Coping with the complexity of economics

New Economic Windows Series Editors M F, M G, A K Series Editorial Board Jaime Gil Aluja Departament d’Economia i Organització d’Empreses, Universitat de Barcelona, Spain

Fortunato Arecchi Dipartimento di Fisica, Università di Firenze and INOA, Italy

David Colander Department of Economics, Middlebury College, Middlebury, VT, USA

Richard H. Day Department of Economics, University of Southern California, Los Angeles, USA

Steve Keen School of Economics and Finance, University of Western Sydney, Australia

Marji Lines Dipartimento di Science Statistiche, Università di Udine, Italy

Thomas Lux Department of Economics, University of Kiel, Germany

Alfredo Medio Dipartimento di Scienze Statistiche, Università di Udine, Italy

Paul Ormerod Directors of Environment Business-Volterra Consulting, London, UK

Peter Richmond School of Physics, Trinity College, Dublin , Ireland

J. Barkley Rosser Department of Economics, James Madison University, Harrisonburg, VA, USA

Sorin Solomon Racah Institute of Physics, The Hebrew University of Jerusalem, Israel

Pietro Terna Dipartimento di Scienze Economiche e Finanziarie, Università di Torino, Italy

Kumaraswamy (Vela) Velupillai Department of Economics, National University of Ireland, Ireland

Nicolas Vriend Department of Economics, Queen Mary University of London, UK

Lotfi Zadeh Computer Science Division, University of California Berkeley, USA

Editorial Assistants: Giovanna Bimonte Dipartimento di Studi Economici, Università degli Studi di Napoli “Parthenope”, Italy

Anna Parziale Dipartimento di Scienze Economiche e Statistiche, Università degli Studi di Salerno, Italy

Marisa Faggini · Thomas Lux (Eds.)

Coping with the Complexity of Economics

123

MARISA FAGGINI Dipartimento di Scienze Economiche e Statistiche Università degli Studi di Salerno, Italy THOMAS LUX Department of Economics University of Kiel, Germany

The publication of this book has been made possible thanks to the financial support of the project GIACS n° 012380

Library of Congress Control Number: 2008936993

ISBN-978-88-470-1082-6 Springer Milan Berlin Heidelberg New York e-ISBN ---108 - This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the Italian Copyright Law in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the Italian Copyright Law. Springer is a part of Springer Science+Business Media springer.com c Springer-Verlag Italia 9  Printed in Italy Cover design: Simona Colombo, Milano c www.pixed.org Cover figure:  Typesetting: by the Autors using Springer Macro package Printing and binding: Grafiche Porpora, Segrate (MI) Springer-Verlag Italia – Via Decembrio  –  Milano Printed on acid-free paper

Preface

Throughout the history of economics, a variety of analytical tools have been borrowed from the so-called exact sciences. As Schoeffler (1955) puts it: “They have taken their mathematics and their deductive techniques from physics, their statistics from genetics and agronomy, their systems of classification from taxonomy and chemistry, their model-construction techniques from astronomy and mechanics, and their methods of analysis of the consequences of actions from engineering”. The possibility of similarities of structure in mathematical models of economic and physical systems has been an important factor in the development of neoclassical theory. To treat the state of an economy as an equilibrium, analogous to the equilibrium of a mechanical system has been a key concept in economics ever since it became a mathematically formalized science. Adopting a Newtonian paradigm neoclassical economics often is based on three fundamental concepts. Firstly, the representative agent who is a scale model of the whole society with extraordinary capacities, particularly concerning her capability of information processing and computation. Of course, this is a problematic reduction as agents are both heterogeneous and boundedly rational and limited in their cognitive capabilities. Secondly, it often confined itself to study systems in a state of equilibrium. But this concept is not adequate to describe and to support phenomena in perpetual motion. As it soon turned out, the equilibrium paths of even very standard economic models are much richer than the saddle-point variety initially favored by neoclassical theory allows for. In fact a dynamic economic system might not approach a steady state, but could end in limit cycles, in which variables endlessly repeat cyclical movements, or even in chaotic paths of a highly irregular kind. Lastly, linear models or at least the linearization of models in the neighborhood of an

VI

Preface

equilibrium have been traditionally preferred by economists. But linear models do not appropriately cover the asymmetries of phenomena like depressions and recessionary periods, stock market price bubbles and corresponding crashes, persistent exchange rate movements, or the occurrence of regular and irregular business cycles. Furthermore, the mechanical character of many economic models does not allow them to treat evolution or structural changes in the systems. More fundamentally, the reductionist approach, applied by traditional economic theory, often overlooks the dependencies or interconnections among elements and their influence upon macroeconomic behaviour. Its focus is not to study the unfolding of the patterns its agents create, but rather to simplify its questions in order to seek analytical solutions. As progress has been made in the exact sciences and new mathematical and statistical tools have become available, it has became impossible for the economists to ignore that many important and interesting economic phenomena cannot be understood using the approach of traditional linear representative agent models. Economies are complex adaptive systems, that is, composed of a large number of interacting components and of the relationships between them. “The goal of complex systems research is to explain in a multidisciplinary way how complex and adaptive behaviour can arise in systems composed of large numbers of relatively simple components, with no central control, and with complicated interactions”(Mitchell 1998; Crutchfield 1994). This awareness and consequently the requirement of more realistic models have lead to powerful new concepts and tools to detect, analyze and cope with them. The new perspective opened by non-equilibrium thermodynamics and the advances in nonlinear dynamic systems theory has spawned significant changes in mainstream economic theory. Economists have begun to study the general emergence of structures and unfolding of patterns in the economy, recognizing the complexity of economics and applying insights gained from Complexity Theory. Admittedly, the notion of Complexity Theory offers a diverse set of new conceptual tools to help explain the diversity of and change of economic systems and so far lacks a unifying framework. This book collects some of these approaches trying to give the reader an overview of recent developments in theory and empirical research that view economic systems as complex phenomena whose aggregate dynamics can often not be inferred from its microscopic (microeconomic) building blocks. This volume covers a collection of peer-reviewed papers on various facets of economic complexity selected from the presentations at the Ecople Conference 2006. It is dedicated to the memory of Massimo Salzano, who has been such a

Preface

VII

fervent and eloquent advocate of the complexity approach, and a most enjoyable companion and amiable colleague.

