression). The null hypothesis of two unit roots is then seen to correspond to (2) (1) H0 : γ1 = γ2 = 0; whereas the null hypothesis of one unit root is H0 :γ1 = 0 and, implicitly, |γ2 | < 0. The generalisation of these models is straightforward. For example, in the case of the error dynamics framework, consider:

Testing for Two (or More) Unit Roots 395

generally more practical to obtain the augmented HF (AHF) regression and base the test statistic on this regression. To see how this is done in general, the next section introduces the HF decomposition of a lag polynomial.

10.2 The Hasza-Fuller (HF) decomposition of a lag polynomial

A(L) = (1 − αL) − C(L)(1 − L)

(10.19)

where C(L) is a polynomial of one lower order than A(L) if p is ﬁnite. Recall that application of the DF decomposition is useful because it means that the AR representation A(L)yt = εt has the following equivalent ADF representation: ∆yt = γyt−1 + C(L)∆yt + εt

(10.20)

where γ = α − 1, implying that the null hypothesis of a single unit root corresponds to γ = 0. The term C(L)∆yt can be regarded as ‘whitening’ the errors. In general, (10.20) is a much more useful ‘vehicle’ for testing a single unit root than the corresponding AR(p) model. In an analogous way, the HF decomposition leads to an AHF regression, which allows the focus to be centred on two coefﬁcients (because it is now two roots that are being tested) with, additionally, a lag polynomial in ∆2 yt that can be viewed as whitening the errors. It may be helpful to state the general AHF regression, which is established in detail by the subsequent analysis; that is: ∆2 yt = γ1 yt−1 + γ2 ∆yt−1 + D(L)∆2 yt + εt

(10.21)

In this set-up, the ﬁrst unit root corresponds to γ1 = 0 and the second to γ2 = 0, with the lag polynomial term D(L)∆2 yt viewed as whitening the errors. Hence, what is required are explicit expressions for γ1 , γ2 and D(L). A particularly simple example of this regression was given in (10.14) and the more general case is now considered. 10.2.1 Derivation of the HF decomposition The ﬁrst step in the HF decomposition is to apply the DF decomposition again, j but to C(L) and note that C(L) is deﬁned as ∑m j=1 cj L so that it differs from A(L) in not having the ﬁrst element as 1. Then the DF decomposition applied to C(L) is: C(L) = C(1)L + D(L)(1 − L)

(10.22)

where D(L) is a polynomial of one lower order than C(L), and hence of two lower orders than A(L). What is required is C(L)(1 − L), but this can be obtained

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The HF decomposition of a lag polynomial is an extension of the DF decomposition described in Chapter 3. It applies the DF decomposition twice. Recall that for an autoregressive lag polynomial A(L) of order p, the DF decomposition is:

396 Unit Root Tests in Time Series

by multiplying C(L), once obtained, by (1 − L). The orders of the polynomials are denoted p for A(L), m = p – 1 for C(L) and n = p – 2 for D(L). The polynomial D(L) is obtained in the same way that C(L) was obtained by expanding the polynomials and equating coefﬁcients on like powers of L: C(L) = C(1)L + ∑i=1 di Li (1 − L) n

∑

∑

(10.23b) (10.23c)

Hence, equating coefﬁcients on like powers of L implies: dn = − cn + 1

(10.24)

dn − dn−1 = cn ⇒ dn−1 = − (cn + 1 + cn ) dn−1 − dn−2 = cn−1 ⇒ dn−2 = − (cn + 1 + cn + cn−1 ) .. .

.. .

.. .

d2 − d1 = c2 ⇒ d1 = − (cn + 1 + cn + . . . + c2 ) n+1 C(1) = ∑j=1 cj

(10.25)

The dj coefﬁcients can be expressed in terms of the underlying AR coefﬁcients ai as follows (using the results of answer A3.5, Chapter 3): dn = − cn + 1 = an + 2

(10.26)

dn−1 = − (cn + 1 + cn ) = an + 1 + 2an + 2 dn−2 = − (cn + 1 + cn + cn−1 ) = an + 2an + 1 + 3an + 2 .. .

.. .

.. .

d2 = − (cn + 1 + cn + cn−1 ) = a4 + 2a5 + . . . + (n − 1)an + 2 d1 = − (cn + 1 + cn + . . . + c2 ) = ∑j=1 jaj + 2 n

n+1 n+1 n+2 n+1 C(1) = ∑j=1 cj = − ∑j=1 ∑i = j + 1 ai = − ∑j=1 jaj + 1

(10.27)

Having obtained D(L), then C(L)(1 − L) = C(1)L(1 − L) + D(L)(1 − L)2 is ﬁrst substituted into the ADF model: ∆yt = γ1 yt−1 + C(L)(1 − L)yt + εt = γ1 yt−1 + C(1)L(1 − L)yt + D(L)(1 − L)2 yt + εt = γ1 yt−1 + C(1)∆yt−1 + D(L)∆2 yt + εt

(10.28)

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∑

(10.23a)

n n = C(1)L + ( i=1 di Li − i=1 di Li + 1 ) n = (C(1) + d1 )L + i=2 (di − di−1 )Li − dn Ln + 1

Testing for Two (or More) Unit Roots 397

p

where γ1 = ∑i=1 ai − 1. In the second step, ∆yt−1 is subtracted from both sides of the equation, noting that ∆2 yt = ∆yt – ∆yt−1 , to obtain the AHF(p – 2) model: ∆2 yt = γ1 yt−1 + γ2 ∆yt−1 + D(L)∆2 yt + εt

(10.29)

where: γ2 = C(1) − 1

10.2.1.i Example 1 Let C(L) = (c1 L + c2 L2 + c3 L3 ) then: (c1 L + c2 L2 + c3 L3 ) = C(1)L + (d1 L + d2 L2 )(1 − L) = C(1)L + [(d1 L + (d2 − d1 )L2 − d2 L3 )] = [(C(1) + d1 ]L + [(d2 − d1 )L2 − d2 L3 ] Hence, equating coefﬁcients on like powers of L implies: d2 = − c3 , (d2 − d1 ) = c2 ⇒ d1 = − (c2 + c3 ) Using the relationships between the C(L) and A(L) polynomials, then (see answers A3.5 and A3.6 in Chapter 3): d2 = − c3 = a4 d1 = − (c2 + c3 ) = a3 + 2a4 C(1) = − (a2 + 2a3 + 3a4 )

10.2.1.ii Numerical example (continuation) Consider the following numerical values of the coefﬁcients: a1 = 0.3, a2 = 0.2, a3 = 0.3, a4 = 0.1 c1 = − 0.6, c2 = − 0.4, c3 = − 0.1 These imply that: d1 = 0.5 and d2 = 0.1, with C(1) = –1.1. Note that these coefﬁcients give rise to roots of 1.05, –0.51 ± 1.70i and 3.03; the modulus of the complex roots is 1.78, so that there is one root moderately close to unity, with the others not close to unity. The corresponding models written as AR(4), ADF(3) and AHF(2), respectively, are: yt = 0.3yt−1 + 0.2yt−2 + 0.3yt−3 + 0.1yt−4 + εt

AR(4)

∆yt = − 0.1yt−1 − 0.6∆yt−1 − 0.4∆yt−2 − 0.1∆yt−3 + εt

ADF(2)

∆2 yt = − 0.1yt−1 − 2.1∆yt−1 + 0.5∆2 yt−1 + 0.1∆2 yt−2 + εt

ADH(2)

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(10.30)

398 Unit Root Tests in Time Series

These three equivalent representations show that the one of interest depends on the focus of the analysis; the AR(4) indicates the dependence of yt on lagged yt ; the ADF(3) would be used for testing for a single unit root, the key coefﬁcient being that on yt−1 ; the AHF(2) is relevant for the joint test of two unit roots, the coefﬁcients of interest being those on yt−1 and ∆yt−1 , but note that the coefﬁcient on yt−1 is the same as in the ADF(3) representation.

The coefﬁcients are now changed to induce the following: a unit root, a root close to unity and two relatively distant roots. The model is constructed as: (1 − L)yt = zt

(10.31)

(1 − λ1 L)(1 − λ2 L)(1 − λ3 L)zt = εt

(10.32)

with λ1 = 0.98, λ2 = 0.5, λ3 = 0.2. Hence, by construction the roots are: 1, 1/0.98 = 1.02, 2 and 5, respectively. The convolution of the lag polynomials gives rise to the following AR(4) model: yt = 2.48yt−1 − 2.01yt−2 + 0.57yt−3 − 0.04yt−4 + εt First notice that A(1) = 0, so that there is at least one unit root. The ADF(3) model is: ∆yt = 0.00yt−1 + 1.48∆yt−1 − 0.53∆yt−2 + 0.04∆yt−3 + εt The coefﬁcient on yt−1 is zero as there is a unit root; note that c1 = 1.48, c2 = −0.53, c3 = 0.04, so that C(1) = 0.99, which is indicative of a second unit root. Finally, the AHF(2) model is: ∆2 yt = 0.00yt−1 − 0.01∆yt−1 + 0.49∆2 yt−1 − 0.04∆2 yt−2 + εt The coefﬁcient on ∆yt−1 is γ2 , which is now close enough to zero to suggest a second unit root.

10.3 DF test as a second unit root is approached The issue to be considered in this section is how DF test statistics designed for testing for a single unit root behave as a second unit root is approached. To get some insight into this situation, consider the starting model DGP comprising (1 − ρL)yt = zt and (1 − ϕ1 L)zt = εt . Interest ﬁrst focuses on the sensitivity of unit root tests as ϕ1 → 1, given that ρ = 1. Such concern has to some extent already been considered in the chapter on lag selection (see, for example, Chapter 9, section 9.1.2); however, here, the question of lag selection is put to one side, so that the ADF is correctly augmented with one lag.

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10.2.1.iii Example 2

Testing for Two (or More) Unit Roots 399

0.08 0.07 0.06 0.05 size 0.04

0.02 0.01 0

0

0.1

0.2

0.3

0.4

0.5 0.6 ϕ1

0.7

0.8

0.9

1

Figure 10.4a Empirical size of DF τˆ µ lower tail.

Haldrup and Lildholt (2002) derive the limiting DF distributions for the case where the starting model DGP has two unit roots, ρ = 1 and ϕ1 = 1, and the maintained regression is the standard DF regression with alternately a constant and a constant and a trend. The test statistics were τˆ µ and τˆ β . They found that the distributions differed from their single unit root counterparts, but largely in the right tail; the left tail, especially for the usual quantiles used for hypothesis testing for a single unit root, is not much affected. The impact of using the (1) (1) wrong null distribution therefore depends on the nature of HA ; if HA is ρ < 1, that is, a stationary alternative for which the left tail is used, then the nominal size will be reasonably well maintained. However, if the explosive alternative is (1) speciﬁed, HA : ρ > 1, the right tail of the distribution is used and the test will not be robust. To illustrate, the results of a small simulation study are presented in Figures 10.4a, 10.4b, 10.5a and 10.5b, the former two for µt = µ and the latter two for µt = β0 + β1 t. The DGP is as in (10.1) with T = 200, and a nominal size of 5%, and the test statistics are τˆ µ and τˆ β . The AR(1) coefﬁcient ϕ1 is varied from 0 through to 1 (some results for other sample sizes in the case of ϕ1 = 1 are reported in Haldrup and Lildholt, 2002). The ﬁgures are comprised of two parts: one for the lower tail, appropriate against a stationary alternative, and one for the upper tail, appropriate for an explosive alternative. (Multiply the left-hand scale by 100 to obtain the % size.) Considering Figure 10.4a ﬁrst, there is a modest oversizing for τˆ µ as ϕ1 → 1; for example, an empirical size of 6.0% at ϕ1 = 0.9; however at ϕ1 = 1, the empirical

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0.03

400 Unit Root Tests in Time Series

1 0.9 0.8 0.7 0.6 size 0.5 size: 42.4% 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5 0.6 ϕ1

0.7

0.8

0.9

1

Figure 10.4b Empirical size of DF τˆ µ upper tail.

0.08 0.07

size 6.5%

0.06 0.05 size 0.04 0.03 0.02 0.01 0

0

0.1

0.2

0.3

0.4

0.5 0.6 ϕ1

0.7

0.8

0.9

1

Figure 10.5a Empirical size of DF τˆ β lower tail.

size of 4.4% is a close match to the nominal size. It is evident from Figure 10.4b that a problem with size is more apparent against an explosive alternative; the empirical size is just over 42.4% at ϕ1 = 1, but τˆ µ starts to become oversized as ϕ1 > 0.5. A similar picture emerges if the data are detrended and the test statistic

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0.4

Testing for Two (or More) Unit Roots 401

1 0.9 0.8 0.7 0.6 size 0.5

size = 26%

0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5 0.6 ϕ1

0.7

0.8

0.9

1

Figure 10.5b Empirical size of DF τˆ β upper tail.

is τˆ β (see Figures 10.5a and 10.5b). There is a slightly greater oversizing against the stationary alternative; for example, an empirical size of 6.5% for ϕ1 = 1 (see Figure 10.5a); but the oversizing is less severe using the upper tail of the (wrong) null distribution; for example, 26.0% at ϕ1 = 1 (see Figure 10.5b). Thus, the nominal size and empirical size are not grossly dissimilar when the alternative hypothesis requires quantiles from the left tail of the null distribution and ϕ1 → 1; however, a problem arises when ϕ1 = 1 because the null and alternative hypotheses are inappropriate. If application of, say, τˆ µ , leads to nonrejection of the null hypothesis, the implied conclusion is that there is a single unit root, but there are two unit roots; if application of τˆ µ leads to rejection of the null, a feasible alternative is not stationarity but one unit root and a second stationary root. What is required is a test designed to assess whether the DGP is I(2).

10.4 Testing the null hypothesis of two unit roots This section outlines three parametric methods of testing the null hypothesis of two unit roots due to Hasza and Fuller (1979), Sen and Dickey (1987) and Dickey and Pantula (1987). A convenient source for the asymptotic distributions for the test statistics considered here is Rodrigues and Taylor (RT) (2004a) (see also the original sources, and Haldrup, 1998; Haldrup and Lildholdt, 2005).

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0.4

402 Unit Root Tests in Time Series

A rather natural way forward to test for the presence of two unit roots, is to compute the F-test of the restrictions implied by two unit roots; for example, in the case of an AR(3) model, the restrictions are given by Equations (10.16)– (10.18). This can be done directly or by specifying the regression in AHF form. Either way, the test is an F-type test for two restrictions with an implicit twosided alternative. Note, however, that some power will be lost in using an F-test for testing against two-sided alternatives if the alternative is one-sided. A critical issue in the case of the HF (and other) tests for two unit roots relates to the process of demeaning or detrending of the data and the role of initial conditions (see RT, 2004a) on which this discussion is based. In the case of a DGP with a single unit root, with an arbitrary initial condition, what we would like to achieve in the small sample and the limiting distributions of a test statistic for the unit root is invariance with respect to the initial condition; otherwise, the distributional dependence on the initial condition makes the test procedure generally impractical. It may be that we would settle for invariance of the limiting distribution if invariance of the small sample distribution could not be achieved, so that the distribution is similar but not exactly so. This is the case if the initial condition is an Op (1) random variable, whereas exact similarity is achieved if the data are ﬁrst demeaned or detrended by a prior regression, a procedure that will be described as direct demeaning, or a constant or constant and linear trend are included in the maintained regression, a procedure that will be described as indirect demeaning. In contrast to the single unit root DGP, when the DGP contains two unit roots, neither method of demeaning results in HF (or SD) test statistics that are exactly similar, and therefore the test statistics have to be distinguished by the method of demeaning. 10.4.1.i Direct demeaning Consider the DGP given by: φ(L)(yt − µt ) = εt

(10.33)

ˆ t, As in the case of testing for a single unit root, let y˜ t ≡ yt − µt and yˆ˜ t ≡ yt − µ ˆ t is a consistent estimator obtained from a prior regression of yt on the where µ components in the deterministic function µt . For simplicity, adjusting the data ˆ t will be referred to as ‘demeaning’, even though the mean may by removing µ contain a trend. Then the resulting HF regression (see Equation (10.21)) is: ∆2 yˆ˜ t = γ1 yˆ˜ t−1 + γ2 ∆yˆ˜ t−1 + D(L)∆2 yˆ˜ t + ε˜ t

(10.34)

ˆ t ); assuming consistency of µ ˆ t for µt , the term in where ε˜ t = εt + φ(L)(µt − µ ˆ t ) is asymptotically negligible. Estimation of (10.34) is by LS, and φ(L)(µt − µ ˆ t are consistent at the rates T2 , T and T1/2 , in passing note that γˆ 1 , γˆ 2 and the µ

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10.4.1 The HF test

Testing for Two (or More) Unit Roots 403

respectively; the former two are sometimes referred to as being hyper-consistent and super-consistent, respectively. (2) The null hypothesis of two unit roots corresponds to H0 : γ1 = 0 and γ2 = 0; (2)

the alternative hypothesis is HA : γ1 = 0 and/or γ2 = 0. Under the restricted model the coefﬁcients on yˆ˜ t−1 and ∆yˆ˜ t−1 are set equal to zero, so that: (r)

∆2 yˆ˜ t = D(L)∆2 yˆ˜ t + ε˜ t

(10.35)

where ˆ indicates a residual from the corresponding regression. Assuming that the available data on yt run from t = 1, . . . , T, then p observations are ‘lost’ due to the lagging operation and p AR coefﬁcients or their HF equivalents are estimated; additionally, the deterministic function includes k coefﬁcients, so that T# = T − (2p + k) is the denominator degrees of freedom. In the case that p = 2, D(L) = 0 and the restricted residual sum of squares is just the sum of squares of ∆2 yˆ˜ t . As to deterministic terms, there are four cases of interest, but in practice not all equally so: a zero mean, µt = 0; a constant (nonzero) mean, µt = µ; a linear trend, µt = β0 + β1 t; and a quadratic trend, µt = β0 + β1 t + δt2 . In order to distinguish these cases, the notation for the HF test is: HF˜ OLS for µt = 0 and HF˜ OLS , for i = µ, i β, δ. If indirect demeaning is adopted the deterministic variables are included in the AHF regression, so that the maintained regression is: ∆2 yt = µ∗t + γ1 yt−1 + γ2 ∆yt−1 + D(L)∆2 yt + εt

(10.37)

where, for example, if µt = µ, then µ∗t = φ(1)µ and if µt = β0 + β1 t, then µ∗t = φ(1)β0 = φ(L)β1 . Practically all that is required is to include a constant or a constant and a linear trend or a constant and linear and quadratic trends in the maintained regression. The restricted regression is: ∆2 yt = µ∗t + D(L)∆2 yt + εt

(r)

(10.38)

Otherwise, the F test statistic is of the same form as HF˜ OLS , and is denoted HFˆ OLS ; that is:   (r) # ∑Tt= p + 1 (ˆεt )2 − ∑Tt= p + 1 (ˆεt )2 OLS  T (10.39) HFˆ = 2 ∑Tt= p + 1 (ˆεt )2

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(r)

where ε˜ t is the t-th restricted residual. The HF test statistic has the general form of an F test; that is:   (r) (εˆ˜ )2 − ∑Tt= p + 1 (εˆ˜ t )2 T# ∑T OLS  t = p + 1 t ˜  (10.36) = HF 2 ∑Tt= p + 1 (εˆ˜ t )2

404 Unit Root Tests in Time Series

Now in the case that p = 2, D(L) = 0 and the restricted residual sum of squares is the residual sum of squares from the regression of ∆2 yt on µ∗t . As in the case of direct demeaning, different test statistics result from different speciﬁcations of µt in the indirect demeaning procedure and the test statistics analogous to HF˜ OLS are HFˆ OLS , i = µ, β, δ. For most purposes, for example, estabi i lishing distributional results, it will sufﬁce to assume that p = 2, so that D(L) = 0, in which case the simple HF regressions are given by:

2

∆

yt = µ∗t + γ1 yt−1 + γ2 ∆yt−1 + εt

(10.40) (10.41)

corresponding to direct and indirect demeaning, respectively. Note for crossreference that RT refer to the versions of the HF F-type tests from (10.40) as FOLS 1 and FOLS for the cases of a constant mean and a constant and a linear trend, 3 respectively, and to those from (10.41) as FOLS and FOLS 4 , respectively. 2 10.4.1.ii Initial conditions In the case of an I(2) data-generating process, the role of the initial conditions is not completely analogous to the I(1) case. RT distinguish three cases as follows. Case 1: y0 and y−1 are ﬁxed constants of which there are two cases: Case 1a (as in Hasza and Fuller): y0 = ξ0 , y−1 = ξ−1 ; Case 1b (as in Sen and Dickey): ξ0 = ξ−1 ; Case 2: y0 and y−1 are draws from an Op (1) random variable; [κT] + i j Case 3: yi = ∑j=0 ∑k=0 εk−[κT] , i = –1, 0 and κ > 0; [.] is the integer part of the expression. Case 1 obviously covers the cases where y0 = y−1 = 0 and y0 = y−1 = µ, corresponding to a zero mean and a constant nonzero mean for yt , respectively. An example of Case 2 is where y0 and y−1 are draws from the distribution of εt , so that y0 = ε0 and y−1 = ε−1 . Cases 1 and 2 may be unattractive because they imply that y0 and y−1 are generated from a process that is different from the rest of the observations; under the null of two unit roots {yt }Tt=1 is of stochastic order Op (T3/2 ) not Op (1) or ﬁxed. The idea in Case 3 is to generate y0 and y−1 from the same kind of process that generates the rest of the observations. RT show that the limiting null distributions of the HF F tests for the same deterministic speciﬁcation differ under direct and indirect demeaning; speciﬁcally, they show that the limiting null distributions of HF˜ OLS and HFˆ OLS differ, µ µ OLS OLS ˜ ˆ as do those of HFβ and HFβ (see RT, 2004a, Lemma 4.1 and Theorems 4.1 provided and 4.2). This implies, for example, that the critical values for HFˆ OLS µ in Hasza and Fuller (1979, Table 4.1) should not be used for HF˜ OLS , and vice µ versa. These results are proved for Case 3 of the initial conditions, but apply equally to Cases 1 and 2, which arise as special cases of Case 3. Although the no

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∆2 yˆ˜ t = γ1 yˆ˜ t−1 + γ2 ∆yˆ˜ t−1 + ε˜ t

Testing for Two (or More) Unit Roots 405

(i) (ii) (iii)

exact invariance to the initial conditions is not achieved either by HFˆ OLS µ or by HF˜ OLS µ , except for Case 1b (initial conditions ﬁxed and equal); asymptotic invariance to the initial conditions is achieved by HF˜ OLS and µ by HFˆ OLS under Cases 1 and 2, but not under Case 3; µ exact invariance to the initial conditions is achieved by HF˜ OLS and HFˆ OLS β β for Cases 1, 2 and 3.

By extension, exact invariance follows for HF˜ OLS and HFˆ OLS . Thus, in general, δ δ given that the initial conditions are not known, then ceteris paribus, the HF tests should allow for (at least) a constant and a linear trend. ˆ OLS ˜ OLS and HFˆ OLS , the only By the exact invariance property for HF˜ OLS β , HFβ , HFδ δ issue in generating critical values arises for the versions of these tests with no deterministics or with just a constant. RT report critical values generated with the assumption of Case 1b and then assess these critical values for robustness to randomising the starting values. By generating m + T observations, where the ﬁrst m observations are discarded, the starting values are random. RT found that the effects on the sizes of the tests using the critical values from the zero starting value case were evident but not substantial; for example, an empirical size of about 4% for a nominal 5% test, for T = 50, 100, 1000, with HFˆ OLS being rather µ less sensitive than HFˆ OLS µ . This ﬁnding raises the practical question of whether, in the case where the only deterministic component is a constant (which might be zero), to prefer slightly inaccurate critical values or to lose power by including a linear trend, which is actually superﬂuous except to control invariance. 10.4.2 Critical values for HF F-type tests Table 10.2 provides some critical values for the HF F-type tests distinguishing between the two methods of demeaning; for later reference, it also includes critical values for the SD test, which is outlined in section 10.4.3. RT also include critical values for sample sizes T = 50 and T = 100 for HFˆ OLS and HFˆ OLS , i = µ, β i (and the SD versions of these tests). 10.4.3 Sen-Dickey version of the HF test The Sen-Dickey approach exploits the symmetry of the backward and forward realisations of an AR process, which lead to the same autocovariance structure (see Fuller, 1996, chapters 2 and 8). This symmetry has been used on a number of occasions before; for example, in the simple and weighted symmetric versions of the DF-type tests (see Chapter 6, section 6.6.2), and in the combination of the

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deterministics and quadratic trend cases are not addressed explicitly, it seems straightforward to extend the results to these two cases. A second, separate issue is whether the null distributions for a given generation of the starting values, are similar, exactly or otherwise. The results in RT show that:

406 Unit Root Tests in Time Series

Table 10.2 Critical values for HF F-type tests.

No deterministics T = 250 HFˆ OLS = HF˜ OLS HFˆ SSLS = HF˜ SSLS

95%

99%

Demeaning

2.42 2.76

3.15 3.49

4.78 5.20

None None

2.40 2.75

3.13 3.47

4.78 5.14

None None

3.90 5.27 3.64 3.69

4.83 6.24 4.49 4.54

6.82 8.41 6.44 6.42

Indirect Direct Indirect Direct

T = 1,000 HFˆ OLS µ HF˜ OLS µ HFˆ SSLS µ HF˜ SSLS µ

3.87 5.22 3.67 3.66

4.77 6.17 4.52 4.47

6.76 8.23 6.40 6.27

Indirect Direct Indirect Direct

Linear trend T = 250 HFˆ OLS β HF˜ OLS β HFˆ SSLS β HF˜ SSLS

8.10 7.11 5.27 5.80

9.27 8.30 6.30 6.85

11.76 10.74 8.49 9.18

Indirect Direct Indirect Direct

T = 1,000 HFˆ OLS β HF˜ OLS β HFˆ SSLS β HF˜ SSLS

8.01 7.02 5.18 5.87

9.10 8.12 6.22 6.93

11.47 10.56 8.25 9.07

Indirect Direct Indirect Direct

Quadratic trend T = 250 HFˆ OLS δ HF˜ OLS δ HFˆ SSLS δ HF˜ SSLS δ

10.83 10.07 6.91 9.30

12.19 11.41 8.00 10.56

15.06 14.23 10.36 13.23

Indirect Direct Indirect Direct

T = 1,000 HFˆ OLS δ HF˜ OLS δ HFˆ SSLS δ HF˜ SSLS δ

10.69 9.86 6.73 8.92

11.94 11.13 7.83 10.17

14.52 13.91 10.08 12.74

Indirect Direct Indirect Direct

T = 1,000 HFˆ OLS = HF˜ OLS HFˆ SSLS = HF˜ SSLS Mean T = 250 HFˆ OLS µ HF˜ OLS µ HFˆ SSLS µ HF˜ SSLS µ

β

β

Sources: Hasza and Fuller (1979, Table 4.1); Rodrigues and Taylor ˆ OLS , HF˜ OLS , HFˆ SSLS and HF˜ SSLS , own calcu(2004a, Table 1); HF˜ SSLS µ , HFδ δ δ δ lations based on 50,000 replications. Other sample sizes are available in the cited sources.

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90%

Testing for Two (or More) Unit Roots 407

yt = φ1 yt−1 + φ2 yt−2 + . . . + φp yt−p + εt

(10.42)

yt = φ1 yt + 1 + φ2 yt + 2 + . . . + φp yt + p + et

(10.43)

These representations have the same autocovariance function. Consider the forward version of the AR(2) model at t – 2, then: yt−2 = φ1 yt−1 + φ2 yt + et−2

(10.44)

With some elementary manipulation this can be expressed as: ∆2 yt = (φ1 + φ2 − 1)yt−1 + (φ2 + 1)∆yt + et−2 = γ1 yt−1 − γ2 ∆yt + ξt

(10.45)

where ξt = et−2 and, as before, γ1 = (φ1 + φ2 − 1) and γ2 = (φ2 + 1). The backward version of the AR(2) model can be parameterised as: ∆2 yt = γ1 yt−1 + γ2 ∆yt−1 + εt

(10.46)

Hence (10.45) and (10.46) can be estimated as a pair imposing the restrictions that the coefﬁcients on yt−1 are the same and the coefﬁcient on ∆yt in (10.45) and the coefﬁcient on ∆yt−1 in (10.46) are equal but opposite in sign. One way of doing this is to treat (10.45) and (10.46) as a two-equation seemingly unrelated regression (SUR) model, and impose the two cross-equation constraints. Alternatively, a double-length regression is created (see section 10.4.3.i). If the basic model needs to be augmented, as would be the case if the underlying model is AR(p), p > 2, then (10.45) and (10.46) become: ∆2 yt = γ1 yt−1 + γ2 ∆yt−1 + ∑j=1 ϕj ∆2 yt−j + εt p−2

∆2 yt = γ1 yt−1 − γ2 ∆yt + ∑j=1 ϕj ∆2 yt + j + ξt p−2

t = p + 1, . . . , T t = 3, . . . , T − p + 2

(10.47) (10.48)

10.4.3.i Double-length regression for the SD symmetric test statistic The model may also be estimated by creating a double-length regression (assuming t = 1 is the start of the effective sample) and estimating by least squares. Once again the treatment of deterministic components is critical to the analysis; however, in order to lay out the general structure of the estimator and test statistic, initially it is assumed that these are not present, they are then taken into account explicitly.

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test statistics from the backward and forward recursions that form the DF-max tests (see Chapter 6, section 6.6.1). The aim is to exploit the known increase in the power of unit root tests based on symmetric estimation. The backward and forward versions of an AR(p) model are (see Fuller, 1996, p.414):

408 Unit Root Tests in Time Series

First deﬁne the following data vectors and matrices: Y = (∆2 yp + 1 , . . . , ∆2 yT ; ∆2 yT−p + 2 , . . . , ∆2 y3 )

(10.49)

X1 = (yp , . . . , yT−1 ; yT−p + 1 , . . . , y2 )

(10.50) (10.51) (10.52)

Zj = (∆2 yp + 1−j , . . . , ∆2 yT−j ; ∆2 yT−p + 2 + j , . . . , ∆2 y3 )

Z = Z1 . . . Zr where r = p − 2 ≥ 0

(10.54)

E = (ε3 , . . . , εT ; ξT , . . . , ξ3 )

(10.55)

(10.53)

Each of the two sub-vectors of Y is of length T# = T − p, so that the overall length of Y is 2T# ; hence there are 2T# observations in the (double-length) regression. The data can also be arranged in ‘tableau’ form (see SD, 1987, Table 2; Haldrup, 1998, Table 1; and Q10.3). A general weighted symmetric estimator can be obtained by ﬁrst deﬁning a diagonal weighting matrix W of dimension 2T# ×2T# , with diagonal elements that reﬂect the weight on the corresponding backward and forward regressions; note that as W is a diagonal matrix, it will have a square root matrix such that ΨΨ = W, where Ψ is the diagonal matrix with the ii-th element being the square root of the ii-th element of W. To start, set p = 2, which is the simplest case. Consider the regression model given by: ΨY = ΨX1 γ1 + ΨX2 γ2 + E = ΨXγ + E

(10.56)

where γ = (γ1 γ2 ) . The LS estimator of γ and associated quantities in this doublelength regression are: γˆ = (X ΨΨX)−1 X ΨΨY = (X WX)−1 X WY

(10.57)

σˆ (γˆ ) = σˆ s2 (X WX)−1

(10.58)

ˆ ∗ σˆ s2 = Eˆ E/T

(10.59)

ˆ (ΨY − ΨXγ)/T ˆ = (ΨY − ΨXγ)

∗

∗ ˆ W(Y − Xγ)/T ˆ = (Y − Xγ)

where T∗ will be deﬁned below as it requires a note of explanation. The last ˆ WY (see Fuller, 1996, p.415), which line can also be written as (T∗ )−1 (Y − X γ)

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X2 = (∆yp , . . . , ∆yT−1 ; −∆yT−p + 2 , . . . , −∆y3 ) X = X1 X2

Testing for Two (or More) Unit Roots 409

follows from LS algebra whereby Y WX γˆ = γˆ X WXˆγ on substitution on the left ˆ however, practically, it is usually easier to work directly in the second term for γ; with the residual sum of squares.

This is the general weighted symmetric case, with diagonal weight matrix W; however, in the SD case the weights are those of the simple symmetric estimator, so that all of the diagonal elements of W are 0.5; thus W = ωI2T# , where ω = 0.5 ˆ the and I2T# is the identity matrix of order 2T# . Substituting for W = ωI2T# in γ, weights cancel, so that γˆ = (X WX)−1 X WY = ω−1 (X X)−1 ωX Y = (X X)−1 X Y. ˆ = σˆ s2 (X X)−1 where: Similar substitutions show that σˆ (γ) ˆ ∗ σˆ s2 = Eˆ E/T ˆ (Y − Xˆγ)/T∗ = (Y − Xγ)

(10.60)

The SD F-type test may be calculated in one of two ways. First, directly as: 1 2 −1 −1 ˆ γˆ σ (γ) γˆ 2 −1 1 = γˆ σˆ s2 (X X)−1 γˆ 2

HFˆ SSLS =

(10.61)

where the superscript SSLS refers to simple symmetric least squares. Alternatively, the test statistic can be calculated as the scaled difference between the residual sums of squares under the null and alternative hypotheses, noting that if γ = 0, as under the null hypothesis, and p = 2, then the restricted residual sum of squares is the sum of squares of Y, so that:

Y Y − Eˆ Eˆ /2 SSLS ˆ HF = ˆ ∗ Eˆ E/T

Y Y − Eˆ Eˆ T∗ (10.62) =

2 Eˆ Eˆ The corresponding test statistic calculated using direct demeaning is denoted HF˜ SSLS . The divisor T∗ needs to be deﬁned. If the double-length regression is estimated using econometric software, with a standard LS routine to calculate the F (type) test, say Fˆ OLS , the package will use T∗ = 2T# − 2, where 2T# is the (effective) number of observations in the regression and 2 coefﬁcients are estimated; hence the regression apparently has T∗ degrees of freedom. Using T∗ results in the SD version of the F-type test, the percentiles of which for the deterministic mean case are provided in Table 7 of Sen and Dickey (1987). However, as noted in Chapter 6 and see Fuller (1996, p.415), there are really only T# observations and T# – 2 degrees of freedom. Hence, using T∗ = T# – 2 implies the correction factor

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10.4.3.ii SD F-type test, p = 2

410 Unit Root Tests in Time Series

CF = (2T# − 2)/(T# − 2) ≈ 2, which is applied to the denominator of Fˆ OLS as it would be outputted from a standard LS program; thus the effect of the correction factor is to divide Fˆ OLS by (approximately) 2 to obtain HFˆ SSLS . Percentiles of the null distribution for this case for the constant mean and the linear trend subcases are provided by Rodrigues and Taylor (2004a, Table 1). It is this version of the SD F test that is used in this chapter.

In the case that the basic model needs to be augmented, as would be the case if the underlying model is AR(p), p > 2, then the general form of the SD regression in double-length form is: Y = X1 γ1 + X2 γ2 + Zω + E

(10.63)

Where Z is the matrix containing the augmented terms; it is of dimension r ×2T# , with r = (p – 2) ≥ 1 and ω = (ω1 . . . ωr ) . Notice that each Zj contains two vectors, each of which has T# = [(T − j) − (p + 1 + j) + 1] = (T − p) observations. Estimation is by LS, as before. The changes of note are that the restricted regression is not null, but comprises Zω; that is, Y = Zω + Er The SDF test statistic is now computed as:

Eˆ r Eˆ r − Eˆ Eˆ /2 SSLS ˆ = HF ˆ ∗ Eˆ E/T

(10.64)

(10.65)

where Eˆ r Eˆ r and Eˆ Eˆ are the residual sums of squares from the restricted and unrestricted regressions, respectively. As before T∗ needs to be deﬁned. Allowing for estimation of the additional coefﬁcients in ω, then T∗ = T# – p, p > 2. If the SD F test is obtained via a standard LS routine using the double-length regression, then the package will use T∗ = 2T# − p for the degrees of freedom. This implies the correction factor CF = (2T# − p)/(T# − p) ≈ 2, so that the outputted F statistic should be divided by CF ≈ 2 (if p = 2, then this is the special case dealt with earlier with no augmentation). 10.4.3.iv Deterministic components As in the HF regression, deterministic terms can be accounted for by direct demeaning or indirect demeaning. In the case of direct demeaning, the basic variable becomes y˜ t or practically yˆ˜ t rather than yt , otherwise the deterministic variables are included directly in the double-length regression. The inclusion of a constant is straightforward, but some care has to be taken with the trend term(s). Automatic setting of the trend variable would result in it being set

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10.4.3.iii SD F-type test, p > 2

to run from 1 to 2T# in the double-length regression; this is incorrect. The time trend should be set to have the sequence of subscript values as in the Y vector of (10.49), or start at 1 but reverse accordingly with the second subvector. Also, in the case of direct demeaning, the program will not recognise the implicit degrees of freedom of demeaning in obtaining yˆ˜ t , although in the case of indirect demeaning there will be an adjustment. As in the case of the standard HF F tests, a ˆ or ˜ above indicates the use of indirectly demeaned or directly demeaned data, respectively, and different speciﬁcations are distinguished by subscripts i = µ, β, δ, with no subscript if no deterministic components are included. The resulting test statistics are denoted HF˜ SSLS and HFˆ SSLS for direct and indirect demeaning, respectively. i i Analogous to the invariance results for the HF F-type tests, Rodrigues and Taylor (2004a) establish the following results: (i) (ii) (iii) (iv) (v)

exact invariance to the initial conditions is not achieved by HFˆ SSLS or µ HF˜ SSLS , except for Case 1b (initial conditions ﬁxed and equal); µ asymptotic invariance to the initial conditions is achieved by HF˜ SSLS and µ SSLS ˆ HFµ under Cases 1 and 2, but not under Case 3; exact invariance to the initial conditions is achieved by HF˜ SSLS for Cases β

1, 2 and 3; HFˆ SSLS is not invariant to y0 and y−1 and Rodrigues and Taylor (2004a) do β not recommend its use; HFˆ SSLS is asymptotically similar only in Case 2 (starting values are draws β from an Op (1) random variable).

Some insight on these results can be obtained from (10.3); recall that in the AR(2) case, yt depends on the starting values such that: yt = t(y0 − y−1 ) + y0 + ∑i=1 ∑j=1 εj t

i

It is the inﬂuence of the ﬁrst two terms that the structure of the test statistics ˜ SSLS ˆ OLS and HF˜ SSLS HFˆ OLS cannot ‘shake off’. Hence Rodrigues and Taylor µ , HFµ , HFµ µ (2004a) note that exact invariance can be obtained using data that removes the contribution of the starting values to yt ; that is, use data deﬁned as: y∗t ≡ yt − t(y0 − y−1 ) − y0

(10.66)

Thus y∗t , ∆y∗t , and so on, are used in the HF and SD regressions. Practically, to compute the data on y∗t , the ﬁrst two observations in the sample are renumbered as y−1 and y0 , respectively, renumbering the remaining observations as t = 1, . . . , T + , where T + = (T – 2). Some intuition onto the importance of the starting values on the invariance or otherwise of the test statistics can be obtained from (10.66). Note that if y0 = y−1 , which is Case 1b, then y∗t = yt – y0 , so that the data just need to be adjusted by a constant, which can be done either directly

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Testing for Two (or More) Unit Roots 411

412 Unit Root Tests in Time Series

10.5 The Dickey-Pantula (DP) approach The DP approach is a rather natural and intuitive extension of the standard DFtype approach. In effect, it assumes that there is one unit root and then tests for the presence of a second unit root, and hence apart from a change of variable, the maintained regressions are as those of the DF or ADF approach. 10.5.1 An extension of the DF procedure Initially suppose that there is deﬁnitely one unit root, and the question of interest is whether there is a second unit root. In the context of the following model: (1 − L)yt = zt

(10.67)

(1 − ϕ1 L)zt = εt

(10.68)

where (10.68) is a special case of ϕ(L)zt = εt . The null hypothesis of interest is (2) (2) H0,1 : ϕ1 = 1 against the alternative hypothesis H1,A : |ϕ1 | < 1. This hypothesis can be tested by ﬁrst deﬁning wt ≡ ∆yt and then, with this deﬁnition, substitute (10.67) into (10.68), resulting in the following simple DF regression for wt : wt = ϕ1 wt−1 + εt

(10.69)

⇒ ∆wt = ϕwt−1 + εt

(10.70) (2)

where ϕ = ϕ1 −1 and H0,1 corresponds to ϕ = 0. This much is familiar from testing for one unit root. If ϕ(L) = 1 + ∑ki=1 ϕi Li with k ﬁnite, the ADF formulation is then exactly ADF (k – 1), that is: ∆wt = ϕwt−1 + ∑j=1 αj ∆wt−j + εt k−1

(10.71)

Otherwise, for example, if zt contains an invertible MA component, such that: (1 − L)yt = zt

(10.72)

ψ(L)zt = εt

(10.73)

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or indirectly (by the inclusion of a constant); if y0 = y−1 , as in Cases 1a, 2 and 3, then there is a trend term, so that adjustment of the data by a constant is insufﬁcient to achieve invariance. The case with p > 2 is not dealt with explicitly by Rodrigues and Taylor (2004a); however, one may surmise that invariance now has to be achieved with respect to p starting values, which changes the deﬁnition of y∗t as p changes; some justiﬁcation for this argument is provided by way of Q10.1. It may be easier, in the general case, to choose an invariant test statistic.