July 2008

Marisa Faggini Thomas Lux

Dedication to Massimo Salzano July 2nd 1946 - October 13rd 2007

Advocates of the complex systems approach to economics are comfortable with the proposition that human social systems are rarely, if ever, in equilibrium. This acceptance of the permanence of change makes us no more immune to emotional shock than other human beings, however, as we realised when we received the unexpected news of the death of one of complexity’s champions, Massimo Salzano. Someone we always thought would be with us was suddenly gone. It did not seem fair-and it was not. Massimo’s intellectual dynamism and outstanding personal bonhomie helped establish many of the intellectual partnerships in modern econophysics. He should have lived a long life to enjoy seeing that research take economics on a new, dynamic and realistic path. Unfortunately, this was not to be, and we are left only with the memory of a generous man who did much to foster a sense of unity of purpose amongst European researchers into complexity. Massimo had a long and distinguished career in Italian public economics before his interest in the then nascent field of complexity brought him to international prominence, when he organized the “New Economic Windows”conference in Salerno in September 2001. He had a knack for knowing which economists would be receptive to the very different techniques of physicists. Whereas previous meetings of economists and physicists were frequently characterized by cognitive dissonance between the two camps-as with the famous meeting at Sante Fe-NEW 2001 was marked by harmony, and plans for future collaboration. Those plans came to fruition, again thanks to Massimo’s organization skills and infectious vitality. He established a European Masters in complexity program, played an important role in the development of the COST P10 Action “The Physics of Risk”, edited the New Economic Windows Series for Springer-Verlag, and all the while lived life to the full- and

IX

encouraged those about him to do likewise. His personal contributions to complexity analysis were notable, though tragically cut short by his early death. Massimo’s starting position on the relevance of complexity for economic theory was that the multiple, often unstable equilibria of complex systems “does not allow one to use the tool of optimization employed in the traditional [economic] approach to the decision theory” (Massimo Salzano 2007, “Economic Policy Hints from Heterogeneous Agent-Based Simulation”, in Colander, D. & Salzano, M, [eds.] Complexity and Policy Analysis, Springer Verlag, Berlin, pp. 244-271, p. 246). His primary passions in complexity was for heterogeneous agent modelling-in contrast to the “Representative Agent” fiction that dominates modern neoclassical economics-and network theory. Both these approaches necessarily involved the rejection of the “micro-macro” distinction that has become synonymous with conventional economic theory. His work of the interaction of heterogeneous agents in macroeconomic also challenged the conventional policy focus on optimality, arguing instead that “in the case of complex systems, we need to obtain economic policies that are “robust” rather than “optimal” as the concept of optimality is illogical for such systems” (ibid, p. 270). “Robust” itself is an appropriate word to associate with Massimo, and his engagement with the world; we only regret that his robust humanity and intellect were cut short by the vissisitudes of health. Massimo will be fondly remembered for his role in nurturing econophysics and the complex systems approach to economics, and lamented for not being with us to celebrate both its maturity, and life itself. Extracts of messages on the news of Massimo’s death “We’ll miss his volcanic attitude toward life.”Mauro Gallegati “With the clear image of Massimo’s vital personality in my mind, it appears hard to believe in the message of his death. I have very lively memories of last year’s 60th birthday celebration at Capri and his intoxicating engagement for our research field. Marisa and I am about to finish the editorial process of the proceedings volume for the Capri meeting. We will dedicate it to Massimo’s memory as someone who devoted so much energy to complexity research.”(Thomas Lux); “I will miss Massimo very much. I will miss in particular his open and reassuring smile notwithstanding the complexity of life that he rightly urged us to study” (Alessandro Roncaglia); “I thought he was indestructible and will miss him” (Alan Kirman); “What a shock.. He was one of the most alive people I ever met... I am missing him already” (Sorin Solomon); “Many of us recall meeting Massimo, at various professional meetings, across Europe. One in particular that some of us recall vividly was organised in Capri where we

X

celebrated in some style, with a truly magnificent display of fireworks, his 60th birthday. A truly amazing event! Most recently, in May I had the privilege to visiting Salerno to give a seminar to the students in his department. Massimo was always enthusiastic and supportive of his young students and during the seminar he made active contributions not only of a technical nature but also to ensure that those whose English was not always the best understood what I was saying.”(Peter Richmond) “In 9/2004 Massimo was the first economist to give me a forum to express my views before a gathering of other economists. I liked him and remain grateful to him.”(Joe McCauley) “Mauro is right, volcanic”. I would say “a force of nature,”uno forza natura, just to slaughter some Italian. We especially remember the second complexity conference he hosted at Salerno and the dramatic evening he provided for so many of us at the ruined Greek temples at Paestum. There will be no replacing this volcanic force of nature.”(Barkley Rosser) “We are quite shocked about his much too early death at the age of 61. We will miss him and his summer conference, which he planned again for 2008. We had told him that we are performing Giuseppe Verdi’s Requiem in Paderborn next weekend, he liked it very much. Now we will sing it for him.”(Juergen and Etta Mimkes) Steve Keen

Contents

Part I Evolution of Economic Systems Evolution of Trust in Economic Systems Terry Bossomaier, Mike Harr´e, James Thompson . . . . . . . . . . . . .