Testing for Two (or More) Unit Roots 413

where ψ(L) = θ(L)−1 ϕ(L), then the ADF(k – 1) regression is an approximation; that is: k−1

∞

εt,k = ∑j = k αj ∆wt−j + εt

(10.74) (10.75)

The regression model (10.74) will be referred to as ADP(k – 1) to indicate that its purpose is to test for a second unit root conditional on a ﬁrst unit root. Whilst the test statistics are as in the standard DF case (or extensions thereof) they will be designated here with a DP superscript to assist the distinction among tests ˆ DP for multiple unit roots; thus the test statistics are designated τˆ DP , τˆ DP µ , τ β and DP DP τˆ δ . In practice, τˆ will be rarely used. 10.5.2 Numerical illustration To gain some insight into the DP test, we reconsider the numerical examples used earlier (see Equation (10.32)). In that case, an AR(4) model was constructed as: (1 − λ1 L)(1 − λ2 L)(1 − λ3 L)∆yt = εt

(10.76)

with λ1 = 0.98, λ2 = 0.5, λ3 = 0.2, and corresponding roots 1.02, 2 and 5, together with the unit root implied by ∆. The resulting AR(4) model is: yt = 2.48yt−1 − 2.01yt−2 + 0.57yt−3 − 0.04yt−4 + εt First notice that A(1) = 0, so that there is at least one unit root. The ADF(3) model is: ∆yt = 1.48∆yt−1 − 0.53∆yt−2 + 0.04∆yt−3 + εt There is no term in yt−1 because of the unit root, which induces a zero coefﬁcient. Similarly, this can be formulated as an AHF(2) model, so that: ∆2 yt = − 0.01∆yt−1 + 0.49∆2 yt−1 − 0.04∆2 yt−2 + εt Again, there is one term missing relative to the general AHF(2); that is, there is no term in yt−1 because of the unit root, and the focus is now on the coefﬁcient on ∆yt−1 to see how close that is to zero. In terms of the notation wt ≡ ∆yt , the AHF(2) restricted by a single unit root, is: ∆wt = − 0.01wt−1 + 0.49∆wt−1 − 0.04∆wt−2 + εt This is an ADP(2) in wt , the coefﬁcient of interest being that on wt−1 , which is, by construction in this example, close to zero. Although the original DP procedure used the DF unit root test statistics, other test statistics that are more powerful in the standard case may be used and offer

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∆wt = ϕwt−1 + ∑j=1 αj ∆wt−j + εt,k

414 Unit Root Tests in Time Series

10.5.3 Start from the highest order of integration The important point about the DP procedure is that it starts from the highest order of integration, say d∗ , being considered (rather than the lowest order), (d∗ ) (d∗ ) (d∗ −1) say H0,d∗ −1 : yt ∼ I(d∗ ); the usual alternative hypothesis HA = H0,d∗ −2 is that (d∗ )

there are d∗ − 1 unit roots, with the remaining roots stationary; if H0,d∗ −1 is (d∗ −1)

rejected, then H0,d∗ −2 becomes the new null hypothesis. The testing procedure is a sequence starting with d∗ unit roots and stopping when the null of d∗ − j unit roots, j = 0, . . . , d∗ − 1, is not rejected. The speciﬁcation of the deterministic components in the DP procedure may not always follow that in the HF method. Suppose we start with the hypothe(2) (2) (1) sis of two unit roots, H0,1 , against one, HA = H0,0 ; then according to the null hypothesis, the order of variation in the dependent variable is not adequately captured by a process with a single unit root combined with a stationary root: hence the second unit root is necessary. Equally, the alternative hypothesis needs to add something to a single unit root to capture the perceived behaviour of yt , and the usual choice is a unit root combined with a linear trend. For example, in the case that yt is a price level in a situation where hyperinﬂation has been taking place, the alternative is that the inﬂation rate, embodying the unit root, follows a linear trend. In contrast to possible alternatives using the HF test, a quadratic trend is not entertained as sufﬁcient variation is captured by the unit root plus linear trend. (For examples of hyperinﬂation, see Georgoutsos and Kouretas, 2004, for the case of post-World War I hyperinﬂation; see Petrovi´c et al., 1999, for Yugoslavian hyperinﬂation; and for the possible importance of explosive roots, see Juselius and Mladenovic, 2002.) The HF test also starts from the highest order being considered, but because of the two-sided nature of the F-type, the alternative is implicitly wider than in the DP case, including explosive alternatives. Some loss of power can be anticipated against an alternative where one of the unit roots in the null hypothesis is replaced by a stationary root under the alternative. (The DP test can be used against explosive alternatives by considering right-tailed critical values.) DP (1987) compared the sequential F procedure with the sequential DP procedure based on the pseudo t statistic in a small simulation study. They considered a number of conﬁgurations of the roots of an AR(3) process and started the

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the potential of power gains in this case as well. For example, simple alternatives are available using the recursive mean adjustment procedure (see Shin and So, 2002) and the weighted symmetric (WS) version of the DF statistics. It is less clear how GLS detrending would be used as the nature of the alternative hypothesis is necessarily more complex when the null hypothesis is of two unit roots, but if the alternative was of a DGP with a single unit root and a linear trend, then the quasi-differencing could be applied to wt .

Testing for Two (or More) Unit Roots 415

10.6 Power A power analysis of tests for multiple unit roots has to take into account the nature of the alternative hypothesis, which is more complex than in tests designed for a single unit root. Section 10.1.3 outlined a structure for alternatives to the hypothesis that the DGP generates an I(2) series. The key point is that the DGP under the alternative hypothesis must match the variational characteristics of yt when there are two unit roots; for example, comparing two unit roots with two stationary roots make little sense as the latter cannot generate the growth in yt ; for example, what would be needed is a quadratic trend. 10.6.1 Asymptotic local power Haldrup and Lildholt (2005) undertook a comparison of the asymptotic local power functions for the doubly-integrated case. Whilst analogous to the single unit root case, it also takes into account the ‘near’ alternatives in two dimensions. Consider the starting model DGP deﬁned in terms of y˜ t : (1 − ρL)y˜ t = zt

(10.77)

(1 − ϕ1 L)zt = εt

(10.78)

The alternatives local to two unit roots are: ρT = 1 + c1 /T

(10.79)

ϕ1,T = 1 + c2 /T

(10.80)

So that under the local-to-unity alternative for each root, the model is: ∆2 y˜ t = (ρ + ϕ1 − ρϕ1 − 1)y˜ t−1 + (ρϕ1 − 1)∆y˜ t−1 + εt c c c1 + c2 c1 c2 = − 1 22 y˜ t−1 + + 2 ∆y˜ t−1 + εt T T T

(10.81)

Hence, under the null of two unit roots, the bracketed terms are zero. Also note that, as might be anticipated, (10.81) is symmetric in c1 and c2 , so that an analysis of power need only consider the upper triangular matrix formed from an array of values C1 × C2 , where ci ∈ Ci .

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sequence at three unit roots; when the alternative had two unit roots and a stationary root, the sequential τˆ testing procedure was more powerful, with power increasing as the third root approached unity; this was also the case when the alternative had one unit root and two stationary roots and, alternatively, three stationary roots. Although not extensively studied, DP also considered the case where one or two roots were (just) explosive, a case that might be thought to favour the sequential F test; however, the sequential t test was again more powerful.

As a reminder, δ1 = ρ−1 is the root of (1 − ρL) and δ2 = ϕ1−1 is the root of (1 − ϕ1 L); then under the alternative hypothesis, the roots are local to unity in the sense that δi,T = T/(T + ci ), is in the stationary region for ci < 0 and in the nonstationary region for ci ≥ 0 and δi → 1 as T → ∞. Haldrup and Lildholt (2005) undertook a simulation comparison of asymptotic power in the case of no deterministic components and under the local alternatives indexed by (c1 , c2 ), or equivalently (δ1,T , δ2,T ), with ci varying from 0 through to ±10 and T = 5,000 for the asymptotic case. They found that the ranking of the tests depended on the pairing of regions. The DP test had the best power when one of the roots was (locally) explosive and the other was either explosive or a unit root, and when there was one unit root and one stationary root for larger values of c2 , whereas the SD test was better for smaller values of c2 and when both roots were stationary. A comparison of the DP and HF tests suggested that there was often little to choose between them except when one of the roots was close to unity, in which case the DP test had better power. Some ﬁnite sample results for different speciﬁcations of the deterministic terms are reported in section 10.6.2.ii.

10.6.2 Small sample simulation results 10.6.2.i SD comparison of the HF and SD versions of the F test SD (1987) report some small sample (T = 50 and T = 100) simulations to evaluate power comparing the SD and HF versions of the F test. In the case of one unit root and a second stationary root, they found that the SD F test was more powerful than the HF version the further the second root was into the stationary region. Allowing both roots to be stationary under the alternative, they found that their version of the test was again more powerful than the HF test, although marginally so at the unit root boundary; when one root or both roots were explosive, there was often little to choose between the two tests. SD also undertook a sensitivity analysis of their results to varying the ﬁxed initial condition. The analysis of tests for a single unit root (see Chapter 7, section 7.4.10, and Equation (10.3)), suggests that power is likely to be affected by the initial condition. Indeed, developments along the lines suggested for I(1) tests could be undertaken for I(2) tests to minimise the effects of the initial conditions on the test outcome. SD (1987) randomised the initial conditions by starting the process for 130 observations and then discarding 80 of these observations, to leave 50 (random) start-up values before starting the regression computations. In this case, the tendency was for power to be reduced and the SD F test was not always superior to HF, although it became so in the interior of the stationary region.

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416 Unit Root Tests in Time Series

Testing for Two (or More) Unit Roots 417

To illustrate some of these results we consider the case with T = 200 and the DGP as in (10.77) and (10.78), combined with (10.79) and (10.80) where c1 , c2 = (0, –4, . . . , –40), resulting in ρ, ϕ1 = (1, 0.98, . . . , 0.80) and δ1 , δ2 = (1, 1.02, . . . , 1.25), although the tables of results are truncated when power = 100%. (Other values of T generated the same qualitative rankings, but the power differences became less marked as T was increased.) As in Rodrigues and Taylor (2004a) and Sen and Dickey (1987), the starting values for the DGP are assumed to be zero; however, it would also be of interest (as in Chapter 7, section 7.4.10) to assess the sensitivity of the tests to variations in this assumption. The results are summarised in Tables 10.3–10.5 for the cases where the data are demeaned by a constant (Table 10.3), by a constant and a linear trend (Table 10.4) and by a constant, linear trend and quadratic trend (Table 10.5). The latter two cases are likely to be of greater empirical relevance, the ﬁrst being included for completeness. The ﬁrst row in each table relates to the situation where there is one unit root, so that the column entries show power as the second root moves further into the stationary region. Then moving down the rows, both roots are in the stationary region, becoming more stationary as the entries move down and across the table. The directly demeaned versions of the tests are used throughout in the comparison. First, comparing the HF and SD versions of the F test, the SD version is virtually uniformly more powerful throughout in all three cases considered (mean, linear trend, quadratic trend). The increase in power is generally quite marked; for example at the pairing (0.96, 0.96) in the with-trend case the empirical power of the HF and SD tests are approximately 41% and 51%, respectively. Next, comparing each test across different deterministic speciﬁcations, note that, as in the case of tests for a single unit root, power decreases with the number of deterministic components. Next consider the ranking when the DP test is included. Here the situation is not as straightforward. In the ﬁrst instance, the deterministic components are taken to be the same in the comparison above. In the with-constant case, the ˜ OLS and HF˜ SSLS test statistic τˆ DP µ is more powerful than both HFµ µ , see Table 10.3. However, this is not uniformly the case for the with-trend and with-quadratic trend cases, where the SD version of the HF test tends to be best when both roots are close to the unit circle, but the DP tests are better – sometimes much better – as the roots become more stationary. Of course, it is only in the ﬁrst row that the alternative hypothesis is correctly speciﬁed for the DP test, since the other rows relate to two stationary roots; notwithstanding, the DP test does have power in the interior of the stationary region. One further point to mention relates to the question, what is the appropriate comparison between test statistics for the DP test? In the case of the DP test,

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10.6.2.ii Illustrative simulations: F test and DP tests

418 Unit Root Tests in Time Series

ρ↓

ϕ1 →

1.0

0.98

0.96

0.94

0.92

0.90

0.88

0.86

0.84

0.82

1.0

HF SD DP HF SD DP HF SD DP HF SD DP HF SD DP

5.0 5.0 5.0

13.4 12.2 20.8 39.4 42.8 57.2

27.0 25.8 52.0 59.4 70.4 84.6 83.4 92.6 98.2

52.6 52.2 85.8 83.6 90.6 98.4 95.8 99.0 99.8 95.0 98.8 99.4

64.2 66.6 95.4 92.6 98.2 99.6 99.0 1.00 1.00 99.4 99.8 1.00 1.00 1.00 1.00

86.6 90.2 99.2 99.6 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

93.8 94.8 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.98

0.96

0.94

0.92

˜ SSLS ; DP: τˆ DP , for i = µ, β Note: The test statistics referred to in Tables 10.2–10.4 are: HF: HFˆ OLS µ , SD: HFi i and δ.

one unit root is assumed, which, for example, supplants the need for a linear trend in the alternative DGP; this implies a comparison as far as the ﬁrst row of Table 10.4 and 10.5 is concerned, between, for example, HF˜ SSLS and τˆ DP µ and β SSLS DP and τˆ . In this case, the DP versions of the test are nearly between HF˜ δ

β

uniformly more powerful. In summary, as far as the F versions of the HF test are concerned, the SD test is generally to be preferred for the region of the parameter space considered here; where HF is superior, it is only marginally so. As far as the SD and DP tests are concerned, which is superior depends in part on the region of the parameter space being considered. In practice, it is likely that computing both tests will be helpful, but the researcher should be aware of the multiple testing issue highlighted elsewhere (see Chapter 9).

10.7 Illustrations of testing for two unit roots This section illustrates applications of the test statistics and concepts of the previous sections to two of the series introduced in section 10.1.2 as possible I(2) variables. 10.7.1 Illustration 1: the stock of US consumer credit The ﬁrst illustration uses the series for the stock of consumer credit in the US, graphed in Figure 10.3b; the data are monthly for the period 1943m1 to

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Table 10.3 Empirical power of tests for two unit roots; data directly demeaned assuming a constant mean.

Testing for Two (or More) Unit Roots 419

ρ↓

ϕ1 →

1.0

0.98

0.96

0.94

0.92

0.90

0.88

0.86

0.84

0.82

1.0

HF SD DP HF SD DP HF SD DP HF SD DP HF SD DP HF SD DP HF SD DP

5.0 5.0 5.0

7.8 9.4 7.4 14.2 20.4 8.8

13.4 19.4 16.4 25.4 32.8 28.6 40.6 50.8 47.6

27.6 36.6 36.4 32.6 44.4 45.4 56.4 72.2 69.6 78.8 91.8 89.0

31.6 44.2 52.4 53.0 67.4 72.4 74.8 88.2 86.4 92.2 97.8 97.4 96.2 99.8 98.6

55.4 67.8 79.0 70.2 82.8 87.4 90.2 97.6 96.0 97.4 1.00 99.6 99.8 1.00 1.00 99.4 1.00 99.8

68.8 83.0 89.8 79.4 90.4 95.6 95.2 99.0 99.6 99.8 99.6 1.00 1.00 1.00 1.00 99.8 1.00 1.00 1.00 1.00 1.00

82.6 91.2 98.0 92.4 97.8 99.2 99.2 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

90.6 96.4 99.6 97.6 99.8 1.00 99.2 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

96.6 99.0 1.00 99.4 1.00 1.00 99.8 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.98

0.96

0.94

0.92

0.90

0.88

2007m12, giving an overall sample of 780 observations, and the data is in natural logs. It is clear that the series is trended, which must be taken into account in the speciﬁcation of the alternative hypothesis. The test will be illustrated with, alternately, a linear trend and a quadratic trend. The model to be ﬁtted is an example of: A(L)y˜ t = εt

(10.82)

where A(L)y˜ t is a p-th order lag polynomial. Interest centres on three possible formulations: AHF(p – 2) for two unit roots: ∆2 y˜ t = γ1 y˜ t−1 + γ2 ∆y˜ t−1 + D(L)∆2 y˜ t + εt

(10.83)

(2)

H0 : γ1 = γ2 = 0

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Table 10.4 Empirical power of tests for two unit roots; data directly demeaned by a constant and linear trend.

420 Unit Root Tests in Time Series

ρ↓

ϕ1 →

1.0

0.98

0.96

0.94

0.92

0.90

0.88

0.86

0.84

0.82

1.0

HF SD DP HF SD DP HF SD DP HF SD DP HF SD DP HF SD DP HF SD DP

5.0 5.0 5.0

7.0 8.6 4.8 8.2 10.4 6.0

11.0 12.6 10.8 12.8 17.4 11.8 16.2 24.4 20.6

12.2 20.2 17.2 17.2 25.0 20.2 27.2 39.2 32.8 40.6 57.6 51.0

23.2 29.8 27.2 30.0 41.8 35.0 40.0 54.2 52.4 60.4 75.2 69.6 72.4 87.6 87.0

29.8 45.2 50.8 37.8 51.4 53.6 55.4 73.0 72.6 71.2 84.2 86.8 90.4 96.4 94.8 96.2 98.8 99.2

48.6 57.4 68.0 54.4 67.4 73.4 69.2 81.8 86.2 81.8 93.8 84.8 59.2 99.4 99.0 99.2 99.8 99.8 99.8 1.00 1.00

58.0 71.8 81.4 65.4 80.4 89.0 83.6 93.2 93.8 94.0 99.0 99.2 97.8 99.8 99.8 99.4 99.8 99.8 1.00 1.00 1.00

72.4 83.6 94.4 82.6 88.8 94.6 91.0 97.2 99.2 98.4 99.8 99.6 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

80.4 90.0 94.4 85.6 95.0 94.6 96.6 98.8 99.2 99.4 1.00 99.8 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

0.98

0.96

0.94

0.92

0.90

0.88

ADP(p – 2) for a second unit root conditional on a ﬁrst unit root: ∆2 y˜ t = γ2 ∆y˜ t−1 + D(L)∆2 y˜ t + εt

(10.84)

⇒ ˜ t = γ2 w ˜ t−1 + D(L)∆w ˜ t + εt ∆w

(10.85)

˜ t ≡ ∆y˜ t w (2)

H0,1 : γ2 = 0 ADF(p – 1) for a single unit root: ∆y˜ t = γ1 y˜ t−1 + C(L)∆y˜ t + εt

(10.86)

(1)

H0 : γ1 = 0 Note that AHF(p – 2) and ADF(p – 1) are equivalent, whereas ADP(p − 2) restricts AHF(p – 2) by setting γ1 = 0, as testing for a second unit root is conditional on the existence of one unit root. Some attention is ﬁrst directed to the speciﬁcation of the DGP that is an alternative to two unit roots. To emphasise the practical difﬁculty in specifying the

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Table 10.5 Empirical power of tests for two unit roots; data directly demeaned by a constant, linear trend and a quadratic trend.

Testing for Two (or More) Unit Roots 421

16 linear trend

15 14

actual

13 12

10 9

quadratic trend

data linear trend quadratic trend

8 1940 1950 1960 1970 1980 1990 2000 2010 Figure 10.6a US consumer credit actual and trends (logs).

0.4 from linear trend 0.2 0 –0.2

from quadratic trend

–0.4 –0.6 –0.8 –1 1940 1950 1960 1970 1980 1990 2000 2010 Figure 10.6b Residuals from trends, US credit.

alternative, the estimated linear and quadratic trends, with associated residuals, are shown in Figures 10.6a and 10.6b, respectively; both trends provide a reasonable ‘eyeball’ ﬁt to the data with, perhaps, an advantage to the quadratic trend because of the slight curvature in the time series, which is reﬂected in the residuals shown in Figure 10.6b. Under the alternative, the likely possibilities are stationarity around a quadratic trend, stationarity around a linear trend or a trended growth rate (one unit root imposed). A linear trend ﬁtted to ∆yt shows a slight negative slope (see Figure 10.6c), with the residuals shown in Figure 10.6d. The HF F test results are ﬁrst reported for the linear trend and then the quadratic trend. Initial estimation, with µt ﬁtted by a linear trend, suggested that p = 6 yielded ‘white’ residuals (p = 11 was an alternative, but the results are not altered by this choice). The sum of the AR coefﬁcients was stable at 0.996 across a wide

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11

422 Unit Root Tests in Time Series

0.08 0.06 0.04 0.02

–0.02 –0.04 1940 1950 1960 1970 1980 1990 2000 2010 Figure 10.6c US credit, monthly growth rate and trend.

0.08 0.06 0.04 0.02 0 –0.02 –0.04 –0.06 1940 1950 1960 1970 1980 1990 2000 2010 Figure 10.6d Residuals for linear trend for growth rate.

range of values for p. The roots of the lag polynomial are in three complex pairs: 1.037 ± 0.019i, 0.429 ± 1.581i, −1.697 ± 0.712i, with moduli of 1.04, 1.64 and 1.84, respectively; hence there are two near-unit roots with the others quite distant from the unit circle, and the time series is a candidate for I(2) behaviour. There was no suggestion of any explosive or near-explosive roots. Some estimation details are summarised in Table 10.6. Consideration is ﬁrst given to the LS test statistics. The initial model of interest was AHF(4), summarised in the ﬁrst part of the table (linear trend and then quadratic trend). The HF test statistics were HF˜ OLS = 23.23 and HF˜ OLS = 27.65; both considerably β δ exceed their respective 95% quantiles of 8.15 and 11.14, and the null hypothesis of two unit roots is ﬁrmly rejected on this basis.

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0

Testing for Two (or More) Unit Roots 423

AHF(4) Trend

yˆ˜ t−1

∆yˆ˜ t−1

∆2 yˆ˜ t−1

∆2 yˆ˜ t−2

∆2 yˆ˜ t−3

∆2 yˆ˜ t−4

∆2 yˆ˜ t ‘t’ AHF(4) Quad.trend ∆2 yˆ˜ t ‘t’ ADP(4) Trend ∆2 yˆ˜ t ‘t’ ADP(4) Mean ∆2 yˆ˜ t ‘t’

–0.0039 –4.31

–0.166 –5.62

–0.584 –14.10

–0.296 –6.68

–0.252 –5.89

–0.102 –2.90

–0.0062 –4.61

–0.177 –5.69

–0.580 –13.79

–0.297 –6.66

–0.253 –5.92

–0.104 –2.95

– –

–0.155 –5.22

–0.576 –13.76

–0.281 –6.28

–0.237 –5.51

–0.092 –2.58

– –

–0.058 –3.06

–0.649 –16.81

–0.333 –7.69

–0.277 –6.52

–0.113 –3.17

ADF(5) Trend ∆yˆ˜ t ‘t’

yˆ˜ t−1 –0.0039 –4.31

∆yˆ˜ t−1 0.245 5.62

∆yˆ˜ t−2 0.288 7.91

∆yˆ˜ t−3 0.044 1.18

∆yˆ˜ t−4 0.149 4.11

∆yˆ˜ t−5 0.102 2.90

SSLS Trend estimate ‘t’

γ˜ 1SSLS –0.0033 –3.72

γ˜ 2SSLS –0.125 –4.39

ϕ˜ 1SSLS –0.595 –14.25

ϕ˜ 2SSLS –0.302 –6.75

ϕ˜ 3SSLS –0.266 –6.14

ϕ˜ 4SSLS –0.113 –3.17

The ADP(4) regression restricts the coefﬁcient on yˆ˜ t−1 to zero, the test statistic being the pseudo t statistic on ∆yˆ˜ t−1 , which at –5.22 is compared to the DF critical value of τˆ β = –3.41 (the sample is large enough to ignore a degrees of freedom adjustment). Conditional on a single unit root, the trend ﬁtted in the previous regression may not be necessary and the ADP(4) regression was also estimated where yˆ˜ t is the residual from a regression of yt on just a constant. The pseudo t statistic on ∆yˆ˜ t−1 is now –3.06, compared to the 5% DF critical value of τˆ µ = –2.87; thus conditional on one unit root, the DP procedure leads to rejection of a second unit root, and the next stage is to test for a single unit root. The trend should now be reinstated, otherwise there is no mechanism under the alternative to generate the trended behaviour evident in the empirical series. Thus, referring to the ADF(5) regression, which is equivalent to the AHF(4) regression if the deterministic components are unchanged, τˆ β = –4.31 compared to a 5% critical value of –3.41. At this stage, the conclusion from the LS-based test statistics is therefore in favour of stationarity, although whether that is best described about a linear trend or a quadratic trend is a matter for further investigation. The maintained regression for the SD symmetric procedure is now of the following general form:

Y = X1 γ1 + X2 γ2 + Zω + E

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Table 10.6 Number of unit roots; estimation details for log of consumer credit, US.

424 Unit Root Tests in Time Series

(see Equation (10.63)). In this illustration, Z contains four columns, with the j-th column, j = 1, . . . , 4, given by (see Equation (10.53)):

with p = 6. The associated SSLS coefﬁcients are denoted γ˜ iSSLS and ϕ˜ jSSLS (see the lower part of Table 10.6), and note that the estimated coefﬁcients are close to their LS counterparts (compare with the coefﬁcients in the ﬁrst row). The SD test statistic is HF˜ SSLS = 17.85 compared to the 95% quantile of 6.93, so that the β null hypothesis of two unit roots is rejected on this basis as well. In order to obtain the t statistics, an adjustment has to be made, as suggested in Chapter 6 and see section 10.4.3; in this case, the outputted t statistics are √ divided by CF = (2T# − p)/(T# − p)1/2 , and these are reported beneath the estimated coefﬁcients. As in the AHF regression, the SD regression can also be used to test for a single unit root (providing that the speciﬁcation of the deterministic components is unchanged); the pseudo t statistic associated with γ˜ 1SSLS is –3.72. The critical values are available in Fuller (1996, Table 10.A.3, T = ∞); the 5% and 1% critical values are –3.27 and –3.80, so the null is rejected using the former but not using the latter. Finally, ﬁtting a quadratic trend, as in the standard LS case, does not change the material conclusions except to reinforce the rejection of two unit roots; the value of the test statistic is HF˜ SSLS = 20.98 compared with the 95% quantile of δ 10.17 and the pseudo t for γ˜ 1SSLS is –6.02 compared to 5% and 1% critical values of –3.73 and –4.26. Thus, on this criterion, the series is more clearly stationary when ﬁtted by a quadratic trend. In summary, despite the proximity of two of the roots, and the sum of the AR coefﬁcients, to unity, the tests do not support the view that this time series is I(2); indeed, on the basis of the tests reported here, there is also some doubt that the series is I(1) rather than stationary about a linear or quadratic trend. 10.7.2 Illustration 2: CPI Denmark The second illustration uses the time series for the CPI for Denmark; the series details were given in section 10.1.2, with the series graphed in Figure 10.2b. The overall sample comprises 108 annual observations and the data is in logs. An initial empirical analysis suggested that p = 5, which we take here, with p = 4 a close alternative. The estimated roots corresponding to p = 5 are: 1.102 ± 0.124i, 0.188 ± 1.374i and 3.749, with moduli for the complex roots of 1.11 and 1.39; hence there are two roots quite close to the unit circle and three quite distant, thus the series is a candidate for I(2) behaviour. The sum of the AR coefﬁcients is 0.965 and there are no indications of explosive behaviour. A linear trend is included as an option in the demeaning, since the alternative hypothesis under stationarity must include a mechanism capable of generating the evident upward movement in the CPI. A quadratic trend also competes with

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Zj = (∆2 yp + 1−j , . . . , ∆2 yT−j ; ∆2 yT−p + 2 + j , . . . , ∆2 y3 )

two unit roots and so may, on occasions, be of interest, even at the risk of over-specifying the polynomial trend. The point is that if the DGP under the alternative comprises a quadratic trend or a unit root and a linear trend, then, when solved for the levels, the series under both representations will have a quadratic trend. To illustrate, the series and the estimated linear and quadratic trends are shown in Figure 10.7a and the residuals from these trends in Figure 10.7b. The quadratic trend, viewed as a local approximation could not be ruled out (although one may have some doubts about projecting the quadratic trend outside the sample period). Imposing a unit root, so that the resulting series is the inﬂation rate, ∆yt , and ﬁtting a linear trend results in Figure 10.7c; note the trend increase in inﬂation, which creates the suspicion associated with an I(2) series. The residuals from this regression are shown in Figure 10.7d, and the question then is whether the residuals have a further unit root. The estimation details for the linear trend and quadratic trend cases are included in Table 10.7. The HF test statistic, HF˜ OLS β , was 8.03, compared with the 95% critical value of 8.63 (from Rodrigues and Taylor, 2004a, Table 1, T = 100); thus the null hypothesis of two unit roots is (marginally) not rejected on this criterion. Fitting a quadratic trend resulted in HF˜ OSLS = 9.40, compared to the δ 95% critical value of approximately 10.30, so that non-rejection of the null of two unit roots also results from using a quadratic trend. Similar results were obtained from the ADP(3) regression, where the test statistic is the pseudo t statistic on ∆yˆ˜ t−1 , which (for the linear trend case) is –3.22, compared to the 5% critical value of –3.46. When the ADP(3) regression is estimated for the case with yˆ˜ t the residual from a regression of yt on just a constant,

10 quadratic trend 9 actual 8 7

linear trend

6 5 4 1900

1920

1940

1960

1980

2000

2020

Figure 10.7a Denmark CPI (logs), actual and trends.

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Testing for Two (or More) Unit Roots 425

426 Unit Root Tests in Time Series

0.8 0.6 0.4 residuals from quadratic trend

0.2 0

–0.4 residuals from linear trend –0.6 1900

1920

1940

1960

1980

2000

2020

Figure 10.7b Residuals from trends, Denmark CPI.

0.25 0.2 0.15 0.1 0.05 0 –0.05 –0.1 –0.15

actual trend

–0.2 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

Figure 10.7c Denmark CPI, annual inﬂation rate and trend.

the pseudo t statistic on ∆yˆ˜ t−1 is –2.30, compared to the 5% critical value of –2.89. In neither case is the null hypothesis of a second unit, conditional on a ﬁrst unit root, rejected. Turning to the SD test regression, in the linear trend case, the SD test statistic is HF˜ SSLS = 7.58 compared to the 95% quantile of 6.89, (from Rodrigues and β Taylor, 2004a, Table 1, T = 100) so that the null hypothesis of two unit roots is rejected on this basis. The simple symmetric test statistic for a single unit root was –2.12 compared to a 5% critical value of –3.31 (Fuller, 1996, Table 10.A.3, T = 100). Thus (simple) symmetric estimation with linear detrended data suggests one unit root rather than two; however, this conﬂicts at the margin with the result when a quadratic trend is ﬁtted. The SD test statistic in this case was HF˜ SSLS = 10.62, compared to a 95% critical value of approximately 11.34, the δ

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–0.2

Testing for Two (or More) Unit Roots 427

0.2 0.15 0.1 0.05 0 –0.05

–0.15 –0.2 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010

Figure 10.7d Residuals from linear trend for growth rate.

Table 10.7 Number of unit roots; estimation details for log of CPI, Denmark. AHF(3) Trend

yˆ˜ t−1

∆yˆ˜ t−1

∆2 yˆ˜ t−1

∆2 yˆ˜ t−2

∆2 yˆ˜ t−3

∆2 yˆ˜ t ‘t’ AHF(3) Quad. trend ∆2 yˆ˜ t ‘t’ ADP(3) Trend ∆2 yˆ˜ t ‘t’

–0.035 –2.28

–0.284 –2.62

0.039 0.33

–0.358 –3.65

–0.113 –1.12

–0.066 –2.93

–0.178 –1.55

0.055 0.43

–0.413 –4.13

–0.161 –1.57

–

–0.346 –3.22

0.114 0.96

–0.320 –3.24

–0.070 –0.70

ADP(3) Mean ∆2 yˆ˜ t ‘t’

–

–0.186 –2.30

0.008 0.074

–0.400 –4.25

–0.122 –1.21

ADF(4) Trend ∆yˆ˜ t ‘t’

y˜ t−1 –0.035 –2.28

∆y˜ t−1 0.756 7.61

∆y˜ t−2 –0.397 –3.21

∆y˜ t−3 0.244 2.00

∆y˜ t−4 0.113 1.12

SSLS Trend ‘t’

γ˜ 1SSLS –0.033 –2.12

γ˜ 2SSLS –0.275 –2.53

ϕ˜ 1SSLS 0.035 0.29

ϕ˜ 2SSLS –0.360 –3.67

ϕ˜ 3SSLS –0.114 –1.13

SSLS Quad. trend estimate ‘t’

γ˜ 1SSLS –0.070 –3.17

γ˜ 2SSLS –0.200 –1.80

ϕ˜ 1SSLS –0.048 –0.38

ϕ˜ 2SSLS –0.404 –4.14

ϕ˜ 3SSLS –0.156 –1.59

p-value of the sample value is approximately 8%; with a fairly small sample, there may be no immediate resolution to the marginal conﬂict. The evidence points to at least a nearly doubly-integrated time series.

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–0.1

428 Unit Root Tests in Time Series

Some economic time series are smoother than those characterised as generated from a process with a single unit root. A second unit root can account for such smoothness as it involves the ‘smoothing’ action of summing irregular stochastic inputs not once but twice. A number of tests have been suggested to test for two unit roots, the principles of which are easily extended to test for a larger number of unit roots. One approach, due to Hasza and Fuller (1979), is to specify the parametric restrictions on an AR(p) model, p ≥ 2, implied by two unit roots. This leads naturally to a joint testing principle, such as the F test, although as may be anticipated from single unit root tests, the distribution of the resulting test statistic is not standard. The HF test is easy to apply by virtue of the manner of the test regression, involving only zero restrictions. Sen and Dickey (1987) exploited the advantage of symmetric estimation using the backward and forward AR(p) difference equations, which generally results in a more powerful test. Dickey and Pantula (1987) have suggested a sequential procedure using an otherwise standard DF test statistic, such as one of the τˆ family. In the DP procedure, the DF test is applied to the highest order of integration and a decision is made whether to reduce the order of integration by one at each stage. These testing principles are capable of simple extensions, just as in the single unit root case. For example, the DP procedure could use one of the more powerful tests, such as the weighted symmetric test, the recursive mean adjusted test or the DF-max test. It may also be worth assessing whether a weighted symmetric version of the SD simple symmetric estimator, and associated test statistic, leads to an improvement in power. An issue that has been the subject of some interest, but on which more remains to be done, is the importance of the initial conditions, a problem previously highlighted for tests for a single unit root. Sen and Dickey (1987) noted that randomising the starting values affected the power of the SD and HF tests, and Rodrigues and Taylor (2004a) noted that a trend would generally be required in the maintained regression of the HF and SD tests to ensure invariance of the small sample or limiting null distribution. Interest in I(2) processes is not just a theoretical nicety. There has been some empirical work where the possible I(2)-ness of the component time series is a key matter of enquiry. In a multivariate setting, higher-order integration is important in polynomial cointegration and vector autoregressive models including vector error correction models (see, for example, Haldrup and Salmon, 1998; Johansen, 1995, 1997, 2006; Paruolo, 1996). (For applications involving possible I(2) variables, see Fanelli and Bacchiocchi, 2005; and Juselius, 2009, on purchasing power parity; Kongsted, 2003, on import prices; and Tahai et al., 2004, on ﬁnancial integration.)

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10.8 Concluding remarks

Testing for Two (or More) Unit Roots 429

Questions Q10.1 Consider the two unit root model, with the errors generated by an AR(1) process; that is: ∆2 yt = zt

with |ϕ1 | < 1. This is an implied AR(3) model for yt . Obtain and comment upon the solution for yt as an extension of Equation (10.4). A10.1 The solution for yt depends on how the error process is viewed as starting. Note that the error generation process (EGP) implies (1 − ϕ1 L)∆2 yt = εt , so that yt = (2 + ϕ1 )yt−1 – (1 + 2ϕ1 )yt−2 + ϕ1 yt−3 + εt and three starting values, y0 , y−1 and y−2 , are required that will feed through into subsequent values of yt . A somewhat simpler approach is to start the process with y−1 = Y−1 , y0 = Y0 + z0 and z1 = ϕ1 z0 + ε1 ; thus y1 = 2y0 – y−1 + z1 , but together with z1 = ϕ1 z0 + ε1 , t−1 j the EGP implies zt = ϕ1t z0 + ∑j=0 ϕ1 εt−j . Then, continuing the sequence, the following are obtained: y1 = 2y0 − y−1 + z1 y2 = 2(y0 − y−1 ) + y0 + 2z1 + z2 y3 = 3(y0 − y−1 ) + y0 + 3z1 + 2z2 + z3 .. . yt = t(y0 − y−1 ) + y0 + ∑j=1 (t − j + 1)zj t

= t(y0 − y−1 ) + y0 + ∑i=1 ∑j=1 zj t

i

= t(y0 − y−1 ) + y0 + z0 ∑i=1 ∑j=1 ϕ1 + ∑i=1 ∑j=1 ∑k=0 ϕ1k εj−k t

i

j

t

i

j

= t(y0 − y−1 ) + y0 + z0 ∑j=1 (t − j + 1)ϕj + ∑i=1 ∑j=1 ∑k=0 ϕ1k εj−k t

t

i

j

= t(y0 − y−1 ) + y0 + tϕ1 z0 + z0 ∑j=2 (t − j + 1)ϕj + ∑i=1 ∑j=1 ∑k=0 ϕ1k εj−k t

t

i

j

where the following have been used:

∑i=1 ∑j=1 ϕ1 t

i

j

=

∑j=1 (t − j + 1)ϕ1 t

j

= tϕ1 + (t − 1)ϕ12 + . . . + ϕ1t The simplest case follows from assuming that z0 = 0, in which case: yt = t(y0 − y−1 ) + y0 + ∑i=1 ∑j=1 ∑k=0 ϕ1k εj−k t

i

j

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(1 − ϕ1 L)zt = εt

430 Unit Root Tests in Time Series

The inﬂuence of the starting values is as before (see Equation (10.5)), otherwise z0 has an inﬂuence through t and ϕ1 ; that is, from the term tϕ1 z0 . Q10.2.i Show that the AHF(0) regression can also be formulated as an ADF(1) regression and hence the test statistic for a single unit root can also be read from an AHF regression. Q10.2.ii Generalise the result of Q10.2.i.

∆2 yt = γ1 yt−1 + γ2 ∆yt−1 + εt Note that ∆2 yt ≡ ∆yt − ∆yt−1 ; hence, by substitution and rearrangement of terms, the ADF(1) results as follows: ∆yt = γ1 yt−1 + (γ2 + 1)∆yt−1 + εt Therefore the ADF coefﬁcient γ1 and associated test statistic are unchanged and can be read from the AFH regression. Moreover, note (from the development below 10.14) that (γ2 + 1) = −(φ2 + 1) + 1 = −φ2 , which is just as in the ADF(1) model. A10.2.ii In effect, this just reverses the development leading to (10.29) and the HF decomposition of a lag polynomial. Starting from the following (see (10.29)): ∆2 yt = γ1 yt−1 + [C(1) − 1]∆yt−1 + D(L)∆2 yt + εt then the derivation can be reversed to obtain the ADF representation: ∆yt = γ1 yt−1 + C(L)(1 − L)yt + εt As in the simple case of Q10.2.i, the coefﬁcient on yt−1 is γ1 and its t statistic, which is the test statistic for a single unit root, can be read from the AHF regression. Q10.3.i Extend the HF test procedure to test for 3 unit roots in an AR(3) model. Q10.3.ii Indicate how this result could be generalised to an AR(p) model, with p > 3. A10.3.i Consider the AR(3) example given by Equation (10.15); that is: yt = φ1 yt−1 + φ2 yt−2 + φ3 yt−3 + εt Then directly by setting the third root to unity in Equations (10.16)–(10.18), obtain the following: φ1 = 3 φ2 = − 3 φ3 = 1

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A10.2.i The simplest AHF regression is:

Testing for Two (or More) Unit Roots 431

The restricted AR(3) model is: yt = 3yt−1 − 3yt−2 + yt−3 + εt Whilst the F statistic is in principle easy to construct, it would be convenient if, as in testing for two unit roots, the regression could be formulated such that the restrictions were of the zero form. A reasonable surmise is that the structure of the HF test regression generalises to: (10.87)

This is indeed the case, with the joint null of three unit roots corresponding to γ1 = γ2 = γ3 = 0, which can be shown by generalising the HF decomposition of a lag polynomial to the next level. However, in this simple case the direct route is easier. First, factorise the AR(3) lag polynomial and then write it in AR(3) form and equate coefﬁcients on like powers of L to obtain the φi coefﬁcients in terms of the roots of the characteristic polynomial: (1 − λ1 L)(1 − λ2 L)(1 − λ3 L)yt = εt yt = φ1 yt−1 + φ2 yt−2 + φ3 yt−3 + εt φ1 = (λ1 + λ2 + λ3 ) φ2 = − (λ1 λ2 + λ1 λ3 + λ2 λ3 ) φ3 = λ 1 λ 2 λ 3 Then, on expanding (10.87) collect terms on like powers of L: yt = (3 + γ1 + γ2 + γ3 )yt−1 − (3 + γ2 + 2γ3 )∆yt−1 + (1 + γ3 )∆y2t−1 + εt Hence we may infer that λ1 = λ2 = λ3 = 1 ⇔ γ1 = γ2 = γ3 = 0 A10.3.ii The generalisation is to apply the DF lag decomposition twice, so that starting from C(L), apply the decomposition to D(L); thus: C(L) = D(1)L + D(L)(1 − L) D(L) = D(1)L + E(L)(1 − L) p−3

where E(L) = ∑j=1 ej Lj . Now substitute for D(L) in C(L): C(L) = C(1)L + D(1)L(1 − L) + E(L)(1 − L)2 and substitute for C(L) in the ADF(p – 1) representation, and simplify: ∆yt = γ1 yt−1 + C(L)(1 − L)yt + εt ∆yt = γ1 yt−1 + [C(1)L + D(1)L(1 − L) + E(L)(1 − L)2 ](1 − L)yt + εt ∆yt = γ1 yt−1 + C(1)∆yt−1 + D(1)∆2 yt−1 + E(L)∆3 yt + εt

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∆3 yt = γ1 yt−1 + γ2 ∆yt−1 + γ3 ∆y2t−1 + εt

432 Unit Root Tests in Time Series

Next, note that ∆3 yt ≡ ∆2 yt − ∆2 yt−1 ≡ ∆yt − ∆yt−1 − ∆2 yt−1 , hence the last equation can be expressed as follows:

where γ1 = (φ(1) − 1), γ2 = (C(1) − 1) and γ3 = (D(1) − 1) From the form of the symmetry in this representation, the generalisation is clear: under the joint null of three unit roots, φ(1) = C(1) = D(1) = 1. Note that γ1 and γ2 are as in testing for one unit root and two unit roots, so these tests can also be obtained form this extended regression. Q10.4.1 Arrange the data for the SD double-length regression in the form of a regression ‘tableau’ that speciﬁes the dependent and independent variables for the augmented version of the regression (see Sen and Dickey, 1987; Haldrup, 1998). A10.4.1 The inputs to the data tableau were given in (10.49)–(10.55). On arrangement of the data into tableau form we obtain Table 10.8. Q10.4.2 Explain how to modify the tableau to compute the weighted symmetric version of the SD test. A10.4.2 The tableau in the previous answer effectively assumes that the estimator that results is a version of the weighted symmetric estimator, where equal weights have been used throughout and will cancel in the calculation of the LS estimator. Given that the backward and forward weights sum to one, this implies that the simple symmetric estimator uses weights of 0.5. To generalise this idea, introduce a set of weights, which are wt for the backward recursion Table 10.8 Dependent and explanatory variables in the simple symmetric test of Sen and Dickey. Dependent variable

Explanatory variables

∆2 yp + 1

yp

∆yp

∆2 yp−1

∆2 yp + 2 .. . ∆2 yT

yp + 1 .. . yp−1

∆yp + 1 .. . ∆yT−1

∆2 yp .. . ∆2 yT−2

∆2 yT−(p−2)

yT−(p−1)

−∆yT−(p−2)

∆2 yT−(p−1)

∆2 yT−(p−1) .. . ∆2 y3

yT−p .. . y2

−∆yT−(p−1) .. . −∆y3

∆2 yT−p .. . ∆2 y4

... .. . .. . ... ... .. . .. . ...

∆2 y3 ∆2 y4 .. . ∆2 yT ∆2 yT−1 ∆2 yT−2 .. . ∆2 yp + 1

Note: See Haldrup (1998, Table 1).

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∆3 yt = γ1 yt−1 + γ2 ∆yt−1 + γ3 ∆2 yt−1 + E(L)∆3 yt + εt

Testing for Two (or More) Unit Roots 433

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and (1 − wt ) for the forward recursion. The LS estimator uses wt = 1, so that it gives no weight to the forward recursion. For the general case, the weights were given in Chapter 6 (see section 6.6.2). In the notation of this chapter, the weights are: wt = (t − p)/[T − 2(p − 1)], wt = 0 for 1 ≤ t ≤ p and wt = 1 for T ≥ t ≥ (T – p). The weighting procedure can be viewed as premultiplying the regression model of (10.63) by a diagonal matrix Ψ comprising two blocks; the ﬁrst √ √ set (block) of diagonal elements is given by wt , and the second by 1 − wt .

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11

Introduction So far the null hypothesis has been that of nonstationarity due to a unit root, the (usual) alternative being stationarity about a constant or a linear trend. The asymmetry in classical hypothesis testing favours the null hypothesis, which is the unit root in the set-up so far. There are, however, a number of tests that reverse the roles of the hypotheses, so that the null hypothesis is of stationarity (about a constant or a linear trend) and the alternative hypothesis is of a unit root, which is one example of nonstationarity; other forms of nonstationarity have also been considered (see for example, Lo, 1991; Lee and Schmidt, 1996; and see Wright, 1999, for the fractional integrated alternative). Most of the tests for stationarity have a simple rationale, the general basis being to view a series as, apart from deterministic components, the sum of a stationary component and an integrated component. The null hypothesis then restricts the second component to be absent. Typical macroeconomic time series are trended, so that the deterministic components will be a constant and a (linear) time trend and, under the null, the series grows (or declines) systematically, but with stationary deviations about the trend. Several stationarity tests examine the behaviour of the residual sum or residual partial sum process (psp), appropriately normalised and scaled. The residual is the usual one, as in unit root tests, which is obtained from a prior regression of the series, either just on a constant for ‘level’ stationarity, or a constant and a linear trend for ‘trend’ stationarity. Only if the stationarity is about a long-run mean of zero are the observations on the unadjusted series used. Let {et }T1 be a sequence of random variables, which is thought to be stationary, and St = ∑ts=1 es be the associated psp. There are several tests for stationary that are based on the growth in the series St , the basic idea being that the ﬂuctuations in St are bounded if the underlying process is stationary but unbounded in the case

434

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Tests with Stationarity as the Null Hypothesis

of a nonstationary process. The leading example is a test due to Kwiatkowski, Phillips, Schmidt and Shin (1992), referred to as the KPSS test. Some difﬁculties immediately spring to mind in this approach: for example, there is the actual size of the resulting test as the stationary component may well have quite a high degree of persistence, which could ‘confuse’ a test for stationarity into rejecting the null hypothesis when the null hypothesis is correct – that is, the test over-rejects. This is potentially quite an acute problem as a series generated by yt = ρyt−1 + εt with εt ∼ iid(0, σε2 ) is (weakly) stationary for |ρ| < 1. One approach to this problem is to make an adjustment via the longrun variance, referred to as the lrv (or its standard deviation, lrsd), which is the scaling factor in several of the proposed tests. (For example, the lrv in the AR(1) model is σε2 (1 − ρ)−2 .) The general form of the semi-parametric stationarity tests in this chapter is a ratio in which the numerator represents a measure of the growth or ﬂuctuations in a series and the numerator is the lrv, or its standard deviation, which acts as a scaling factor. As usual in test design, the test statistic should be of different orders of probability under the null and alternative hypotheses, being Op (1) under the former and Op (Tκ ) with κ > 0 under the latter and, ceteris paribus, larger values of κ are desirable for more powerful tests. The problem for stationarity tests of this ratio form is that both the numerator and the denominator diverge under the alternative; therefore, for a consistent test, the denominator must diverge more slowly than the numerator. To take an example, under the alternative, the numerator of the KPSS test is Op (T2 ), whereas the denominator, which uses a semi-parametric estimator with a Bartlett kernel and bandwidth denoted m, is Op (mT); hence the ratio is Op (T2 /mT) = Op (T/m); thus, minimally, m = Op (Tκ ) with κ < 1 is required, otherwise the test statistic will not diverge, with the rate of divergence better for smaller κ. On the other hand, κ must also be chosen to be large enough to achieve desirable properties under the null hypothesis. Generally, it is required that m → ∞ as T → ∞, such that m/T → 0. There are a number of tests in addition to the KPSS test; for example, the rescaled range, (RS) test, the Kolmogorov-Smirnoff (KS) test and a DurbinWatson-type test, referred to as the SBDH (Sargan-Bhargava, Durbin-Hausman) test. These tests are usually combined with a semi-parametric estimator of the long-run variance, a concept that is familiar from the PP versions of unit root tests. Although the tests differ in the metric used to assess the size of the deviations from stationarity, they have in common that they are right-tail tests in the sense that sample values of the test statistic exceeding the (1 – α)% quantile lead to rejection of the null hypothesis of stationarity. As usual, the structure of the weak dependence is not generally known and has to be estimated and, whether this is done semi-parametrically or parametrically, there are two generic problems. First, in the former case, a kernel estimator

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Tests with Stationarity as the Null Hypothesis 435

and a bandwidth parameter have to be chosen and, in the latter case, a lag length parameter has to be chosen; the results can be quite sensitive to these choices, perhaps more so in the semi-parametric case. Second, there are wellknown biases in applying standard methods of estimation to processes with AR (and MA) components (see Chapter 4), and even relatively small biases are important because the lrv is a nonlinear function of the underlying parameters and this accentuates apparently small biases. Any oversizing due to the misinterpretation of persistence as nonstationarity will lead to tests that look more powerful than they are, and substantially different actual sizes for a given nominal size make size-adjusted power comparisons generally difﬁcult to interpret. As stationarity tests are designed to lead to rejection when the integrated component is present, then the more important this component is, the more likely a test should lead to rejection. Conversely, a small integrated component could be present without being detected by the test. The ‘importance’ of the integrated component is usually measured by the ‘signal-tonoise’ ratio; that is, within a structural time series components interpretation, this is the ratio of the variance of the integrated component to the variance of the stationary component; and, generally, it is easier to detect nonstationarity as this ratio increases. Equivalently, in terms of second-order properties, the structural time series model has a reduced form representation, that is, ARIMA(0, 1, 1), with the null hypothesis corresponding to θ1 = – 1 and the alternative hypothesis to 0 ≥ θ1 > −1; thus a test of stationarity can be based on testing θ1 = –1, which implies a non-invertible MA root, against the alternative of invertibility (see, for example, Tanaka, 1990). This chapter is organised as follows. Section 11.1 introduces structural time series models, which are the basis for most tests of stationarity. Four semiparametric stationarity tests, including the much-used KPSS test, are introduced in section 11.2 and a parametric test is described in section 11.3. Section 11.4 considers the long-run variance, which is used to rescale the ﬂuctuation measure used in the denominator of a typical stationarity test and is a key parameter. A simulation-based evaluation of the various tests is presented in section 11.5. Section 11.6 illustrates the use of stationarity and nonstaionarity tests in the context of some consumer price indices.

11.1 A structural time series model interpretation It is useful to set the scene by way of introducing some simple structural time series models (see, especially, Harvey, 1989). In this framework the ARMA or ARIMA models of previous chapters are reduced form representations of structural time series models. As in the econometrics of simultaneous equations, the

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436 Unit Root Tests in Time Series

Tests with Stationarity as the Null Hypothesis 437

structure is at a higher theoretical or fundamentally behavioural level compared to the reduced form. 11.1.1 The local level model Suppose that a time series is modelled as a level component plus an irregular component, then: yt = µt + ςt

where the level is µt and the irregular component is ςt . The speciﬁcation of the level should match the observed characteristics of the time series being modelled. The simplest speciﬁcation is that the level is constant, thus µt = µ, so the series is a constant ‘contaminated’ by a random component; that is: yt = µ + ςt

(11.2)

More relevant is the case where the level is changing over time. One possibility is an evolution according to a simple random walk: µt = µt−1 + ηt

(11.3)

At a minimum, the random inputs ςt and ηt are assumed to have a zero mean, with variances, σς2 and ση2 , respectively, and be uncorrelated with each other. Initially it is convenient to assume that ςt is iid, but this can be relaxed to allow ςt to be generated by a linear process, and this generalisation is taken up below. If σς2 = 0 and ση2 = 0, then yt = µt and µt = µt−1 + ηt , implying that yt = yt−1 + ηt , and therefore yt is a random walk. Thus (11.1) and (11.3) are the structural equations and the (familiar) AR(1) model of the random walk is the reduced form. If ση2 = 0 and σς2 = 0, then µt = µt−1 , µt−1 = µt−2 , and so on, so that µt is just a constant, say µ, and yt = µ + ςt , which is just a constant plus an irregular component and this is the model stated as (11.2). To generalise this model, suppose that the random walk has a drift component, β1 , then the structural model is: yt = µt + ςt

(11.4)

µt = β1 + µt−1 + ηt

(11.5)

Also, note that by back-substitution, using the recursion for µt , yt can be expressed as: yt = β0 + β1 t + ∑i=1 ηi + ςt t

(11.6)

where β0 is the ‘starting’ value of µt (that is, µ0 = β0 ) and can be viewed as the intercept in a graph of µt against t. The stochastic trend, and hence the induced

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(11.1)

438 Unit Root Tests in Time Series

nonstationarity, is now clearly present through the term ∑ti=1 ηi ≡ Λt . The model simpliﬁes in the case of no drift to: yt = β0 + ∑i=1 ηi + ςt t

(11.7)

yt = β0 + β1 t + δΛt + ςt

(11.8) (β)

(µ)

Then the possible null hypotheses are: H0 : δ = 0 ⇒ trend stationarity; H0 : (0)

δ = 0 and β1 = 0 ⇒ level, or mean, stationarity; H0 : δ = 0, β0 and β1 = 0 ⇒ stationarity about a zero mean; under the respective alternatives, all have in (i) common that HA : δ = 1, i = 0 (no deterministic components), µ (constant), β (linear trend) and, implicitly, that β0 + ςt is stationary. However, this is a conceptual rather than a practical framework because Λt is not directly observed, rather its presence is assessed indirectly. For example, consider the following model that generates trend behaviour: yt = β0 + β1 t + et

(11.9a)

et = δΛt + ςt

(11.9b)

The presence of a stochastic trend can then be inferred by the behaviour of the et process; under the null of δ = 0, the properties of et are just those of ςt , so, for example, its ﬂuctuations are bounded; however, under HA , et contains a stochastic trend and will have a variance that increases over time. Whilst the sequence {et } is not directly observed, it can be consistently estimated as the residual sequence {eˆ t } from a regression of yt on a constant and a linear trend. The particular measure of the deviation of the ﬂuctuations from the case under H0 distinguishes several tests of stationarity, and is considered in more detail in section 11.1.3; for the moment it is useful to consider some further aspects of the structural time series approach, partly as it suggests an alternative strategy of testing for stationarity through the presence of a negative unit root in the MA component. Returning to (11.4), note that: ∆yt = ∆µt + ∆ςt

(11.10)

and substituting for ∆µt = β1 + ηt from (11.5) results in: ∆yt = β1 + ηt + ∆ςt

(11.11)

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Thus one way of viewing the testing framework for the null hypothesis of stationarity is to introduce the coefﬁcient δ so that (in the former, with drift, case):

Tests with Stationarity as the Null Hypothesis 439

In this model the variance, autocovariances and autocorrelations of ∆yt are: γ(0) = ση2 + 2σς2

(11.12a)

γ(1) = − σς2 ρ(1) =

− σς2 /(ση2 + 2σς2 )

γ(k) = 0 for k > 1

(11.12b)

ρ(k) = 0 for k > 1

(11.12c)

∆yt = β1 + wt

(11.13a)

wt = εt + θ1 εt−1

(11.13b)

The variance, autocovariance and autocorrelations of wt are given by: γ(0) = σε2 (1 + θ21 )

(11.14a)

γ(1) = θ1 σε2

γ(k) = 0 for k > 1

(11.14b)

ρ(1) = θ1 /(1 + θ21 )

ρ(k) = 0 for k > 1

(11.14c)

This ARIMA model is the reduced form of the structural components model (in the sense that the second-order moments are the same); the separate random components in the latter are combined into a single random term in the ARIMA model. By equating the nonzero autocovariances and autocorrelations, then: θ1 σε2 = − σς2

(11.15a)

⇒ σε2 = − σς2 /θ1 θ1 /(1 + θ21 ) = − σς2 /(ση2 + 2σς2 )

(11.15b) (11.16)

The latter can be solved for θ1 as: θ1 = {[ψ(ψ + 4)]1/2 − (ψ + 2)}/2

(11.17)

where ψ = ση2 /σς2 = − (1 + θ1 )2 /θ1

(11.18)

The coefﬁcient ψ can be interpreted as the signal-to-noise ratio, with 0 ≤ ψ < ∞ corresponding to –1 ≤ θ1 < 0. Harvey (1989) notes that the admissible region for θ1 does not include θ1 > 0 if the reduced form model is to be related back to the local level structural model (note that θ1 > 0 in (11.18) results in ψ < 0).

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The cut-off at k = 1 indicates that this pattern is that of an MA(1) process in ∆yt , which is the same as the ARIMA(0, 1, 1) model given by:

440 Unit Root Tests in Time Series

To illustrate the connection, suppose that σς2 = 1 and ση2 = 2, then ψ = 2, θ1 =–0.268, σε2 = σς2 /0.268 = 3.713 and ρ(1) = –0.25. Thus the ARIMA(0, 1, 1) model is: ∆yt = β1 + εt − 0.268εt−1 If, to continue the example, ση2 = 10, then ψ = 10, θ1 = –0.084, σε2 = σς2 /0.084 = 11.919 and ρ(1) = –0.083. The reduced form of this model is given by:

Finally, let ση2 = 0.1, then the reduced form model is: ∆yt = β1 + εt − 0.73εt−1 This last model has the ‘large’ (negative) MA coefﬁcient thought to be characteristic of some economic time series (see Schwert, 1987). 11.1.2 The importance of the limits Noting that as θ1 → 0 then ψ → ∞, and as θ1 → –1 then ψ → 0. The limits θ1 = 0 and θ1 = –1 are of particular interest. In the latter case, ση2 = 0 implies ψ = 0, so that ηt = 0 for all t. The structural model is: yt = µt + ςt µt = β1 + µt−1

(11.19a) (because all ηt = 0)

(11.19b)

The trend is deterministic rather than stochastic, with constant slope given by ∆µt = β1 ; at time t = 0, µ0 = β0 , say, and the level of the deterministic trend is µt = β0 + β1 t, leading to the familiar trend stationary model: yt = β0 + β1 t + ςt

(11.20)

This is just the model previously referred to as (11.9a) and (11.9b), with δ = 0, and hence et = ςt . The reduced form of the model (11.19a) and (11.19b) is: ∆yt = β1 + ∆εt

(11.21)

This has the common factor (1 – L), which can be cancelled and the level of yt solved for as: yt = β0 + β1 t + εt

(11.22)

This is as in the structural model with σε2 = σς2 . In the case of the other limit, if θ1 → 0 then ψ → ∞, and in the limit the resulting reduced form model is the (pure) random walk with drift: ∆yt = β1 + εt

(11.23)

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∆yt = β1 + εt − 0.084εt−1

Tests with Stationarity as the Null Hypothesis 441

However, note that as long as ση2 > 0, then ψ > 0 and there is a stochastic trend; in the limit ψ → ∞, the MA(1) component in εt is removed. In contrast, if ση2 = 0 then the trend is deterministic. Thus the null hypothesis of stationarity can be (i) (i) expressed as either H0 : ση2 = 0 or, equivalently, H0 : ψ = 0 against the alternative (i)

11.1.3 The ‘ﬂuctuations’ testing framework The framework we adopt is as in three possible specialisations of Equation (11.8), that is the respective null hypotheses are: (β)

trend stationarity

(11.24)

(µ)

level stationarity

(11.25)

(0)

zero mean stationarity

(11.26)

H0 : yt = β0 + β1 t + et H0 : yt = β0 + et H0 : yt = et et = δΛt + ςt

and

(i)

δ = 0 under H0

(11.27)

(i) The LS residuals, denoted eˆ t , are deﬁned according to H0 . Where particular (i)

clarity is required we will write eˆ t , so that the form of the regression from which the residuals are deﬁned matches the alternative hypothesis. Note that (i) under the null et = ςt , and under HA , the common component is δ = 1. Also, as noted in the previous section, equivalent ways of formulating the null and alternative hypotheses, motivated by the structural time series approach, are: (i)

H0 : ση2 = 0 or ψ = 0

(i)

HA : ση2 > 0 or ψ > 0

Temporal dependence can be allowed in the structure of ςt ; for example, let ∞ ∞ j 2 2 ςt = b(L)vt , with b(L) = ∑∞ j=0 bj L , b(1) = ∑j=0 bj = 0, ∑j=0 j bj < ∞ and vt ∼ iid(0, σv ); then (see, for example, Phillips and Solo, 1992) such a scheme is permitted provided that the partial sum process satisﬁes the functional central limit [rT] theorem (FCLT); that is, T−1/2 ∑t=1 ςt ⇒ Wς (r), which is a Brownian motion 2 = b(1)2 σ2 . In terms of the standardised Brownian motion, with variance σζ,lr v 2 is familiar as the ‘long-run’ variance, here B(r), then Wς (r) = σζ,lr B(r) and σζ,lr of the process ςt , which is equal to 2πfς (0), where fς (0) is the spectral density of ςt evaluated at the zero frequency. The long-run variance is a key concept

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(i)

HA : ση2 > 0 or equivalently HA : ψ > 0, respectively. Considering the reduced form of the model, one strategy in testing for stationarity is to test for a negative unit root in the MA component, with ∆yt as the dependent variable (see Equation (11.21)). The null hypothesis in this case of stationarity is θ1 = – 1, whereas the alternative is –1 < θ1 ≤ 0, so that the MA component is invertible. Tanaka (1990), for example, suggests a score-type test for this null, which is locally best, invariant and unbiased (and can be applied when the error process is quite general). In the present context, the test can be viewed as testing that the series has been over-differenced.

442 Unit Root Tests in Time Series

underlying the PP tests introduced in Chapter 6; estimation will be considered 2 by σ 2 . Of further below; for the moment denote a consistent estimator of σζ,lr ˜ ζ,lr particular interest in the tests outline below are the two partial sum processes (i) (i) (i) (i) of eˆ t , given by St = ∑tj=1 eˆ j and ∑tk=1 (Sk )2 = ∑tk=1 (∑kj=1 eˆ j )2 , where the superscript (i) indicates whether the original series has been used; that is, i = 0, or whether the data have been demeaned, i = µ, or detrended, i = β.

This section outlines four semi-parametric tests for stationarity and one test using a parametric adjustment. All tests reject the null of stationarity for ‘large’ values of the test statistic, so the typical quantiles for the critical values are 90%, 95% and 99%. 11.2.1 The KPSS test As noted in the introduction to this chapter, the essence of a number of tests for stationarity is to assess, in a particular metric, the magnitude of the ﬂuctuations in the partial sum process of eˆ t , which is the observable analogue of et , relative to the case of stationarity. The generic form of the KPSS test statistic is: T−2 ∑Tt=1 (St )2 (i)

KPSS(i) =

2 σ˜ ζ,lr

(11.28)

2 is a consisThroughout the tests, the superscript (i) indicates i = 0, µ, β; and σ˜ ζ,lr

2 = ∞ tent estimator of σζ,lr ∑j = −∞ E(ςt ςt−j ). As KPSS (1992) note, their test statistic is a special case of the test suggested by Nabeya and Tanaka (1988) for random coefﬁcients in a regression model. 2 = σ2 , and a consistent In the simplifying case where ςt ∼ iid(0, σζ2 ), then σς,lr ζ

(i) 2 , under the null, is the familiar LS quantity σ estimator of σζ,lr ˆ ς2 = ∑Tt=1 (eˆ t )2 /T.

In this case the KPSS(i) statistic simpliﬁes to: T−1 ∑Tt=1 (St )2 (i)

KPSS(i) =

(i)

∑Tt=1 (eˆ t )2

(11.29)

2 are available and are a common part of tests for staSeveral estimators of σζ,lr tionarity; for example, KPSS (1992) use the semi-parametric estimator outlined in Chapter 6. This has the general form: 2 ˆ ˆ σ˜ ζ,lr = γ(0) + 2 ∑κ = 1 ω(κ)γ(κ) m

(11.30)

(i) ˆ where γ(κ) is the k-th order autocovariance based on eˆ t ; γˆ (0) = ∑Tt=1 (eˆ t )2 /T,

(i) (i) ˆ = ∑Tt= κ + 1 eˆ t eˆ t−κ /T; ω(κ) is a weighting, or kernel, function satisfying |ω(z)| γ(κ) 2 < 1, ω (z)dz < ∞; m is the ‘bandwidth’ parameter; m → ∞ as T → ∞, such that 2 depends on the kernel m/T → 0, and the order of m for consistency of σ˜ ζ,lr function.

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11.2 Tests for stationarity

The limiting distribution of KPSS(i) depends on the deterministic terms in the null speciﬁcation and consistency of the test requires that m increase with T, but not ‘too’ quickly. (The limiting distributions are collected together in Table 11.1 in section 11.5.2.) KPSS (1992) show that the numerator of their test is Op (T2 ) and the denominator is Op (mT); hence, KPSS(i) is Op (T/m), so that m/T → 0 implies T/m → ∞, ensuring consistency of the test. Part of the proof to establish the limiting distribution under the alternative uses m = o(T1/2 ). In their simulations, KPPS (1992) use the rule m(j) = [j(T/100)1/4 ], for j = 4, 12, and the Bartlett kernel; for the quadratic spectral (QS) kernel, m = o(T1/2 ) is sufﬁcient for consistency (Andrews, 1991). The selection of the kernel and m is discussed further below (see section 11.4). Some problems, highlighted by the simulations reported in KPSS (1992), which are typical for stationarity tests, are indicated here and considered in more detail below. First, note that under the null, KPSS(i) is invariant to β0 and β1 , and the scale factor σς2 cancels in the numerator and denominator of the test. As a result, the size of the test just depends on T and m; under this speci2 = σ2 , then m = 0, so choosing m > 0 implies ﬁcation of the null, such that σζ,lr ς superﬂuous lags. The effect on actual size is moderate for j = 4 in the m(j) rule, but the test becomes undersized for j = 12, although this effect is mitigated as T increases and is absent for T = 500. In the more realistic case of serially correlated errors, m > 0 is required. If ςt = ρςt−1 + vt , where vt is iid, then the series yt will look more like a nonstationary series as ρ → 1; hence the general problem is of over-rejection of the null of stationarity. For example, KPSS (1992) report that when ρ = 0.8, a 5% nominal size for KPSS(µ) translates to an empirical size of 25% for T = 100, with m(4); using m(12) reduces the empirical size to about 8%. Power increases with the signal-to-noise ratio, ψ = ση2 /σς2 ; but the choice of j in the m(j) rule is critical to the level of power. As j increases power is reduced, usually very substantially. Hence there is the usual problem in the case of a data-dependent parameter, in this case the selection of the bandwidth, m: if ςt is serially correlated it is necessary to increase m to mitigate size distortions, but increasing m when ςt is, in fact, iid has a critical effect on power. 11.2.2 Modiﬁed rescaled range test statistic (MRS) (Lo, 1991) One basis for an evaluation of stationarity is to consider the range of St over t = 1, · · · , T, scaled by the series standard deviation; and one such test on this principle is the RS, or rescaled range, statistic (see Hurst, 1951; Mandelbrot, 1970, 1972; Mandelbrot and Wallis, 1969). The basic RS statistic, illustrated for the demeaned case, is given by: T−1/2 RT sy

(µ)

RS(µ) =

(11.31)

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Tests with Stationarity as the Null Hypothesis 443

444 Unit Root Tests in Time Series

where:

(µ) (µ) (µ) RT = max St − min St t

t

¯ 2 sy = T−1 ∑t=1 (yt −y) T

1/2

y¯ = T−1 ∑t=1 yt T

(µ)

(µ) (µ) (µ) ¯ As ST = 0, this = ∑ti=1 eˆ i = ∑ti=1 (yt − y). (µ) (µ) is the lower limit to max St , and thus max St ≥ 0. Similarly, the minimum of (µ) (µ) St over t must have an upper bound of zero, and hence RT , and RS(µ) , must

be ≥ 0. The RS statistic is robust to non-Gaussian time series and processes with inﬁnite variance (Lo, 1991). The limiting distribution of RS(µ) under the null that ςt is iid is given by: RS(µ) ⇒D V = sup V(r) − inf V(r) r

r

(11.32)

where V is the range of the Brownian bridge on [0, 1], V(r) = B(r) – rB(1) and the original data has been demeaned. The distribution function for V is: ∞ (11.33) FV (ν) = 1 + 2 ∑k=1 (1 − 4k2 ν2 ) exp (−2kν)2 It can be shown that E(V) = (π/2)1/2 and E(V2 ) = π/6. The distribution is graphed in Lo (1991, Figure 1), from which it is apparent that there is a positive skew, with less mass in the left-tail compared to a normal distribution with the same mean and variance. The problem with the rescaled range test statistic is it does not take account of weak dependence in the errors, so that if yt has ‘short-range’ dependence then the critical values using the distribution of V are misleading. The essential point is that the variance of the partial sum is no longer just the sum of the variances of the components of the sum, but includes the autocovariances. This is a problem that has occurred in several contexts already and the solution is to scale the range by a consistent estimator of the long-run standard deviation σζ,lr . Thus the modiﬁed rescaled range, MRS, test statistic is: T−1/2 RT σ˜ ζ,lr

(µ)

MRS(µ) =

(11.34)

This normalisation ensures that MRS(µ) has the same limiting distribution when the errors are weakly dependent as RS(µ) when the errors are iid. The test can also be applied to the ‘raw’ data and to data that has been detrended, with these test statistics denoted MRS(0) and MRS(β) , respectively. Lo (1991) shows that the modiﬁed rescaled range statistic is consistent against 2 , and hence several interesting alternatives using a Bartlett kernel to estimate σζ,lr

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RT is the range of the partial sums, St

Tests with Stationarity as the Null Hypothesis 445

11.2.3 Komogoroff-Smirnoff-type test (KS) (Xiao, 2001) Xiao (2001) suggested a test for stationarity also based on assessing the ﬂuctuations in the time series, appropriately demeaned or detrended. In this case the metric used leads to a limiting distribution of the Komogoroff-Smirnoff form, referred to as the KS test. Speciﬁcally, the general form of the test statistic is: (i)

(i) ST (i) −1/2 t St KS = max T (11.35) − σ˜ ζ,lr t T t = 1,...,T =

T−1/2 max t y¯˜ rt − y¯˜ σ˜ ζ,lr t = 1,...,T

where y¯˜ rt = t−1 ∑ti=1 y˜ i is the recursive mean at time t of y˜ t , interpreted as either the unadjusted/demeaned/detrended observations, for which i = 0, µ, β, respectively, in (11.35); y¯˜ = T−1 ∑Ti=1 y˜ i is the corresponding ‘global’ mean. If the LS (i)

estimator is used to demean or detrend the data, then y˜ t = eˆ t ; that is, the LS (i) (i) residuals are used, so that y˜¯ rt = t−1 ∑tj=1 eˆ j ; and, as ∑Tt=1 eˆ t = 0 by construction (i)

(0)

for i = µ, β, then the T-th term in ST is absent, otherwise ST = ∑Tt=1 yt . 2 (and T1/2 ) renders the As in the case of the MRS statistic, normalising by σ˜ ζ,lr test statistic asymptotically invariant to weak dependence in the errors. The limiting distributions are given in Table 11.1 (section 11.5.2) and depend upon the detrending process. The general form of the limiting distribution is: KS(i) ⇒ sup |V(i) (r)|

(11.36)

0≤r≤1

where V(i) (r), is a ﬁrst-level Brownian bridge for i = 0, i = µ, or a second-level Brownian bridge for i = β (see Xiao, 2001, Theorem 1 and Remark 2).

Xiao (1991) shows that T−1/2 max t y¯˜ rt − y¯˜ diverges at rate T under the t = 1,...,T alternative of a unit root; however, as noted above, σ˜ ζ,lr also diverges, so it is vital to the consistency of KS(i) that σ˜ ζ,lr diverges at a slower rate; that slower rate (Xiao, 1991, Theorem 2) requires m = op (T1/2 ). In a simulation analysis, Xiao (1991) considered ﬁxed bandwidths of m = 1, 2, and the m(j) rule with j = 4, 12, with the Bartlett kernel used in estimating

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σζ,lr , with m increasing such that m = o(T1/4 ). For example, the MRS statistic is consistent not only against the unit root alternative, but also against fractional integrated alternatives (in which case it could become a two-sided test depending on which side of zero the fractional differencing parameter is placed). Some simulations reported by Lo (1991) indicate size problems with MRS; ﬁrst, over-parameterisation of the bandwidth lag in the iid error case can lead to undersizing (for example, m = 5) and oversizing (for example, m = 50); to some extent the size inﬁdelity can be controlled by Andrews’ (1991) data-dependent criterion.

446 Unit Root Tests in Time Series

2 , and for data generated under the null that y is a sequence of iid variσ˜ ζ,lr t

11.2.4 A Durbin-Watson-type test (SBDH) (Choi and Ahn, 1999) An alternative strategy is to approach the problem of testing for stationarity through an AR framework and this underlies the work of Choi and Ahn (1999) and Choi and Yu (1997). To take a very simple example to illustrate the general principle let yt = et , where et is stationary, then a unit root can be induced by summing the process: Se,t = ∑ti=1 ei so that Se,t ≡ Se,t−1 + et . Presently, this is simply an identity, which will satisfy Se,t = ρSe,t−1 + et , with ρ = 1, and by hypothesis Se,t is I(1). If et is I(1), then Se,t is I(2) and et can be interpreted as an I(1) ‘error’. The presence of the I(1) error will cause standard unit root tests, such as the DF τˆ -type tests, to diverge in probability, which will lead to rejection of the null hypothesis that yt is stationary. Choi and Ahn (1999) derive two tests within this framework and also consider the multivariate extension. The ﬁrst is an LM test and the second is a test based on what is referred to as the SBDH principle. Cappuccio and Lubian (2006) show, the LM test is (substantially) inferior in terms of power to the SBDH test, so only the latter is considered here. In the case that a nonzero mean or trend is ﬁtted under the null, Choi and Ahn (1999,) suggest two ways to make a prior adjustment of the data. In the ﬁrst, the data are demeaned or detrended as in the usual LS regression and the residuals are used in place of the original data to form the partial sum. In the second, used here, the estimate of the partial sum is obtained by estimating the partially summed regression. An example will make the point. Suppose the DGP is yt = µ + et , then ∑ti=1 yi = µt + ∑ti=1 ei ; hence µ can be estimated by regressing (µ) ˆ will be Sy,t ≡ ∑t yi on t, and the residual from this regression Sˆ = Sy,t – µt, e,t

i=1

an estimate of the partial sum of the errors. In the case of detrended data, Sy,t is regressed on t and ∑tj=1 j = 12 t(t + 1), with estimated coefﬁcients βˆ 0 and βˆ 1 , (β)

respectively, and residual Sˆ e,t , where: 1 (β) Sˆ e,t = Sy,t − [(βˆ 0 t + βˆ 1 t(t + 1)] 2

(11.37)

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ables, that KS(µ) and KS(β) were undersized, rather more so in each case than the corresponding version of KPSS(i) (this may have been due to the use of the asymptotic critical values). As noted above, size distortions increased with m. Serial correlation in the form of an AR(1) process for the errors, ςt = ρςt−1 + vt , had predictable effects, namely that as ρ → 1, the KS(i) tests over-reject; and whilst this can be controlled by increasing the bandwidth parameter, the problem being attenuated with m(12) compared to m(4), there is a noticeable loss of power.

Tests with Stationarity as the Null Hypothesis 447

(β) The quantity Sˆ e,t is then used as the partial sum process. (Note that, unlike the (µ) case of LS demeaned or detrended data, it does not follow that Sˆ ≡ 0 and e,T

SBDH(i) = SBDH(i) =

2 T−2 T ∑t=1 Se,t 2 σ˜ ∆S e ,lr T−2 σ˜ 2

∑t=1 T

(i)

(i) Sˆ e,t

2

i=0

(11.38)

i = µ, β

(11.39)

∆Sˆ e ,lr

2 The parameters σ˜ ∆S and σ˜ 2 e ,lr 2 ances σ∆S and σ2 e ,lr

(i)

∆Sˆ e ,lr

(i)

∆Se ,lr

are consistent estimators of the long-run vari-

, respectively, which are just as in the development of (i)

(11.30), but with ∆St and ∆Sˆ e,t replacing eˆ t . Note that when the unadjusted data are used then Se,t = ∑ti=1 yi and ∆Sˆ e,t = eˆ t ; therefore the SBDH test is the same as the KPSS test; for example, observe that the quantiles reported in Choi and Ahn (1999, Table 1) for n = 1, the number of variables in their multivariate version of the test, are also those of the KPSS test using unadjusted data. Furthermore, note that if standard LS demeaning or detrending is used, then the SBDH test using these data is the same as the corresponding KPSS test. As in the KPSS test, the bandwidth is required to satisfy m → ∞ as T → ∞, such that m/T → 0, with m = o(T1/2 ), giving a rate of divergence under the alternative of O(T1−κ ), 0 < κ < 1/2 (for stochastic versions of these rules use op (.) and Op (.), respectively). Finite sample size depends on y0 , although ﬁnite sample power does not; the assumption that y0 = 0 may be critical and we assess this below. Choi and Ahn (1999) (see also Choi, 1994), note that over-rejection is likely as |y0 | increases, and the simulations reported in section 11.6 conﬁrm this observation. Choi and Ahn (1999) use the m(j) rule to select the bandwidth, with j = 12, and Andrews’ (1991) method of automatic selection, assuming an AR(4) approximation, with an upper limit, to ensure consistency, of m = 2 if m ≥ Tξ , where ξ = 0.7 and 0.65 for the raw series and detrended series, respectively. It should be noted that the m(12) rule results in quite a long lag; for example, m(12) = 12 for T = 10 and 16 for T = 400, and size is reasonably well maintained for that choice. The m(12) rule is better on balance than the automatic selection method. As to power, the automatic lag selection results in higher power than the m(12) rule, but this is likely to be due in part to the over-parameterisation resulting from the m(12) rule when the errors are iid.

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(β) Sˆ e,T ≡ 0.) The SBDH test statistics are:

448 Unit Root Tests in Time Series

It is clear that stationarity tests have their counterparts of problems associated with unit root tests. In particular, since it is unlikely that most economic time series are well characterised by iid deviations around a constant or deterministic trend, any testing method has to be able to cope with persistence in the deviations; applying a test statistic that has assumed iid errors will lead to misleading inference when they are, in fact, persistent. The four stationarity tests of section 11.2 (referred to here collectively as the semi-parametric tests) outlined so far deal with this problem by a scale factor based on the long-run variance; estimation of this factor is constrained by the need to ensure that the test statistic still diverges under the alternative hypothesis. An alternative is to try to remove the serial correlation by a parametric method. This section considers the test in that framework due to Leybourne and McCabe (LBM) (1994). 11.3.1 The Leybourne and McCabe (1994) test The local-level model of (11.4) is generalised to allow an autoregressive structure to yt ; that is: ϕ(L)yt = β1 + µt−1 + ηt + ςt

(11.40)

p

where ϕ(L) = 1 − ∑j=1 ϕj Lj . This model has the following reduced form: ϕ(L)(1 − L)yt = β1 + (1 + θ1 L)εt

(11.41)

which gives rise to the same second-order moments as the structural model. The relationships stated in (11.15) and (11.16) also hold here, so that (11.41) has an invertible MA component for ση2 ≥ 0. As before, θ1 = –1 implies a root in common in the AR and MA lag polynomials, and hence yt is (trend) stationarity, which constitutes the null hypothesis in this context. The basis of the LBM test is quite intuitive. If ϕ(L) was known, then the testing framework does not differ under the null from the standard case, apart from the change of variable from yt to ϕ(L)yt . Instead of the usual ﬁrst step, which is to detrend the data (the trend case is used as an example) based on obtaining an estimator of et = (yt − β0 − β1 t), one seeks an estimator of et = y∗t −(β0 + β1 t), where: y∗t = yt − ∑i=1 ϕi ∆yt−i p

(11.42)

If this can be obtained, then the KPSS (and other tests) can be applied in their standard form as the serial correlation has been modelled separately. Augmenting the standard LS detrending equation with p lags of yt is not recommended by LBM as it results in residuals that are close to being stationary (which is what one wants to test) and hence a test based on these will lack

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11.3 A test with parametric adjustment

Tests with Stationarity as the Null Hypothesis 449

power. Instead, LBM recommend the (exact) ML estimators from estimation of the reduced form, that is, from estimation of: ∆yt = β1 + ∑i=1 ϕi ∆yt−i + εt + θ1 εt−1 p

(11.43)

Denote the ML estimators of ϕi and θ1 as ϕ˜ i and θ˜ 1 , respectively; these are consistent even if θ1 = –1 (which is the null value) and hence the MA component is non-invertible. Then form the AR adjusted dependent variable as: y˜ ∗t = yt − ∑i=1 ϕ˜ i ∆yt−i p

(β)

Next, form the residuals e˜ t trend, thus:

from the regression of y∗t on a constant and a time

(β) e˜ t = (y˜ ∗t − βˆ 0 − βˆ 1 t)

(11.45)

where ˆ above indicates the LS estimator. In practice, the ML estimates can be obtained from a grid search over θ1 ∈ [0, –1), possibly with a coarse starting interval followed by a ﬁner interval to locate the maximum to avoid numerical errors that can occur in unrestricted estimation (especially in a simulation context). (i) The LBM test is the KPSS test using e˜ t to form the partial sum process; that is: LBM(i) =

(i) T−2 ∑Tt=1 (S˜ t )2 (i)

∑Tt=1 (e˜ t )2 /T

(11.46)

(i) (i) where i = 0, µ, β and S˜ t = ∑tj=1 e˜ j . The asymptotic distribution of LBM(i) is

the same as that for KPSS(i) ; however, the ﬁnite sample distributions differ (see Table 11.2, section 11.5.2). The ﬁnite sample distribution is skewed more to the right for LBM(i) , implying that the right-tail quantiles – for example, the 95% quantile and, particularly, the 99% quantile – are larger. Some of the variation in the distribution of LBM(i) can be controlled by restricting the estimated values of θ1 to be contained in the region θ˜ 1 ∈ [0, –1); for example, by searching over a grid. Of course, a choice has to be made to determine the lag length p, which matches the element of choice in determining the bandwidth in the semiparametric estimator and, as LBM (1994) show, the quantiles of the null distribution also depend on p. The balance of evidence from unit root tests is that the AR-based estimator, even if it involves an element of approximation to an MA process, gives better properties to the test statistics than the semi-parametric estimator; hence, it is a matter of practical importance to know whether that is also the case with stationarity tests; and if, say, over-ﬁtting of the AR component is costly in terms of size and power (see section 11.4.2 for further discussion).