3

Keynes, Hayek and Complexity Paul Ormerod . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 A look at the relationship between industrial dynamics and aggregate fluctuations Domenico Delli Gatti, Edoardo Gaffeo, Mauro Gallegati . . . . . . . . . 33 Part II Quantum Physics and Financial Markets Quantum-like Viewpoint on the Complexity and Randomness of the Financial Market Olga Choustova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Classical and Quantum-Like Randomness and the Financial Market Andrei Khrennikov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Part III New Empirical Approaches to Complex Economic Dynamics Examination of the Cost-of-Carry Formula for Futures Contracts on WIG20. Wavelet and Nonlinear Cointegration Analysis

XII

Contents

Joanna Bruzda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 A Complex Approach to Estimate Shadow Economy: The Structural Equation Modelling Roberto Dell’Anno, Friedrich Schneider . . . . . . . . . . . . . . . . . . . . . . . 111 Will house prices rise in 2007? A comparative assessment of house prices in London and Dublin Peter Richmond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 Part IV Models of Risk Management and Decision Support Risk Measuremant and Fair Valuation Assessment in the Life Insurance Field Mariarosaria Coppola, Valeria D’Amato, Emilia di Lorenzo, Marilena Sibillo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Analysis of Qualitative and Quantitative Rankings in Multicriteria Decision Making Livia D’Apuzzo, Gabriella Marcarelli, Massimo Squillante . . . . . . . 157

List of Contributors

Bossomaier NameTerry Centre for Research in Complex Systems Charles Sturt University, Australia [email protected]

D’Apuzzo Livia Dipartimento di Costruzioni e Metodi Matematici in Architettura, Universit` a di Napoli, Italy [email protected]

Bruzda Joanna Dell’Anno Roberto Nicolaus Copernicus University Dipartimento di Economia, of Toru˜ n, Poland Matematica e Statistica, [email protected] Universit` a di Foggia, Italy [email protected] Choustova Olga International Center for Delli Gatti Domenico Mathematical Modeling in Universit` a Cattolica di Milano, Physics, Engineering and Italy Cognitive Science, [email protected] V¨axj¨ o University, Sweden di Lorenzo Emilia [email protected] Universit` a degli Studi di Napoli Federico II, Italy Coppola Mariarosaria [email protected] Universit` a degli Studi di Napoli Federico II, Italy [email protected] D’Amato Valeria Universit` a degli Studi di Napoli Federico II Italy [email protected]

Gaffeo Edoardo Universit` a di Trento, Italy [email protected] Gallegati Mauro Politecnico delle Marche, Italy [email protected]

XIV

List of Contributors

Harr´ e Michael School of Information Technologies and The Centre for the Mind University of Sydney, Australia [email protected] Khrennikov Andrei International Center for Mathematical Modeling in Physics, Engineering and Cognitive Science, V¨axj¨ o University, Sweden [email protected] Marcarelli Gabriella Dipartimento di Analisi dei Sistemi Economici e Sociali, Universit` a del Sannio, Italy [email protected] Ormerod Paul Volterra Consulting, London, UK [email protected]

Richmond Peter School of Physics Trinity College Dublin 2, Ireland [email protected] Schneider Friedrich Department of Economics, Johannes Kepler University Linz, Austria [email protected] Sibillo Marilena Universit` a degli Studi di Salerno [email protected] Squillante Massimo Dipartimento di Analisi dei Sistemi Economici e Sociali, Universit` a del Sannio, Italy [email protected] Thompson James Centre for Research in Complex Systems Charles Sturt University, Australia [email protected]

Part I

Evolution of Economic Systems

Evolution of Trust in Economic Systems Terry Bossomaier1 , Mike Harr´e1 and James Thompson3 1 2

Centre for Research in Complex Systems - Australia School of Information Technologies and The Centre for the Mind Australia

1 Introduction When ET in the eponymous Spielberg movie follows a trail of Smarties to meet his new human friends, we find this entirely plausible – that a baby alien will quickly form a bond with human kids and for each to quickly learn to trust the other. This surely reflects the way trust is such a fundamental part of human nature that its absence would be more remarkable than its presence, even where one of the protagonists is a not overly attractive alien species! Another, less plausible, fictional character which has often been employed in economic settings is Homo Economicus. The character played by this economic actor is that of the perfectly informed, perfectly rational, self interested individual trying to maximise their personal utility. This view of the economic actor, particularly as it is used in decision theory, inspired Herbert Simon in his Nobel Prize award speech to criticise models of economic microphenomena [29]: “Thus economists who are zealous in insisting that economic actors maximize turn around and become satisficers when the evaluation of their own theories is concerned. They believe that businessmen maximize, but they know that economic theorists satisfice.” However, in the three decades since Simon’s speech there has been a large body of evidence accumulating which demonstrates the lack of universality of the assumptions of economic man [6] and has contributed to the advancement of more reasonable models, as described by Daniel Kahneman [14], another Nobel Laureate: “. . . proposed that an automatic affective valuation – the emotional core of an attitude – is the main determinant of many judgements and behaviours.” As a sample of the varied problems that are tackled by economists today modern research includes: neuro-economics which looks to understand the activity within the neural substrate during economic de-