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(11.44)

450 Unit Root Tests in Time Series

random walk 1, 0) process, The ﬁrst stage 1. To illustrate

∆yt = β1 + ϕ1 ∆yt−1 + εt + θ1 εt−1

(11.47)

In this case, ϕ1 and θ1 are not identiﬁed, and there is a positive probability that ϕ˜ 1 is indistinguishable from unity (and θ˜ 1 from minus unity); in this case, if ϕ˜ 1 = 1, then y∗t = yt − ϕ˜ 1 yt−1 = ∆yt , and the unit root, under the alternative, has been removed at the ﬁltering stage, which should only remove the non-unit roots. This argument is due to Hobijn, Franses and Ooms (hereafter HFO) (2004), who also illustrate the practical importance of this case in their simulation analysis. LBM (1994) were aware of this possibility as it is implied by ruling out θ1 = 0 under the alternative. Nevertheless, it could be argued that whilst the simple random walk model is a useful prototype, in practice it is random walk-type behaviour, implied by an integrated process with stationary but not iid errors, rather than the pure random walk, which is likely to be of importance. This view ﬁnds support in the representation of many economic time series as ARIMA(p, 1, q) processes rather than ARIMA(0, 1, 0) processes. 11.3.2 The modiﬁed Leybourne and McCabe test Leybourne and McCabe (LBM, 1999) suggest a simple modiﬁcation to their 1994 test, which they report results in better power properties. The essence of the modiﬁcation is to use an alternative (ML) estimator of σς2 . Initially, reconsider the simple version of the local-level model (see Equations (11.4) and (11.5)): yt = µt + ςt µt = β1 + µt−1 + ηt which is (second-order) equivalent to: (1 − L)yt = β1 + (1 + θ1 L)εt

(11.48)

Then, for any of the stationarity tests so far considered, the required scale factor, that is, the denominator of the KPSS/LBM test statistic, is a consistent esti2 of σ2 . Rather than use a semi-parametric estimator, LBM (1999) mator σ˜ ζ,lr ζ,lr suggest exploiting the link between the structural and reduced forms, such that 2 = −σ2 θ (see Equation (11.15a)). Thus, given consistent estimators of σ2 and σζ,lr ε 1 ε θ1 , distinguished by ˜ above, the following can be used: 2 = − σ˜ ε2 θ˜ 1 σ˜ ζ,lr

(11.49)

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The LBM test is not, however, consistent against simple alternatives; that is, where yt is generated by an ARIMA(0, yt = β1 + yt−1 + εt , with or without drift, and εt ∼ white noise. of the LBM test requires ﬁltering out any non-unit roots, so p ≥ for the case that p = 1, the ﬁrst-stage model is:

(Note that the convention used here is to write the MA lag polynomial with + signs preceding the coefﬁcients; if – signs are used, then the – sign on the right-hand side of (11.49) is removed.) In the LBM framework, σ˜ ε2 and θ˜ 1 are available from the preliminary stage of ML estimation, so that no additional esti2 in (11.49) requires −θ ˜ 1 > 0, which mation is required; note that positivity of σ˜ ζ,lr will not necessarily be satisﬁed unless estimation is restricted by, for example, a grid search over the admissible range of θ1 to ensure that θ˜ 1 ∈ [0, –1). 2 Let σ˜ ML,ς,lr be the estimator from (11.49), then variants of the KPSS, MRS, KS and SBDH tests can be deﬁned using this estimator, and the resulting test statistic will have the same asymptotic distribution as in the standard case. For example, the modiﬁed KPSS test is: T−2 ∑Tt=1 (St )2 (i)

MKPSS(i) =

(11.50)

2 σ˜ ML,ζ,lr

(i)

where St is as deﬁned above (see Equation (11.28)). LBM (1999) also show that use of this estimator has implications for power as under the alternative the modiﬁed test statistic diverges at a faster rate, (i) speciﬁcally at the rate Op (T2 ) under HA rather than Op (T) for the standard version. The generalisation to the LBM(i) test, where the local-level model is 2 = −σ2 θ also holds in ϕ(L)yt = β1 + µt−1 + ηt + ςt , is straightforward because σζ,lr ε 1 ˜ this case. As (ϕ˜ i − ϕi ) and (θ1 − θ1 ) are both op (1), then σ˜ 2 = −˜σε2 θ˜ 1 = σ2 + op (1) ζ,lr

ζ,lr

(see LBM, 1999; and Potscher, 1991, for the former result). Thus, to continue, the modiﬁed LBM test statistic is: MLBM(i) =

(i) T−2 ∑Tt=1 (S˜ t )2 2 σ˜ ML,ζ,lr

(11.51)

and the corresponding asymptotic distributions are as in the standard case. 11.3.3 Data-dependent lag selection for the LBM test(s) From a practical point of view, a choice has to be made at the ﬁltering stage to determine the lag length p. Under the alternative hypothesis (of the invertibility of the MA component), the distribution of ϕ˜ i is asymptotically normal, with the √ usual T-consistency; simulations undertaken by LBM (1994) suggest that this is also the case under the null that θ1 = 1. In their simulation analysis, LBM (1994) use ﬁxed values for p, rather than a data-dependent method, partly to illustrate whether their test is robust to different values of p. LBM (1999) suggest a data-dependent method for lag selection, which is in the nature of an application of a general-to-speciﬁc (G-t-S) methodology, starting from a maximum lag length and then sequentially deleting lags according to some criterion. The selection criterion is as follows. LBM (1999, Theorem 2)

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Tests with Stationarity as the Null Hypothesis 451

452 Unit Root Tests in Time Series

show that if ϕp = 0, that is, the p-th coefﬁcient in the lag polynomial ϕ(L) of Equation (10.40), and –1 ≤ θ1 < 0, then: (11.52)

Thus Z(p) may be used to test for the inclusion or deletion of ϕp in an estimated ARIMA(p, 1, 1) model. Note that Z(p) differs from the conventional ‘t’ statistic on ϕ˜ p ; LBM (1999) suggest that the former may have an advantage over the latter as θ1 → –1, because of numerical problems in computing standard errors at or near the non-invertible region, whereas Z(p) does not require standard errors. The test is consistent because if ϕp = 0, then Z(p) ∼ Op (T1/2 ). Thus the suggested G-t-S criterion is Z(p), so that starting from an ARIMA(p∗ , 1, 1) model, where p∗ is an upper lag length, the quantities Z(p∗ − j), j = 0, 1, p – 1, are computed sequentially, stopping at pˆ = p∗ − j if |Z(p∗ − j)| > |zα/2 | where |zα/2 | is the α/2 (upper) quantile from N(0, 1); for example, |z(0.05) | = 1.645. LBM (1999) suggest α = 0.1 to lessen the chance of under-ﬁtting, which is the more serious error compared to over-ﬁtting. Then, with pˆ determined at this auxiliary stage, the revised pre-ﬁltering stage uses y˜ ∗t deﬁned as: pˆ

y˜ ∗t = yt − ∑i=1 ϕ˜ i ∆yt−i

(11.53) (i)

Otherwise the residuals, e˜ t , are formed as before as is the test statistic. Provided pˆ ≥ p, then the asymptotic distribution of the test statistic using the data-dependent procedure is the same as if p was known. Simulations conducted by LBM (1999) suggest that the loss of power resulting from the search procedure is quite small relative to the known p case. Carrion-i-Silvestre and Sanso´ (2006) also consider standard information-based criteria (for example, BIC and AIC), for selecting the lag and the simulation results reported below (see Table 11.4, section 11.5.3.i), and ﬁnd that p = p(AIC) performs quite well.

11.4 The long-run variance The long-run variance and its estimation by semi-parametric methods are key parts of several tests with stationarity as the null hypothesis. (This topic has already been partly considered, as it arises in the construction of the DF and PP unit root tests; see Chapter 6.) As the long-run variance is an important input into these tests, the opportunity is taken here to illustrate some of the issues in its estimation. 11.4.1 Estimation of the long-run variance As a reminder, a consistent estimator of the long-run variance of ζt (under 2 , based on the estimated the null, see Equations (11.24)–(11.27)), denoted σ˜ ζ,lr

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Z(p) ≡ − T1/2 ϕ˜ p θ˜ 1 ⇒D N(0, 1)

Tests with Stationarity as the Null Hypothesis 453

ˆ autocovariances, γ(κ), κ ≥ 0, is given by: 2 ˆ ˆ = γ(0) + 2 ∑κ = 1 ω(m, κ)γ(κ) σ˜ ζ,lr T

ˆ = T−1 ∑Tt= κ + 1 eˆ t eˆ t−κ ; eˆ t is an estimator of the random variable et ; where γ(κ) ω(m, κ) = {ωm,κ }m κ = 1 is a kernel function that assigns weights to the autocovariances at the lags 1 through to m; and m is referred to as the bandwidth parameter. The kernels are distinguished by the different patterns of weights; several are in frequent use, but they have in common that they ensure posi2 . The ones to be considered in this chapter are the tivity of the estimator σ˜ ζ,lr quadratic spectral (QS) and the Bartlett (Newey-West) (BW) kernels, given by: ωm,κ =

25 [sin(x)/x − cos(x)] 12π2 (κ/m)2

the QS kernel

where x = 6π(κ/m)/5 ωm,κ = 1 − κ/(m + 1)

the BW kernel

where κ = 1, ..., m. The QS and, more familiar, BW kernels are plotted in Figure 11.1, for m = 8. The QS kernel is a nonlinear weighting scheme, with relatively more weight on the earlier lags compared to the Bartlett kernel. Whichever kernel function is chosen, a decision has to be made about m, the bandwidth parameter. This could be ﬁxed somewhat arbitrarily; alternatively, it could be determined by a rule. A deterministic rule in common use is m(j) = [j(T/100)1/4 ], where j = 4, 12 are frequent choices (see Chapter 9, section 9.1.4.ii). 1 0.9 0.8

QS weights (nonlinear)

0.7 0.6 weight 0.5

Bartlett weights (linear)

0.4 0.3 0.2 0.1

1

2

3

4

5

6

7

8

lag Figure 11.1 Barlett and quadratic spectral kernels.

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(11.54)

454 Unit Root Tests in Time Series

1.

2.

Initially choose a bandwidth parameter, n(qs) of order o(T2/25 ) for the QS kernel and n(bw) = o(T2/9 ) for the BW kernel; for example, n(qs) = int[j(T/100)2/25 ] and n(bw) = int[j(T/100)2/9 ], respectively, with j = 4. The starting value n(qs) is relatively insensitive to T; for example, n(qs) = 4 for T ∈ [100, 1627]; whereas n(bw) = 4 for T ∈ [100, 272] and n(bw) = 5 for T ∈ [272, 620]. (i) (i) 2 ˆ j , sˆ (2) = 2 ∑n ˆ j, Then, using this value of n(i) , calculate sˆ (0) = γˆ 0 + 2 ∑n j=1 γ j=1 j γ and select the bandwidth as:  2 1/5 ˆ (2) s (qs) (qs) 1/5 (qs) ˆ cab (j) = min[T, λˆ m T ] λˆ = 1.3221  (0)  sˆ ˆ (bw) T1/3 ] ˆ (bw) m cab (j) = min[T, λ (qs)

 2 1/3 (2) ˆ s λˆ (bw) = 1.1447  (0)  sˆ

(bw)

ˆ cab (j) and m ˆ cab (j) are the ‘automatic’ bandwidths under these selecThus m tion rules for the QS and BW kernels, respectively. The ‘cab’ subscript refers to constrained automatic bandwidth, the constraint being reﬂected in the min[.] function determining the bandwidth. 11.4.2 Illustrative simulation results 2 It is clear that σζ,lr is an important nuisance parameter, whether in the context of tests for nonstationarity or stationarity. Some illustrative results for estima2 tion of σζ,lr in the AR(1) and MA(1) cases are presented here using different values of the bandwidth parameter. For comparison, the following are included: (i) the unweighted estimator – that is, simply setting all weights in the kernel to unity; (ii) the LS estimator in the AR(1) model (or the conditional ML estimator in the MA(1) case); (iii) the QS and the BW kernels for different values of m, and ˆ (qs) the QS kernel with (constrained) automatic bandwidth selection, m cab (4). The results, in terms of root mean squared error (rmse), are shown in Figures 11.2–11.5 for T = 100 (illustrative of a wider set of values for T) based on 10,000 simulations. Figure 11.2 is concerned with the AR(1) case yt = ρyt−1 + εt , where εt is niid(0, 1). It shows the rmse for each estimator as a function of ρ (for the more relevant case where ρ > 0), with one function for each of the bandwidths m = 2, 3, 4. Note that in each case, rmse unambiguously increases with ρ. What is not shown, because of the effect it would have on the scale of the graphs, is that the rmse for the LS estimator increases sharply as ρ → 1. For example, at ρ = 0.95, it is over ten times the rmse of the other estimators; this reﬂects the

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A data-dependent rule due to Newey and West (1994) is also in frequent use (see also Andrews, 1991), and for an application in the current context see HFO (2004). The rule is as follows.

Tests with Stationarity as the Null Hypothesis 455

10

10

m=2 m=3 m=4

8

8 6

6 rmse

rmse 4

4

QS kernel

0

10

0.2

0.4 ␳

0.6

0

0.8

10

m=2 m=3 m=4

8

0.2

0.4 ␳

0.6

0.8

0.4 ␳

0.6

0.8

m=2 m=3 m=4

8 6

6 rmse

rmse

Equal weights

4

BW kernel

4 2

2 0

0

0

0.2

0.4 ␳

0.6

0.8

0

0

0.2

Figure 11.2 Estimators of long-run variance: rmse, AR(1).

sharp increase in the bias of the LS estimator as the unit root is approached (see Chapter 4). As to sensitivity to the bandwidth, m, in the case of the QS and BW kernels, m = 2 leads to the smallest rmse for small values of ρ (ρ < 0.3); however, thereafter, rmse is smallest for m = 4; m = 2 is best for the equally weighted estimator, except as ρ > 0.6. Thus, as far as the kernel estimators are concerned, and for typical economic time series, a larger value of m is preferred. This raises the question of what happens in the case of the (constrained) automatic bandwidth selection method, illustrated here for the QS kernel, which does not require a prior view on the magnitude of ρ. In fact, the results can ˆ (qs) be anticipated from the average bandwidth chosen by m cab (j), referred to as QSCAB on Figure 11.3; this function is quadratic in ρ (see the lower-right subplot of Figure 11.3), and never much below 4; thus it will tend to dominate when a larger value of m results in a lower rmse, that is when ρ is (approximately) greater than 0.5. Figure 11.3 also shows that there is little to choose in terms of rmse between the QS and BW kernel estimators for the same value of m in the AR(1) case. The results for the MA(1) case yt = εt + θ1 εt−1 , where εt is niid(0, 1), are shown in Figures 11.4 and 11.5. In the case of θ1 < 0, and for the kernel estimators, smaller rmse is achieved for m = 3 and m = 4, with the QS kernel better than

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2

2 0

LS estimator

456 Unit Root Tests in Time Series

10

10

QS, m = 2 BW, m = 2 QSCAB

8

QS, m = 3 BW, m = 3 QSCAB

8 6

6

rmse 4

4

2

2

0

0

10

0.2

0.4 ρ

0.6

0.2

0.4 ρ

0.6

0.8

0.6

0.8

QSCAB: ave lag

5 CAB average 4.5 lag

6

rmse 4

4

2 0

0

5.5

QS, m = 4 BW, m = 4 QSCAB

8

0

0.8

0

0.2

0.4 ρ

0.6

0.8

3.5

0

0.2

0.4 ρ

Figure 11.3 QS, BW and QS (CAB) estimators: rmse, AR(1).

the BW kernel. The additional lags make no difference to the ML estimator, or to the equally weighted estimator for θ1 < 0. The comparison by bandwidth is shown in Figure 11.5. In each case, the QS kernel, with ﬁxed bandwidth, tends to dominate, although there are some small parts of the parameter space where ˆ (qs) m cab (j) is better in rmse terms.

11.5 An evaluation of stationarity tests This section is concerned with illustrating some of the practical issues in assessing the relative merits of the different tests for stationarity outlined in previous sections. Issues of concern relate to selection of the bandwidth parameter, size and power. Sections 11.5.1 and 11.5.2 consider issues relevant to bandwidth selection and critical values; and size and power comparisons are reported in Sections 11.5.3 and 11.5.4. 11.5.1 Selection of the bandwidth parameter One key practical aspect for those tests based on kernel estimation is selection of the bandwidth parameter, m. As in HFO (2004), the following criteria are considered:

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rmse

Tests with Stationarity as the Null Hypothesis 457

2.5 2

3

m=2 m=3 m=4

2.5 2

rmse 1.5 1

rmse 1.5 QS kernel

1 0.5

0.5 0 –1

ML estimator

–0.5

0

0

0.5

–1

–0.5

3 2.5 2 rmse 1.5 1

3

m=2 m=3 m=4

2.5

0.5

2

0

0.5

m=2 m=3 m=4

rmse 1.5

unweighted estimator

1

BW kernel

0.5

0.5 0 –1

0 θ1

θ1

–0.5

0

0 –1

0.5

–0.5

θ1

θ1

Figure 11.4 Estimators of long-run variance: rmse, MA(1). 2.5 2 rmse

2.5

QS, m = 2 BW, m = 2 QSCAB

2

1.5

rmse

1

0.5

0.5

–0.5

0

0.5

θ1 2.5 2 rmse

1.5

1

0 –1

0 –1

–0.5

θ1

0

0.5

12

QS, m = 4 BW, m = 4 QSCAB

10

1.5

QSCAB: ave lag

average 8 lag

1

6

0.5 0 –1

QS, m = 3 BW, m = 3 QSCAB

–0.5

0 θ1

0.5

4 –1

–0.5

θ1

0

0.5

Figure 11.5 Comparison by bandwidth criteria: rmse, MA(1).

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3

458 Unit Root Tests in Time Series

deterministic: m(bw) (j) = [j(T/100)1/4 ]

Bartlett weights

m(qs) (j) = [ 32 j(T/100)2/9 ]

quadratic spectral

data dependent: (bw)

ncab (j) = [j(T/100)2/9 ] (bw)

ˆ cab (j) ⇒m (qs)

ncab (j) = [j(T/100)2/25 ] (qs)

ˆ cab (j) ⇒m

quadratic spectral

where j = 4, and in the case of the data-dependent rules n(.) indicates the starting ˆ (k) value and m cab (j), k = qs, bw, is the second-stage estimate. Note that the upper limit for m is capped at T1/3 to avoid problems with consistency of the test (see Choi and Ahn, 1994, 1999). 11.5.2 Summary tables of limiting distributions and critical values The limiting distributions under the null are collected together in Table 11.1, where the following notational conventions are used: V(1) (r) = B(r) − rB(1) V(2) (r) = V(1) (r) + 6r(1 − r)

1 B(1) − 2

1 0

= B(r) + (2r − 3r2 )B(1) + 6(r2 − r)

B(s)ds

(11.55a) (11.55b)

1 0

B(s)ds

(11.55c)

V(1) (r) is the standard (ﬁrst-level) Brownian bridge and V(2) (r) is the second-level Brownian bridge. Table 11.2 reports some critical values from simulation of the ﬁnite sample distributions with yt = εt , where εt ∼ niid(0, 1); these are obtained from published sources (see notes to the table) and by simulation using 50,000 replications. Note that the quantiles for the semi-parametric tests are based on the assumption, which is standard in this context, that no adjustment is required to estimate the long-run variance, and, in the case of the LBM tests, that p = 1. In practice, this is rarely the case, as this information is not usually known, and the long-run variance is estimated, for example, by a kernel estimator. (This issue is considered further below.) A similar problem arises in the case of the LBM test statistics, where the use of a search procedure induces greater variation in the ﬁnite sample distribution, as does over-ﬁtting the lag length, for example estimating with p = 4 when p = 1 is the correct (but unknown) lag length. To illustrate, some simulations with

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Bartlett weights

459

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0≤r≤1

sup |V(2) (r)|

0≤r≤1

sup |V(1) (r)|

sup |V(1) (r)|

0≤r≤1

KS

SBDH

1 2 0 B (r)dr − 3

0

0

2

1 0 rB(r)dr

2 1 2 B (r)dr − 3 1 rB(r)dr

1 2 0 B (r)dr

+6

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1 0 rB(r)dr

1 2 0 B (r)dr

Sources: KPSS: KPSS (1992) and HFO (2004); MRS: Lo (1991) KS: Xiao (2001, Theorem 1, Remark 2), and see also Cappucio and Lubian (2006).

0≤r≤1

0≤r≤1

sup V(2) (r) − inf V(2) (r)

1 2 0 V(2) (r)dr

0≤r≤1

(β)

H0

0≤r≤1

sup V(1) (r) − inf V(1) (r)

0≤r≤1

sup B(r) − inf B(r)

MRS

1 2 0 V(1) (r)dr

1 2 0 B (r)dr

KPSS

(µ)

H0

(0) H0

Table 11.1 Limiting distributions of test statistics for stationarity.

460 Unit Root Tests in Time Series

Table 11.2 Quantiles of null distributions for stationary tests (critical values).

KPSS 100 200 ∞ MRS 100 200 ∞ KS 100 200 ∞ SBDH 100 200 ∞ LBM 100 200 ∞

Mean adjusted

Mean and trend adjusted

90%

95%

99%

90%

95%

99%

90%

95%

99%

1.20 1.93 1.15

1.63 1.63 1.61

2.69 2.73 2.74

0.35 0.34 0.35

0.46 0.46 0.46

0.73 0.72 0.74

0.12 0.12 0.12

0.15 0.15 0.15

0.21 0.22 0.22

2.12 2.15 2.20

2.36 2.40 2.47

2.86 2.90 2.95

1.50 1.54 1.62

1.62 1.66 1.75

1.84 1.91 2.01

1.40 1.43 1.50

1.51 1.55 1.63

1.74 1.79 1.89

1.16 1.18 1.22

1.29 1.31 1.36

1.53 1.58 1.63

1.17 1.18 1.22

1.30 1.30 1.36

1.54 1.58 1.63

0.78 0.79 0.83

0.85 0.87 0.90

0.98 1.00 1.04

1.20 1.20 1.19

1.63 1.63 1.66

2.69 2.73 2.77

0.19 0.19 0.19

0.25 0.25 0.25

0.38 0.38 0.38

0.09 0.09 0.09

0.12 0.11 0.11

0.16 0.16 0.16

1.07 1.12 1.15

1.70 1.65 1.61

3.39 2.78 2.74

0.35 0.35 0.35

0.52 0.47 0.46

0.78 0.75 0.74

0.14 0.12 0.12

0.18 0.15 0.15

0.28 0.23 0.22

Notes: Based on 50,000 replications, where not available in published sources. All of the tests reject for sample values exceeding the value shown in the table. The DGP is yt = εt , where εt ∼ niid (0, 1) and the standard (that is assuming no weak dependence) forms of the respective test statistics are used; in the case of the LBM statistics, p = 1. Other sources: KPSS: KPSS (1992, Table 1); KS: Xiao (2001, Table 1); MRS: Lo (1991, Table II); SBDH: Choi and Ahn (1999, Table 1).

LBM(µ) were undertaken using the following lag selection variations: p = 1, p = 4, 2 p = p(AIC), p = Z(p) and p chosen by Z(p) combined with σ˜ ML,ς,lr (see Equation (11.49)), with T = 100. The results are reported in Table 11.3. Of course, these variations could be multiplied further, but this set is enough to illustrate the effects involved. For example, the effect of searching by p(AIC), relative to setting p = 1, is to increase the 95% quantile marginally from (approximately) 0.52 to 0.54 and to 0.75 for p = 4; the 95% quantile is about 0.6 if Z(p) is used and 2 0.61 if Z(p) is combined with σ˜ ML,ς,lr . Generally, the use of a lag selection procedure results in a rightward shift in the null distribution of LBM(µ) , so that some oversizing occurs in ﬁnite samples when the standard quantiles are used. 11.5.3 Empirical size The general problem for stationarity tests when the errors are generated by an AR process is that they tend to be oversized, especially as the dominant root approaches unity. Nearly nonstationary series are mistaken to be nonstationary

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Unadjusted data

Tests with Stationarity as the Null Hypothesis 461

Table 11.3 Effect of lag selection on quantiles of LBM(µ) . Selection method

90%

95%

99%

p=1 p=4 p = p(AIC) p = Z(p) p = Z(p) + ML variance

0.35 0.54 0.39 0.43 0.434

0.52 0.75 0.54 0.60 0.614

0.78 1.30 0.92 1.05 1.219

series; hence the test statistic leads to rejection, and this is the case even when an adjustment has been made by using a semi-parametric estimator of the longrun variance. Generally, the empirical size can be made closer to the nominal size by increasing the bandwidth parameter; however, there is the usual tradeoff between size and power given that the structure of the weak dependence it is not known ex ante.

11.5.3.i Illustrative simulations The simulations reported here consider AR(1) and MA(1) DGPs, with εt ∼ niid(0, 1) and y0 = 0. Variations on the zero initial condition are considered in Chapter 12, section 12.1. The nominal size of the tests is 5%, and a summary of the results is presented in Tables 11.4 and 11.5 for sample sizes T = 100 and T = 500, and Figures 11.6–11.9 for T = 100. The test statistics use the versions scaled by a consistent long-run variance (or standard error) estimator; thus, size at ρ = 0.0 will not necessarily be 5%, but any deviations will be a good indicator of which tests hold size better even when there is no serial correlation. Note that QSCAB and BWCAB refer to the kernel bandwidth chosen by the automatic bandwidth procedure of section 11.5.1. Table entries in bold indicate best or good performance. Although the simulations cover the MA(1) case with θ1 > 0, it should be borne in mind that, if the reduced form model is to be related back to the local-level structural model, then the admissible region for θ1 does not include θ1 > 0, as this results in the signal-to-noise ratio ψ being negative. The region θ1 > 0 is included because this case does sometimes occur in practice.

AR(1) errors: summary (Table 11.4) In this case, the DGP is yt = ρyt−1 + εt . Taking each test statistic in turn: • KPSS(µ) becomes more oversized as ρ → 1, and is best when combined with ˆ (bw) m cab (j), although it remains oversized, but less so as T increases; there is a noticeable improvement in empirical size as T increases.

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Note: Based on 50,000 replications; other details as Table 11.2.

462 Unit Root Tests in Time Series

Table 11.4 Stationarity tests: empirical size for 5% nominal size AR(1) errors.

0.0 0.3 0.6 0.9

QS 4.7 10.3 21.7 71.6

T = 500

KPSS(µ)

KPSS(µ)

QSCAB 5.0 6.2 10.2 30.6

BW 4.7 6.2 12.8 48.8

BWCAB 4.6 4.9 9.2 17.1

QS 5.0 5.0 15.3 63.9

QSCAB 5.4 4.9 7.4 23.4

MRS(µ) 0.0 0.3 0.6 0.9

4.8 12.9 30.9 86.2

5.2 7.0 5.7 8.8

3.7 6.3 10.1 52.5

4.5 10.4 23.4 75.1

4.3 6.1 6.7 23.8

4.1 6.2 10.5 46.4

4.0 5.1 3.0 0.9

4.3 5.7 20.0 87.8

4.7 4.7 5.0 22.8

4.4 9.7 23.7 77.7

4.4 5.9 9.0 28.7

4.1 6.6 12.5 49.0

3.0 4.4 4.6 5.3

4.6 4.9 15.9 73.3

4.7 4.9 6.2 21.9

p=1 5.0 6.0 6.9 21.0

p=4 11.1 12.9 15.0 30.9

p(AIC) 6.7 7.8 8.5 18.5

2.9 3.2 2.1 0.8

4.6 4.6 11.3 55.1

4.3 3.8 4.6 4.2

SBDH(µ) 3.9 4.8 8.4 13.2

5.2 5.3 16.0 75.0

5.2 5.2 7.0 27.8

LBM(µ) 0.0 0.3 0.6 0.9

4.2 4.5 12.0 69.8 KS(µ)

SBDH(µ) 0.0 0.3 0.6 0.9

BWCAB 4.8 4.5 5.4 8.6

MRS(µ)

KS(µ) 0.0 0.3 0.6 0.9

BW 4.9 4.9 11.4 47.4

5.6 5.1 12.9 58.7

4.7 3.9 5.1 8.9

LBM(µ) Z(p) 8.0 8.6 10.6 24.2

p=1 5.0 5.0 5.3 9.3

p=4 6.0 6.2 6.5 12.2

p(AIC) 5.2 5.2 5.6 8.6

Z(p) 6.1 6.4 7.2 16.6

ˆ (qs) • MRS(µ) is best with m cab (j), and maintains its size quite well as ρ → 1, although the larger sample size does not improve size retention at ρ = 0.9. ˆ (qs) • KS(µ) combined with m cab (j) gives a reasonably sized test except as ρ → 1, (bw)

ˆ cab (j), KS(µ) is slightly when it becomes oversized; when combined with m undersized for ρ = 0, but otherwise maintains its nominal size. ˆ (bw) • SBDH(µ) is similar to KS(µ) , and is best combined with m cab (j), but it is (µ) dominated by KS in terms of size retention as ρ → 1. • LBM(µ) shows relatively moderate departures from its nominal size when combined with p(AIC), provided that ρ is not too close to 1; for example, an empirical size of 8.5% for ρ = 0.6 when T = 100 and 5.6% for T = 500.

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ρ

T = 100

Tests with Stationarity as the Null Hypothesis 463

θ1 –0.9 –0.6 –0.3 0.0 0.3 0.6 0.9

QS 0.0 0.0 1.7 4.7 6.5 7.9 6.6

T = 100

T = 500

KPSS(µ)

KPSS(µ)

QSCAB 0.1 1.4 3.4 5.0 5.6 7.8 5.7

BW 0.0 0.2 2.5 4.7 5.4 7.2 5.0

BWCAB 0.0 0.2 1.6 4.6 4.6 6.4 5.1

QS 0.0 0.5 2.6 5.0 5.2 6.1 5.0

QSCAB 0.1 2.3 3.3 5.3 5.3 6.0 5.6

MRS(µ) –0.9 –0.6 –0.3 0.0 0.3 0.6 0.9

0.0 0.0 1.3 4.8 5.2 7.1 6.8

14.3 10.8 5.7 5.2 4.1 6.1 4.3

0.0 0.1 2.1 4.5 6.4 7.2 7.4

3.8 5.0 5.1 4.3 4.8 5.9 5.4

0.0 0.1 1.8 3.7 3.3 4.8 2.7

0.0 0.1 2.2 4.0 3.5 3.3 4.1

0.0 0.0 3.2 4.7 5.8 6.6 5.9

5.1 4.4 4.7 4.6 6.3 7.7 6.8

0.0 0.0 1.7 4.4 7.2 7.1 9.2

1.5 1.6 4.1 4.4 6.2 8.3 7.5

0.0 0.2 2.8 4.1 4.4 5.5 4.8

0.0 0.2 2.2 3.0 4.3 3.9 4.4

0.0 0.2 2.4 4.8 5.4 5.9 5.5

–0.9 –0.6 –0.3 0.0 0.3 0.6 0.9

0.0 0.0 2.9 5.0 3.8 2.6 1.7

0.0 0.3 2.2 3.8 5.0 5.3 5.6

0.9 2.9 3.5 5.0 5.4 6.6 5.8

0.0 0.1 2.1 4.7 5.4 6.2 5.5

0.0 0.3 2.1 4.3 5.6 6.0 5.3

SBDH(µ)

0.0 0.0 2.3 4.1 6.1 6.3 7.7

0.0 0.0 2.2 3.9 5.5 6.8 7.1

0.0 0.2 3.1 4.8 5.1 6.7 6.3

LBM(µ) p=1

0.0 0.0 2.7 4.6 6.0 6.7 6.2

KS(µ)

SBDH(µ) –0.9 –0.6 –0.3 0.0 0.3 0.6 0.9

BWCAB 0.0 0.6 2.8 5.2 5.7 5.9 5.0

MRS(µ)

KS(µ) –0.9 –0.6 –0.3 0.0 0.3 0.6 0.9

BW 0.0 0.4 2.5 5.2 5.4 6.3 5.4

0.3 1.9 4.1 4.8 5.5 6.9 6.4

0.0 0.2 2.8 4.7 5.3 6.7 6.2

0.0 0.6 3.1 4.8 5.3 6.5 5.9

LBM(µ)

p=4

P(AIC)

Z(p)

0.0 6.0 10.8 11.8 12.6 15.7 22.0

0.0 2.9 5.6 6.8 7.1 9.7 14.8

0.0 3.6 7.0 7.9 9.1 10.6 14.4

p=1 0.0 0.0 2.2 5.0 3.8 3.0 3.8

p=4

p(AIC)

Z(p)

0.0 2.4 6.1 7.3 7.0 11.6 36.0

0.0 1.7 4.1 5.1 4.9 6.0 25.2

0.0 1.9 4.1 5.9 6.1 9.0 34.4

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Table 11.5 Stationarity tests: empirical size for 5% nominal size, MA(1) errors.

464 Unit Root Tests in Time Series

Overall, LBM(µ) is slightly better when combined with p(AIC) than with Z(p). Some of the key points are illustrated in Figures 11.6–11.7. To avoid repetition, the KS(µ) test is taken as illustrative for the semi-parametric tests, with

0.7 0.6

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QS BW

0.5 0.4

QSCAB

size 0.3

BWCAB

0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5 ρ

0.6

0.7

0.8

0.9

1

Figure 11.6a KS(µ) , empirical size for AR(1) errors: 4 kernels.

0.45 0.4 0.35 p=4 0.3 0.25

p = Z(p)

size p = p(AIC)

0.2 0.15 0.1 0.05

5% p=1

0

0

0.1

0.2

0.3

0.4

0.5 ρ

0.6

0.7

0.8

0.9

1

Figure 11.6b LBM(µ) , empirical size for AR(1) errors.

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Tests with Stationarity as the Null Hypothesis 465

0.35 0.3

LBM(µ)

0.25 KPSS(µ) 0.2 size

SBDH(µ) MRS

(µ)

0.1 5%

0.05 (µ)

0

KS 0

0.1

0.2

0.3

0.4

0.5 ρ

0.6

0.7

0.8

0.9

1

Figure 11.7 Best of each test, empirical size for AR(1) errors.

Figure 11.6a showing the empirical size for each of the four bandwidth criteria. The empirical size for the LBM(µ) test is shown in Figure 11.6b for each of the four lag selection criteria, p = 1, p = 4, p = p(AIC), p = Z(p). Finally, the best combination for each test is then shown in Figure 11.7. For brevity, only the case for T = 100 is shown; however, the same conclusions follow for T = 500. The tests combined with automatic bandwidth selection are better than those with a ﬁxed bandwidth. Provided that ρ is not too close to unity, say ρ < 0.5, there is not much to choose between the semi-parametric tests with automatic bandwidth selection; thereafter, the best combination of the semi-parametric (µ) + m ˆ (bw) ˆ (qs) tests is KS(µ) + m cab (j), with MRS cab (j) a reasonable second, but not as good when T increases; this test is competitive with LBM(µ) + p(AIC). MA(1) errors: summary (Table 11.5) In this case the DGP is yt = εt + θ1 εt−1 . For T = 100, with one exception, all combinations of the tests and bandwidth selection criteria have poor size properties as θ1 → –1. This case is well known to be problematic for tests of stationarˆ (qs) ity and unit root tests. The exception is KS(µ) with m cab (j), which maintains a reasonably accurate size throughout (perhaps remarkable given the notable problems in this case). Many combinations have poor size properties even as ˆ (qs) θ1 →− 0 for θ1 < 0, tending to be undersized, although MRS(µ) with m cab (j) is an exception, being oversized. The problems are less acute throughout when θ1 > 0, and several combinations lead to quite accurate empirical size; for exam(µ) and m (µ) with m ˆ (bw) ˆ (qs) ˆ (qs) ple, KPSS(µ) and m cab (j), MRS cab (j); however, KS cab (j) is the best combination throughout the range of θ1 .

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0.15

As far as LBM(µ) is concerned, the empirical size becomes less accurate as θ1 → − 1 for all lag selection methods, but the problem is generally less acute in comparison with the other tests for moderately negative values of θ1 . Overall, p = p(AIC) is marginally the best criterion at maintaining size. As in the AR(1) case, some of the key points are illustrated graphically, in this case in Figures 11.8–11.9. For example, the effects of combining KPSS(µ) with different kernel selection methods are illustrated in Figure 11.8a, which shows the undersizing for θ1 < 0 and oversizing for θ1 > 0. The combinations of LBM(µ) with the four lag selection criteria are shown in Figure 11.8b, with a mixture of undersizing and oversizing except for p = 1. The ‘best’ of the various combinations is then shown in Figure 11.9 (for brevity, only the combinations for T = 100 are shown). These are the semi(µ) with p(AIC); although in ˆ (qs) parametric tests combined with m cab (j) and LBM the case of MRS(µ) , the choice of ‘best’ really depends on the sign of θ1 . The best test combination, in the sense of retaining size across a wide range of values ˆ (qs) of θ1 , is KS(µ) with m cab (j); others are only competitive for parts of the range of θ1 . Recall that in the case of an AR(1) DGP, KS(µ) was robust if combined ˆ (bw) ˆ (qs) with m cab (j), but not quite as robust if combined with mcab (j); nonetheless, no other test was dominant. Size is, of course, only one aspect of test evaluation; some simulations to assess power are considered in section 11.6.

0.08 QS

0.07 0.06 0.05 size 0.04

5% QSCAB

BWCAB

0.03 BW 0.02 0.01 0 –1 –0.8 –0.6 –0.4 –0.2

0 θ1

0.2

0.4

0.6

0.8

1

Figure 11.8a KPSS(µ) , empirical size for MA(1) errors: 4 kernels.

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466 Unit Root Tests in Time Series

Tests with Stationarity as the Null Hypothesis 467

0.25

0.2 p=4 0.15 size p = p(AIC) p = Z(p) 0.05

5% p=1

0 –1 –0.8 –0.6 –0.4 –0.2

0 θ1

0.2

0.4

0.6

0.8

1

Figure 11.8b LBM(µ) , empirical size for MA(1) errors.

0.16 0.14

(µ)

LBM

0.12 SBDH(µ)

0.1 size 0.08 0.06

KS(µ) 5%

0.04 0.02 0 –1 –0.8 –0.6 –0.4 –0.2

MRS(µ)

KPSS(µ) 0 θ1

0.2

0.4

0.6

0.8

1

Figure 11.9 Best of each test, empirical size for MA(1) errors.

11.5.3.ii Looking at the quantiles: comparing empirical distribution functions Another way of looking at the sensitivity of inference concerning the null hypothesis of stationarity (or, indeed, nonstationarity) is to assess how the 95% quantile, say, for a particular test statistic, varies with ρ (or θ1 in the MA(1) case).

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0.1

468 Unit Root Tests in Time Series

1 0.9

KPSS(µ)

0.8 0.7

(µ)

LBM

ρ = 0.0

0.6 cumulative probability

0.5

0.3

finding the 95% quantile(s)

0.2 0.1 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

1.1 1.2 1.3

Figure 11.10 EDFs for KPSS(µ) and LBM(µ) .

Provided that |ρ| < 1, then yt is still stationary; however, as noted, the problem for stationarity tests is that they tend to over-reject as ρ → 1. This is because the correct value for the 95% quantile shifts to the right for ρ > 0 as ρ → 1, so that a sample value that is not actually signiﬁcant appears signiﬁcant judged by the incorrect 95% quantile. The relative merit of different kernel estimators, and bandwidth selection criteria, is how good they are at returning the null distribution to the one that assumes ρ = 0, so that the ‘standard’ quantile provides accurate inference. To illustrate the issues, consider KPSS(µ) and LBM(µ) when the DGP is alternately AR(1) and MA(1); the selection criterion mqs (4) was used to illustrate the case for KPSS(µ) and p = 1 for LBM(µ) , and the simulations were with T = 100. Some of the results are presented in Figures 11.10–11.14, which show a number of (cumulative) empirical distribution functions (EDFs). In Figure 11.10, the opportunity is taken to show that when ρ = 0, although the 95% quantiles for KPSS(µ) and LBM(µ) are close, the EDF for the latter indicates that there is a higher probability of large values for LBM(µ) . Next ρ = 0.9 for Figures 11.11 and 11.12, which is the real point of the comparison. In this case the selection criterion m(qs) (4) is used for KPSS(µ) and p = 1 for LBM(µ) . In principle, the implied adjustments should ensure that the distribution function is not distinguishable from the standard case; however, as shown in Figure 11.11, this is not so in the case of KPSS(µ) . For example, whilst the standard 95% quantile is approximately 0.46, the 95% quantile for KPSS(µ) , with ρ = 0.9 and mqs (4), is approximately 2. The actual size, if the standard 95% quantile is used, can be read from Figure 11.11 by projecting a vertical line at

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0.4

Tests with Stationarity as the Null Hypothesis 469

1 0.9

EDF assuming ρ = 0.0

0.8 EDF using QS kernel estimation

0.7 0.6

ρ = 0.9

0.5 0.4 0.3

acual size = (1 – 0.29)% = 71% using standard quantile

0.2 0.1 0

95% quantile assuming ρ = 0.0 0

0.2 0.4

0.6

0.8

1

1.2 1.4

1.6

1.8

2

2.2

2.4

Figure 11.11 EDFs for KPSS(µ) , AR(1) errors.

1 0.9

LBM(µ) with ρ = 0.0 LBM(µ) with ρ = 0.9

0.8 0.7 0.6

actual size = (1 – 0.79)% = 21% using standard quantile

cumulative probability 0.5 0.4 0.3 0.2

95% quantile of 'standard' distribution

0.1 0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1 1.1 1.2 1.3 1.4 1.5

Figure 11.12 EDFs for LBM(µ) AR(1) errors.

that value onto the EDF for KPSS(µ) , with ρ = 0.9 and mqs (4); this shows that about 71% of the empirical distribution is to the right of this point, not 5% as given by the nominal size.

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cumulative probability

470 Unit Root Tests in Time Series

1 0.9

actual size of 1.63% using standard quantile when θ1 = –0.9

0.8 0.7 0.6 cumulative probability 0.5 0.4 0.3 0.2

EDF assuming θ1 = 0.0

0.1 0

0

0.1

0.2

0.3

0.4

95% quantile assuming θ1 = 0.0 0.5

0.6 0.7

0.8

0.9

1

1.1

1.2

Figure 11.13 EDFs for KPSS(µ) , MA(1) errors.

In the case of the LBM test Figure 11.12 shows that the parametric AR(1) adjustment is more effective at returning the empirical distribution function to that in the standard case, but still leaves noticeable oversizing. The MA(1) case is likely to be more demanding for the LBM test and should present more of an advantage to the kernel-based methods. This is because the AR correction will now only approximate the MA(1) error, whereas kernel-based methods should be able to exploit the lag-one cut-off in the autocovariances. The DGP is yt = εt + θ1 εt−1 , with θ1 = –0.9. Initial simulations indicated that the actual size for both methods was close to zero using mqs (j) for KPSS(µ) , with j = 4 or j = 8, and p = 4 or p = 8 for LBM(µ) ; this is despite the apparent theoretical advantage to the former method. It was necessary to set j = 12 and p = 12 to achieve an empirical size noticeably above zero. The results are illustrated in Figures 11.13–11.14, where the former shows that the EDF for KPSS(µ) with m(qs) (12) still has not shifted enough to preserve the 95% quantile and the empirical size is about 1.6%. Figure 11.14 shows a similar problem for LBM(µ) with p = 12, and an empirical size of about 1.3% if the 95% quantile with θ1 = 0.0 is the reference point. This comparison uses the simulated 95% quantile for LBM(µ) with θ1 = 0.0 and T = 100, which is slightly to the right of that for KPSS(µ) ; if the latter is used, there is a marginal improvement in size.

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EDF using QS kernel estimation

Tests with Stationarity as the Null Hypothesis 471

1 0.9 actual size = 0.3% using standard quantile when θ1 = –0.9

0.8 0.7 cumulative probability 0.5 0.4 LBM(µ) with θ1 = –0.9

0.3 0.2

standard 95% quantile assuming θ1 = 0.0

0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 11.14 EDFs for LBM(µ) , MA(1) errors.

11.5.4 The power of tests for stationarity against the unit root alternative Consider the structural model comprising Equations (11.2) and (11.3), which embodies a random walk component; that is: yt = µt + ςt µt = µt−1 + ηt Together these imply that yt has a stochastic trend, so that: yt = µ0 + ∑i=1 ηi + ςt t

where ηt ∼ niid(0, ση2 ) and εt ∼ niid(0, 1). The magnitude of ση2 is a measure of the ‘size’ of the random walk component, with the tests expected to increase in power as ση2 increases. The DGP for the power simulations can be represented in one of two equivalent ways. In the ﬁrst, the simulation is designed with ση2 varying; for example, ση2 = (0.01, 0.1, 1.0, 10). Equivalently, the simulation is designed with θ1 varying in an ARIMA(0, 1, 1) model: ∆yt = εt + θ1 εt−1 Noting that the null hypothesis is H0 : θ1 = –1, whereas HA : –1 < θ1 ≤ 0, then, for example, the values of θ1 corresponding to ση2 above are (–0.905, –0.730, –0.382, –0.084). As ση2 → ∞, then θ1 → 0 and ∆yt → εt ; that is, yt is a pure random walk. As the reduced form representation of the model is probably the more familiar way of writing a time series model, the power calculations are presented in terms

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LBM(µ) with θ1 = 0.0

0.6

472 Unit Root Tests in Time Series

of a set of values of θ1 , but the translation back to ψ = ση2 /σς2 is simple enough (see, for example, Q11.1 and Q11.2).