4

Terry Bossomaier et al.

cision making [21], bounded rationality where agents are assumed to experience limits in formulating, processing and solving complex problems [30] and reference based utility which concerns the way in which our ever increasing standard of living is not reflected in a subjective sense of our increasing utility [11]3 . Of particular interest here is the nascent field of neuroeconomics which through the use of fMRI is starting to reveal the underlying neural dynamics of trust which seems to be driven by old parts of the brain, common to all mammals. It now appears, is indeed a deeply innate process. As such it becomes interesting to ask to what extent it has evolved to match the expectations from game theory or perhaps to subserve other non-economic functions. For example, others have argued that trust is an essential prerequisite for learning as a child. There is so much to learn in such a short time, that trust in what one is told by parents and others may be the key to mastering such a huge body of knowledge. This scaffolding for early learning may subsequently prejudice economic decision making. Physiological evidence for this comes in studies of the development of oxytocin receptors in children. Closely related to trust are two ideas now current in evolutionary theory: indirect reciprocity and altruistic punishment. Game theory in its simplest form is dyadic, but many of our social interactions involve once-off interactions. Yet because of social communication, good or bad outcomes on one occasion may prejudice or enhance subsequent interactions with different individuals, a process referred to as indirect reciprocity, recently encaptured in a Hollywood tear jerker Pay It Forward. Conversely altruistic punishment, which seems to be prevalent not only in human but also in various animal societies [8], involves punishing rule breakers or adverse behaviour where the punisher has not suffered at the hands of the transgressor. In both cases reputation becomes an attribute which individuals need to build and maintain. Cox [7] in a series of experiments asserts that three dimensions need to be considered in any game theory or investment game studies: altruism, the unsolicited giving to others, usually less fortunate4 ; reciprocity in which a gift or monetary transfer is made on the assumption that it will be returned, either by the recipient or some other group member; and trust 3

4

This is simply an example of those aspects of quality of life which Easterlin considers, just as the topics mentioned here are a small subset of the new directions in economics. In an interesting analysis of the evolution of altruism, Lotem et al [20] suggest that altruism may be sexually selected in the manner of the peacock’s tail. Altruism handicaps the giver and thus demonstrates power and control.

Evolution of Trust in Economic Systems

5

in which the giver renders themselves vulnerable without any immediate evidence of a future reward. But although the latter might seem like a propensity to take risks, the neuroeconomic evidence (section 3) demonstrates that this is not the same. The outline of the paper is as follows. Section 2 outlines the ideas of indirect reciprocity and altruistic punishment from an evolutionary perspective and the extent to which these ideas have permeated economics. Section 3 gives an overview of the current neuroeconomic results on trust, reciprocity and altruistic punishment. Section 4 describes a model of trust in the relationship between clients and wealth management advisors (WMAs) inspired by neuroeconomic findings and section 5 describes some of the findings of the model.

2 Group Mechanisms in Evolution Many studies of cooperation in groups reveal a variety of different mechanisms, some destructive such as Hardin’s tragedy of the commons [12]. Cooperation can arise in a variety of situations where there is direct, repeated interaction between agents. Populations may exist with mixtures of cooperators and others who exploit but do not contribute to the common good [10]. But currently of considerable interest are group behaviours where interactions with particular individuals might be infrequent and often not repeated. In the last year or so evidence has been emerging that although primates in general cooperate with each other, humans might be the most cooperative. Melis et al. demonstrated that chimpanzees not only will seek collaborators in a task they cannot accomplish alone, they also remember, and act upon, who will be a good collaborator. Warneken and Tomasello demonstrate that collaboration is apparent in children as young as 18 months, but that young chimpanzees, although they will collaborate, seem to do so less readily. This recent set of results prompted a provocative article by Tomasello suggesting that the whites of human eyes (actually almost unique amongst over 80 species of primates) evolved specifically to assist collaborative action! 2.1 Indirect Reciprocity Trust and reciprocity may differ from game theory expectations in a subtle way. Game theory would normally consider only the outcomes and their likelihoods. The intentions of others are not relevant, so the probability distribution of a player’s actions may be imposed or it may

6

Terry Bossomaier et al.

result from decisions to trust to some extent. McCabe et al [22] untangle these effects in an elegant series of experiments. Outcome based models are not sufficient to explain the results, the degree of reciprocity being based in part by the perceived level of trust by another player. Nowak and Sigmund [26] argue that the complexity of indirect reciprocity leads to sufficiently complicated outcomes that it may have been a driving force in the expansion of the brain in human evolution. Extensive simulations show that cooperation within a population is stable even if only local interactions occur and there are a relatively small number of “accidental” deviations from fair play. But some form of indirect or altruistic punishment, or action by the community is required. Game theory shows that if individuals get labelled (i.e. their reputation is reduced in some way) for selfish behaviour, then indirect reciprocity will survive providing everybody punishes (i.e. doesn’t cooperate with) them for some period of time [26, 27]. This leads to the notion of altruistic punishment. 2.2 Altruistic Punishment Many people experience anything from discomfort through to outrage when they witness somebody else exploit a third person (ironically there is often considerably greater tolerance to the abuse of large impersonal organisations!) and will often go as far as to intervene in someway. This is altruistic punishment: taking action against somebody who is not actually doing you any harm, even though they might then do you some harm! Recent neuroeconomic experiments have shown distinct localisation of brain activity during altruistic punishment events (section 3).

3 Neuroeconomics The advent of brain imaging techniques, such as functional Magnetic Resonance Imaging (fMRI) has led to increasingly precise insights into the way tasks are subdivided and processed in different parts of the human brain. The types of experiments that are now being conducted are incredibly diverse with results providing fuel to the fire of new and more robust models of human behaviour. Most notable from the view point of economics are the imaging experiments conducted while the subject is involved in some financial task such as gambling, computer vs. human trust games, status related product identification, charitable donations [24] and many more [15]. This field has naturally enough come to be known as neuroeconomics.

Evolution of Trust in Economic Systems

7

Typically subjects in these studies play some sort of game while brain activity is monitored [16]. One of the more popular experiments involves the trust game: one player, the trustor (TR), has the opportunity to give some portion of a fixed amount, say $10, to another player, the trustee (TE). TE is now able to triple this sum of money, say from $10 to $30, and can retain an arbitrary fraction of this $30 and return the rest to TR. This game is repeated N times. Now if we fix N in advance and if both TE and TR were of the species H. Economicus, TE has an incentive to keep everything on the last game as there would be no consequences for this choice. However, TR is able to reason equally accurately and realises what TE will do on the last round and as such decides to give TE zero money to invest, essentially reducing the number of profit generating games to N − 1. This reasoning continues until both players reach the same conclusion, TE is going to keep all the money in the first game and therefore TR is not going to trust TE with any money. This is clearly not a good outcome as there is obvious potential for both players to be better off through trusting each other, but the most important question is how well does this compare with experimental results? In fact the experimental results compare very poorly for this game and others like it. McCabe and Smith [23] showed that in experiments based on the trust game described above5 , TR opts to trust TE 50% of the time and that TR reciprocates 75% of the time, a significant difference in outcome when compared to the game-theoretic prediction. King-Casas et al [16] used fMRI to study what happens in the brain. The basic result is that the act of trust is associated with activity in a brain area called the caudate nucleus, part of the basal ganglia. In some ways this is not surprising since it is these areas that control dopamine release. Dopamine is the neurotransmitter strongly associated with motivation and implicated in reinforcement learning mechanisms [25]. Also, not surprisingly, these areas are an older (non-cortical) part of the mammalian brain. Perhaps ET has them too. A second salient finding from this study is that the amount of money in each iteration of the game on either side was a good indicator of trust (as measured by cortical activity). Note that both TR and TE essentially require trust. TE wants further investment from TR. If TR were not to invest again, there would be no point in returning anything. We use this result directly in the simulation model in section 4 to determine how much money a client will invest at each time step. 5

In the McCabe-Smith experiments N = 1.

8

Terry Bossomaier et al.

A similar and very popular experiment called the Ultimatum game has a similar disparity between the game theoretic outcome and the experimental observations. In this game one player has the opportunity to divide a pie and assign one portion to the second player. The second player is then allowed to either accept the offer or reject it, and if rejection is the choice then neither player receives anything. In this case the game-theoretical outcome is for the first player to offer the second player the smallest possible piece of the pie and the second player to accept it on the grounds that any offer is better than no offer. However experiments show [28] that about 50% of offers below 20% are rejected, and that most offers are around 50% of the total, a much fairer result than traditional economic game-theory would suggest. This rejection has a strong emotional (rather than rational) component to it as Sanfey et al. found that unfair offers activated the anterior insula an area heavily involved in emotional processing. Fairness, it appears, also plays a significant role for monkeys when deciding what a fair deal is. Brosnan and de Waal [4, 9] studied the behaviour of two brown capuchin monkeys when they are receiving a reward for work done. It was shown that capuchins will co-operate even in the case where only one of them will be rewarded and which one will be rewarded is known to the monkeys. After successful co-operation the sharing of rewards increases, demonstrating that in at least these monkeys there are some behavioural parallels with the way in which humans treat some economic situations. While it is interesting to note where co-operative behaviour has come about in other species, it is also worth mentioning when such behaviour is not demonstrated in humans. Camerer et al. [5] discuss the effect of autistic ‘mind blindness’ on economic decision making. In particular they draw attention to the fact that in the ultimatum game more than half of the participants will offer very low or zero values to the other player, much more like the classical game-theory results we have discussed above. Another use of the trust game appeared in a paper by Kosfeld et al. [19]. The neuropeptide oxytocin functions in various mammals to increase social attachments, particularly relating to pregnancy and infancy. But it also functions in the trust game to increase levels of trust on the part of the investor! The effect seems quite specific to trust, and does not arise from an increase in risk taking behaviour or from increased social ability. The role of oxytocin has also been studied in the social interactions of other primates [13, 18], suggesting a common neuro-biochemical substrate. An intriguing relationship, dis-

Evolution of Trust in Economic Systems

9

covered within the last year, of which the full implications are still to be worked out, is that donation strongly activates the subgenual area, which plays a major role in the release of oxytocin. One way of looking at the trust game draws parallels with reciprocity and altruistic punishment. A decision to cooperate in trust games requires delayed gratification. The trustee could take all the money and run, but would then get no further investment. McCabe et al [21] use fMRI to show that the way this works is that low level pressures for immediate acceptance of a reward are blocked by inhibition generated in the prefrontal cortex6 . The neural underpinnings of altruistic punishment are also becoming clearer. De Quervain et al. show that again the caudate nucleus is activated during altruistic punishment, indicating reward in the punishment of non-conformers. Singer et al. [31] show that men have reduced empathic brain activity when an offender is punished in a typical neuroeconomic game. Two brain areas, the frontal insular and anterior cingulate cortex, show increased activity through empathy with the pain of others. But this activity falls in the just punishment case. These brain areas one might consider more evolutionary advanced areas, than the basal ganglia location of basic trust mechanisms, perhaps reflecting the emergence of altruistic punishment in more complex animal societies.