Assessing the various tests by a power comparison has to take into account the complications caused by the difﬁculties posed in ﬁnding a combination of test statistic and bandwidth or lag selection criterion that maintains the empirical size of a test close to nominal size (see Lee, 1996). At the simplest level, power can be presented for the ‘standard’ versions of the tests; that is, those that assume no corrections are needed for short-range dependence. However, this situation is of limited practical relevance as it is untypical of economic time series, although it may act as a benchmark. More realistically, the test statistics in their adjusted versions will be used, with the implication that the empirical size may depart from the nominal size if standard critical values are used (see Tables 11.4 and 11.5). If, ceteris paribus, the errors are not weakly dependent, but a test statistic incorporates an adjustment mechanism for this case, there will be an advantage to the mechanism that leads to the least number of superﬂuous terms. Additionally, the test combinations differ quite markedly in how well size is retained in the presence of short-range dependence, with the implication that size-adjusted (s-a) power has to be carefully interpreted, especially if the DGP involves some serial correlation in addition to the unit root, because it does not reﬂect achievable power. To set a benchmark, power was calculated for the standard forms of the test statistics (for the LBM tests this assumes p = 1) in their versions using mean adjusted data, and T = 100. The DGP was given by ∆yt = εt + θ1 εt−1 , where θ1 ∈ [–1.0, –0.4], with the quantiles at θ1 = –1.0 providing the critical values for other values of θ1 ; power and s-a power was calculated for a nominal size of 5%. Power in this case is shown in Figure 11.15a and is the same as s-a power. The ranking in descending order of power is: MRS(µ) , KS(µ) , SBDH(µ) , KPSS(µ) , LBM(µ) ; and power is close to 100% for all tests at θ1 = –0.5. To illustrate the differences, power at θ1 = –0.8 is, in the same order as the ranking, 91.3%, 87%, 86.9%, 86.2% and 68.7%. The next comparison is of the different bandwidth selection criterion for each of the four semi-parametric tests. The results are shown in Figures 11.15b– 11.15e, where the comparison is in terms of s-a power as otherwise differences in size are not controlled. The ranking is the same across the four semi-parametric ˆ (qs) tests; that is, in terms of descending power: m(qs) (4), m(bw) (4), m cab (4) and (µ) with p = 1, p = 4, p(AIC) and ˆ (bw) m cab (4). A similar comparison, but for LBM Z(p), is shown in Figure 11.15f. As expected, p = 1 is best, but this is because it minimises the number of superﬂuous terms; next best is p(AIC), which is then used in an overall comparison with m(qs) (4) and the previous four tests. This

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11.5.4.i Power against a ﬁxed alternative

Tests with Stationarity as the Null Hypothesis 473

1 0.9

SBDH

(µ)

(µ)

MRS

KS

0.8 KPSS

0.7 0.6

power

LBM

(µ)

(µ)

(µ)

0.5 0.4 0.3

0.1 0 –1

–0.9

–0.8

–0.7

–0.6

–0.5

θ1 Figure 11.15a Power of test statistics in standard versions (T = 100).

1 0.9

MRS

(µ)

SBDH

(µ)

0.8

KS

(µ)

KPSS

(µ)

0.7 0.6

power 0.5 0.4 semi-parametric tests use m(qs)(4)

0.3 0.2 0.1 0

–1

–0.9

–0.8

–0.7

–0.6

–0.5

–0.4

–0.3

θ1 Figure 11.15b Size-adjusted power of test statistics (T = 100, semi-parametric tests use m(qs) (4)).

comparison is shown in Figure 11.16a. MRS(µ) is generally most powerful, with little to choose amongst the next three, and then LBM(µ) with p(AIC). However, MRS(µ) is not the dominant test because it is very sensitive to superﬂuous terms in the bandwidth and loses power quickly in the automatic procedures. This is a pertinent factor when power and size are brought together. Another comparison involves choosing the combinations that have good size properties for this sample size (or at least the best in the set, referring back to Tables 11.4 and 11.5) and then comparing power. For example, KPSS(µ) with

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0.2

474 Unit Root Tests in Time Series

0.8

(µ)

KPSS

0.7

SBDH

(µ)

0.6 KS

0.5

(µ)

power 0.4 0.3

(µ)

MRS

(qs)

semi-parametric tests use mcab

0.1 0

–1

–0.9

–0.8

–0.7

–0.6

–0.5

θ1 Figure 11.15c Size-adjusted power of test statistics (T = 100, semi-parametric tests use (qs) mcab ).

0.9 (µ)

MRS

0.8

(µ)

0.7

KPSS

SBDH

(µ)

KS

(µ)

0.6

power

0.5 0.4 0.3

semi-parametric tests use m

(bw)

(4)

0.2 0.1 0 –1

–0.9

–0.8

–0.7 θ1

–0.6

–0.5

Figure 11.15d Size-adjusted power of test statistics (T = 100, semi-parametric tests use m(bw) (4)).

(µ) with m (µ) with m (µ) with m ˆ (bw) ˆ (qs) ˆ (qs) ˆ (bw) m cab (4), MRS cab (4), KS cab (4), SBDH cab (4); to (µ) which is added LBM with p(AIC). The ranking (see Figure 11.16b) (which is the same whether power or s-a power is used) is now: LBM(µ) , KS(µ) , KPSS(µ) , SBDH(µ) , MRS(µ) . Notice also that, in contrast to the standard versions of the tests, power does not approach unity as θ1 → 0 from below. This shows how difﬁcult it is to choose a test (or combination) that has both good size and power properties.

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0.2

Tests with Stationarity as the Null Hypothesis 475

0.7 0.6

(µ)

KPSS

SBDH

0.5

(µ)

0.4

power 0.3 MRS

(µ) (bw)

semi-parametric tests use m cab (4) 0.1 0 –1

–0.9

–0.8

–0.7 θ1

–0.6

–0.5

Figure 11.15e Size-adjusted power of test statistics (T = 100, semi-parametric tests use (bw) mcab ).

1 0.9

p=1

p(AIC)

0.8 0.7

p = Z(p)

0.6

power 0.5

p=4

0.4 0.3 0.2 0.1 0 –1

–0.9

–0.8

–0.7 θ1

–0.6

–0.5

Figure 11.15f Size-adjusted power of LBM test statistics (T = 100).

As LBM (1994) note, a number of empirical studies suggest that the ARIMA(p, 1, 1) model, with p = 0, is often too simple to capture the key characteristics of the data; some consideration is therefore also given to the case with p = 1. The DGP is (1− ϕ1 L)∆yt = εt + θ1 εt−1 , with ϕ1 = 0.6 used for illustrative purposes. In this case, the QS kernel m(qs) (4), is now unequivocally the best of the ˆ (qs) bandwidth selection criteria considered here, and m cab (4) is the better of the two automatic procedures. On a size-unadjusted power basis, the choice of lag selection criteria is rather more marginal between p = 4 and p(AIC); however,

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0.2

(µ)

KS

476 Unit Root Tests in Time Series

1

SBDH(µ)

0.9

(µ)

KS(µ)

MRS 0.8

KPSS(µ)

0.7 LBM(µ) with p(AIC)

0.6 0.5

semi-parametric tests use m(qs)(4)

0.4 0.3 0.2 0.1 0 –1

–0.9

–0.8

–0.7

–0.6

–0.5

–0.4

–0.3

θ1 Figure 11.16a Best power combinations (T = 100).

0.9 0.8

LBM(µ) KS

0.7

(µ)

0.6 power

KPSS(µ)

0.5 0.4

SBDH

(µ)

0.3 (µ)

0.2

MRS

(For combinations see Figure 11.18b)

0.1 0 –1

–0.9

–0.8

–0.7

–0.6

–0.5

–0.4

–0.3

θ1

Figure 11.16b Power comparison with best size combinations (T = 100).

the latter is favoured as the former is oversized (size at θ1 = –1 is larger than the nominal 5%).

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power

Tests with Stationarity as the Null Hypothesis 477

1 0.9

MRS(µ) KS(µ) (µ)

SBDH

0.8 KPSS(µ)

0.7 0.6 power 0.5

LBM(µ) with p = 1

LBM(µ) with p(AIC)

0.3 semi-parametric tests use m(qs)(4)

0.2 0.1 0 –1

–0.9

–0.8

–0.7

–0.6

–0.5

–0.4

–0.3

θ1 Figure 11.17a Size-adjusted power, AR(1) DGP (T = 500, semi-parametric test use m(qs) (4)).

Generally, power is increased for all tests when the DGP is ARIMA(1, 1, 1). The s-a power is shown on Figure 11.17a, using the m(qs) (4) kernel for the semiparametric tests. The semi-parametric tests dominate the LBM version, with, as before, MRS(µ) the most powerful test on this basis, and power does approach 100% for this test as θ1 → − 0. Note that the power of the LBM tests dips noticeably at θ1 = – 0.6; this is because at this point there is a common factor in the AR(1) and MA(1) lag polynomials, and the DGP is just a random walk, against which the LBM tests are known not to be inconsistent, although they still have some power in ﬁnite samples. The power for the best combinations for size (as for Figure 11.16b) results in a different ranking, as shown on Figure 11.17b. In this case, MRS(µ) suffers most and is least powerful; the empirical (s-a) power functions for the other tests are bunched together for –1 < θ1 < –0.9, thereafter, KS(µ) and LBM(µ) are more powerful than the others. Finally, to gauge the sensitivity of the previous results to increasing the sample size, some of the variations are also illustrated for T = 500. Figure 11.18a graphs the semi-parametric tests using m(qs) (4), which correspond to those in Figure 11.16a (there for T = 100); LBM(µ) is also graphed for p = 1 and p = p(AIC). MRS(µ) is the most powerful of the tests, as in Figure 11.16a, with little to distinguish the remaining tests provided θ1 < –0.6 (and, as a result the other tests are not separately distinguished, except when possible), with 100% power achieved for all the tests when θ1 = – 0.85; however, note that the power of LBM(µ) , with

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0.4

478 Unit Root Tests in Time Series

0.8 LBM(µ) with p(AIC) (µ)

0.7

KS

0.6 0.5 power 0.4

KPSS(µ)

SBDH(µ)

MRS(µ)

0.2 0.1 0 –1

(For combinations see Figure 11.18b) –0.9

–0.8

–0.7

–0.6

–0.5

–0.4

–0.3

θ1 Figure 11.17b Best power and size, AR(1) DGP (T = 100).

p(AIC), declines to about 93% when θ1 = – 0.45, whilst the others tests maintain their power. A second comparison is shown in Figure 11.18b, and is analogous to Figure 11.16b, where power is presented for the combination that gives good size properties (the same combinations are used). Despite the later dip in power, LBM(µ) has relatively good power characteristics, being more powerful than its closest rivals, MRS(µ) and KS(µ) , until θ1 = –0.6. The tests KPSS(µ) and SBDH(µ) (bw) suffer a noticeable loss of power; these combinations suffer as they use mcab (4), which selects a bandwidth that is too long; however, in the case of T = 500, (qs) using mcab (4) gives comparable size properties, and when this criterion is used, (µ) KPSS and SBDH(µ) are comparable. Figures 11.19a and 11.19b are analogous to Figures 11.17a and 11.17b, respectively, but for T = 500, so that the DGP is ARIMA(1, 1, 1), with AR(1) coefﬁcient ϕ1 = 0.6. Referring to Figure 11.19a, MRS(µ) is again the most powerful test (when combined with m(qs) (4)), whereas the LBM tests are least powerful. As to the latter, it is evident that power again dips noticeably at θ1 = –0.6. Figure 11.19b indicates that making a choice also based on size characteristics (qs) still leaves MRS(µ) as the most powerful test (combined with mcab (4)) in this (µ) (µ) (µ) case), followed by KS ; KPSS and SBDH lose power substantially when (bw) (qs) used with mcab (4); however, power is restored when mcab (4) is used, as shown in Figure 11.19c, with MRS(µ) still the best of the tests. 11.5.4.ii Power against local alternatives A related study of particular interest is that by Cappuccio and Lubian (CL) (2006), who undertook a comparison of power in the nearly-stationary case,

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0.3

Tests with Stationarity as the Null Hypothesis 479

1 0.9 MRS(µ)

0.8

LBM(µ) with p(AIC)

0.7 (remaining tests not distinguishable)

0.6 0.5

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power

0.4 T = 500

0.3 0.2

semi-parametric tests use m(qs)(4)

0.1 0

–1

–0.9

–0.8

–0.7 θ1

–0.6

–0.5

Figure 11.18a Best power combinations (T = 500).

1 0.9 KPSS(µ)

0.8

MRS(µ)

KS(µ)

LBM(µ)

0.7 SBDH(µ)

0.6 power

Combinations:

0.5

KPSS

0.4

T = 500

MRS

(µ)

(µ)

0.3

KS

0.2

SBDH

0.1

LBM

(µ)

(qs)

with m(cab)(4) (qs)

with m(cab)(4) (µ)

0 –1

–0.9

–0.8

–0.7

–0.6

–0.5

(bw)

with m(cab)(4)

(µ)

(bw)

with m(cab)(4)

with p(AIC)

–0.4

–0.3

θ1 Figure 11.18b Power comparison with best size combinations (T = 500).

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480 Unit Root Tests in Time Series

1 0.9 0.8

(µ)

MRS

LBM

(µ)

with p = 1

0.7 0.6 power 0.5

LBM

0.4

(µ)

T = 500 (qs)

semi-parametric tests use m

0.2

(4)

(other tests not distinguishable)

0.1 –0.9

–0.8

–0.7

–0.6

–0.5

–0.4

–0.3

θ1

Figure 11.19a Size-adjusted power, AR(1) DGP (T = 500, semi-parametric tests use m(qs) (4)).

1 0.9 0.8

KS

(µ)

(µ)

MRS

(µ)

LBM with p(AIC)

0.7 0.6 SBDH

KPSS

(µ)

(µ)

power 0.5 0.4 0.3

T = 500

0.2 (Combinations as Figure 11.18b)

0.1 0 –1

–0.9

–0.8

–0.7

–0.6

–0.5

–0.4

–0.3

θ1

Figure 11.19b Best power and size, AR(1) DGP (T = 500).

which is the analogue of the nearly-integrated case in unit toot testing. This is a ‘local’ power comparison, with a sequence of local stationary alternatives (rather than the ﬁxed alternative framework of the previous section). Referring back to the structural time series representation then under the null ηt = 0, which implies ση2 = 0, whereas under the alternative ηt = 0, which implies ση2 = 0 and ψ = ση2 /σς2 = 0; hence the idea is to let there be a function such that ηt will tend to zero with T and so be asymptotically stationary. Initially assume that there is no weak dependence in ςt , so that (trivially) ςt = vt , where vt is iid(0, σv2 ).

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0.3

0 –1

with p(AIC)

Tests with Stationarity as the Null Hypothesis 481

1

(µ)

KS

0.9

(µ)

MRS

0.8 SBDH

0.7

(µ)

KPSS

(µ)

LBM

(µ)

with p(AIC)

0.6 power 0.5 0.4 T = 500 0.3

0.1 0

–1

–0.9

–0.8

–0.7

–0.6

–0.5

–0.4

–0.3

θ1

Figure 11.19c Best power and size, AR(1) DGP (T = 500, semi-parametric tests use (qs) mcab (4)).

√ √ Then let ηt = ( ψ)σv ht , where ht is iid(0, σh ), σh = 1, and let ψ = T−1 c, c ≥ 0, where the parameter c controls the near-stationarity of yt , so that ηt = T−1 cσv ht . For a given T the importance of the nonstationarity component increases with c, but as T grows the process is stationary: yt is a nearly-stationary series. In the case that ςt is not iid, but a linear process, say ςt = b(L)vt , where b(L) is deﬁned √ above (see section 11.1.3), then σς = b(1)σv . In this case, let ψ = T−1 cb(1), so that ηt = T−1 c[b(1)σv ]ht , and then ηt is controlling the number of units of the long-run standard deviation of ςt . CL (2006) consider three rules for determining the bandwidth: m = 0, which is the correct value under iid errors; m(j) = [j(T/100)1/4 ], for j = 4, 12; and the automatic lag selection method due to Andrews (1991), where the bandwidth is capped at 2 whenever it exceeded T0.7 to ensure that the resulting test statistic is consistent under the alternative hypothesis (see also Choi and Ahn, 1999); ˆ (And) this is referred to as m cab . Under this local-to-stationarity framework, CL (2006) considered the four semi-parametric tests of the previous sections (an LM-type test was also considered but is omitted here as it was markedly inferior to the other tests). With the rules m = 0 and m(4) = [4(T/100)1/4 ], CL found that MRS(µ) is generally more powerful than the other tests but, as noted earlier, it loses power quickly when the bandwidth is over-parameterised relative to the sample size, as with m(12) (µ) tends to have ˆ (And) and m cab . When the tests are combined with m(12), KPSS greater power for small and moderate sample sizes (T = 50, 100) and to be a bit more powerful with T = 200; when T = 500, MRS(µ) becomes competitive again. However, in all cases power increases slowly with the sample size, as was also evident from the ﬁgures presented in section 11.5.4, and can be a long way from

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(qs)

semi-parametric tests use mcab (4)

0.2

482 Unit Root Tests in Time Series

100% as θ1 → 0 from below, or as c increases in the local-to-stationary framework (for example, c = 60); for example, even with T = 500 and c = 60, power is around 80–90%.

To illustrate the combined application of tests of this and the previous chapters, the methods are applied to a sample of monthly observations on some US consumer price indices (CPI). The ﬁrst illustration considers the aggregate series, whereas the second illustration considers the relation between the CPIs in two regions of the US. 11.6.1 US CPI (aggregate series) In its aggregate form, the US CPI has been used in a number of studies; for example, LBM (1999) and Schwert (1987). The notation is as follows: yt refers to the log of the original series and xt ≡ ∆yt . The latter is important because the empirical analysis suggests that the tests should also be applied to the rate of inﬂation of the CPI. It is important to note the nature of the alternative hypothesis in applying tests for stationarity. If the reduced form ARIMA (p, 1, 1) model, p ≥ 0, is to be related back to an underlying local-level structural model, with a random walk component, it must be the case that θ1 lies in the admissible region –1 ≤ θ1 ≤ 0, otherwise the implication is that ψ, interpreted as the signal-to-noise ratio in the structural model, is negative. The time series for yt and xt are graphed in Figures 11.20 and 11.21, from which it is evident that if yt is stationary, then it is best characterised as trend stationary (the estimated linear trend is also shown on the ﬁgure); whereas if xt (shown as the monthly log difference at an annualised rate) is stationary, then it does not appear trended (the sample mean inﬂation rate of 3.75% p.a. is shown as a horizontal line). The sample period for estimation is 1950m1 to 2006m12, so that T = 684. The ﬁrst stage in the testing procedure for the LBM test statistic, in this case LBM(β) , is to apply an AR(p) ﬁlter to yt from estimation of the following regression: ∆yt = β1 + ∑i=1 ϕi ∆yt−i + εt + θ1 εt−1 p

(11.56)

To illustrate, we start with an ARIMA (2, 1, 1) model, where p = 2 was selected by p(AIC), with estimates as follows: ∆yt = 0.00005 + 1.236∆yt−1 − 0.252∆yt−2 + ε˜ t − 0.850˜εt−1 It is evident from this estimation that ∑2i=1 ϕ˜ i = 0.984, implying that one of the roots is close to unity; indeed, the roots are 3.88 and 1.02, so the polynomial

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11.6 Illustrations: applications to some US consumer price indices

Tests with Stationarity as the Null Hypothesis 483

5.5

5

4.5

4

3.5

3

2.5 1946 1951 1956 1961 1966 1971 1976 1981 1986 1991 1996 2001 2006 Figure 11.20 US CPI and linear trend (logs).

0.25 0.2 0.15 0.1 ∆yt 0.05

sample mean

0 –0.05 –0.1 –0.15 1946 1951 1956 1961 1966 1971 1976 1981 1986 1991 1996 2001 2006 Figure 11.21 US CPI inﬂation rate.

factors to (1 – 0.2756L)(1 – 0.9785L). In this situation it is sensible to consider yt ∼ I(1) and the unit root could be incorporated at this stage, implying that the empirical analysis is in terms of ∆yt , with the ﬁltering regression in terms of ∆2 yt . Both are illustrated and the conclusions drawn together below. The estimated

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yt

484 Unit Root Tests in Time Series

AR(1, 1, 1) model for the inﬂation rate, xt , was (note that the AR polynomial order is reduced by one to allow for the unit root): ∆xt = 0.282∆xt−1 + ε˜ t − 0.882˜εt−1 The ﬁltered series, y˜ ∗t and z˜ ∗t , are, respectively: y˜ ∗t = yt − (1.236yt−1 − 0.252yt−2 )

The ﬁltered series are then regressed on the deterministic components, which are a constant and a linear time trend for LBM(β) , and a constant for LBM(µ) , in the cases of yt and xt , respectively: Considering the tests for the stationarity of yt ﬁrst, the appropriate null (β) hypothesis is that of trend stationarity, that is H0 . The value of LBM(β) is 4.03 (see Table 11.6a) which, compared to the ‘standard’ 95% quantile of 0.146, indicates clear rejection; this quantile assumes that the AR correction has sufﬁced to restore the distribution of the test statistic to the stationary case. To assess robustness of the quantiles the null was imposed, that is θ1 = –1, and the value of the estimated AR coefﬁcients, ϕ˜ 1 = 1.62011 and ϕ˜ 2 = 0.62178, were used in the DGP to simulate the empirical distribution of LBM(β) ; the proximity to the unit root is evident and is likely to lead to an increased right-tailed skew to the null distribution. The resulting 95% quantile was 2.43 which, whilst clearly much greater than the standard quantile, still suggests rejection of the null hypothesis. The other test statistics are reported in Table 11.6b. The bandwidth parameter ˆ (qs) ˆ (bw) is alternately selected by mqs (j), m(bw) (j), m cab (j) and mcab (j), for j = 4 and j = 12; the ﬁrst two being the deterministic rules and the latter two being the automatic selection method, and the kernels are QS and BW. The (log) level of the series is considered ﬁrst. There is a substantial variation in the values of each test statistic, depending on the bandwidth parameter and the kernel estimation method (see Table 11.6b for the test statistics and Table 11.7 for the bandwidth parameter). Generally, the test statistics indicate rejection of stationarity with reference to the standard 95% quantile, but the substantial serial correlation indicated by the near-unit root in the ML estimation reported for calculation of the LBM test, implies that most of these tests will be oversized (see Figure 11.7). Turning next to the inﬂation rate, xt , the appropriate null hypothesis is now (µ) (β) H0 ; that is, ‘level’ stationarity (of the inﬂation rate), rather than H0 , as a trend in the inﬂation rate seems implausible under the null of stationarity. The test statistics therefore have a µ superscript. The sample value of LBM(µ) is 2.71, which exceeds the ‘standard’ 95% quantile of 0.462, suggesting rejection. As to the other test statistics, the general pattern is that they exceed the standard 95% quantile; the exception being when the test statistics are combined with ˆ (bw) (j). m

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x˜ ∗t = xt − 0.282xt−1

Tests with Stationarity as the Null Hypothesis 485

Table 11.6a LBM tests, US CPI (logs). yt LBM(β) LBM(µ)

xt

95% quantile

2.71

0.146 0.462

4.03

yt KPSS(β)

QS

QSCAB

BW

BWCAB

95% quantile

j=4 j = 12 j=4 j = 12 j=4 j = 12 j=4 j = 12

1.81 0.62 4.45 2.59 2.47 1.44 1.90 0.65

0.74 0.29 2.84 1.79 1.58 0.99 0.77 0.31

1.36 0.49 3.86 2.30 2.15 1.28 1.43 0.51

0.18 0.15 1.37 1.29 0.77 0.72 0.18 0.16

0.146

KPSS(β) MRS(β) MRS(β) KS(β) KS(β) SBDH(β) SBDH(β) xt KPSS(µ) KPSS(µ) MRS(µ) MRS(µ) KS(µ) KS(µ) SBDH(µ) SBDH(µ)

j=4 j = 12 j=4 j = 12 j=4 j = 12 j=4 j = 12

1.14 0.49 3.21 2.10 2.07 1.35 1.13 0.49

0.62 0.31 2.35 1.66 1.52 1.07 0.61 0.31

0.94 0.42 2.90 1.95 1.87 1.26 0.93 0.42

0.23 0.19 1.43 1.29 0.93 0.83 0.23 0.18

0.462

1.622 0.900 0.112

1.750 1.360 0.248

Table 11.7 Long-run variance estimates and bandwidth parameters. QS

QSCAB

BW

BWCAB

j=4 0.312 (12)

0.106 (4) 0.656 (26)

0.261 (10) 0.140 (6) 0.396 (19)

1.091 (59) 1.268 (71*)

j=4 j = 12

0.366×10−4 (4) 0.855×10−4 (12)

0.678×10−4 (9) 0.136×10−3 (6)

0.447×10−4 (6) 0.981×10−4 (19)

yt j = 12 xt

0.183×10−3 (52) 0.225×10−3 (71*)

0.65 ˆ (bw) Notes: * indicates constrained m ]; bandwidth parameters are shown in (.) parentheses; see cab (j) = int[T section 11.4.1 for details of the estimation procedure.

To assess the robustness of the LBM test statistics to the order of the AR(p) correction, the value of p was increased. (In the case of yt , lag selection by Z(p)

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Table 11.6b Stationarity tests, US CPI (logs).

486 Unit Root Tests in Time Series

led to p = 4.) The values of LBM(β) for p = 3 and p = 4 were 7.36 and 7.32, respectively, compared to LBM(β) = 4.03 for p = 2; thus, increasing p by one results in a more stable test statistic. In the case of xt , setting p = 2, and p = 3 as selected by Z(p), resulted in LBM(µ) = 2.71 and 2.85, respectively.

This section illustrates the combined use of tests for stationarity and nonstationarity. Busetti, Fabiani and Harvey (BFH) (2006) consider the joint use of tests for stationarity and nonstationarity in the context of testing for (pairwise) convergence and stability of regional price levels and inﬂation rates in Italy. This is a theme also taken up in the next chapter, where it is argued that constructing a conﬁdence interval is the best way of summarising the information about the stationarity/nonstationarity issue. For now, however, the combination of the two types of test is illustrated in use. 11.6.2.i Framework for tests The general framework is as follows. Let yt = y1t − y2t , where yit is a nonstationary time series; for example, the yit could be the time series on the price of the same good, or the price index of a bundle of goods, in different geographic locations and, therefore, if yit is in logs, then yt is the log of relative prices. Also likely to be of interest is ∆yt = ∆y1t − ∆y2t , which is the differential in inﬂation rates. The variables yt and ∆yt are often referred to as ‘contrasts’; for example, if yit is the price level of the i-th region, then yt is a price contrast and ∆yt is an inﬂation contrast. Other applications include GDP in different regions or states within a country or between countries. The economic concepts of interest are convergence and stability and they can be applied to levels (or indices) and rates; for example, to price levels and inﬂation rates or to GDP levels and growth rates. BFH (2006) suggest that stationarity tests can be used to assess stability, whereas unit root tests can be used to assess convergence. The idea is that if yt is a stationary process, then there is a stable relationship between y1t and y2t , whereas rejection of the null hypothesis of stationarity implies that y1t and y2t have not settled to a stable relationship; if each yit is I(1) then stationarity implies that y1t and y2t are cointegrated with cointegrating vector (1, –1), and a cointegrating test would be an alternative way of testing for stability. On the other hand, convergence is part of a dynamic adjustment, where the limiting result of a convergent process is a stable relationship. Thus it is possible that y1t and y2t are converging as part of an asymptotically stationary process, although yt is not yet (strictly) stationary. Convergence (to a constant) is an asymptotic property captured by the following deﬁnition. The process generating yt is said to be convergent (here to a

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11.6.2 US regional consumer prices: using tests of nonstationarity and stationarity in testing for convergence

Tests with Stationarity as the Null Hypothesis 487

constant) if: lim E(yt + τ |yt , yt−1 , . . .) = µ τ > 0

τ→∞

(11.57)

Convergence is absolute if µ = 0 and relative if µ = 0. A possible model for convergence is the familiar AR(1) with mean µ: yt − µ = ρ(yt − µ) + zt zt = εt

(11.58a)

The null hypothesis corresponding to no convergence is in essence just that of the unit root, H0 : ρ = 1, whereas the alternative is HA : |ρ| < 1, which implies asymptotic stationarity. Taking y1t as the ‘base’ region (or country) and y2t as a contrast region (or country), a rejection of the unit root hypothesis for yt implies convergence of y1t and y2t , and non-rejection of the hypothesis of stationarity implies stability; together they imply convergence to a stable relationship between the base and contrast regions; rejection of the unit root combined with rejection of the stationarity hypothesis implies convergence is taking place, but has not yet achieved a stable relationship. There are, of course, a number of other possibilities and these are enumerated in BFH (2006, especially Figure 2). 11.6.2.ii Consumer prices for two regions of the US The methodology is illustrated with data for the US consumer price indices for the two regions North-East and West; for present purposes, the logs of these series are denoted y1t and y2t , respectively. The data are monthly for the period 1987m1 to 2007m5, so that T = 245 and the common base period is 1984 = 100, (quarterly data is available for a slightly longer period). The individual series, y1t and y2t , and their contrast, yt = y1t − y2t , are plotted in Figures 11.22 and 11.23, respectively. From the ﬁrst of these ﬁgures, it seems that, in general terms, the series follow a similar trend and each is consistent with being generated by an I(1) process; the second ﬁgure suggests that the stationarity of yt is not as clear. The two inﬂation rates, ∆y1t and ∆y2t , and their contrast, xt ≡ ∆yt = ∆y1t − ∆y2t , are shown in Figures 11.24 and 11.25, respectively (as annualised log monthly differences). These ﬁgures are broadly suggestive of an inﬂation rate contrast that is serially correlated, but stationary. A unit root test on the components of yt ﬁrst assesses whether each is consistent with being an I(1) process. If a linear trend is ﬁtted over the whole period, each of y1t and y2t is well below trend in the early part of the sample and, as a result, the initial condition, is relatively large; however, there is a strong visual indication that the trend is split (about 1991), which will have an effect on the initial condition. The estimated scaled initial condition is denoted ξˆc,i for the i-th series. This point can be illustrated by moving the start of the sample forward and assessing its effect on the initial condition and the test statistics, τˆ β ,

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(11.58b)

488 Unit Root Tests in Time Series

5.5 5.4 5.3 5.2

y1t: North-East Region

5

y2t: West Region

4.9 4.8 4.7

1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Figure 11.22 US regional CPIs (logs).

0.055 0.05 0.045 0.04 0.035 yt

0.03 0.025 0.02 0.015 0.01 0.005

1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Figure 11.23 CPI contrast (log).

glsc

τˆ β and τˆ DFG , as in Chapter 7, section 7.4.10. The unit root tests are reported β in Table 11.8. Although starting the sample at the beginning 1987m1 (t = 1) does result in non-rejection of the unit root null for both series, more secure

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yit 5.1

Tests with Stationarity as the Null Hypothesis 489

0.15

0.1

0.05

0

–0.05

–0.1

North-East West

1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Figure 11.24 Regional CPI inﬂation rates.

0.1

0.05

0 ∆yt –0.05

–0.1

–0.15

1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 2008 Figure 11.25 Contrast of inﬂation rates.

inference as to their likely nonstationarity is obtained by starting the sample at 1991m1 (t = 48).

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∆yit

490 Unit Root Tests in Time Series

Table 11.8 Unit root tests on US regional CPIs; (log) levels, N-E and W. Start t =

1

12

24

36

48

5% cv

–2.93 –2.87 –0.49 –2.14

–3.31 –3.55 –0.62 –2.77

–3.73 –3.79 –0.80 –3.19

–3.60 –3.50 –1.08 –2.93

–0.81 –1.59 –1.62 –1.61

–3.44 –2.64 –3.13

–2.85 –2.47 –0.54 –1.85

–3.08 –3.14 –0.69 –2.42

–3.93 –3.86 –0.81 –3.32

–4.43 –4.45 –1.06 –3.87

–1.11 –2.53 –2.22 –2.33

–3.44 –2.64 –3.13

–2.36 –8.61 –1.89 –5.99

0.30 –8.50 –7.33 –7.45

0.26 –8.30 –7.44 –7.52

1.84 –8.40 –2.10 –5.38

0.52 –7.99 –5.38 –5.96

–2.88 –2.08 –2.33

–2.05 –2.56 –0.69 –1.74

–1.87 –2.16 –0.78 –1.51

–0.49 –1.61 –1.45 –1.48

–0.22 –1.52 –1.50 –1.50

–0.16 –1.44 –1.46 –1.46

–2.88 –2.08 –2.33

ξˆc,y1 τˆ β glsc τˆ β τˆ DFG β y2t ξˆc,y 2

τˆ β glsc τˆ β τˆ DFG β xt

ξˆc,x τˆ µ glsc τˆ µ τˆ DFG µ yt

ξˆc,y τˆ µ glsc τˆ µ τˆ DFG µ

Notes: The test statistics are based on an ADF lag of 3 as indicated by marginal-t selection and MAIC.

Conditional on y1t and y2t each being I(1), a unit root test on the inﬂation contrast xt should result in rejection of the null hypothesis (as ∆yit ∼ I(0) is implied by yit ∼ I(1)); non-rejection of this null would, anyway, end the testing process. In this case, the testing framework does not include a trend and may not necessarily include a constant; in any case, the results are virtually indistinguishable and are reported for the ‘with constant’ case in Table 11.8 for xt . The glsc unit root is rejected except for τˆ µ starting at t = 1, which is associated with the relatively large initial condition. The next stage is a unit root test on the levels; that is, on the price contrast, yt ; rejection of this test would lead to a stationarity test on yt to see whether a converging process had converged, whereas non-rejection leads to a stationarity test on xt to see whether the ﬁrst differences have converged. Referring to the last part of the table, the unit root tests uniformly indicate non-rejection. (This is also the case for the BFH variation of the DF test, τˆ µ , which uses the last

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y1t

Tests with Stationarity as the Null Hypothesis 491

LBM(µ)

p=1

p=4

p(AIC)

Z(p)

Z(p) + ML

t=1 t = 12 t = 24 t = 36 t = 48 95% cv

0.16 0.11 0.05 0.05 0.07 0.47

0.33 0.20 0.09 0.09 0.16 0.75

0.31 0.15 0.11 0.11 0.12 0.54

0.31 0.21 0.11 0.11 0.17 0.60

0.31 0.21 0.11 0.11 0.16 0.61

Note: 95% quantiles from Table 11.3.

complete year of the sample, rather than the complete sample, to estimate the mean, with values of the test statistic that differ only marginally from those reported.) Given non-rejection of the unit root with the series yt , the next stage is therefore stationarity testing on xt . The relevant LBM test is LBM(µ) . The ﬁrst stage is to estimate the coefﬁcients to ﬁlter the series xt into x∗t , which is then the basis for the KPSS test. Both p(AIC) and z(p) led to a lag of 3 for t = 1, so that the initial model, in this case, was ARIMA(3, 1, 1); and, of note, the estimated value of θ1 was –0.952, which is close enough to –1 to be suggestive of stationarity. The value of LBM(µ) was 0.306, which is below the standard 95% quantile of approximately 0.47; other variations – for example, moving the start date for the sample period forward – are reported in Table 11.9. The results of other test statistics are reported in Table 11.10. The magnitude of the initial condition was not indicative of potential problems with an interpretation of the test statistics for stationarity. The balance of the stationarity tests favour non-rejection of the null for xt , especially as the sample starting date is moved forward. The various versions of KPSS(µ) and KS(µ) suggest non-rejection, whereas the decision is more marginal with MRS(µ) and SBDH(µ) (although not for the later sample starts). Overall, the testing sequence suggests that the inﬂation rates have converged, but that the price levels have not. Of course, in any sequence of tests that are not completely dependent, the type I error cumulates and is not the p-value associated with a single test.

11.7 Concluding remarks In contrast to a test for a unit root, a test for stationarity is not naturally a test that a parameter takes a single value. For example, suppose that the data are generated by yt = ρyt−1 + εt , where εt is stationary; if |ρ| < 1, then yt is (asymptotically) stationary and, thus, for example, the null hypothesis that ρ ∈ (–1, 1)

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Table 11.9 US regional inﬂation contrast, xt , LBM(µ) tests for stationarity.

492 Unit Root Tests in Time Series

KPSS(µ) t=1 t = 12 t = 24 t = 36 t = 48 MRS(µ) t=1 t = 12 t = 24 t = 36 t = 48 KS(µ) t=1 t = 12 t = 24 t = 36 t = 48 SBDH(µ) t=1 t = 12 t = 24 t = 36 t = 48

No lags

mqs (j)

ˆ (qs) m cab (j)

m(bw) (j)

(bw) ˆ cab m (j)

95% quantile

0.25 0.19 0.11 0.10 0.14

0.42 0.30 0.17 0.14 0.17

0.33 0.24 0.14 0.12 0.14

0.37 0.27 0.15 0.13 0.16

0.29 0.22 0.14 0.13 0.14

0.46

1.54 1.47 1.50 1.50 1.78

2.15 1.85 1.88 1.84 2.00

1.89 1.66 1.71 1.68 1.82

2.02 1.76 1.80 1.76 1.93

1.79 1.58 1.70 1.76 1.77

1.66

0.89 0.75 0.88 0.91 1.10

1.16 0.94 1.10 1.11 1.23

1.02 0.84 1.00 1.01 1.12

1.09 0.89 1.05 1.06 1.19

0.97 0.80 1.00 1.06 1.09

1.30

0.22 0.18 0.11 0.09 0.13

0.38 0.28 0.17 0.13 0.16

0.29 0.23 0.14 0.11 0.13

0.33 0.25 0.15 0.12 0.15

0.26 0.21 0.14 0.12 0.12

0.25

is of interest in testing for stationarity, whilst ρ = 1 characterises the unit root hypothesis. In order to ‘capture’ the stationarity hypothesis into a single parameter, one approach is to start from ∆yt = (1 + θ1 L)εt , then θ1 = –1 implies that there is a common factor in the AR and MA lag polynomials; cancelling the common factor results in yt = εt , which is stationary. The null hypothesis of stationarity is therefore θ1 = –1. This set-up is easily generalised to allow for deterministic terms and more complex dynamics; for example, consider ϕ(L)∆yt = (1 + θ1 L)εt , then θ1 = –1 implies ϕ(L)yt = εt . An interpretation of this approach is that a test can be based on whether over-differencing has occurred inducing a common term in the differencing polynomial (see, for example, Tanaka, 1990; McCabe and Leybourne, 1998). A complementary approach is to exploit the duality between a structural time series interpretation and the (second order in moments) equivalent reduced form. In the former, the time series is viewed as being built up from components; for example, an unobservable level plus an irregular component, with variance

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Table 11.10 US regional inﬂation contrast, xt , tests for stationarity.

σς2 , where the level is evolving over time potentially subject to a random element, with variance ση2 . The ratio of ση2 to σς2 is the signal (of the level)-to-noise (of the irregular component) ratio and this can be shown to link to the MA(1) coefﬁcient in the reduced form representation of the model, such that a value of zero for the signal-to-noise ratio implies θ1 = –1. Given σς2 = 0, the hypothesis of stationarity can be expressed as ση2 = 0, which forms another basis for a test of stationarity There are a number of tests of stationarity built upon the observation that the partial sum process of a stationary series behaves quite differently from that of a nonstationary series, so that in some unit of measurement the growth of a nonstationary series should exceed that of a stationary series. Thus such tests are right-tailed, with large values of the test statistic leading to rejection. A generic problem with such tests is that a time series may well be quite persistent but nevertheless stationary; indeed, this is more likely to be the case than not with economic time series, and tests for stationarity tend to be oversized as the persistence increases. This problem is usually dealt with by estimation of the long-run variance by semi-parametric methods, which takes the persistence into account. Alternatively, Leybourne and McCabe (1994, 1999) suggest dealing with the persistent but short-run dynamics in a ﬁrst stage involving standard AR modelling, and then using the ﬁltered series to test for stationarity. A cost of the latter procedure is that it leads to a test that is not consistent against a pure random walk, although such a process is unlikely to characterise many economic time series. The simulation evidence presented here and elsewhere, highlights the extent of oversizing as the degree of persistence approaches that of a process with a unit root. This can be (partially) controlled in the semi-parametric tests by the choice of kernel and the bandwidth, although the different tests have different characteristics. However, the cost to power can be substantial. Indeed, whilst the tests in their ‘standard’ form have power that approaches 100% as θ1 → 0 from below, this falls away to noticeably less than 100% once the tests are adjusted for possible short-run persistence, even for reasonably large sample sizes (for example, T = 500; Cappuccio and Lubian, 2006). This suggests that the subject is far from closed. Some avenues of further development include: as in the case of unit root tests (Elliott, Rothenberg and Stock, 1996), modifying the stationarity tests to estimate the deterministic components more efﬁciently along GLS lines; and using bias-adjusted AR approximations to estimation of the long-run variance. The next chapter continues the development of ideas in this chapter, including alternative approaches to the dichotomy of tests for stationarity versus tests for nonstationarity.

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Tests with Stationarity as the Null Hypothesis 493

494 Unit Root Tests in Time Series

Questions Q11.1 Given ση2 = (0.01, 0.1, 1.0, 10) and σς2 = 1, conﬁrm that θ1 = (−0.905, −0.730, −0.382, −0.084), and write the general form of the implied ARIMA model. A11.1 Note that: (11.59)

ψ = ση2 /σς2 = − (1 + θ1 )2 /θ1

(11.60)

where the coefﬁcient ψ is the signal-to-noise ratio. As σξ2 = 1, ψ = ση2 /1 = ση2 and (11.59) is then solved for θ1 ; for example, ση2 = 1 (so that ψ = 1) gives: θ1 = {[1(1 + 4)]1/2 − (1 + 2)}/2 = {[5]1/2 − (3)}/2 = − 0.382 This implies the ARIMA(0, 1, 1) model: ∆yt = β1 + εt − 0.382εt−1 The general form of which is: ∆yt = β1 + εt + θ1 εt−1 Q11.2 Given θ1 = (–0.8, –0.6, –0.4, –0.2, –0.1), ﬁnd the corresponding values of ψ in the structural time series model. A11.2 In this case (11.60) is used. For example, if θ1 = –0.8, then: ψ = − (1 + θ1 )2 /θ1 = (0.2)2 /0.8 = 0.05 In a similar way, θ1 = (–0.8, –0.6, –0.4, –0.2, –0.1) ⇒ ψ = (0.05, 0.27, 0.9, 3.2, 8.1). Q11.3 Show that the local-level model given by: yt = µt + ςt µt = β1 + µt−1 + ηt

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θ1 = {[ψ(ψ + 4)]1/2 − (ψ + 2)}/2

Tests with Stationarity as the Null Hypothesis 495

can be represented as: yt = µ0 + β1 t + ∑i=1 ηi + ςt t

and interpret this representation (the text equations are (11.4a), (11.4b) and (11.6), respectively).