4 Modelling the Evolution of Trust in Wealth Management The discussion above suggests that a simplified view of trust is a single deep rooted characteristic of the human brain. Yet as we discuss elsewhere [2], trust in human society has many dimensions: we trust people, we rely on strong, established brands and organisations, we rely on accreditation mechanisms and so on. Thus for the present purposes we consider these sociological dimensions of trust to operate as control functions (within prefrontal cortex) conditioning the low level reinforcement mechanisms. Thus in the simulation work described in the following sections, we use a single trust parameter, but is also useful to think of how the behaviour of wealth advisors relates to these cognitive dimensions. One plausible cognitive model is that of Mayer et al., tested on both individuals and teams. They define just three independent components: competence, integrity and benevolence. 6

The prefrontal cortex is the large part of the front of the brain assumed to be the location of many uniquely human skills such as language.

10

Terry Bossomaier et al.

4.1 The Challenge of Wealth Management Many countries in the developed world foresee a looming crisis in pensions and superannuation and are adopting various strategies to try to avert it. One important recent change within Australia was the deregulation of super funds, allowing anybody to choose, or even construct, their own fund as opposed to being committed to the one operated by their employer. This major change necessitated many people taking advice from financial planners or wealth management advisors (WMAs). With increase in demand for such services, the possibility of a decline in average quality of service was a distinct possibility. As part of a large scale project involving extensive analysis of real data, we attempted to examine how trust evolves between WMAs and their clients. Agent based modelling (ABM) is a tool now widely used for studying social systems. The complexity of outcomes from even simple social groups is just to great to make analytical prediction feasible, so one resorts to computer simulation. Each entity with flexible behaviour (an agent) is modelled explicitly including its interactions with other agents and the environment. The behaviour of the whole system of agents and the environment is then simulated over time. The model seeks answers to two main questions. Firstly, we want to know if trust evolves naturally between WMAs and their clients. Secondly does the trust show abrupt or rapid changes or the phase transitions common in many complex systems. Explicit details of the model will be found in [2, 3, 32]. We provide here a qualitative overview before showing some illustrative results. 4.2 Model Components There are three components to the model 1. The clients receive a steady income at each time step and determine a fraction to invest with a WMA of this income plus any money they have in the bank. As in the neuroeconomic studies in section 3, the amount invested is proportional to the client’s trust for their WMA. If their trust falls to a sufficiently low value, they will abandon their WMA and choose another, transferring all their assets to the bank. At this stage an important feature of the model is the way clients are connected and exchange information with their others. Following substantial recent work, as for example described in the books by Watts and Strogatz [33] on small world networks and

Evolution of Trust in Economic Systems

11

Barabas´ı [1] on scale free networks, clients are linked to others by these two types of networks found in social systems. 2. WMAs invest funds from clients in one of two types of investment. These are proxies for the wide variety of financial instruments available, two capture two strategic options for the WMA: a) equitites are a risky investment, with potentially high returns, requiring a fee for service. WMAs receive no advantage or further compensation from this investment. Thus equity investment requires research, is totally transparent to the client and thus requires competence and integrity. b) funds, which are aggregates of equities are a conservative choice for the WMA with trailing commissions. Thus they do not capture so strongly the above two aspects of trust and may in fact exhibit a loss of integrity. In fact in the last two years there have been notable concerns within Australia in the way WMAs have not fully disclosed commissions received on investments. They choose the investment based partly on their own policies and partly on the requirements of the client. WMAs set their own fee structures for equity investment and receive trailing commissions from investment in funds. WMAs charge fees for services which reflect their level of benevolence towards their clients. 3. An artificial stockmarket, comprising a range of equities and funds. 4.2.1 Client Trust The key element of client behaviour is the way trust evolves. Many complex economic systems will exhibit lock-in effects, as shown by Kirman, for example, in the Marseilles fish markets. In their agent-based model they show that loyalty emerges in the population of sellers and buyers, even though there may be little direct cost to churn. Trust is something more, in that it involves vulnerability. When a client gives money to an advisor, it might disappear. In fact, some WMAs early in evolution do set their fees impossibly high, fleecing (and losing) their clients. In a situation where information flow between clients was low, this would be easy to do. The communication network allows trust to develop. Different clients have different expectations, some requiring high gain over a long period, others requiring smaller gains with minimal short term fluctuations, which we encapsulate in the client’s caution, ζ. Thus clients exchanging investment performance with each other would be rather complicated. It probably does not reflect the real world very

12

Terry Bossomaier et al.

well either, since detailed financial information is often shared only amongst the closest friends. But people are more willing to exchange (honestly) their opinions as to how well their WMAs are performing. In this model, this performance is captured by their trust. If the client’s desired return is represented by ξ, then their trust, ψ between 0 and 1 is given by equation 1. 1 (1) 1 + e−ωξ where ω determines how rapidly the client’s trust changes as a function of the returns they are getting on their investment. If the client’s desired return is met exactly, their trust has the value 0.5. Clients also need a point, the churn threshold, η, where their trust has fallen so low that they abandon their WMA and find another. When they switch to another WMA, they need an initial trust value, ψnew . These three parameters, ω, η, ψnew are under evolutionary control, i.e. values are found which maximise the clients’ wealth. ψ=