µ1 = β1 + µ0 + η1 y1 = µ1 + ς1 = β1 + µ0 + η1 + ς1 µ2 = β1 + µ1 + η2 = β1 + β1 + µ0 + η1 + η2 y2 = 2β1 + µ0 + η1 + η2 + ξ2 .. . yt = β1 t + µ0 + ∑i=1 ηi + ςt t

This structure is familiar as it contains a stochastic trend component ∑ti=1 ηi and a drift component β1 that cumulates into a deterministic trend term, β1 t. Notice that µ0 can regarded as an intercept when yt is plotted as a function of t. (Note that we could equally write µt = β0 + β1 t, and therefore µ0 = β0 .) Hence, unless it is known that µ0 = 0, the data should either be demeaned or a constant included in any regression-based representations. Q11.4 Consider the ARIMA(0, 1, 1) model of Q11.2: what is special about the cases θ1 = 0 and θ1 = –1? A11.4 The limits of θ1 = 0 and θ1 = –1 are of interest because the former corresponds to the limit ψ → ∞ and the latter to ψ = 0. In the former case, the variance of ηt is inﬁnitely large compared to the variance of ςt , and the random walk component is dominant; whereas in the latter case, all ηt = 0, resulting in ∆µt = β1 , so that there is no random walk component; equivalently, refer to the answer to Q11.3 to observe that the stochastic trend is absent from yt . The interpretation in terms of the reduced form model is particularly simple as the two cases correspond to ∆yt = β1 + εt , a random walk (with drift if β1 = 0) and ∆yt = β1 + ∆εt ⇒ yt = ∆−1 β1 + εt , a stationary series, respectively. Q11.5 Why do tests for stationarity tend to over-reject when the errors are weakly dependent in the form of, say, an AR(1) process? What can be done to mitigate the oversizing?

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A11.3 Start the process with µ0 given, then sequentially generate µt and substitute it into yt for t = 1, 2, . . . ; thus:

A11.5 If ςt = ρςt−1 + vt , where vt is iid, then the series yt will look more like a nonstationary series as ρ → 1; and in the limit ρ = 1, so there is a unit root. In practice, economic time series tend to exhibit weak dependence in the errors, representing the presence of dynamic reactions and the persistence of shocks; rarely is it the case that empirical modeling will suggest that ρ is close to zero. Thus, without any adjustments, a test for stationarity will tend to reject the null because of the presence of persistent shocks, even though such shocks are stationary. Lo’s (1991) modiﬁcation of the rescaled range statistic was motivated by the empirical observation that asset returns could have substantial ‘shortrange’ dependence as well as ‘long-range’ dependence. Indeed, once the shortrange dependence is taken into account, long-range dependence is often harder to ﬁnd. The ‘solution’ to the problem of incorrect sizing is to modify the test statistic to make it robust to short-range, or weak, dependence. The generic solution to a number of tests for stationarity is to scale by the ‘long-run’ variance (or standard deviation), where the latter embodies an adjustment for serial correlation in the errors. However, this adjustment is not ‘foolproof’ and different tests have different size characteristics as ρ → 1 (see, especially, Figure 11.7).

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496 Unit Root Tests in Time Series

12

Introduction The previous chapter outlined a number of tests for stationarity, but left some problems to be considered. The ﬁrst issue parallels the problem with unit root tests, where the power of the tests was known to be sensitive to the initial condition. In the case of tests of stationarity, the roles are reversed so that the initial condition becomes important to the null hypothesis, with the problem now being that size is dependent on the initial condition. This problem is considered in section 12.1. Sections 12.2 and 12.3 address some of the problems in applying stationarity tests to persistent series. As persistence seems likely in macroeconomic time series, the question of how close a root is unity can be quite critical in terms of the appropriate analytical framework. In a local-to-unity framework, ‘close’ is ρ = 1 + c/T < 1, so that c = −T(1 − ρ) with, for example, 0 ≤ –c < 20 being regarded as close; thus, if T = 200, then ρ = 1 – 20/200 = 0.9, whereas if T = 500, then ρ = 1 – 20/500 = 0.96. M¨ uller (2005) notes from previous studies that estimates of T(1 − ρ) are frequently in this region, so that the size and power properties of particular tests have to be quite robust to be useful. The ‘weak’ spot in tests for stationarity is the need to adjust the scale of the test statistic to account for serial correlation, which is usually achieved by dividing a measure of the magnitude of ﬂuctuations in the times series by the long-run error variance or its standard deviation. Whilst traditional large-T asymptotics suggest that the bandwidth employed in an estimator of the long-run variance can be op (T1/2 ), or o(T1/2 ) if a deterministic rule is applied, local-to-unity asymptotics suggests that it needs to be Op (T2 ), and even then that may not sufﬁce, depending on the structure of the estimator, to provide a well-behaved test statistic. The local-to-unity asymptotic analysis explains some of the simulation results in the literature in general and in Chapter 11 in particular.

497

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Combining Tests and Constructing Conﬁdence Intervals

Given these problems, rather than design a test for stationarity based in some way on a measure of the relative ﬂuctuations of stationary and nonstationary time series, M¨ uller (2005) suggests using the point-optimal testing framework, which was outlined in Chapter 7, simply reversing the roles of the null and alternative hypotheses, with ‘large’ values of the same test statistics leading to rejection of the null hypothesis of stationarity. Section 12.2 is concerned with these issues. Finally, in section 12.3, with the same motivation of removing the distinction of whether a test is for stationarity or for nonstationarity, the conﬁdence interval (CI) approach of Chapter 5 is revisited. The reason for this is that focusing on a single test for stationarity or a single test for nonstationarity does not exploit all of the information available from a data set, whereas the CI approach is dual to hypothesis testing in the sense of summarising all of the ‘not rejected’ null hypotheses. A CI can be obtained by inverting a test statistic, but for time series that are highly persistent, it would be better to focus on inverting a test that has good power near the local-to-unity alternative (see M¨ uller, 2005). Elliott and Stock (ES) (2001) show how this can be done for the PT and QT tests of Chapter 7; an alternative, illustrated here, uses the DF pseudo t test corresponding to the zero initial condition. Inverting t tests to obtain conﬁdence intervals is a familiar method and is easily generalised to the other pseudo t tests described in Chapter 7 that are designed to deal with the case where the initial condition is not known. The method of inverting one of the more powerful unit root test statistics is illustrated by revisiting the example of Chapter 5.

12.1 The importance of the initial condition Just as in Chapter 7, section 7.4.10, for unit root tests, stationarity tests do not avoid an initial condition problem. Indeed, as much can be inferred from Chapter 6, where it was noted that under the unit root null the initial condition acts as a nuisance parameter, so that different values result in simple mean shifts in the data. Thus differences in the initial condition can be accommodated by including a constant in the regression model, or prior demeaning of the data, so that the distribution of a unit root test statistic under the null will be invariant to the unknown initial condition. The problem for a unit root test is that the value of the initial condition affects the distribution under the alternative and so the power of a test. However, the alternative hypothesis for a unit root test is the null hypothesis for a stationarity test; hence, in the latter case, size will be affected by the value of ξ = (y1 − µ1 ), where µ1 is the deterministic trend component of y1 . Choi and Ahn (1999) also note that power will be affected in the case of zero-mean series. The size of the initial condition may well be important in a number of empirical contexts. For example, Busetti, Fabiani and Harvey (2006) considered the

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498 Unit Root Tests in Time Series

Combining Tests and Constructing Conﬁdence Intervals 499

1/2

and σˆ e = [(T − 1)−1 ∑Tt=2 eˆ 2t ] ; for the notational background see Chapter 11, Equations (11.24)–(11.26). Let e1 be a draw from a distribution that is niid(0, (κσe )2 ), with σe2 = 1, and κ ≥ 0 is a parameter governing the size of the variance; thus, κ = 0 corresponds to a zero initial condition, otherwise the initial condition will be a draw from a normal distribution with variance (κσe )2 . (In this case, one might equally write σε2 for σe2 .) The aim of the simulation set-up is to assess the sensitivity of stationarity tests to variations in κ. (For a similar exercise for unit root tests see Chapter 7, sections 7.4.9–7.4.10.) (µ) ˆ (qs) Each of the semi-parametric tests is combined with m cab (j), whereas LBM is combined with p(AIC). The impact of the initial condition is assessed by calculating the empirical size assuming a zero initial condition, whereas e1 is actually a draw from niid(0, (κσe )2 ), with κ2 = (4, 8, 12, 16); the 5% critical values from the ‘standard’ versions of the tests are used, so empirical size may not equal nominal size. The results are illustrated for T = 100 (from a wider range of values of T), see Table 12.1. The sensitivity to ρ has already been considered and is an aspect of the oversizing noted from the results in Chapter 11, Table 11.4. The additional complication here is to note how the actual size of a test changes as κ varies for a given value of ρ. All of the tests exhibit some sensitivity to the value of κ, with empirical size (generally) increasing with κ, and (generally) increasing with ρ ˆ (qs) for given κ. Of the semi-parametric tests (all are in combination with m cab (j)), KPSS(µ) and MRS(µ) show rather greater variation than KS(µ) and SBDH(µ) ; and LBM(µ) is the least sensitive test to variations in κ. For example, with ρ = 0.2 √ and κ = 8, the actual sizes are: KPSS(µ) , 13.2%; MRS(µ) , 13.2%; KS(µ) , 8.8%; SBDH(µ) , 12.2%; and LBM(µ) , 8.4%. Since it is only exceptionally the case that the initial condition is zero, oversizing can be expected on this count as well as when yt has a stationary AR component. On the basis of these results, LBM(µ) protects most against these two problems.

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stability and convergence of regional prices in Italy, and the data indicated the existence of substantial initial conditions for both regional relative prices and inﬂation rates; for example, some regional relative prices start a long way from each other, and an initial condition ten times the error standard deviation was not unusual. The notation generally follows Chapter 11 as the issues here relate to tests of stationarity (the translation to the notation of unit root framework is straightforward: see Chapter 7, section 7.4.9). Thus ξc refers to the scaled ˆ 1 )/ˆσe initial condition; that is, ξc = (y1 − µ1 )/σe , with estimator ξˆc = (y1 − µ

500 Unit Root Tests in Time Series

(κσe )2

of

empirical

0

4

8

KPSS(µ) ρ = 0.0 ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8

5.4 6.4 7.4 10.4 14.2

6.2 8.2 9.6 12.6 18.8

MRS(µ) ρ = 0.0 ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8

6.0 8.2 6.8 6.6 4.6

KS(µ) ρ = 0.0 ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8

size

to

the

initial

12

16

11.2 13.2 8.6 13.6 19.0

11.8 12.4 13.8 19.0 23.0

11.2 16.4 17.6 16.0 29.0

8.0 9.8 13.0 9.2 8.6

10.8 13.2 9.8 8.6 11.4

13.8 16.0 15.4 13.0 15.8

15.8 17.4 19.0 12.0 15.8

3.8 5.2 6.0 6.4 10.4

5.4 5.4 8.6 10.0 13.6

9.2 8.8 7.0 10.4 12.6

10.4 9.6 11.6 12.2 15.6

10.2 14.2 15.2 12.4 23.0

SBDH(µ) ρ = 0.0 ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8

3.6 4.8 8.0 8.0 9.8

5.8 6.8 9.6 11.8 16.4

10.2 12.2 9.0 15.0 19.0

10.4 13.2 14.8 17.0 25.8

10.4 13.2 14.8 17.0 25.8

LBM(µ) ρ = 0.0 ρ = 0.2 ρ = 0.4 ρ = 0.6 ρ = 0.8

6.2 7.2 8.2 9.4 13.6

5.8 7.6 7.4 9.4 14.8

8.0 8.4 5.0 8.6 12.2

8.8 8.8 8.8 9.2 17.0

5.6 8.0 9.2 9.0 16.0

Note: Table entries are empirical size for a nominal 5% test.

12.2 Problems with stationarity tests for highly correlated series M¨ uller (2005) notes a number of drawbacks of standard tests of stationarity, with the implication that they are designed for situations that are unlikely to occur with economic time series or lead to inherent difﬁculties in obtaining good test properties. Consider the framework summarised as yt = β0 + β1 t + et , where et = δΛt + ςt (see Chapter 11, Equations (11.9a) and (11.9b)), so that yt is the sum of a linear

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Table 12.1 Sensitivity condition.

trend, an integrated component with the property that Λt = Λt−1 + ηt and a stationary component ςt . The question then is, can the integrated component be detected in the sample analogue eˆ t of et ? If so, the null hypothesis of stationarity is rejected. Intuitively, this becomes more difﬁcult the more highly serially correlated is the stationary component ηt . In the starting case, which is admittedly unrealistic, ηt is independent Gaussian, leading to the simplest version of the KPSS test (see Chapter 11, Equation (11.29)). To be more practical, the test has to allow for ηt to be highly serially correlated, which it does through an adjustment due to the long-run variance, the aim of which is to neutralise the serial correlation without prejudicing the ability of the test statistic to reject the null if the serial correlation is, in fact, due to an integrated process. In order to make progress in such a case, the better starting point would recognise that the appropriate framework is local-to-unity rather than local-tostationary. This is familiar ground from Chapter 7 and from tests designed to exploit the ‘closeness’ to unity of the hypothesis of stationarity. To consider the problems further, the starting point is familiar from previous chapters, with the set-up as follows: yt = µt + ut

t = 1, . . . , T

(12.1a)

ut = ρut−1 + zt

t = 2, . . . , T

(12.1b)

where, as usual, µt comprises the deterministic part of yt . If |ρ| <1 in (12.1b), then this is a special case of Equations (11.4) and (11.5) of Chapter 11, with ση2 = 0 and ut = ςt ; and if ρ = 1, then this is the special case with zt = ηt and ςt = 0. 2 is ﬁnite and nonzero and It is assumed that the long-run variance of zt , σz,lr [rT]

that zt satisﬁes the usual condition on its scaled partial sum, namely T1/2 ∑t=1 zt ⇒D σz,lr B(r). Interest then centres on the properties of the KPSS and similarly constructed tests in a local-to-unity framework in which ρT = 1 + c/T. Using a local-to-unity asymptotic framework, M¨ uller (2005) is able to explain how problems with the KPSS test statistic arise and what can be done to minimise their effects. To elaborate on these conclusions, ﬁrst consider semiparametric estimators of the general form given by Chapter 11, Equation (11.54); that is: 2 ˆ ˆ σ˜ z,lr = γ(0) + 2 ∑κ = 1 ω(mT , κ)γ(κ) T

(12.2)

There is a difference in notation here compared to Chapter 11. An explicit T subscript has been added to the bandwidth parameter, which is now mT , in order to emphasise the central role of the rate of expansion of the bandwidth with respect to the sample size. Previously considered were deterministic rules of the form mT (j) = [j(T/100)1/4 ] or data-driven stochastic rules, as in the automatic ˆ (QS) ˆ (BW) bandwidth procedures of m T,cab (j) and mT,cab (j). Estimators of the type given 2 are consistent under standard asymptotics if m = o(T) or o(T1/2 ); for by σ˜ z,lr T

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Combining Tests and Constructing Conﬁdence Intervals 501

example, a popular choice is the mT (j) rule above, together with the Bartlett kernel, which is o(T1/4 ); and the data-dependent rule of Hobijn, Franses and Ooms (1994) outlined in Chapter 11, section 11.4.1, which leads to mT = op (T). However, these orders are not sufﬁciently fast under a local-to-unity framework to protect against over-rejection with probability one for the KPSS statistic, and it can reasonably be conjectured that this is also the case for test statistics of a similar form. M¨ uller (2005, Proposition 1) shows that for any c = −T(1 − ρT ) and mT = op (T), and for any nominal size, the probability of rejection of the null hypothesis of stationarity tends to unity as T → ∞. The KPSS test statistic is consistent because this proposition applies to c = 0 as well as to c < 0, but at the cost of a test that will reject with probability one if the sample is large enough even though the time series process does not contain a unit root. M¨ uller’s Proposition 1 explains some of the simulation ﬁndings reported in Chapter 11, Table 11.4. For example, note that the rejection rates for the semiparametric tests that use the m(j) = [j(T/100)1/4 ] rule are always larger for ρ > 0 than the automatic versions of the rule (which are expanding mT at a faster rate), substantially so for ρ = 0.9. Moreover, these high rejection rates are only marginally affected by increasing the sample size from T = 100 to T = 500, whereas such an increase is usually more than enough to rectify any size inﬁdelity. Indeed, in some cases the rejection rates worsen with the increase in sample size, in line with the proposition – that is as T increases the oversizing increases! 2 have the property that m = O (T2 ). These A second class of estimators of σ˜ z,lr p T include: (i) (ii) (iii)

(iv)

2 the autoregressive estimator of the form σˆ z,lr,AR (see Chapter 6, section 6.8.2.ii); the pre-whitened estimator of Andrews and Monohan (1992); estimators of the form given by Equation (11.54), with the automatic bandwidth suggested by Andrews (1991), based on a spectral density estimator, with either a Bartlett/Newey-West or quadratic spectral kernel; the estimator suggested by Kiefer and Vogelsang (2002).

However, all of the resulting KPSS tests, based on the respective estimators, are inconsistent; that is they fail to reject an integrated process, c = 0, with probabil2 ity one. Correct size is achieved asymptotically for σˆ z,lr,AR , and conditionally on underlying parameters for the other estimators, but only for values of −c > 50, indicating a series with a high degree of mean reversion. Given these conclusions, can stationarity tests be improved? One possibility is to note that the point-optimal unit root tests within the Elliott, Rothenberg and Stock (ERS) (1996) framework (see Chapter 7) are optimal whether the null is the unit root and the alternative is some stationary value (point) or if the null is one

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502 Unit Root Tests in Time Series

stationary value and the alternative is some other value; for example, the unit root. Indeed, one can just reverse the null and alternative hypotheses compared to the unit root case and a test of stationarity results. In this case, using the same test statistics as in the unit root case, the null hypothesis of stationarity is rejected for large values of the test statistic. In effect, this is just a reinterpretation of the information provided by the unit root test, so reversing the roles of the null and alternative hypotheses does not lead to a separate test with value added compared to the unit root test; nevertheless, generalising the point-optimality approach is valuable both for the conceptual framework it provides and for the practical contribution it enables for constructing conﬁdence intervals. To pursue these ideas, let ρ∗ = 1 + c∗ /T, c∗ < 0, be a minimum threshold level for the degree of mean reversion under the null hypothesis, so that H0 : ρ∗ ≤ ρ < 1. For example, suppose that c∗ = –10 and T = 200, then ρ∗ = 1 − 10/200 = 0.95 and H0 implies 0.95 ≤ ρ < 1. The null hypothesis implies mean reversion close to, but distinguished from, the unit root. A point-optimal test for H0 : ρ = 1 against HA : ρ = ρ∗ < 1 is equally a point-optimal test for H0 : ρ = ρ∗ < 1 against HA : ρ = 1, where critical values are obtained from the other tail of the null distribution. Thus the null and alternative hypotheses are reversed, with the null of stationarity tested against the alternative of a unit root. As in the case of unit root tests (see Chapter 7), in principle the point optimality of the test relies on constructing the test statistic and its distribution, and hence quantiles for each possible value of c∗ . However, analogously with unit root tests, it sufﬁces to select a single value of c∗ , which we will again denote c˜ (not necessarily the same as c∗ ), as in Chapter 7, at which to evaluate the tests. The optimal test statistics are relatively insensitive to the choice of c˜ . As to the value of c˜ , the choice depends on the particular test statistic, the rule being to set the value as in the unit root version of the test statistic. Thus, for example, c˜ = –7 and –13.5, respectively, for PT,µ (c˜ ) and PT,β (c˜ ), and c˜ = –10 for QT,µ (c˜ ) and QT,β (c˜ ), although in the latter case c˜ = –15 is also a valid choice. Some illustrative upper quantiles for the test statistics of Chapter 7 (see Chapter 7, Tables 7.5, 7.6 and 7.7) are given in Table 12.2 for T = 200 (the lower quantiles are also given for ease of cross-reference). For example, from Table 7.6, the 5% quantile for QT,µ (−10) with T = 200 is 4.66, with values of the test statistic less than this indicating rejection of the null hypothesis of a unit root. On the other hand, the 95% quantile is approximately 63.7 and test values exceeding this value indicate rejection of the null hypothesis of stationarity.

12.3 Conﬁdence intervals revisited Given that many macroeconomic time series are highly persistent, with the largest autoregressive root close to unity, a more informative outcome than a reject or non-reject decision or p-value of the test statistic, comprises the ‘outer’ limits of the set of null hypotheses with which the realised test statistic is consistent. This information is available from a conﬁdence interval, which is dual to

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Combining Tests and Constructing Conﬁdence Intervals 503

504 Unit Root Tests in Time Series

Table 12.2 Upper quantiles of unit root test statistics when used for stationarity (T = 200).

PT,µ (−7) 200 200

10%

5%

1%

90%

95%

99%

1.91 62.21

3.17 85.24

10%

5%

1%

90%

95%

99%

5.97 48.31

4.66 63.71

3.11 102.6

4.33 143.5

QT,µ (−10) 200 200

–3.44 –0.53

–2.79 –0.21

–2.47 0.36

3.47 15.74

2.91 19.71

2.14 28.8

–3.78 –1.34

–3.20 –1.15

–2.94 –0.77

glsu

–2.69 0.66

–2.08 1.05

–1.76 1.78

τˆ µ (−10) 200 200

PT,β (−13.5) 200 200

4.05 35.70

5.66 45.88

6.86 72.25

QT,β (−10) 200 200 τˆ β (−10) 200 200

glsc

glsu

τˆ β (−13.5) 200 200

–3.46 –0.93

–2.93 –0.70

–2.64 –0.26

ˆ (µ) (10, 3.8) Q 200 200

–5.16 27.78

–6.20 39.50

–7.48 68.97

ˆ (β) (15, 3.968) Q 200 –8.26 200 13.84

–9.33 20.63

–10.79 38.44

Sources: Own calculations based on 50,000 replications. For cross-references, see Chapter 7, Tables 7.5, 7.6 and 7.7.

a hypothesis test, and therefore it makes sense to construct a conﬁdence interval using the most powerful (or asymptotically or locally most powerful) test available. α Recall that a conﬁdence set (or interval) CIα12 (y) is a function of the data y at a conﬁdence level (1 – α)%, where α = α1 + α2 . It is deﬁned in the following way: consider testing H0 : ρ = ρ∗ against HA : ρ = ρ∗ at size α for ρ∗ ∈ Θ, where Θ is the α set of admissible values under the null hypothesis; then CIα12 (y) is the set of ρ∗ in Θ that cannot be rejected by a size α test of H0 . There would be nothing much α of novelty in this observation if the construction of CIα12 (y) was just as in the straightforward ‘textbook’ case familiar from the usual approach of inverting the t statistic: given the assumed constant quantiles and an estimated standard α error, some simple calculations would then give CIα12 (y). However, the message of Chapter 5 is that the construction of conﬁdence intervals assuming constant quantiles across the parameter space is generally misleading in AR(p) models, especially as the dominant root approaches unity. One possible solution to this problem is to adopt a grid approach and allow the varying nature of the quantiles of the t statistics to be incorporated into the construction of a conﬁdence interval. However, this approach is still based on

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glsc

τˆ µ (−7) 200 200

Combining Tests and Constructing Conﬁdence Intervals 505

H0 : value speciﬁed under null HA : value speciﬁed under alternative Value used to obtain quantiles

c

ρ

c∗

ρ∗ = 1 + c∗ /T ρ¯ = 1 + c¯ /T ρ˜ = 1 + c˜ /T

c¯ c˜

conventional t statistics but, in the unit root case, the optimality of standard t statistics no longer holds (at the unit root, the t test is, in effect, a DF test) as they are not uniformly most powerful invariant even asymptotically; indeed, we know that a UMP test of the unit root hypothesis does not exist (see ERS, 1996, and Chapter 7). We also know that the DF or ADF pseudo t tests, which are standard t tests in construction, are generally far from the Gaussian power envelope and are less powerful than a number of other tests (although this statement has to be qualiﬁed by reference to the initial condition, which does affect the power of the various tests; see Chapter 7, section 7.4.10). Thus, there is a possibility that basing conﬁdence intervals on more powerful unit root t-type tests will result in conﬁdence intervals that are better, in a well deﬁned sense, than those based on the conventional t statistic; for example, accuracy is usually taken to refer to the agreement between nominal and actual coverage rates and length of the (1 – α)% conﬁdence interval. 12.3.1 Inverting the PT unit root test statistic to obtain a conﬁdence interval One possibility is to base a conﬁdence interval on inverting one of the more powerful unit root test statistics described in Chapter 7. Recall that the tests based on quasi-differencing (QD) the data involve the local-to-unity framework in which the null hypothesis is H0 : ρ = 1, corresponding to the unit root, in effect c∗ = 0, and the (stationary) alternative is HA : ρ¯ = 1 + c¯ /T, with c¯ < 0. A family of tests is mapped out as c¯ varies but, to avoid the dependence on a range of values for c¯ , one value, denoted c˜ , is chosen as representative in order to obtain the quantiles for hypotheses testing; the standard choice for c˜ being where the power function of the test as a function of c is approximately 0.5. An example of such a test is the PT (c¯ ) test of Chapter 7 (see Equation (7.76)), which is based on the assumption that u1 = z1 so that var(u1 ) = σz2 ; when this is evaluated at c˜ it is referred to as PT (c˜ ), an example being PT,µ (−7), so that c˜ = –7. In this section the aim is to extend tests, such as PT,µ (c¯ ) and QT,µ (c¯ ), to allow the null hypothesis to be that of stationarity and the alternative can then either be another stationary value of ρ or the nonstationary alternative ρ = 1. This aim requires an extension of the notation to recognise that there are three values of c, with associated values of ρ, which are involved in the testing framework, summarised in Table 12.3.

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Table 12.3 The roles of different values of c and ρ.

The value of ρ under H0 is ρ∗ , whilst according to HA it is ρ = ρ¯ = ρ∗ , so that these hypotheses imply that c∗ = c¯ ; however, rather than construct a test statistic and its distribution for each possible value of c¯ ; for example, say c¯ = –1, –2, . . . , –15, and so on, the test statistic is constructed using the single value c˜ ; for example c˜ = –7, and, hence, ρ˜ is used for a quasi-differenced based test statistic. The difference from the set-up in Chapter 7 is that ρ∗ is not necessarily unity under the null; for example, if T = 100 and H0 : ρ = ρ∗ = 0.9 against HA : ρ = ρ¯ = 0.85, then c∗ = –10 and c¯ = –15. In an obvious extension of the notation of Chapter 7, the PT statistic, with mean-adjusted data, becomes PT,µ (c∗ , c¯ ) so that, for example, the notation for a test of a unit root against a local stationary alternative is PT (0, c¯ ), with c¯ < 0. Similar notational extensions apply to other test statistics. The practical counterpart of this test statistic replaces c¯ by c˜ , with c˜ = –7 for the ‘with mean’ case and c˜ = –13.5 for the ‘with trend’ case. In this notation, the PT test statistic is: ρ¯ 1 PT,i (c∗ , c¯ ) = 2 S(¯ρ) − ∗ S(ρ∗ ) i = µ, β (12.3) ρ σ˜ z,lr 2 is a consistent estimator of the long-run variance of z (see Equation where σ˜ z,lr t 2 ). Further, S(¯ ρ) = ∑Tt=1 ω ˆ 2t,¯ρ (12.1b) for zt , and Chapter 7, section 7.4.2.ii for σ˜ z,lr

and S(ρ∗ ) = ∑Tt=1 ω ˆ 2t,ρ∗ , where ω ˆ t,¯ρ and ω ˆ t,ρ∗ are the LS residuals from the GLS detrending regression (see below) under the null and alternative hypotheses, respectively. As an example of the notation, PT (−10, −15) is the case where c∗ = –10 and c¯ = –15, so that if T = 100, then ρ∗ = 0.9 and ρ¯ = 0.85. The residual sums of squares required in (12.3) are obtained from GLS detrending. For example, S(¯ρ) is obtained as follows. The trend coefﬁcients λ are estimated from: Yρ¯ = Xρ¯ λ + ω where, in the case of PT,β (c∗ , c¯ ), the data are as follows: Yρ¯ ≡ (y1 , y2 − ρ¯ y1 , y3 − ρ¯ y2 , . . . , yT − ρ¯ yT−1 )

Xρ¯ ≡ Pρ¯ X 1 X= 1

1 2

... ...

1 t

... ...

1 T

Pρ¯ is the quasi-differencing matrix deﬁned explicitly in Chapter 7 (see Equation (7.57)). Let λ˜ ρ¯ denote the LS estimator of λ given ρ¯ , then the residual sum of squares S(¯ρ) is: S(¯ρ) = (Yρ¯ − Xρ¯ λ˜ ρ¯ ) (Yρ¯ − Xρ¯ λ˜ ρ¯ ) = ∑t=1 ω ˜ 2t,¯ρ T

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506 Unit Root Tests in Time Series

Combining Tests and Constructing Conﬁdence Intervals 507

The analogous residual sum of squares under H0 , that is, S(ρ∗ ), follows on replacing ρ¯ by ρ∗ , resulting in: S(ρ∗ ) = ∑t=1 ω ˜ 2t,ρ∗ T

The unit root case of Chapter 7 follows on setting ρ∗ = 1, so that: 1 (S(¯ρ) − ρ¯ S(1)) 2 σ˜ z,lr

(12.4)

This is, of course, just the PT test statistic of Chapter 7, section 7.4.2.i, Equation (7.76). Speciﬁcation of the alternative hypothesis The alternative hypothesis is that ρ differs from the value under the null by the distance c¯ U , say, for an upper-tail test and by c¯ L for a lower-tail test. To simplify, ES (2001) suggest keeping these distances ﬁxed as c∗ varies. Thus let these distances be given by: c¯ (U) = c∗ + c¯ U ∗

c¯ (L) = c + c¯ L

c¯ U > 0

upper-tail test

c¯ L < 0

lower-tail test

So the ﬁxed distances, c∗ , from the null are, respectively, c¯ U and c¯ L ; and the implied values of ρ under the alternative are: ρ¯ U = 1 + c¯ (U)/T for an upper-tail test and ρ¯ L = 1 + c¯ (L)/T for a lower-tail test. As to setting c¯ L and c¯ U , the unit root case corresponds to c∗ = 0 and ES (2001) suggest being guided by this case. For example, for test statistics based on the assumption of a ﬁxed initial condition, such as the PT test statistic or corresponding DF τˆ -type test statistic, set c¯ L = –7 and c¯ L = –13.5 for the mean and trend cases; and, based on simulations, for the upper-tail test they suggest c¯ U = 2 and c¯ U = 5 for the mean and trend cases, respectively. (The same general procedure applies to the QT test and its corresponding DF τˆ -type test; see Chapter 7, section 7.4.4, in which the initial condition comes from its stationary distribution.) Based on the asymmetry of the power function, being steeper for alternatives greater than the null compared to less than the null, ES (2001) also suggest an asymmetric setting of the signiﬁcance levels; for example, an α% overall significance level is split as αL = 0.6α and αU = 0.4α, so that if α = 0.10, then αL = 0.06 and αU = 0.04, whereas if α = 0.05, then αL = 0.03 and αU = 0.02. It might be helpful to lay out a sequence of such values under the null and alternative hypotheses for T = 100 and the mean adjusted case, which is shown in Table 12.4. The general principle of test inversion to obtain a conﬁdence interval was illustrated in Chapter 5. The test statistic PT (c∗ , c¯ ), its corresponding DF τˆ -type test statistic or, indeed, any of the unit root test statistics, can be inverted for a

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PT (0, c¯ ) =

508 Unit Root Tests in Time Series

H0 : ρ = ρ ∗ ρ∗ = 0.95 ρ∗ = 0.9 ρ∗ = 0.85

HA : ρ = ρ¯ U

HA : ρ = ρ¯ L

ρ¯ U = 0.97 ρ¯ U = 0.92 ρ¯ U = 0.87

ρ¯ L = 0.88 ρ¯ L = 0.83 ρ¯ L = 0.78

c = c∗

c¯ (U)/T = (c∗ + c¯ U )/T

c¯ (L)/T = (c∗ + c¯ L )/T

c∗ = −5 c∗ = −10 c∗ = −15

0.03 0.08 0.13

−0.12 −0.17 −0.22

sequence of values of c∗ ∈ C∗ , with the resulting test statistic referred to as the sequence test; for example, ERS (1996) illustrate the asymptotic power of their tests with C∗ = [–40, + 10]. However, they also ﬁnd that keeping c∗ = 0, that is, with the test statistic PT (0, c¯ ), and setting c∗ equal to the values in ERS (1996), as described in Chapter 7 (especially section 7.4.2), results in a test with barely distinguishable power from the sequence test. This suggests a considerable economy in just inverting the PT -type test, or other form of test statistic, for just one value of c∗ . 12.3.2 Constructing a conﬁdence interval using DF tests with GLS detrended data 12.3.2.i Test inversion Inverting a t test to form a conﬁdence interval is familiar from introductory econometrics and offers a readily interpretable framework, hence it is a natural approach to take in the case being considered here. Stock (1991) has considered the inversion of the DF/ADF unit root test statistics, and the construction of conﬁdence intervals to take account of quantiles that are not constant across the parameter range was developed by Hansen (1999). Further, in the case that the initial conditions of the PT or QT tests are satisﬁed, there is little loss of power in using the corresponding DF/ADF test as amended for QD data; however, as the initial condition is not generally known, a number of weighted average t-type tests have been suggested that balance the merits of alternative tests (see Chapter 7, section 7.4.9). Thus it would also be possible to invert one of these to obtain a corresponding conﬁdence interval. glsc The general principle is illustrated with the τˆ i (c˜ ), i = µ, β, versions of the pseudo t test using QD data (in the notation of Chapter 7). This is the DF-type test derived under the assumption of a ﬁxed initial condition, which is a more powerful test than the standard ADF test; hence it should be better able to disglsc criminate against a true alternative. Recall that, for example, τˆ µ (−7) is the t statistic from an ADF regression using quasi-differenced data; that is, the data is of the form yt − ρ˜ yt−1 , apart from the initial observation, where ρ˜ = 1 + c˜ /T, with c˜ = –7 in the mean-adjusted case. In testing the unit root null hypothesis, glsc the test statistic τˆ µ (−7) is used as a one-sided (lower-tailed) test, so that large negative values lead to rejection. In that case, large positive values were not of

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Table 12.4 H0 : ρ = ρ∗ against HA : ρ = ρ¯ or, equivalently, c = c∗ against c = c¯ .

interest; however, in the general case now considered they will be relevant as alternatives of interest will lie to the right-hand side of ρ∗ when ρ∗ < 1. The ﬁrst practical question to be addressed is how to construct the quasidifferenced data for the situation where a two-sided test is required. That is, in the more general set-up considered here, the hypotheses are H0 : ρ = ρ∗ and HA : ρ = ρ¯ , and tests on both sides of ρ∗ are required. According to the lower-tailed alternative, ρ¯ L < ρ∗ , so that c¯ (L) = c∗ + c¯ L , c¯ L < 0; and according to the uppertailed alternative, ρ¯ U > ρ∗ , so that c¯ (U) = c∗ + c¯ U , c¯ U > 0. The data are quasidifferenced under the local alternative, with the quasi-differencing parameters ρ¯ L = 1 + c¯ (L)/T and ρ¯ U = 1 + c¯ (U)/T, so that the QD parameter for ρ¯ < ρ∗ differs from that for ρ¯ > ρ∗ . Let ρ˜ now implicitly denote either ρ¯ L or ρ¯ U depending on the alternative hypothesis, so that the resulting QD parameter can, for simplicity, still be referred to generically as ρ˜ , with the resulting data distinguished by the subscript ρ˜ . In the unit root set-up ρ∗ = 1 or, equivalently, c∗ = 0, and implicit in the notaglsc tion τˆ i (c˜ ) was the assumption that the test corresponded to the unit root null. However, this is now a special case and a more explicit notation is necessary, as noted in the previous section and illustrated for the PT (.) test. In this case, glsc glsc the test statistic is referred to as τˆ i (c∗ , c¯ (L)) or τˆ i (c∗ , c¯ (U)), depending on whether reference is to the lower-tail or upper-tail test statistic. The general form of the ADF regression using QD data is given by: y˜ ρ˜ ,t = ρy˜ ρ˜ ,t−1 + ∑j=1 cj ∆y˜ ρ˜ ,t−j + ερ˜ ,k,t k−1

(12.5)

where y˜ ρ˜ ,t is the QD data based on the demeaned or detrended data, y˜ t (see Chapter 7, Equation (7.80)), with the QD parameter ρ˜ , which will be either ρ¯ L or ρ¯ U depending on the alternative hypothesis. The t-type test statistic for the alternative ρ¯ L < ρ∗ is given by (illustrated here glsc for the µ version of the test statistic τˆ i ): τˆ µ (c∗ , c¯ (L)) = glsc

ρˆ L − ρ∗ σˆ (ˆρL )

for ρ¯ L < ρ∗

(12.6)

where ρˆ L denotes the usual LS estimator of ρˆ , but in this case from (12.5) with QD parameter ρ¯ L . In the unit root case ρ∗ = 1.0, c∗ = 0 and c¯ (L) = –7; otherwise c¯ (L), and c¯ (U) for the upper-tail test, vary according to the null hypothesis. For example, if ρ¯ L < ρ∗ , then the sequence in Table 12.3, with T = 100, gives the following corresponding sequence of test statistics for lower-tailed alternatives: glsc

τˆ µ (−5, −12) = glsc

τˆ µ (−10, −17) = glsc

τˆ µ (−15, −22) =

ρˆ L − 0.95 σˆ (ˆρL ) ρˆ L − 0.9 σˆ (ˆρL ) ρˆ L − 0.85 σˆ (ˆρL )

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Combining Tests and Constructing Conﬁdence Intervals 509

510 Unit Root Tests in Time Series

In the case of the alternative ρ¯ U > ρ∗ , the general form of the regression is unchanged, but as the value of ρ¯ differs from the case ρ¯ L < ρ∗ , then even though ρ∗ is unchanged, the actual QD parameter and so the QD data differs. The t-type test statistic for the alternative ρ¯ U > ρ∗ is, however, of the same form, and is given by: glsc

ρˆ U − ρ∗ σˆ (ˆρU )

for ρ¯ U > ρ∗

(12.7)

where ρˆ U denotes the usual LS estimator of ρˆ , but in this case from (12.5) with QD parameter ρ¯ U . For example, if the sequence in Table 12.3 was for the alternative ρ¯ U > ρ∗ , then with c¯ (U) = c∗ + c¯ U and c¯ U = 2.0 for the mean adjusted case, the sequence of test statistics is: gslc

τˆ µ (−5, −3) = glsc

τˆ µ (−10, −8) = gslc

τˆ µ (−15, −13) =

ρˆ U − 0.95 σˆ (ˆρU ) ρˆ U − 0.9 σˆ (ˆρU ) ρˆ U − 0.85 σˆ (ˆρU )

In general, each test statistic in the sequence, which is determined by a grid of values for ρ∗ , or equivalently c∗ , can be simulated, either conventionally or by a bootstrap, to obtain the quantiles of the corresponding ﬁnite sample distribution; thus, the method parallels that of the grid-bootstrap described in Chapter 5, due to Hansen (1999). ES (2001) note that the power of the sequence of PT (c∗ , c˜ ) tests does not differ much from the single test PT (0, c¯ (L)), which refers to the unit root case using the lower-tail value; hence, one simpler possibility glsc in this context is to invert just the single test τˆ µ (0, c¯ (L)) for both tails in order to obtain a two-sided conﬁdence interval. The construction of conﬁdence intervals was outlined in Chapter 5, but a brief review in the present context may be helpful here as there are some variations with practical implications. For the alternative that ρ is less than the value under the null, that is HA : ρ¯ L < ρ∗ , the corresponding one-sided conﬁdence interval is: ˆ L − τˆ µ,α1 (c∗ , c¯ (L))ˆσ(ˆρL )] CI0.00 α1 = (−∞ < ρ < ρ glsc

glsc

(12.8) glsc

where τˆ µ,α1 (.) is the α1 %-quantile of the distribution of τˆ µ (c∗ , c˜ L (L)); and ρˆ L

glsc is the estimate of ρ using QD data with c¯ (L). Note that τˆ µ (c∗ , c˜ L (L)) will be negative, and hence multiplying it by −ˆσ(ˆρL ) means that the right limit of the conﬁdence interval will be greater than ρˆ L . Next, consider the (point) alternative with HA :¯ρU > ρ∗ ; then the corresponding one-sided conﬁdence interval is: α

2 = [ˆρU − τˆ µ,1−α (c∗ , c¯ (U))ˆσ(ˆρU ) < ρ < + ∞) CI0.00

glsc

2

(12.9)

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τˆ µ (c∗ , c¯ (U)) =

Combining Tests and Constructing Conﬁdence Intervals 511

The intersection of these two intervals is the two-sided conﬁdence interval. That is: 2

(12.10)

Thus, as the notation makes clear, there is no need to set either α1 = α2 or |c¯ (L)| = |c¯ (U)|. However, an implication of the latter is that whilst conventional conﬁdence intervals assume that the quantiles come from the same distribution, that is not the case here; also, as the QD data differs for c¯ (L) and c¯ (U), the point estimates and estimated standard errors may also differ; hence, for example, ρˆ L and ρˆ U may differ. Of course, if c¯ (L) is used in both cases, this complication will not be present. Although illustrated for the conditional version of the ADF test using QD data, a similar conﬁdence interval can be obtained by using the unconditional version of the test or some other t-type test. Otherwise, for practical purposes, the method is as for the grid-bootstrap described in Chapter 5. In particular, a sequence of test statistics is obtained by deﬁning a grid of values of ρ under the null hypothesis, H0 : ρ = ρ∗G , where ρ∗G is an element from Gρ∗ = [ρLow , ρUp ], where ρLow and ρUp are the likely lower and upper limits of ρ, respectively; these glsc

imply values for c¯ (L) and c¯ (U), and hence the distributions of τˆ µ (c∗ , c¯ (L)) and glsc τˆ µ (c∗ , c¯ (U)) are obtained by simulation. 12.3.2.ii Illustration using time series on US GNP glsc

To illustrate, an application of inverting the τˆ i (.) tests to obtain a conﬁdence interval, the example used in Chapter 5 is revisited using QD data. The data are annual for US GNP over the period 1929–2000, giving 72 observations and are analysed in natural logarithms. As a trend is evident in the data, the relevant test statistic will have a β subscript. By way of a reminder, LS estimation, using conventional estimation of the trend, resulted in: yt = 0.943 + 0.858yt−1 + 0.588∆yt−1 + 0.005t + εˆ t

(12.11)

The standard ADF(1) test statistic was τˆ β = –3.42. The 5% critical value is –3.46, and hence the decision on the unit root is marginal as the graphical analysis in Chapter 5, Figures 5.3 and 5.10, made evident. The semi-parametric tests statistics suggested rejection of the null hypothesis of stationarity. Some of the test results are reported in Table 12.5; these are for j = 4, but the results are not changed materially for j = 12. The 5% critical values (cv) are from Chapter 11, Table 11.2; however, we know that these become increasingly compromised as the dominant root approaches unity (see, for example, Chapter 11, Table 11.4), as is likely to be the case with this time series. It therefore seems sensible to consider constructing a conﬁdence interval using one of the more powerful unit root tests designed to have good power close to the local-to-unity alternative hypothesis.

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α glsc glsc CIα21 = [ˆρU − τˆ µ,1−α (c∗ , c¯ (U))ˆσ(ˆρU ) < ρ < ρˆ L − τˆ µ,α1 (c∗ , c¯ (L))ˆσ(ˆρL ))]

512 Unit Root Tests in Time Series

Table 12.5 Stationarity tests for US GNP. kernel method yt KPSS(β) MRS(β) KS(β) SBDH(β)

QS 0.50 2.41 1.48 0.33

QSCAB 0.30 1.86 1.14 0.20

BW 0.37 2.07 1.27 0.24

BWCAB 0.23 1.63 1.00 0.15

5% cv 0.146 1.511 0.848 0.117

The detrending regression, using QD data for the ﬁxed initial condition, is of the general form given in section 12.3.1, but here with ρ˜ (as a generic notation, actual values depending on the ‘tail’ of the test); that is: Yρ˜ = Xρ˜ λ + v

(12.12)

Yρ˜ ≡ (y1 , y2 − ρ˜ y1 , y3 − ρ˜ y2 , . . . , yT − ρ˜ yT−1 )

(12.13a)

Xρ˜ ≡ Pρ˜ X 1 X= 1

(12.13b) 1 2

... ...

1 t

... ...

1 T

(12.13c)

where Pρ˜ is deﬁned in Chapter 7, Equation (7.57). The estimated coefﬁcients are: λ˜ 1 = 6.63 and λ˜ 2 = 0.035. These coefﬁcients are then used to obtain detrended data (rather than the LS detrended data used in the standard ADF regression), resulting in: y˜ ρ˜ ,t = yt − (6.63 − 0.035t) The ADF regression using QD detrended data, with c¯ (L) = –13.5, was: y˜ ρ˜ ,t = 0.864y˜ ρ˜ ,t−1 + 0.586∆y˜ ρ˜ ,t−1 + εˆ ρ˜ ,k,t

(12.14a)

The point estimates of ρ, based on actual and QD data, differ slightly, but not in glsc this case very markedly; the ADF test statistic using QD data is τˆ β (0, – 13.5) = glsc

–3.50, with σˆ (ˆρL ) = 0.0389. The 5% critical values for τˆ β and τˆ β are approximately –3.46 and –3.10, respectively; thus rejection of the unit root null is glsc somewhat marginal using τˆ β , but clearer for τˆ β , with the implied p-value of the latter much lower than for the former. When detrended QD are used with c¯ U = 2 (the results with c¯ U = 5 were similar and are not therefore reported), which is the value appropriate for detrended data and a right-sided test, the estimated equation is: y˜ ρ˜ ,t = 0.877y˜ ρ˜ ,t−1 + 0.581∆y˜ ρ˜ ,t−1 + εˆ ρ˜ ,k,t

(12.14b)

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Note: 5% critical values are for T = 100 from Chapter 11, Table 11.2.

Combining Tests and Constructing Conﬁdence Intervals 513

gslc

and τˆ β (0, 5) = –3.34, with σˆ (ˆρU ) = 0.0367. The differences between (12.14b) and (12.14a) are relatively slight. The two-sided conﬁdence interval is constructed as: α

CIα21 [c¯ (L), c¯ (U)] = [ˆρU − τˆ β,1−α (c∗ , c¯ (U))ˆσ(ˆρU ) < ρ < ρˆ L − τˆ β,α (c∗ , c¯ (L))ˆσ(ˆρL ))] glsc

glsc

2

1

(12.15)

α glsc glsc CIα21 [c¯ (L), c¯ (L)] = [ˆρL − τˆ β,1−α (c∗ , c¯ (L))ˆσ(ˆρL ) < ρ < ρˆ L − τˆ β,α1 (c∗ , c¯ (L))ˆσ(ˆρL ))] 2

(12.16) The right-hand limit is the same for the two conﬁdence intervals, but the lefthand limit differs. Note that if a single test statistic is inverted, in this case it glsc glsc would be τˆ β (0, – 13.5), so that τˆ β ,α (0, – 13.5) is just the lower 5% quantile 1 1 1 of the distribution of the unit root test statistic for the with trend case. The conﬁdence intervals, (12.15) and (12.16), are obtained by the gridbootstrap method of Hansen (1999), illustrated in Chapter 5; previous estimation suggested a set Gρ∗ = [0.75, 1] in terms of ρ∗ , which corresponds to a set in terms of c∗ of Gc∗ = [–18, 0]. The bootstrap routine drew errors from the empirical distribution function corresponding to either (12.14a) or (12.14b), depending on which distribution was being simulated. Figure 12.1 shows how to obtain the 90% symmetric interval (5% in each tail) of (12.15) based on equations (12.14a) and (12.14b). (For reference back, the grid-bootstrap ‘t’ conﬁdence interval was shown in Chapter 5, Figure 5.10.) The ‘t’ function is also plotted on the ﬁgure, which is of the general form t(ρ∗ ) = (ˆρ − ρ∗ )ˆσ(ˆρ)−1 , where the LS estimates are ρˆ = 0.858 and σˆ (ˆρ) = 0.041. The estimates using the QD data depend on whether they are based on c¯ L or c¯ U ; taking c¯ L = –13.5 and c¯ U = 2, resulted in ρˆ L = 0.864 and σˆ (ˆρL ) = 0.0389 and ρˆ U = 0.877 and σˆ (ˆρU ) = 0.0367. (As noted, a variation with c¯ U = 5 was also estimated, but the differences to the estimates and conﬁdence intervals were slight and are not reported.) Using the grid-bootstrap ‘t’, the relevant quantiles were τˆ bs β,0.05 = –3.268 and τˆ bs β,0.95 = 0.975, and hence the symmetric 90% conﬁdence interval was (see Chapter 5): CI0.05,bs 0.05,G = [0.858 − 0.975(0.041), 0.858 − (−3.268)(0.041)] = [0.818, 0.992]

(12.17)

The grid-bootstrap conﬁdence interval of (12.15), with α1 = α2 = 0.05, was: CI0.05 0.05 [−13.5, 2] = [0.877 − 1.253(0.0367) < ρ < 0.864 − (−2.612)0.0389] = [0.831, 0.966]

(12.18)

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If a single test is used, then the conﬁdence interval uses quantiles from the same distribution at each point in the grid and has the following form:

514 Unit Root Tests in Time Series

8 95% quantile using c(U) = 2

6

glsc

t-function for ˆτβ

4 95% quantile using c(L) = –13.5

0 5% quantile using c(U) = 2

–2 5% quantile using c(L) = –13.5

–4 0.7

0.75

0.8

90% confidence interval

0.85 ρ

0.9

0.95

1

Figure 12.1 90% conﬁdence interval using QD data: US GNP.

glsc

This uses quantiles from τˆ β

glsc

(c∗ , c¯ (L)) and τˆ β

glsc

glsc

(c∗ , c¯ (U)); thus τˆ β,0.05 (c∗ , −13.5) =

–2.612 and τˆ β,0.95 (c∗ , + 2) = 1.253. If a single test statistic is inverted, the quanglsc

tiles of τˆ β is:

(c∗ , c¯ (L)) are used for both tails and the conﬁdence interval of (12.16)

CI0.05 0.05 [−13.5, −13.5] = [0.864 − 1.003(0.0389) < ρ < 0.864 − (−2.612)0.0389] = [0.825, 0.966]

(12.19)

The construction of this conﬁdence interval is shown in Figure 12.2. Note that the difference in the left-hand limit of the two conﬁdence intervals (12.18) glsc glsc and (12.19) is slight because τˆ β,0.95 (c∗ , c¯ (L))ˆσ(ˆρL ) ≈ τˆ β,0.95 (c∗ , c¯ (U))ˆσ(ˆρU ) and ρˆ L ≈ ρˆ U . This was found to be the case quite generally; that is, for larger variaglsc tions in |c¯ L | = |c¯ U |. There was little sensitivity of the quantiles of τˆ β (c∗ , c¯ (U)) to the choice of c¯ U , except as ρ → 1, which affects the lower-tail quantiles; for glsc example, τˆ β,α (c∗ , c¯ (U)), but this quantile is not needed for construction of the 0.05 conﬁdence interval. glsc Inverting τˆ β results in shorter conﬁdence intervals than inversion using the standard t statistic (which coincides with τˆ β at the unit root); the lengths of (12.17), (12.18) and (12.19) are: 0.174, 0.134 and 0.141, respectively. The conﬁdence intervals suggest that the time series on US GNP is a highly persistent but stationary time series.

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quantile 2

Combining Tests and Constructing Conﬁdence Intervals 515

4 t-functions ˆτβ ˆτglsc β

3 2

glsc

ˆτβ,0.95 (0, –13.5)

ˆτβ,0.95

95% quantiles

1

–1

ˆτβ,0.05

glsc

ˆτβ,0.05 (0, –13.5)

–2 5% quantiles

–3 90% confidence interval

–4 0.7

0.75

0.8

0.85 ρ

0.9

0.95

1

Figure 12.2 90% conﬁdence interval, QD data, with c(L): US GNP.

Finally, consideration is given to a brief variation on the theme, where the conﬁdence interval(s) is based on asymmetric size allocation. In this case, α = 0.10, as before, but α1 = 0.06 and α2 = 0.04; these values result in the following conﬁdence intervals: CI0.04 0.06 [−13.5, 2] = [0.828, 0.964]

(12.20)

CI0.04 0.06 [−13.5, −13.5] = [0.820, 0.964]

(12.21)

These have lengths of 0.136 and 0.144, respectively, and do not affect the previous conclusions; in particular, the right-hand limit excludes the unit root.

12.4 Concluding remarks This chapter has shown that it is possible to do rather more than just compute either a test for stationarity or a test for nonstationarity by way of a unit root. First, such tests can be used together. Second, the idea that motivated the construction of the (generally) more powerful unit root tests of ERS (1996) and Elliott (1999) can also be applied to tests where the null hypothesis is that ρ takes a value in the stationary part of the parameter space. This approach explicitly recognises that a more realistic alternative to stationarity under the null for strongly autocorrelated time series is a local-to-unity alternative.

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quantile 0

Third, rather than concentrate on a single hypothesis test, more information can be provided by constructing a conﬁdence interval. This almost brings us full circle, since Chapter 5 was concerned with the construction of such intervals using the standard t test, but recognising that the quantiles of the t distribution were not constant especially as ρ → 1. The difference here is that the conﬁdence intervals were constructed by inverting one of the more powerful unit root test statistics in order to obtain a conﬁdence interval. Elliott and Stock (2001) invert the PT and QT tests, whereas in this chapter this approach was illustrated by the relatively familiar process of inverting a pseudo t-type statistic, this case was illustrated with the ERS (1996) version of the ADF test statistic using GLS detrended or quasi-differenced data. As it is known that such test statistics are vulnerable to contradiction of the assumed initial condition, other more robust pseudo t-type tests, as described in Chapter 7, could be inverted.

Questions Q12.1 Is the empirical (ﬁnite sample) size of stationarity tests sensitive to variations in the initial condition? What are the implications of any sensitivity and are such variations likely? A12.1 The simulation results reported in Table 12.1 show that the magnitude of the initial condition does matter. Moreover the problem is compounded when the time series under scrutiny exhibits a temporally correlated structure. First consider the case when ρ = 0 so that the only problem is the scale of the initial condition, then the test statistics for stationarity tend to become oversized; that is, reject more than their nominal size would suggest for a true null hypothesis (that is, of stationarity). The oversizing is moderate for (κσe )2 = 4; that is, κ = 2 (given that σe2 = 1 in the simulations), but is about twice the nomi√ nal size for κ = 8 = 2.83 and maintains about the same degree of oversizing as κ increases further. As ρ increases the tests become oversized on that account, which is then compounded by the increase, in κ. Overall, the best of the tests in this simulation, taking into account the effects from ρ and κ were MRS(µ) and LBM(µ) , although some further research is required to establish the robustness of this conclusion; for example, empirical size is also known to be related to the window length in estimation of the long-run variance. The second part of the question relates to whether variations of the initial condition on such a scale are likely. Recall that ξc = (y1 − µ1 )/σe , where µ1 is the initial value of the deterministic component, so that the scale is in terms of units of the standard deviation of et , an estimate of which can be obtained by regressing yt on the hypothesised deterministic components and using the estimated standard error from such a regression. Referring back to Chapter 7, section 7.4.12, an example of the nature of the problem was shown with US

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516 Unit Root Tests in Time Series

Combining Tests and Constructing Conﬁdence Intervals 517

industrial production in Figure 7.9, where the estimated scaled initial condition varied between + 1 and –5, a range of over six times σˆ e . An important characteristic of this series is the fairly substantial deviations from trend; particularly in the early part of the sample, which tended to indicate that chosen starting point in the sample is likely to be critical. Similar characteristics are present in other time series spanning long periods; for example, the estimated scaled residuals for the Nelson and Plosser unemployment series (see Chapter 7, Figure 7.1) varied between + 3.6 and –3.5.

A12.2 Five steps provide the summary, as follows.

1.

2.

3.

4.

The starting point is the underlying hypothesis testing framework: H0 : ρ = ρ∗ and HA : ρ = ρ¯ , but, in contrast to the set-up in Chapter 7, ρ∗ = 1, usually ρ∗ ≤ 1 and ρ¯ = ρ∗ , with both ρ¯ = ρ¯ L < ρ∗ and ρ¯ U > ρ∗ possible. Thus, the ‘local’ alternative may be either side of the null value. The implication is that the quasi-differencing of the data will differ depending on whether it is based on ρ¯ L or ρ¯ U , even though the null hypothesis is the same in each case. From Chapter 7, the QD data for the dependent variable in the case with a ﬁxed initial condition is of the form: Yρ˜ ≡ (y1 , y2 − ρ˜ y1 , y3 − ρ˜ y2 , . . . , yT − ρ˜ yT−1 ) , where ρ˜ = 1 + c˜ /T, and c˜ is the (single) representative value of c¯ (where ρ¯ = 1 + c¯ /T) under the stationary local alternative; for example, c˜ = – 7, in the case of mean adjusted data, corresponding to the tangency of the power function at approximately 50% power. However, in the generalisation of hypothesis testing to non-unit root null hypotheses, the data is quasi-differenced using ρ¯ L for ρ¯ < ρ∗ and ρ¯ U for ρ¯ > ρ∗ , where ρ¯ L = ρ¯ U , so that the data for the dependent variable differs, as does that for the ‘explanatory’ variables. Thus the data and the implied GLS detrending depend on which side of the null hypothesis the alternative is localised. Now consider a set of possible values for the value of ρ under the null hypothesis; that is, values of ρ∗ ; this was the set denoted ρ∗G ∈ Gρ∗ = [ρLow , ρUp ], and suppose that there are S such values (this is a practical approximation as, in principle, this number is inﬁnite within a ﬁnite interval). glsc This implies that there are S bootstrap distributions of τˆ µ , one for each value of ρ∗ in the set Gρ∗ ; at the point ρ∗ = ρ∗G = 1, this is just the unit root bootstrap distribution. Within the bootstrap, the data are generated (or replicated) under the null, but the QD data, on which the ADF-type

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glsc

Q12.2 Consider constructing a conﬁdence interval by inverting the test τˆ µ for a unit root. Summarise the process by which the QD data is obtained and then used in obtaining the distribution of the test statistic, for each point in a grid of values for ρ.

518 Unit Root Tests in Time Series

gslc

assume that the distribution of τˆ µ is the same at each point, it can and does vary as ρ∗G varies. The standard method of obtaining a conﬁdence interval, familiar from introductory texts on econometrics, can then be used to obtain the grid-bootstrap conﬁdence interval, but allowing the quantiles to vary over the set Gρ∗ .

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5.

model are estimated, are as speciﬁed under the local alternative; thus the nature of the alternative determines whether ρ¯ L or ρ¯ U is used. glsc The S bootstrap distributions of the test statistic, for example, τˆ µ , provide ∗ the relevant quantiles at each of the S points ρG ∈ Gρ∗ . Thus, rather than

13

Introduction Much economic data exhibits a seasonal pattern; for example, consumers’ expenditure is higher at Christmas than at other times and expenditure on restaurants and hotels is higher in the summer. These examples relate to festivals or other cultural activities, whereas some seasonal patterns, whilst also related to the weather, are more fundamentally determined by them; for example, employment in agriculture and construction varies across the seasons and ﬁshery activity is governed by biological processes that have an inherent seasonal cycle. More formally, what is seasonality? Hylleberg (1992) (and see also Franses, 1996) suggested the following characterisation: Seasonality is the systematic, although not necessarily regular, intra-year movement caused by changes of the weather, the calendar, and timing of decisions, directly or indirectly through the production and consumption decisions made by the agents of the economy.

It is clear from this deﬁnition that whilst some of the effects of seasonality might be approximated by deterministic patterns, behavioural decisions also affect the impact of seasonal changes. This introduces the possibility of changing seasonal patterns, as the following examples illustrate: (i) there has been a tendency for supermarkets in Western Europe to source seasonal produce all the year round by accessing overseas markets and air freighting the produce into the home country; (ii) the principal seasonal sale in many countries was for a long time associated with January (the ‘New Year’ sale); however, in recent years, sales have been brought forward, often taking place before Christmas and even in November; (iii) the central holiday season previously associated with August for

519

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Unit Root Tests for Seasonal Data

Northern hemisphere countries has now been changed and extended, supplemented by short breaks associated with budget airline travel. Changing seasonal patterns means that what is ‘winter’ one year can be become ‘autumn’, if not ‘summer’, in due course! Whilst the nature of seasonality means that it is likely that a satisfactory model should allow for changing patterns, it may still be helpful, at least initially, to approximate the seasonal effects by deterministic variables; and the seasonal effects might anyway be a combination of both deterministic and stochastically varying effects. Thus this chapter includes reference to dummy variables as a means of modelling seasonal effects, but also develops a framework in which stochastic variation is allowed. An excellent and comprehensive guide to the issues involved with seasonal data is provided by Ghysels and Osborn (2001). Following Hylleberg’s deﬁnition, cases of particular interest are those associated with the seasonal frequencies within a year. For example, in the case of quarterly data, that is, a frequency of four observations each year, there is the possibility of a semi-annual pattern, which is a cycle for each halfyear within the year, and four patterns, each of which lasts three months and repeats itself each year. In parts of the world with two seasons a year, seasonality associated with changing weather patterns leads to a semi-annual frequency. There are also some patterns of a repeating nature that occur with a longer period than a year, hence one might prefer periodicity rather than seasonality as a general description of the data characteristics. Thus, whilst the focus in this chapter is on intra-year variation, other patterns of a similar nature do occur over a longer period and can be modelled with the same framework; for example, a ‘leap’ year (with 29 February) occurs once every four years; the Olympic Games and other major world sporting festivals occur once every four years. Also some seasonal patterns may occur with a shorter period than a quarter or a month. There may be different daily trading patterns in commodity and ﬁnancial markets within a ﬁve-day working week; for example, Tokihisa and Hamori (2001) ﬁnd ‘seasonal’ integration for trading volumes on the Japanese stock market taking each day as a separate ‘season’. It is not intended that this chapter provides a survey of methods to test for seasonal unit roots as well as testing for the usual long-run root (the ‘conventional’ unit root). Extensive and critical surveys and assessments by Franses (1996), Gyhsels and Osborn (2001) and Brendstrup et al. (2004) complement this chapter. Rather, the aim is to introduce the reader, who may have some familiarity with testing for the conventional unit root and some acquaintance with the idea of seasonal variation, to the central ideas of testing for unit roots at the seasonal frequencies.

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521

This chapter progresses by ﬁrst illustrating some seasonal patterns in economic time series and then assessing informally whether such patterns are present. There is then an introduction to seasonality viewed from the frequency domain which, although brief, is critical to an understanding of the importance of seasonal variation. By way of an introduction to some of the testing issues, the Dickey, Hasza and Fuller (hereafter DHF) (1984) extension of the standard DF test to seasonal series is taken as a starting point for seasonal unit root testing. However, the DHF test fails to distinguish amongst the S possible unit roots in a time series with S seasons, the development of which leads to the HEGY test, which is the dominant test in this area. Once this framework is established it is possible to consider some developments that generally lead to an improvement relative to the standard version of the test. Several examples complement the more formal analysis. The emphasis in this chapter is on nonstationarity in the mean, but the reader should be aware that seasonality may have implications for nonstationarity in the variance. The reader should also be aware that this chapter covers a lot of material and is quite long, so that it may be best read in more than one ‘sitting’. Sections 13.1–13.4 are by way of background, but essential, reading; these are followed by sections 13.5 and 13.6, which introduce two of the most important tests for seasonal unit roots. Section 13.7 considers the use of the DF test when there are seasonal unit roots, and section 13.8 considers the possibility of improving the power of some standard tests, followed by section 13.9 where ﬁnite sample results are illustrated. Section 13.10 contains some empirical illustrations and section 13.11 outlines the periodic AR model as an alternative way of modelling seasonality.

13.1 Seasonal effects illustrated To illustrate seasonality from a time series perspective, we consider time series data for consumers’ expenditure on nondurable goods and services (CND) for the UK in seasonally unadjusted (n.s.a.) and seasonally adjusted (s.a.) form, where the latter are the ofﬁcial data (from the UK’s Ofﬁce for National Statistics, ONS). Figure 13.1a shows the data in s.a. and n.s.a. form; the difference between the two is the seasonal effect, which is shown in Figure 13.1b. Figures 13.2a and 13.2b do the same, but for the logs of the respective series. The data in logs indicate that a linear trend may be ﬁtted, whereas this view is more difﬁcult to sustain for the series in levels. A frequent deterministic model of seasonality uses the dummy variable (DV) approach, which extends the standard demeaning or detrending prior regression to include (S – 1) seasonal dummy variables, where S is the frequency of the data. The s-th seasonal dummy variable is 0 except for season s, when it takes the value 1; more formally, let DVst = 1 for the s-th season and 0 elsewhere. As

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Unit Root Tests for Seasonal Data

522 Unit Root Tests in Time Series

5

2

× 10

level of constant price series

1

not seasonally adjusted seasonally adjusted 0.5 1965 1970 1975 1980

1985 1990 1995 2000 2005

Figure 13.1a Consumers’ expenditure (UK).

10,000 8,000

from level of constant price series

6,000 4,000 2,000 0 –2,000 –4,000 –6,000 –8,000 1965 1970 1975 1980 1985 1990 1995 2000 2005 Figure 13.1b Consumers’ expenditure: seasonal component.

is well known, to avoid perfect linear dependence only (S – 1) seasonal dummy variables are included if a constant is present in the prior regression. To continue the illustration, the data in logs are preferred here as the seasonal pattern is more nearly constant in this case. The average seasonal components from the ONS data are: Q1: –0.0413; Q2: –0.0187; Q3: 0.00845; Q4: 0.0483; whereas the corresponding estimated seasonal effects from the DV regression

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1.5

Unit Root Tests for Seasonal Data

523

12.4 12.2 12 11.8 11.6

11.2 11 10.8 1965

1970

1975

1980

1985

1990

1995

2000

2005

Figure 13.2a Consumers’ expenditure (logs).

0.08 0.06 0.04 0.02 p.a. 0 –0.02 –0.04 –0.06 1965 1970

1975 1980 1985

1990 1995 2000

2005

Figure 13.2b Consumers’ expenditure (logs): seasonal component.

(including a trend) are: –0.0399, –0.0177; 0.00986; 0.0477. The seasonal components estimated by the (simple) DV method are constant by construction and are shown in Figure 13.3. Of course, this is a simple model of seasonality and possible extensions, even in deterministic form, include modelling seasonality in the trend and seasonality in the variance. The DV method is a quick and convenient method to ‘eyeball’ possible seasonality in a time series; however, it implies that the seasonality is deterministic and that shocks to the seasonal pattern are not persistent (although split seasonal

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11.4

524 Unit Root Tests in Time Series

0.05 0.04 0.03 0.02 0.01

–0.01 –0.02 –0.03 –0.04 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 Figure 13.3 Consumers’ expenditure: DV seasonal component.

effects could be introduced, this is not a substantive change). An alternative is to allow the seasonal effects to change over time with, for example, the possibility of persistent shocks in the seasonal components; this latter suggestion indicates the possibility of unit roots at the seasonal frequencies. It would be helpful, therefore, to have a rule of thumb indicator of the constancy of seasonal patterns, an idea that is taken up in the next section, and then a method for testing for seasonal unit roots.

13.2 Seasonal ‘split’ growth rates A graphical way of assessing the constancy of the seasonal pattern is to look at the changes from one season to the next arranged by season (see Franses, 1996). If logarithms are taken, which is often the case, then ∆1 yt = yt – yt−1 is the one-period growth rate, which is then arranged by season (the ﬁrst-difference operator is written explicitly in this chapter with a 1 subscript). In a more explicit notation, let n denote the year and s the season, then: ∆1 yn,s = yn,s – yn,s−1 is the one-period growth rate for season s; for example, and for simplicity, run the n index from 1 to N and s from 1 to 4 – the sequence is shown in Table 13.1. Note that ∆1 yn,1 = yn,1 – yn−1,4 . The idea is then to plot these growth rates for the different seasons. This is done for the log of the CND series and the four seasonal patterns are shown in Figure 13.4. The graph shows that there is a separation in the patterns, which indicates seasonality, with growth lowest in Q1 (in fact, negative relative to the previous quarter). The pattern is reasonably constant for Q4; but there is some

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Unit Root Tests for Seasonal Data

525

Q1

Q2

Q3

Q4

n.a y2,1 – y1,4 y3,1 – y2,4 .. . yN,1 – yN−1,4

y1,2 – y1,1 y2,2 – y2,1 y3,2 – y3,1 .. . yN,2 – yN,1

y1,3 – y1,2 y2,3 – y2,2 y3,3 – y3,2 .. . yN,3 – yN,2

y1,4 – y1,3 y2,4 – y2,3 y3,4 – y3,3 .. . yN,4 – yN,3

Note: Data assumed to be in natural logarithms.

0.1 seasonal split growth rates

quarter 3

quarter 4

0.05

0

quarter 2

∆yn,s –0.05 quarter 1 –0.1

–0.15 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010 Figure 13.4 Consumers’ expenditure.

indication of a changing pattern between Q2 and Q3. Similar data (quarterly for 1963q1 to 2004q4 in constant prices) for UK seasonally unadjusted expenditure on hotels and restaurants are shown in Figure 13.5. Again there is a strong separation in the seasonal patterns, indicating that seasonality is an important feature of the data; and, as to changing patterns, note that there has been a slight decline in the Q2 growth rate (relative to Q1). Thus, in both cases, ﬁtting the seasonal pattern by dummy variables should only be considered a ﬁrst approximation.

13.3 The spectral density function and the seasonal frequencies It is helpful to approach the concept of seasonality from the frequency domain, which involves a slight, but worthwhile, digression in order to deﬁne terms

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Table 13.1 Calculation of seasonal ‘split’ growth rates: ∆1 yn,s = yn,s – yn,s−1 .

526 Unit Root Tests in Time Series

0.4 seasonal split growth rates 0.3 quarter 2

0.2 quarter 3 0.1 ∆yn,s quarter 4 –0.1 –0.2

quarter 1

–0.3 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Figure 13.5 Expenditure on restaurants and hotels (UK).

and illustrate concepts. To start, an alternative but still deterministic representation of seasonality is to use trigonometric functions to capture the seasonal contributions to the time series.

13.3.1 Frequencies, periods and cycles Consider the sine and cosine functions yt,sin = A sin(λt) and yt,cos = B cos(λt), where λ is the angular frequency measured in radians from 0 through to 2π. In the language associated with these functions, yt,sin and yt,cos are often referred to as the signals generated by an input with frequency λ and amplitude A for yt,sin and B for yt,cos , respectively. These are periodic functions in the sense that yt,sin = yt + kP,sin and yt,cos = yt + kP,cos , for k an integer and period of length P = 2π/λ. Initially, consider the case with λ = π, then each of these functions has a period of P = 2π, so that one complete cycle of the process is completed as the index t moves from 0 through to 2π; as t extends past 2π, or before 0, in integer multiples of 2π, a complete cycle is repeated. As noted, the amplitudes of these functions are A and B, respectively; for example, if A = 3, then the limits of yt,sin are ±3. The amplitude controls the importance of the contribution of each frequency to the whole when periodic signals are combined. Different periods are obtained by varying the frequency λ; for example, if λ = π/6 = 2π/12, then the period is P = 2π/(2π/12) = 12 time units. Notice that by writing the frequency in this form, with 2π as the numerator, then the period can be read off as the denominator, so that λ = 2π/P. The period P is sometimes referred to as the ‘fundamental’ period, since for k = 1 it is the smallest integer for

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0

527

which the periodicity condition yt = yt + kP is satisﬁed; and the corresponding λ is sometimes referred to as the ‘fundamental’ frequency. It is possible for a signal to have several different frequency components, and the different frequencies will be distinguished by a subscript; thus, λj is associated with the period Pj = 2π/λj . A time series may have more than one period suggesting a modiﬁcation of the deﬁnition of periodicity, so that yt = yt+kPj indicates a repeating cycle with a period Pj . The period Pj is also the length, or span, of the cycle at the frequency λj . For example, if the time unit is a quarter and λj = 2π/4 then Pj = 2π/λj = 4 quarters (1/4 cycle per quarter), so that a complete cycle takes four quarters = one year to complete; if λj = 2π/2 then Pj = 2π/λj = 2 quarters (1/2 cycle per quarter), so that a complete cycle takes two quarters to complete and there are two cycles in a year. In the case of quarterly data, the two frequencies corresponding to one cycle a year and two cycles a year are the seasonal frequencies. A related concept of frequency can also be deﬁned as the inverse of the period, that is Fj = 1/Pj = λj /2π; the context will usually distinguish this use of frequency, but if emphasis is needed then λj is referred to as the angular frequency. Since a period has units of measurement of time periods, the associated frequency, Fj , has units of measurement that are the reciprocal of the time unit. For example, if the time unit is a quarter and λj = 2π/4, then Fj = λj /2π = 1/4, with the interpretation that a 1/4 of the cycle associated with this frequency is completed in the time unit (a calendar quarter); and, as noted above, λj = 2π/2 implies F = 1/2 cycle per calendar quarter. Note that Pj cannot be less than 2 implying that Fj cannot exceed 1/2; this follows from the observation that at least two time units are required to identify or resolve a cycle. This frequency is known as the Nyquist frequency which, in angular units, is simply λj = 2π/2 = π. 13.3.2 Power spectrum and periodogram One way of assessing the presence and importance of the different frequencies is to plot the power spectrum, sometimes referred to as the spectral density function, sdf (see also Chapter 2, section 2.2.1.ii). The power spectrum is the Fourier transform of the autocovariance function; that is: 1 1 ∞ γ(0) + ∑k=1 γ(k) cos(λj k) 2π π 1 1 ∞ = γ(0) + ∑k=1 γ(k)e−iλj k 0 ≤ λj ≤ π 2π π

f(λj ) =

(13.1)

Where γ(0) is the variance of yt and γ(k) is the k-th order autocovariance of yt . The range of λj assumes a real-valued time series. Anderson (1971, chapter 7) contains an extensive discussion of these functions and the reader may also like to consult Hamilton (1994, chapter 6). The power spectrum can be plotted as f(λj ) against λj ∈ [0, π]; alternatively, f(λj ) may be plotted against Fj using Fj = λj /2π, so that the horizontal axis extends from 0 to 0.5. (The deﬁnition in

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Unit Root Tests for Seasonal Data

(13.1) follows Anderson, 1971, section 7.3.) Some authors alternatively take the spectral density function to refer to the (scaled) Fourier transform of the autocorrelation function; that is, F(λj ) = 2πγ(0)−1 f(λj ), as in Chapter 2, Equation (2.10)). To estimate the power spectrum, γ(0) and γ(k) are replaced by their sample ˆ j) ˆ ˆ counterparts, γ(0) and γ(k), and the sample spectral density function (sdf) f(λ is: T−1 ˆ j ) = 1 γ(0) ˆ ˆ + 2 ∑k=1 γ(k) f(λ cos(λj k) 0 ≤ λj ≤ π (13.2) 2π ˆ − 1). A range of frequenThe last sample autocovariance in this sequence is γ(T ˆ cies for evaluating f(λj ) is the set comprising λj = 2πj/T, j = 0, . . . , int[T/2], where int[.] indicates the integer part of the expression. A related function is the periodogram (indeed, as deﬁned here, they are the same, but they differ with some authors). First, deﬁne the discrete Fourier transform (DFT) of the sequence {yt }Tt=1 : ωy (λj ) =

1 √ 2π

T

∑ yt e−iλj t

(13.3)

t=1

Then the periodogram is: |ω(λj )|2 T 2 T 1 −iλj t = ∑ yt e 2πT t=1

I(λj ) =

(13.4)

= f(λj ) The last line follows as: 2 2 1 T 1 T −iλj t −iλj t ¯ = ∑ (yt − y)e ∑ yt e T t=1 T t=1

(13.5)

∞

= γ(0) + 2 ∑k=1 γ(k) cos(λj k) (see Harvey, 1981, chapter 3). The DFT has here been deﬁned with the divisor √ 2π; however, this is not always the case, so that the spectral density function and the ‘periodogram’ will differ, but only by a constant; for example, the DFT in √ MATLAB does not use the divisor 2π, so care has to be taken in programming implementations. A seasonal time series has peaks in the spectrum at the seasonal frequencies, λj = 2πj/S, j = 1, . . . , S/2 for S even and [S/2] for S odd; for example, if S = 4, then the seasonal periodicities are j = 1, λ1 = π/2, P1 = 4, with one cycle per year; and j = 2, λ2 = π, P2 = 2, with two cycles per year. In practice, many economic time

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Unit Root Tests for Seasonal Data

529

series also have a peak in the spectrum as λj → 0 + ; this is now easy to interpret as it corresponds to fj → 0 + and Pj → ∞; that is, the spectrum indicates that there is a very long (possibly) inﬁnite cycle in the time series.

The ﬁrst illustration uses simulated data. The function yt = A1 cos(λ1 t) has an angular frequency of λ1 , an associated period of P1 with frequency F1 and amplitude A1 . In this example, λ1 = 2π/12 so that P1 = 12 and F1 = 1/12; thus all the power of this signal is concentrated at one frequency. To complicate the identiﬁcation of the sole contributory frequency, the signal is subjected to some white noise, such that yt = A1 (cos(λ1 t) + ω) where ω is a random variable, uniformly distributed on (–π, π). The simulated time series comprises 512 observations and the periodogram is plotted in Figure 13.6. The peak is correctly located at F1 = 1/12 even though the noise shows some minor, but not systematic, power at other frequencies (as T is increased, the location of the peak becomes even clearer). In the next example, the observations are generated by a combination of two signals: the ﬁrst as before with a period of 12, while the second adds in a longer cycle with a period of 120; that is, now yt = A1 cos(λ1 t) + A2 cos(λ2 t), where λ1 = 2π/12 and λ2 = 2π/120. In this example, A1 = 1 and A2 = 2, so that the long cycle is more important in its contribution to the total variance. The signals are again randomised and the periodogram for a particular set of realisations is plotted in Figure 13.7a. The periodogram picks up the contributions 12 peak in spectrum at fj = 1/12, Pj = 12 10

8

power 6 4

2

0

0

0.05 0.1 0.15 0.2

0.25 0.3 0.35 0.4 fj

0.45 0.5

Figure 13.6 The periodogram for a single frequency.

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13.3.3 Illustrations (randomising the input)

530 Unit Root Tests in Time Series

10 power at fj = 1/120, Pj = 120

9 8 7 6 power 5

3 2 1 0

0

0.05

0.1 0.15 0.2

0.25 0.3 0.35 0.4 fj

0.45 0.5

Figure 13.7a Periodogram for periods of 12 and 120.

12 power at fj = 1/120, Pj = 120 10

8

power 6 power at fj = 1/4, Pj = 4

4

2

0

0

0.05

0.1 0.15

0.2 0.25 fj

0.3 0.35

0.4 0.45

0.5

Figure 13.7b Periodogram for periods of 4 and 120.

from both frequencies: the long cycle is associated with power at the lower frequency end of the spectrum and the shorter cycle is identiﬁed from the peak at F1 = 1/12. If the data are interpreted as being generated from monthly observations, then the shorter cycle is associated with a seasonal frequency. In a

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power at fj = 1/12, Pj = 12

4

Unit Root Tests for Seasonal Data

0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8

531

detrended, data in logs 0

20

40

60

80

100

120

140

160

180

0.08

long-run cycle peak as fj → 0+, λj → 0+

0.06 power 0.04

seasonal peak at fj = 0.25, λj = π/2

0.02 0

0

0.05

0.1

0.15

0.2

0.25 fj

0.3

0.35

0.4

0.45

0.5

Figure 13.8b Periodogram for levels.

variation of this illustration, if λ1 = 2π/4 then there is a period of 4 and, keeping the long period associated with λ2 = 2π/120, this case is shown in Figure 13.7b. The third illustration in this section uses the log of the quarterly time series for UK seasonally unadjusted expenditure on hotels and restaurants (constant prices). The detrended time series and the periodogram are shown in Figures 13.8a and 13.8b, respectively. Taking the latter ﬁrst, it is evident that there is a peak in the spectrum near to the origin corresponding to a long cycle; there is a second peak at the seasonal frequency of Fj = 1/4; that is, at the quarterly seasonal frequency λj = 2πj/S = π/2 for j = 1 and S = 4; there is some evidence of power at the second seasonal frequency of π and Fj = 1/2, but this is far less pronounced than for Fj = 1/4. As to the power for λj → 0 + , the plot of the detrended series in Figure 13.8a is suggestive of a long cycle, but there is at most only one such cycle within the data period (thus the period is about 40 years). One possible explanation of the spectral peak as λj → 0 + is a (regular) unit root which, if present, can be removed by applying the ﬁrst difference ﬁlter (1 − L), the resulting series being the quarterly growth rate. This series and its associated periodogram are shown in Figures 13.9a and 13.9b, respectively. The peak at the long-run cycle has been removed, leaving the seasonal peak at Fj = 1/4, with some power at Fj = 1/2. (Seasonal integration is considered in detail in section 13.3.5.)

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Figure 13.8a Detrended expenditure on restaurants and hotels.

532 Unit Root Tests in Time Series

0.4

quarterly growth rate: expenditure on restaurants and hotels

0.2 p.a.

0

–0.2 –0.4

0

20

40

60

80

100

120

140

160

180

seasonal peak at fj = 0.25, λj = π/2

0.03

some power at Nyquist frequency of fj = 0.5, λj = π

power 0.02 0.01 0

0

0.05

0.1

0.15

0.2

0.25 fj

0.3

0.35

0.4

0.45

0.5

Figure 13.9b Periodogram for growth rate.

13.3.4 Aliasing (artiﬁcial cycles) Consider the periodic signal generated by yt,cos = cos(λj t) as a continuous function of t; if λj = 6π/12, then this function generates three cycles in each interval of 12 time units, and each has a period of 4 time units. Now suppose that this signal is systematically sampled once every three time units, then what results is shown in Figure 13.10 panel A in two representations: (i) ﬁrst, as a straight line joining each sampled value; (ii) second, as the function yalias t,cos = cos(λm t), where λm is the frequency that generates the same wave form as the original function, but with the period given by the sampled values. In this case there is now (apparently) only one cycle every 12 time units, so that the period is 12 not four. This is an example of aliasing; that is, where the original signal is sampled at intervals that select points from more than one cycle, thus creating an artiﬁcial cycle of longer period than the original. If cycles are generated with the basic time unit of a month and the data are sampled quarterly, at what are referred to as sampling instants, then the original cycles with more than two cycles per year are aliased into longer cycles. The possible frequencies, λj = 2πj/12 for j = 0, . . . , 6, and associated cycles, for monthly data are summarised in Table 13.2. The aliasing has already been illustrated in Figure 13.10, panel A, for λj = π/2 and the remaining graphs in Figures 13.10, panels B–D, complete the set of aliased frequencies. Of particular note is the aliasing of λj = (2/3)π into the zero frequency (see panel B); that is, there

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Figure 13.9a Quarterly growth rate.

Unit Root Tests for Seasonal Data

1

1

B

0

–1

–1 0

10

20 t

10

0 original signal sampled alias

1

C

0

20 t

1

D

0

–1

0

–1 0

10

20

0

10

t

20 t

Figure 13.10 Aliasing of monthly cycles when sampled quarterly.