4.2.2 The Wealth Management Advisors WMAs evolve to set their fees and fund/equity investment balance to maximise their wealth, which in some scenarios could also maximise return to their clients. If the supply of clients is limited, hanging onto, and snapping up churning clients, will be a good strategy. They also have another strategic decision – the investment in research. At each time step in a stock market simulation, the WMA may pay for information on an equity, in fact the quadratic term in equation 2. The more information of this kind the better they can serve their clients. We run simulations for different research costs per equity, while the extent of research carried out by each WMA is an evolvable parameter. 4.2.3 The Stock Market The stock market comprises a set (typically 1000) of hypothetical equities, each described by a quadratic growth function, for the value, vt of an equity at time t. vt = v0 + gt + qt2 + n

(2)

where g is a linear growth function estimable by all WMAs, q is a hidden long term growth factor and n a Gaussian noise term. WMAs

Evolution of Trust in Economic Systems

13

gain access through research for which there is fixed fee per equity. Funds simply consist of a random collection of equities. An ASX like stock market performance indicator is computed, against which clients compare their investment gains. 4.3 Running the Model There are several ABM software packages available. One of the most firmly established is RePast a descendent from the first main ABM software, Swarm. Our model was built using RePast version 3.1. We had to add components for scale-free networks created by the BISON project (which was in turn based on previous network research, and wrote software and used Matlab C source for fuzzy logic client behaviour where required. The model operates on two scales. The clients and WMAs coevolve using a genetic algorithm. At each evolutionary step, the stock market is simulated for one hundred time steps (notionally five years). The best WMAs and clients at the end of the evolutionary phase (several thousand steps), are then retested in a further series of stock market simulations.

5 Results and Discussion Figure 1 shows the coevolution in time of the wealth of clients and WMAs, with clients connected using a small world network. After a starting transient, a fluctuating but roughly even balance of wealth for the top WMAs and clients eventuates. Over a wide range of parameters and model variations, the evolution towards trust above 0.5 occurs unless the bank rate is very high relative to the ASX growth or the research costs are prohibitive. This implies that WMAs are consistently trying to offer clients better than their expected returns. Figures 2 represents the Total amount of clients’ wealth kept by the top 10% of WMAs and Figure 3 represents the average trust of the top 10% of WMAs by their clients. The result is the development of a “trusting strategy” which is a situation where the WMA reduces fees and returns a profit to their client(s) that exceeds expectations, subsequently gaining them more clients through their reputation and making up for the lost revenue stemming from the reduced fees. In this situation the WMAs are in competition for churning and other unattached clients, as they need to get enough to profit more than other WMAs.

14

Terry Bossomaier et al.

Fig. 1. Watts-Strogatz Network

Fig. 2. Cut vs. Fitness

6 Conclusions and Future Work There are two major findings from this model: 1. The agents coevolved toward a more trusting ecology where the most profitable agents displayed the most trusting relationships. 2. The network communication amongst clients leads to WMAs choosing lower fees to retain clients.

Evolution of Trust in Economic Systems

15

Fig. 3. Trust vs. Fitness

Differences appear between different client networks, square lattice, small world and scale free. The reputation of a WMA can flow more rapidly in the scale-free and small world case, but the simulations show relatively small effects. However, this will depend on a range of parameters, such as the costs of research, and larger effects may be apparent at some parameter settings. There are three broad directions of further work. 1. The parameters of the model need to be parametrised against real human data obtained from data mining of investment and other transactions records. This work is currently in progress with upwards of a million records of client data to parametrise the model. The model does not yet look at the advantages gained by clients in their initial trust level and, given the natural propensity to trust, this could be quite revealing. 2. Only one of the trust interactions has been included so far. The WMAs have the potential to aggregate as teams and share clients according to investment specialisation. Here trust is constrained by

16

Terry Bossomaier et al.

the need to maintain and individual competitive edge, a variant of the tragedy of the commons problem. 3. As we discussed above, altruistic punishment is common in human and animal societies, but there is as yet no censorship of bad performance of WMAs. This is an important issue. The last couple of years have several diverse scandals in WMA failings, from the incompetent to the deceptive. Trust is now an important research area across psychology, neuroscience, business and socio/economic systems. We believe that the integration of these diverse disciplines can be accomplished by computer simulation. From such simulations we can gain insight into emergent behaviours not in any way apparent from the individual elements.

7 Acknowledgements Thompson and Harr´e were funded under Australian Research Council Grant LP0453657 in partnership with NCR.

References 1. Barab´ asi, A-L. (2002). Linked, Perseus, Massachusetts. 2. Bossomaier, T.R.J., Jarratt, D., Thompson,J., Anver, M. and Cooper, J. (2005). Optimisation of client trust by evolutionary learning of financial planning strategies in and agent based model. Proc. IEEE Congress on Evolutionary Computation. 3. Bossomaier, T.R.J., Thompson, J. and Jarratt, D. (2005). Modelling Dispositional Trust Amongst Financial Advisors. Proc. IBIMA 2005 International Conference on Information Management in Modern Enterprise. 4. Brosnan, S. and de Waal, F. (2003). Monkeys reject unequal pay. Nature, 425–563. 5. Camerer, C., Loewenstein, G. and Prelec, D. (2005). Neuroeconomics: How Neuroscience Can Inform Economics.Journal of Economic Literature XLIII, 9–64. 6. Camerer, C. and Fehr., E. (2006).When Does “Economic Man”Dominate Social Behaviour? Science 311, 47–52, 7. Cox, J.C. (2004). How to identify trust and reciprocity. Games and Economic Behaviour 46, 260–281. 8. Clutton-Brock, T.H. and Parker, G.A. (1995). Punishment in Animal Societies. Nature 373, 209. 9. de Waal, F. (2003). Payment for labour in monkeys. Science 404, 563.