Table 13.2 Aliased frequencies: monthly data sampled at quarterly intervals. λj Monthly λj Cycles per year Pj (months) Fj (cycle per month) Quarterly λj Cycles per year Pj (quarters) Fj (cycle per quarter) See Figure number

0 0 ∞ 0 0 0 ∞ 0

(1/6)π 1 12 1/12 (1/2)π 1 4 1/4

(1/3)π 2 6 1/6 π 2 2 1/2

(1/2)π 3 4 1/4 aliased 1 4 1/4 13.10A

(2/3)π 4 3 1/3 aliased 0 ∞ 0 13.10B

(5/6)π 5 2.4 5/12 aliased 1 4 1/4 13.10C

π 6 2 1/2 aliased 2 2 1/2 13.10D

is no cyclical behaviour in the function at the aliased frequency, despite there being a cycle in the monthly data. This occurs because the same value of yt,cos is picked out at the systematic sampling instants, resulting in a constant that just depends on the starting point of the sample. The information concerning the original and aliased frequencies is summarised in Table 13.2. Aliasing can be problematic for temporally aggregated data, since we cannot be sure that a periodic cycle is inherent to the original signal or an artifact of the sampling process; however, as elaborated in section 13.8.2.ii, data of different

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A

533

534 Unit Root Tests in Time Series

frequencies can also be used to improve the power of tests for unit roots in seasonal data. 13.3.5 Seasonal integration from a frequency domain perspective Consider the ARMA model given by: φ(L)yt = θ(L)εt

where φ(L) is invertible. Multiplying through by φ(L)−1 gives the inﬁnite MA representation: yt = φ(L)−1 θ(L)εt = w(L)εt

(13.7)

w(L) = φ(L)

−1

θ(L)

The MA representation can be used to obtain the sdf of yt at frequency λj , fy (λj ), which is: fy (λj ) = |w(e−iλj )|2 fε (λj )

(13.8)

where w(e−iλj ) denotes w(.) evaluated at e−iλj and fε (λj ) = (2π)−1 σε2 , a constant, is the sdf of white noise. The function w(.) is the linear ﬁlter that transforms the white noise input into the output stream. Making the substitutions for fε (λj ) and |w(e−iλj )|2 = |θ(e−iλj )|2 |φ(e−iλj )|−2 , then fy (λj ) is: fy (λj ) = |w(e−iλj )|2 =

σε2 2π

(13.9)

|θ(e−iλj )|2 σε2 |φ(e−iλj )|2 2π

Now consider the case where φ(L) is not invertible because of the presence of a conventional unit root, so that φ(L) = (1 − L)φ∗ (L), where φ∗ (L) is invertible. The ﬁlter function associated with the ﬁrst difference operator, (1 − L), is |w(e−iλj )|2 = |(1 − e−iλj )|2 , leading to: fy (λj ) = |(1 − e−iλj )|−2 g(λj )

(13.10)

≈ |λj |−2 g(λj ) Where the second line uses the approximation |1 − exp(−iλj )|−2 ≈ |λj |−2 and: g(λj ) =

|θ(e−iλj )|2 σε2 |φ∗ (e−iλj )|2 2π

(13.11)

As λj → 0 + then g(λj ) → G, a constant, thus fy (λj ) → ∞, so that the spectrum will tend to inﬁnity as λj approaches zero from the right.

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(13.6)

Unit Root Tests for Seasonal Data

535

In the case of d conventional unit roots (or a fractionally integrated process, with d fractional), then (13.10) generalises to: fy (λj ) = |(1 − e−iλj )|−2d g(λj )

(13.12)

where |1 − exp(−iλj )|−2d ≈ |λj |−2d . A process with sdf of the form (13.12) is said to be integrated of order d at the frequency λj = 0. Typically d = 1, but other cases, for example, fractional d and d = 2, have attracted attention. This deﬁnition can be extended to frequencies other than the zero frequency, so that a process is integrated of order d at the frequency λs if fy (λj ) is of the form: fy (λj ) = |(1 − e−i(λj −λs ) )|−2d g(λj − λs )

(13.13)

≈ |(λj − λs )|−2d g(λj − λs ) As λj → λs , g(λj − λs ) tends to a constant, say Gs , so that fy (λj ) → ∞. The case of λs = 0 can be seen as just the special and limiting case of (13.13) where the implicit cycle → ∞, otherwise the cycles associated with the frequencies λs have a ﬁnite period. In the case that the frequencies λs correspond to the harmonic seasonal frequencies then if the sdf has the form (13.13), the series of yt is generated by an integrated seasonal process. Practically, plotting the (sample) periodogram is a useful ﬁrst stage in the analysis of an empirical series as seasonality is associated with peaks in the sdf at the seasonal frequencies.

13.4 Lag operator algebra and seasonal lag polynomials The ﬁrst difference operator applied to a series yt that has a unit root at the zero frequency, will remove the unit root; whereas the seasonal differencing operator (1 − LS ) applied to a series yt that not only has a unit root at the zero frequency, but also unit roots at the seasonal frequencies, will remove S unit roots in total. This section considers the implicit structure of the operator ∆S ≡ (1 − LS ). (See Appendix 2 for the deﬁnition of the lag operator and section A2.4.3 for the seasonal case in particular.) 13.4.1 The seasonal operator in terms of L Recall that the root(s) of a polynomial correspond to the values of z that result in φ(z) = 0. This section considers the roots of φ(z) for the case in which seasonal frequencies are involved. Starting with (1 − L2 ) = (1 − L)(1 + L), this has roots δ1 = 1 and δ2 = –1; it thus has a positive unit root and a negative unit root. Next consider (1 − L3 ) = (1 − L)(1 + L + L2 ). This polynomial has three roots, δi = 1, −0.5 ± 0.866i, where the modulus of both complex roots is 1. The next in this

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≈ |λj |−2d g(λj )

536 Unit Root Tests in Time Series

series is the fourth difference operator (1 − L4 ), which has particular relevance for quarterly data. This can be expanded as follows:

2

(13.14a)

= (1 − L)(1 + L)(1 + L )

(13.14b)

= (1 − L)(1 + L)(1 − iL)(1 + iL)

(13.14c)

From the last line, it is evident that the ﬁrst two factors have roots of δ1 = 1 and δ2 = –1, respectively; and the two other roots are contributed from 1 + L2 , which factors as (1 – iL)(1 + iL) = (1 − i2 L2 ), with roots δ3 = δ4 = ±i, which have a mod√ ulus of 1, where i = −1. Thus the polynomial (1 − L4 ) has four roots that have a modulus of 1. In general, (1 − Lj ) = (1 − L)Sj , where Sj = (1 + L + L2 + . . . + Lj−1 ), has j roots of modulus 1; notice that Sj is the seasonal summation operator that sums the observations over the S seasons. In the case of (1 − L4 ), the + 1 root corresponds to the zero frequency (the ‘long-run’, that is, an ‘inﬁnite’ period); the root –1 corresponds to the frequency Fj = 1/2 cycle per quarter, that is, an angular frequency of λj = π and a semiannual period or cycle of Pj = 2; the root i corresponds to the frequency Fj = 1/4 cycle per quarter, λ = π/2 and an annual cycle P = 4. The root –i cannot j j be distinguished from the root i and is interpreted as the annual cycle. These roots are sometimes written in polar form as exp(±iλj ) for λ0 = 0 and, for the seasonal frequencies, λj = 2πj/S, j = 1, . . . , [S/2]; for example, if S = 4, then λ0 = 0, λ1 = π/2 and λ2 = π and the corresponding roots are e0 = 1, eiπ/2 = i, e−iπ/2 = –i, eiπ = e−iπ = –1. The seasonal differencing operator (1 − L4 ) for quarterly data is therefore somewhat more complex than it seems at ﬁrst sight; tests based on this operator implicitly assume that all the roots with modulus 1 are present, whereas some or none may be present. For example, in each of the following possibilities 1–6, only some of the unit roots are present, with all but the ﬁrst having seasonal components; thus it would not be appropriate to apply the ∆4 operator to remove the seasonal unit roots in these possibilities. Nonseasonal unit root 1. Long-run unit root only: yt = yt−1 + εt , φ(L) = (1 − L) Seasonal DGP (with or without nonseasonal unit root) 2. Nyquist frequency λ2 = π, two cycles a year: yt = −yt−1 + εt , φ(L) = (1 + L) 3. λ1 = π/2, one cycle a year: yt = −yt−2 + εt , φ(L) = (1 + L2 ) 4. Long run + Nyquist: yt = yt−2 + εt , φ(L) = (1 − L)(1 + L) = (1 − L2 ) 5. Long run + λ1 : yt = yt−1 − yt−2 + yt−3 + εt , φ(L)=(1−L)(1 + L2 ) = (1 − L + L2 − L3 ) 6. All seasonal: yt = −yt−1 −yt−2 −yt−3 + εt , φ(L) = (1 + L)(1 + L2 ) = (1 + L + L2 + L3 ) All possible unit roots, seasonal and nonseasonal 7. yt = yt−4 + εt , φ(L) = (1 − L4 ) = (1 − L)(1 + L)(1 + L2 )

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1 − L4 = (1 − L)(1 + L + L2 + L3 )

Unit Root Tests for Seasonal Data

537

6 5 peak at seasonal frequencies fj = 0.25 and fj = 0.5

4 power 3

fj → 0+, λj → 0+

1 0

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 fj Figure 13.11a Periodogram for ∆4 yt = εt .

8 6

quarter 1

4 quarter 2

2 ∆yn,s

0

quarter 4

–2 –4 quarter 3

–6 –8 –10

0

5

10

15 20 t (annual)

25

30

35

Figure 13.11b Seasonal splits for ∆4 yt = εt .

To illustrate some of the seasonal patterns that can arise, ﬁrst consider data simulated from ∆4 yt = εt . There are three peaks in the spectrum: one at the zero frequency, and two at the seasonal frequencies, where the latter generate four distinct quarterly patterns (see Figures 13.11a and 13.11b, respectively). Data simulated from possibilities (2) and (3) above are shown in Figures 13.12a–c; the

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peak at long-run frequency

2

538 Unit Root Tests in Time Series

30 yt = –yt–1 + εt generates peak at seasonal frequency fj = 0.5, λj = π; two cycles per year

25 20

10 5 0

0

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 fj

Figure 13.12a Periodograms for (1 + L)yt = εt and (1 + L2 )yt = εt .

20 quarter 2

15 10 quarter 4

5 ∆yn,s 0 quarter 3

quarter 1

–5 –10 –15 –20

0

5

10

15

20

25

30

35

t (annual) Figure 13.12b Seasonal splits for yt = −yt−1 + εt .

periodogram for both cases is shown in Figure 13.12a, and the seasonal splits are shown in Figures 13.12b (possibility (2)) and 13.12c (possibility (3)). The seasonal patterns can be combined by using the appropriate lag operator either together, as in possibility (6), or with the long-run unit root, possibilities (4) and (5). Finally, possibility (7) has all possible unit roots and is simply ∆4 yt = εt .

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yt = –yt–2 + εt generates peak at seasonal frequency fj = 0.25, λj = π/2; one cycle per year

power 15

Unit Root Tests for Seasonal Data

539

20 15 quarter 1 10 5 ∆yt

quarter 2

0

–10 quarter 3 –15 –20

0

5

10

15 20 n (annual)

25

30

35

Figure 13.12c Seasonal splits for yt = −yt−2 + εt .

13.5 An introduction to seasonal unit root tests: the DHF (1984) test We are now in a position to consider some leading tests for unit roots at the seasonal frequencies. By way of introduction, this section considers the DHF extension of the DF test. This provides a useful basis for considering further developments. The central case is for quarterly data, S = 4; whereas an important secondary case is for monthly data, S = 12. Other cases have been considered in the literature; at a theoretical level, see Smith and Taylor (1999) and for the analysis of daily data within a weekly cycle, S = 5, see Tokihisa and Hamori (2001). 13.5.1 The seasonal DGP Suppose that the data is generated by the following simple seasonal model: yt = ρS yt−S + zt

t = 1, 2, . . . , T

zt = εt

(13.15a) (13.15b)

Initial conditions now relate to the S pre-sample values: y0 , y−1 , ..., y−(S−1) . The null and alternative hypotheses for the seasonal unit root case are: HS,0 : ρS = 1

HS,A : ρS < 1

(13.16)

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quarter 4

–5

540 Unit Root Tests in Time Series

For example, if S = 4 for quarterly data, then this implies a random walk process, but with a time difference of four periods. In this case: yt = yt−4 + εt

(13.17)

(1 − L4 )yt = εt

(13.18)

The LS estimator ρˆ S of ρS is a straightforward extension of the standard (nonseasonal) case for which S = 1: ρˆ S =

∑t=1 y2t−S T

−1

∑t=1 yt yt−S T

assuming pre-sample values exist (13.19)

σˆ (ˆρS ) = σˆ 2

∑t=1 y2t−S T

σˆ ε2 = (T − 1)−1

−1

1/2 (13.20)

∑t=1 (yt − ρˆ S yt−S T

2

δˆ S = T(ˆρS − 1) τˆ S =

(13.21) (13.22)

ρˆ S − 1 σˆ (ˆρS )

(13.23)

Thus the test statistics δˆ S and τˆ S are analogues of the DF test statistics in the nonseasonal model. 13.5.1.ii Seasonal intercepts The model can be extended to allow for S seasonal intercepts as follows: yt = ∑s=1 DVs,t αs + ρS yt−S + εt S

t = 1, 2, . . . , T

(13.24)

where DVs,t is 1 in season s and 0 otherwise. For the s-th season, the model is: yt = αs + ρS yt−S + εt

s = 1, . . . , S

(13.25)

where αs is the DV coefﬁcient associated with the s-th season; thus, in the case that |ρS | < 1, yt is generated as a stationary process about S varying seasonal means. This model can also be reparameterised, as in the standard case, in deviations from (long-run) mean form, but with the means varying according to the season. This will reduce to the standard model in the event that there is no seasonal variation in the means. The intention is to write the model for the s-th season as: (1 − ρS LS )(yt − µs ) = εt

(13.26)

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13.5.1.i No deterministic components

Unit Root Tests for Seasonal Data

541

In the nonseasonal case, S = 1 and µs = µ for all s. A simple rearrangement of (13.26) gives: yt = (1 − ρS )µs + ρS yt−S + εt

(13.27)

By comparison with (13.25), it is evident that: αs = (1 − ρS )µs ⇒

(13.29)

Hence the model of (13.27) can also be written in deviations from mean form as: yt − µs,t = ρS (yt−S − µs,t ) + εt

(13.30)

where µs,t = DVs,t µs , where is DVs,t is 1 in season s and 0 elsewhere. (DHF, 1984, ˆ s , based on replacing αs and note that there are two estimators of µs : ﬁrst, say µ ρS by their LS estimators from (13.25); and, second, the seasonal means of the series.) As in the standard set-up, under the stationary alternative yt will rarely have a zero mean so that allowance should be made for nonzero means, in which case the appropriate maintained regression is (13.24). In order to indicate the deterministic terms included in the maintained regression, a second subscript is introduced on the test statistic. For example, τˆ S,Sµ is the version of τˆ S (see Equation (13.23)), with seasonal means in the maintained regression, as in Equation (13.24). Otherwise the null and alternative hypotheses are unchanged. The DHF test statistic in this case is: τˆ S,Sµ =

ρˆ S − 1 σˆ (ˆρS )

(13.31)

where σˆ (ˆρS ) is the estimated standard error of ρˆ S from (13.24). σˆ ε2 and εˆ t are given by: σˆ ε2 = (T − (S + 1))−1 ∑t=1 εˆ 2t S εˆ t = yt − ∑s=1 DVs,t αˆ s + ρˆ S yt−S T

(13.32) (13.33)

A version of δˆ S , say δˆ S,Sµ , can be deﬁned similarly. A special case of (13.24) occurs when the seasonal means are equal, so that αs = α for s = 1, . . . , S. The model then simpliﬁes to: yt = α + ρS yt−S + εt

t = 1, 2, . . . , T

(13.34)

The DF-type test statistics derived from this regression are denoted δˆ S,µ and τˆ S,µ ; that is, seasonal but without seasonally varying means.

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µs = αs (1 − ρS )

(13.28)

−1

542 Unit Root Tests in Time Series

Table 13.3 Seasonal test statistics distinguished by mean and trend components. Case

Deterministic components

DHF test

1 2 3 4 5

zero mean (none) single mean (constant), µ seasonal means (seasonal intercepts), Sµ single mean and a single trend (constant, trend), µ, β seasonal means and a single trend (seasonal intercepts, single trend), Sµ, β seasonal means and seasonal trends (seasonal intercepts, seasonal trends), Sµ, Sβ

τˆ S τˆ S,µ τˆ S,Sµ τˆ S,µ,β τˆ S,Sµ,β τˆ S,Sµ,Sβ

Note: The possibility of a single mean combined with seasonal trends forces the starting points of the seasonal trends to be equal and is not included.

13.5.1.iii Seasonal deterministic trends The model under the stationary alternative can also be extended to allow for seasonal trends as follows: yt = ∑s=1 DVs,t αs + ∑s=1 DVs,t ηs t + ρS yt−S + εt S

S

(13.36)

Notice that the speciﬁcation in (13.36) also allows for seasonal intercepts, so that the trends are not forced through a single intercept. For the s-th season, s = 1, . . . , S, the speciﬁcation is: yt = αs + ηs t + ρS yt−S + εt

(13.37)

This can be written in deviations form as: yt − µs,t = ρS (yt−S − µs,t ) + εt µs,t = DVs,t µs + DVs,t βs

(13.38) (13.39)

where µs = αs (1 − ρS )−1 and βs = ηs (1 − ρS )−1 . There is a single trend if βs = β for all s and, additionally, a single mean if µs = µ. The DHF t-type test statistic when the regression model includes seasonal means and seasonal trends is denoted τˆ S,Sµ,Sβ , and δˆ S,Sµ,Sβ can be deﬁned analogously. As usual, an alternative to including the deterministic components directly in the maintained regression is to ﬁrst regress yt on the deterministic components and then use the residuals from this prior regression. Table 13.3 summarises the combination of deterministic terms and the notation that is used in this chapter. As shorthand, the combinations are referred to by their case numbers. Those cases of most frequent use are Cases 3, 5 and 6, which allow some seasonal variation under the alternative hypothesis of stationarity; indeed, it seems unlikely that Cases 1, 2 and 4, which do not allow for seasonal variation under HS,A , will have a very wide application.

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6

Unit Root Tests for Seasonal Data

543

13.5.2 Limiting distributions

(i)

(i)

1 Bs (r)dBs (r) ∑S ≡ F(δˆ S,i )DF δˆ S,i ⇒D s=1 0 (i) 1 S ∑s=1 0 Bs (r)2 d(r)

∑Ss=1

1 (i

(i)

Bs (r)dBs (r) 1/2 ≡ F(ˆτS,i )DF 1 (i) ∑Ss=1 0 Bs (r)2 d(r)

τˆ S,i ⇒D

(13.40)

0

(13.41)

(i)

where Bs , s = 1, . . . , S, comprise a set of S independent Brownian motions, either standard, demeaned, seasonally demeaned, detrended or seasonally detrended according to the deterministic speciﬁcation in the maintained regression. ˆ DHF (1984) provided quantiles for the seasonal δ-type and τˆ -type tests, where the maintained regression includes, respectively: no deterministic components, Case 1; a single mean, Case 2; and seasonal means, Case 3, for S = 2, 4, 12. For convenience, some critical values for the 1%, 5% and 10% quantiles are provided in Table 13.4 for Cases 1–6 of Table 13.3 and S = 4, 12. (The rows headed RMA are required later in section 13.8.1 on recursive mean adjusted test statistics.) As noted in section 13.4.1, the seasonal differencing operator (1 − LS ) implies that there are S unit roots, one of which is associated with the ﬁrst-difference operator (1−L) and the remainder are associated with S – 1 seasonal frequencies; thus the DHF test is unable to distinguish whether nonstationarity arises because all S unit roots are present or some lesser, but still positive, number are present. To address this problem, the next section outlines the testing framework due to Hylleberg, Engle, Granger and Yoo (1990), referred to as HEGY. Power and size properties of the various tests are considered later in section 13.9.

13.6 HEGY tests The HEGY test is distinguished from the DHF test as it recognises that the seasonal differencing operator in the quarterly case implicitly contains four roots

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The limiting distributions for the DHF test statistics depend upon the deterministic components and the seasonal parameter S. These are summarised by Taylor (2003) for the leading cases τˆ S , τˆ S,Sµ and τˆ S,Sµ,Sβ , and other cases follow naturally by noting that the appropriate concept of Brownian motion is that for the demeaned or detrended processes as in the speciﬁcation of the maintained regression. Denoting the test statistics generically as δˆ S,i and τˆ S,i , the limiting distributions are:

544 Unit Root Tests in Time Series

Table 13.4 Quantiles for DHF tests. T Case

S=4 1%

5%

10%

200 400

–2.60 –2.49

–1.87 –1.91

–1.54 –1.54

200 400 200 400

–3.08 –3.05 –2.83 –2.80

–2.40 –2.36 –2.15 –2.15

200 400 200 400

–4.67 –4.64 –3.72 –3.76

200 400 200 400

T

S = 12 1%

5%

10%

240 480

–2.46 –2.49

–1.78 –1.81

–1.42 –1.43

–2.00 –1.99 –1.79 –1.80

240 480 240 480

–2.74 –2.75 –2.65 –2.66

–2.02 –2.04 –1.95 –1.96

–1.64 –1.67 –1.58 –1.59

–4.06 –4.08 –3.12 –3.17

–3.75 –3.77 –2.80 –2.83

240 480 240 480

–6.48 –6.46 –5.42 –5.13

–5.82 –5.82 –4.73 –4.49

–5.49 –5.50 –4.37 –4.16

–3.23 –3.17 –3.03 –2.99

–2.47 –2.46 –2.31 –2.33

–2.08 –2.09 –1.93 –1.96

240 480 240 480

–2.74 –2.77 –2.73 –2.73

–2.03 –2.06 –2.01 –2.02

–1.64 –1.68 –1.62 –1.63

200 400 200 400

–4.89 –4.88 –4.03 –4.06

–4.31 –4.30 –3.39 –3.45

–3.99 –4.01 –3.07 –3.13

240 480 240 480

–6.60 –6.57 –5.59 –5.32

–5.94 –5.95 –4.90 –4.66

–5.61 –5.61 –4.54 –4.32

200 400 200 400

–5.89 –5.87 –5.00 –5.12

–5.33 –5.32 –4.42 –4.52

–5.04 –5.02 –4.12 –4.23

240 480 240 480

–8.81 –8.69 –8.22 –7.95

–8.18 –8.13 –7.65 –7.35

–7.86 –7.82 –7.33 –7.03

1

RMA 3: Sµ

RMA 4: µ, β

RMA 5: S, Sµ, β

RMA 6: S, Sµ, Sβ

RMA

Notes: Based on 50,000 replications of the null model yt = ρS yt−S + εt , ρS = 1, where εt is a standard normal white noise input, S = 4, 12. RMA refers to recursive mean adjustment and is considered in section 13.8.1.

of unit modulus, not all of which may be present. This is an important development as a typical ﬁnding of studies of seasonal time series is that there is at least one but less than S unit roots present. Also, as a unit root implies random walk behaviour it may not be theoretically appealing for S random walk components to be unrelated (see section 13.11). The HEGY tests are structured to allow for none, some or all of the unit roots to be present. The plan of this section is to give an introduction to the HEGY testing principles and then elaborate on the empirically important cases of quarterly and monthly seasonal patterns.

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2: µ

Unit Root Tests for Seasonal Data

545

13.6.1 The HEGY regression The starting point is the potentially inﬁnite autoregression, which may arise from an ARMA model with an invertible MA component: A(L)yt = εt

Where A(L) can be an inﬁnite or rational polynomial, but is of order at least equal to the seasonal span S. (Alternatively, one can view the process as being governed by a lag polynomial of order S for yt , with any further dynamics placed into the error process as in an error dynamics model.) The polynomial A(L) is linearised about the zero and seasonal frequency unit roots, exp(±iλs ), as follows (see HEGY, 1990): γ(L) S + R(L) (13.43) A(L) = ∑s=1 κs γs (L) γ(L) = ∏s=1 γs (L) S

1 L λs

(13.45)

A(λs ) S γ (λ ) ∏j =s j s

(13.46)

γs (L) = 1 − κs =

(13.44)

R(L) = γ(L)Ar (L)

(13.47)

Ar (L)

where is a remainder polynomial, with roots outside the unit circle. The idea is to substitute for A(L) in (13.42) using the expansion in (13.43). For example, in the quarterly case, HEGY obtain the following: A(L) = κ1 ( + 1)L(1 + L)(1 + L2 ) + κ2 (−1)L(1 − L)(1 + L2 ) + κ3 (−iL)(1 − L)(1 + L)(1 − iL) + κ4 ( + iL)(1 − L)(1 + L)(1 + iL) + Ar (L)(1 − L4 ) = − Π1 L(1 + L)(1 + L2 ) − Π2 (−L)(1 − L)(1 + L2 ) − (Π4 + Π3 L)(−L)(1 − L2 ) + Ar (L)(1 − L4 ) = − Π(L) + Ar (L)(1 − L4 )

(13.48)

where Π1 = − κ1 , Π2 = −κ2 , 2κ3 = −Π3 + iΠ4 , 2κ4 = −Π3 − iΠ4 ; and the polynomial Π(L) is, apart from sign, everything other than Ar (L)(1 − L4 ) on the right-hand side of (13.48). Let p∗ denote the order of Ar (L), so that p∗ = 0 if the order of A(L) is exactly S, but otherwise p∗ > 0 (for the connections between the Πj and κj coefﬁcients in the monthly case, see Beaulieu and Miron, 1993, p.323). Multiplying the right-hand side of A(L) by yt , using (13.42) and rearranging gives: Ar (L)∆4 yt = ∑s=1 Πs Ys,t−1 + εt 4

(13.49)

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(13.42)

546 Unit Root Tests in Time Series

Y1,t ≡ (1 + L)(1 + L2 )yt = (1 + L + L2 + L3 )yt

(13.50)

Y2,t ≡ − (1 − L)(1 + L2 ) = − (1 − L + L2 − L3 )yt

(13.51)

2

Y3,t ≡ − (1 + L )Lyt = Y4,t−1

(13.52)

Y4,t ≡ − (1 + L2 )yt

(13.53)

Then, taking lagged terms in ∆4 yt to the right-hand side results in the following (augmented) HEGY regression for quarterly data: p∗

∆4 yt = ∑s=1 Πs Ys,t−1 + ∑j=1 αj ∆4 yt−j + εt 4

(13.54)

For general S, the (augmented) HEGY regression is: p∗

∆S yt = ∑s=1 Πs Ys,t−1 + ∑j=1 αj ∆S yt−j + εt S

(13.55)

It is important to allow for deterministic variables in the maintained regression, as in the DHF maintained regression, which can be achieved either by including them directly in (13.55), or adjusting the data by using the residuals from a prior regression of yt on the set of deterministic variables. The set of possible deterministic terms is as for the DHF tests, summarized in Table 13.3, and the same subscript notation is adopted for the HEGY case. 13.6.2 The constructed variables and the seasonal frequencies Before outlining the set of HEGY test statistics, it is helpful to be able to connect the constructed variables in the regression to the seasonal frequencies to which they relate. In order to do so requires a slightly different notational arrangement from the original HEGY set-up. The framework followed, with a sight variation, is due to Smith and Taylor (hereafter ST) (1999, 2001). First consider the case where S is even, which implies λS/2 = π, then the constructed variables are: y1,t ≡ y(S/2) + 1,t ≡ ys + 1,t ≡

∑j=1 cos(jλ0 )yt−(j−1) = ∑j=1 yt−(j−1)

λs = 0

(13.56)

∑j=1 cos(jλS/2 )yt−(j−1)

λS/2 = π

(13.57)

∑j=1 cos(jλs )yt−(j−1)

λs = 2sπ/S

(13.58)

λs = 2sπ/S

(13.59)

S

S

S S

yS + 1−s,t ≡ − ∑j=1 sin(jλs )yt−(j−1) S

where s = 1, . . . , [S/2]. As a check, note that y1,t is associated with the zero frequency and y(S/2) + 1,t with the Nyquist frequency; there are then (S – 2) pairs of variables associated with the harmonic seasonal frequencies. If S is odd, as for daily data with a period of ﬁve or seven days, then S/2 is not an integer and the term associated with π is omitted. For example, if S = 5,

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where the constructed variables are:

Unit Root Tests for Seasonal Data

547

y1,t is associated with the zero frequency, y2,t and y5,t are associated with the frequency 2π/5, and y3,t and y4,t are associated with the frequency 4π/5. The augmented HEGY regression has the same general form, but now with the constructed variables as in (13.56)–(13.59): S

(13.60)

This is the notational framework that is used in the remainder of this chapter, unless otherwise noted; and, for simplicity, S is assumed to be even. Note the coefﬁcients on the constructed variables ys,t−1 are denoted πs to distinguish them in this version from Πs of (13.55). 13.6.3 The structure of hypothesis tests The nature and structure of hypothesis tests of interest follows from identifying the relevant variables and coefﬁcients with the associated frequency, as in (13.56)–(13.59). In each case, the null hypothesis is of a unit root, but there are now S possible unit roots. Several tests are available that fall into three categories: 1. 2. 3.

a t-type test for a unit root at the long-run frequency and a similar test for a unit root at the Nyquist frequency; F-type tests for a unit root at a single frequency, for seasonal frequencies apart from the Nyquist frequency; F-type tests of a joint null hypothesis corresponding to the presence of more than one unit root.

Distributional results for the various test statistics are collected in section 13.6.5. 13.6.3.i Tests for the roots +1, −1 The unit root null hypothesis for the long-run frequency is H0,0 : π1 = 0; the alternative being H0,A : π1 < 0, implied by the stationary alternative A(1) > 0. This can be tested by a t-type test, say t1 , with large negative values leading to rejection. The null hypothesis for the negative unit root, corresponding to the Nyquist frequency λS/2 = π, for S even, is H(S/2),0 : π(S/2) + 1 = 0, to be tested against the alternative H(S/2)A : π(S/2) + 1 < 0, implied by the stationary alternative A(−1) > 0 ; again this can be tested by a t-type test, designated t(S/2) + 1 , with large negative values leading to rejection. 13.6.3.ii Tests for unit roots at the harmonic seasonal frequencies, λs , s = 0, s = S/2 The remaining seasonal frequencies occur in pairs, leading to a set of joint null hypotheses for a unit root at the s-th frequency: Hs,0 : πs + 1 = πS + 1−s = 0 for the s-th seasonal frequency, s = 1, . . . , [S/2], implying |A(λs )| = 0. The alternative hypothesis is two-sided, Hs,A : πs + 1 = 0 and/or πS + 1−s = 0, with |A(λs )| > 0.

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p∗

∆S yt = ∑s=1 πs ys,t−1 + ∑j=1 αj ∆S yt−j + εt

548 Unit Root Tests in Time Series

13.6.3.iii Overall tests All S unit roots An overall test of unit roots at the zero and seasonal frequencies is an Sdimensional joint test with null hypothesis: HS,0 : πs = 0, s = 1, . . . , S, for which an F-type test, denoted FAll , can be used. Although the null hypothesis is the same as in the DHF case, the alternative allows for stationarity arising from the zero or any of the seasonal frequencies; hence it is speciﬁed as HAny,A : πs = 0 for at least one value of s. Whilst the null hypothesis of the overall HEGY test is the same as for the DHF test, the alternative hypotheses differ; that of the HEGY test FAll allows rejection if some but not all of the S unit roots are present. All seasonal unit roots A variation on the joint testing theme is the null hypothesis that all of the seasonal unit roots are absent; that is, say, HAS,0 : πs = 0, s = 2, . . . , S against the alternative, HAS,A : πs = 0 for at least one value of s = 2, . . . , S. An F-type test is again appropriate and is denoted FAS ; this test differs from FAll in not including the zero frequency component. As is usual with F-tests, they are implicitly two-sided; hence, whilst the presumption of a rejected null hypothesis may be stationarity, this is not strictly the case. 13.6.4 The HEGY tests: important special cases The frequencies of data most used for hypothesis testing for seasonal unit roots are quarterly and monthly; this section, therefore, deals with these as two examples within the general framework; a question considers the weekly case, where S is odd. 13.6.4.i Quarterly data In the case of quarterly data, the seasonal differencing operator is (1 − L4 ) with four roots of modulus 1. The two (resolvable) seasonal frequencies in this case are: λ1 = 2π/4 = (1/2)π, λ2 = 4π/4 = π; which correspond to Ps = 4 and 2, respectively, in quarters and Fs = 1/4, 1/2 in fractions of cycles per quarter; λ1 corresponds to the roots ±i and λ2 to the negative unit root; in addition, λ0 = 0 refers to the standard unit root, that is at the zero frequency. The frequencies, periods and roots for the quarterly case are summarised in Table 13.5; Table 13.6

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The F-type test for this null is denoted Fs + 1,S−s + 1 . For example, if S = 4, then a test of a unit root at the frequency λ1 = π/2 is a test of the joint null H1,0 : π2 = π4 = 0, requiring both y2,t and y4,t to be absent from the regression, and implies |A(i)| = 0; the F-test is denoted F2,4 . The alternative hypothesis is twosided, H1,A : π2 = 0 and/or π4 = 0, with |A(i)| > 0. HEGY suggested a pair of t-type tests could be used to test for unit roots at these frequencies, and also provided critical values; however, such tests are not generally pivotal (see Burridge and Taylor, 2001) and are not widely used (see also section 13.6.5.iv).

Unit Root Tests for Seasonal Data

549

λs (angular)

0

(1/2)π

π

Ps (months) Fs (cycle per month) Root(s): exp(±iλs ) Modulus

∞ 0 1 1

4 1/4 ±i 1

2 1/2 –1 1

Table 13.6 Constructed variables and seasonal frequencies: the quarterly case.

y1,t y2,t y3,t y4,t

Constructed variable

Alternative (HEGY) form

πs

λs

∑4j=1 cos(0)yt−(j−1) ∑4j=1 cos(jλ1 )yt−(j−1) ∑4j=1 cos(jλ2 )yt−(j−1) − ∑4j=1 sin(jλ1 )yt−(j−1)

= ∑4j=1 yt−(j−1) = −(1 − L2 )Ly

π1 π2 π3 π4

λ0 = 0 λ1 = π/2 λ2 = π λ1 = π/2

t = −(1 − L + L2 − L3 )yt = −(1 − L2 )yt

Notes: cos(0) = 1, cos(π) = cos(3π) = –1, cos(2π) = cos(4π) = + 1; cos[(1/2)π] = cos[(3/2)π] = 0; and sin[(1/2)π] = 1, sin[(3/2)π)] = –1, sin(π) = sin(2π) = 0.

Table 13.7 Hypothesis tests: the quarterly case. H0 :

H0,0

Unit root

H1,0

H2,0

HAS,0

HS,0

Nonseasonal π1 = 0

Harmonic seasonal π2 = π4 = 0

Nyquist

All seasonal

All unit roots

π3 = 0

HA

π1 < 0

π3 < 0

πs = 0, s = 1, 2, 3, 4 At least one of πs = 0

Test

t1 : ‘t’ left tail

π2 = 0 and/or π4 = 0 F-type, F2,4

π3 = π2 = π 4 = 0 At least one of π2 = 0, π3 = 0, π4 = 0 F-type, FAS

t3 : ‘t’ left-tail

F-type, FAll

deﬁnes the constructed variables on the ST and HEGY deﬁnitions, and shows how they relate to the (angular) frequencies; and the structure of hypothesis tests for quarterly data is shown in Table 13.7. In each case, the null hypothesis is of a unit root, but recall that there are three possibilities for quarterly data, that is, + 1, –1, |i|; the alternative in each case is stationarity. The unit root test at the long-run frequency is of H0,0 : π1 = 0; the alternative being H0,A : π1 < 0, implied by the stationary alternative A(1) > 0. The null hypothesis for the negative unit root, corresponding to the frequency λ2 = π, is H2,0 : π3 = 0 against the alternative H2,A : π3 < 0, implied by the stationary alternative A(−1) > 0; the null hypothesis for the remaining frequency,

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Table 13.5 Seasonal frequencies, periods and roots: the quarterly case.

λ1 = π/2, is the joint null H1,0 : π2 = π4 = 0, as it requires both y2,t and y4,t to be absent from the regression and implies |A(i)| = 0. The alternative hypothesis is two-sided, H1,A : π2 = 0 and/or π4 = 0, with |A(i)| > 0. An alternative testing procedure for H1,0 , leads to the use of two t-type tests: ﬁrst test π4 = 0 against π4 = 0, with a t-type test, say t4 , and if this is not rejected test π2 = 0 against π2 < 0, with a t-type test, say t2 (see HEGY, 1990, for details and justiﬁcation); however, as noted above, such tests are not asymptotically pivotal, a problem discussed further below. A joint test of the absence of any seasonal unit roots is provided by FAS , with the null hypothesis designated H4S,0 : π2 = π3 = π4 = 0, and the alternative hypothesis being the negation of this. The overall test of no unit roots is a joint test of HS,0 : πs = 0, s = 1, 2, 3, 4, for which the F-type test, FAll , can be used. 13.6.4.ii Monthly data In the case of monthly data, the seasonal differencing operator is ∆12 ≡ (1−L12 ), so that: (1 − L12 ) = (1 − L)(1 + L + L2 + . . . + L11 )

(13.61)

The operator ∆12 implies 12 roots of modulus 1. The six seasonal frequencies in this case are: λ1 = ±2π/12 = ±(1/6)π, λ2 = ±4π/12 = ±(1/3)π, λ3 = ±6π/12 = ±(1/2)π, λ4 = ±8π/12 = ±(2/3)π, λ5 = ±10π/12 = ±(5/6)π and λ6 = π. These correspond to the following cycles per year: 1 and 11 for λ1 ; 2 and 10 for λ2 ; 3 and 9 for λ3 ; 4 and 8 for λ4 ; 5 and 7 for λ5 ; and 6 for λ6 . To obtain the period in months, Ps , associated with each cycle, divide 12 by the number of cycles per year – for example, 12 and 12/11 for λ1 ; and to obtain the fraction of a cycle per month, Fs , divide the number of cycles by 12 – for example, 1/12 and 11/12 for λ1 . The seasonal frequencies λs are shown in Figure 13.13. Note that these divide the circle into 12 equal segments in units of (1/6)π, with the frequencies below the horizontal paired with those above the horizontal; these correspond, respectively, to the negative value and positive value in each of the two values of λs . It may be helpful to illustrate the frequencies arising from monthly data by graphing the periodogram for data simulated from ∆12 yt = εt , where T = 30×12 = 360 and εt is drawn from niid(0, 1). The data realisations and the periodogram are shown in Figures 13.14a and 13.14b, respectively; note that the peaks in the spectral density occur, as expected, at the long run and the seasonal frequencies. The frequencies, periods and roots for the monthly case are summarised in Table 13.8; and Table 13.9 deﬁnes the constructed variables on the ST deﬁnition (Beaulieu and Miron (BM) 1993, consider the monthly case using a HEGY type notation). Table 13.9 shows how the seasonal frequencies are related to the constructed variables, and hence how the hypothesis tests are structured

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550 Unit Root Tests in Time Series

Unit Root Tests for Seasonal Data

551

(1/2)π (2/3)π

(1/3)π

(5/6)π

(1/6)π

0

radians

(1/6)π

(5/6)π

(2/3)π

(1/3)π (1/2)π

Figure 13.13 The monthly, seasonal frequencies λj .

10 8 6 4 yt

2 0 –2 –4 –6 –8

0

50

100

150

200

250

300

350

Figure 13.14a Simulated data for ∆12 yt = εt .

to reﬂect the presence or absence of a unit root at a particular frequency. The consequent sub-hypotheses are detailed in the Table 13.10. Notice the alternative hypotheses for H0,0 and H6,0 are (negative) one-sided, implying that they can be tested with a t-type test, with rejection based on

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π

radians

552 Unit Root Tests in Time Series

λ1 = (1/6)π

8

from simulated data

7 λ5 = (5/6)π

6 λ0 → 0+

5 power

λ4 = (2/3)π

λ2 = (1/3)π

λ3 = (1/2)π

4

λ6 = π

2 1 0

0

0.05 0.1 0.15 0.2

0.25 0.3 0.35 0.4 fj

0.45 0.5

Figure 13.14b Periodogram for ∆12 yt = εt .

Table 13.8 Seasonal frequencies, periods and roots: the monthly case. λs (angular)

0

(1/6)π

(1/3)π

(1/2)π

(2/3)π

(5/6)π

π

Ps (months) Fs (cycle per month) Root(s): exp(±iλs )

∞ 0 1

3 1/3 −0.5± 0.866i 1

2.4 5/12 –0.866 ±0.5i 1

2 1/2 –1

1

6 1/6 0.5± 0.866i 1

4 1/4 ±i

Modulus

12 1/12 0.866 ±0.5i 1

1

1

the lower quantiles (for example, 5%); the remaining alternative hypotheses are two-sided, implying that they can be tested with an F-type statistic (the subscripts indicate the coefﬁcients to be set to zero, not the degrees of freedom).

13.6.5 Limiting distributions and critical values The idea in this section is to consider the case of monthly data ﬁrst, then both the general case and other speciﬁc cases – for example, quarterly data – are easily established. The notation thus initially relates to the monthly case. Reference to ‘distribution’ refers to limiting distribution under the null hypothesis; where emphasis is required, an N subscript is added to the test statistic to indicate that this is the limiting null distribution. Osborn and Rodrigues (2002) provide extensive and unifying distributional results for the leading tests.

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3

Unit Root Tests for Seasonal Data

553

ys+1,t

Constructed variable (ST notation)

Associated coefﬁcient: πs

Associated angular frequency: λs

y1,t y2,t y3,t y4,t y5,t y6,t y7,t y8,t y9,t y10,t y11,t y12,t

12 ∑12 j=1 cos(0)yt−(j−1) = ∑j=1 yt−(j−1) cos(jλ )y ∑12 1 t−(j−1) j=1 ∑12 j=1 cos(jλ2 )yt−(j−1) ∑12 j=1 cos(jλ3 )yt−(j−1) ∑12 j=1 cos(jλ4 )yt−(j−1) ∑12 j=1 cos(jλ5 )yt−(j−1) ∑12 j=1 cos(jλ6 )yt−(j−1) − ∑12 j=1 sin(jλ5 )yt−(j−1) − ∑12 j=1 sin(jλ4 )yt−(j−1) − ∑12 j=1 sin(jλ3 )yt−(j−1) − ∑12 j=1 sin(jλ2 )yt−(j−1) − ∑12 j=1 sin(jλ1 )yt−(j−1)

π1 π2 π3 π4 π5 π6 π7 π8 π9 π10 π11 π12

λ0 = 0 λ1 = π/6 λ2 = π/3 λ3 = π/2 λ4 = 2π/3 λ5 = 5π/6 λ6 = π λ5 = 5π/6 λ4 = 2π/3 λ3 = π/2 λ2 = π/3 λ1 = π/6

Table 13.10 Hypothesis tests: the monthly case. H0 :

H0,0

H6,0

H12S,0

HS,0

Unit root→ H0 HA

Nonseasonal

All seasonal

All unit roots

π1 = 0 π1 < 0

Nyquist λ6 = π π7 = 0 π7 < 0

Test

t1

t7

H0 :

H1,0

H2,0

H3,0

H4,0

H5,0

Unit Root→

λ1 = π/6