Evolution of Trust in Economic Systems

17

10. Doebell, M. and Hauert, C. and Killingback, T. (2004). The evolutionary origin of cooperators and defectors. Science 306, 859–862. 11. Easterlin, R. (2004). The Economics of Happiness. Daedalus 133 (2), 26. 12. Hardin, G. (1968) The Tragedy of the Commons Science 162, 1243-1248. 13. Insel, T. and Fernald, R. (2004). HOW THE BRAIN PROCESSES SOCIAL INFORMATION: Searching for the Social Brain.Ann. Rev. Neuroscience27, 697–722. 14. Kahneman, D. (2002). Maps of bounded rationality: a perspective on intuitive judgment and choice Nobel Prize Lecture Dec. 8. 15. Kenning, P. and Plassmann, H. (2005). NeuroEconomics: An overview from an economic perspective. Brain Research Bulletin 67, 343–354. 16. King-Casas, B., Tomlin, D., Anen, C., Camerer, C., Quartz, S. and Montague, P. (2005). Getting to Know You: Reputation and Trust in a TwoPerson Exchange. Science 308, 78. 17. Kirman, A.P. and Vriend, N.J. Learning to be loyal: A study of the Marseille Fish Market. In D. Delli gatti, M. Gallegati and A.P. Kirman, eds., Interaction and Market Structure: Essays on Heterogeneity in Economics. Springer Berlin. 18. Kirsch, P., Esslinger, C., Chen, Q., Mier, D., Lis, S., Siddhanti, S., Gruppe, H., Mattay, V., Gallhofer, B. and Meyer-Lindenberg, A. (2005).Oxytocin Modulates Neural Circuitry for Social Cognition and Fear in Humans.The J. of Neuroscience 25(49), 11489–11493. 19. Kosfeld, M., Heinrichs, M., Zak, P.J., Fischbacher, U. and Fehr, E. (2005). Oxytocin increases trust in humans. Nature 435, 673–676. 20. Lotem, A., Fishman, M.A. and Stone, L. (2003). From reciprocity to unconditional altruism through signalling benefits. Proc. R. Soc. Lond. B 270, 199–205. 21. McCabe, K. A., Houser, D., Ryan, L., Smith, V.L. and Trouard, T. (2001). A functional imaging study of cooperation in two-person reciprocal exchange. pnas 98, 11832–11835. 22. McCabe, K.A., Rigdon, M.L. and Smtih, V.L. (2003). Positive reciprocity and intentions in trust games. J. Econ Behaviour & Organization 2, 267– 275. 23. McCabe, K. and Smith, V. (2000). A comparison of naive and sophisticated subject behaviour with game theoretic predictions. PNAS 97 (7), 3777–3781. 24. Moll, J., Krueger, F., Pardini, M., de Oliveira-Souza, R. and Grafman, J. (2006).Human fronto-mesolimbic networks guide decisions about charitable donations. Proc. Nat. Acad. Sci. 103, 15623–15628. 25. Montague, R., Hyman, S. and Cohen, J. (2004). Computational roles for dopamine in behavioural control. Nature 431, 760–767. 26. Nowak, M.A. and Sigmund, K. I. (2005).Evolution of indirect reciprocity. Nature 437, 1291–12298.

18

Terry Bossomaier et al.

27. Panchanathan, K. and Boyd, R. (2004). Indirect reciprocity can stablize cooperation without the second-order free rider probem.Nature 432, 499– 502. 28. Sanfey, A., Rilling, J., Aronson, J., Nystrom, L. and Cohen, J. (2003). The Neural Basis of Economic Decision-Making in the Ultimatum Game. Science 300, 1755–1758. 29. Simon, H. (1978).Rational decision-making in business organizations.Nobel Memorial Lecture. 30. Simon, H. (1999). Bounded Rationality and Organizational Learning. Reflections: The SoL Journal 1 (2), 17–27. 31. Singer, T., Seymour, B., O’Doherty, J.P., Stephan, K.E., Dolan, R.J. and Frith, C.D. (2006). Empathic neural responses are modulated by the perceived fairness of others. Nature 439, 466–469. 32. Thompson, J. and Bossomaier, T.R.J. (2006). Critical behaviour in the evolution of trust. Proc. The Sixth IASTED International Conference onModelling, Simulation, and Optimisation. 33. Watts, D.J. (1999). Small Worlds. Princeton University Press.

Keynes, Hayek and Complexity Paul Ormerod Volterra Consulting - UK

1 Introduction In the spirit of the overall topic of the conference, in this paper I consider the extent to which economic theory includes elements of the complex systems approach. I am setting to one side here the developments over the past decade in applying complex systems analysis to economic problems. This is not because this recent work is not important. It most certainly is. But I want to argue that there is a very distinct tradition of what we would now describe as a complex systems approach in the works of two of the greatest economists of the 20th century. There is of course a dominant intellectual paradigm within economics, that known as ‘neo-classical’economics. This paradigm is by no means an empty box, and is undoubtedly useful in helping to understand how some aspects of the social and economic worlds work. But even in its heyday, neo-classical economics never succeeded by its empirical success in driving out completely other theoretical approaches, for its success was simply not sufficient to do so. Much more importantly, economics over the past twenty or thirty years has become in an increasing state of flux. All the results in general equilibrium, the brilliant and core achievement of neo-classical economics, were established by the mid-1970s. Since then, all the advances which have been made, from the bounded rationality of Akerlof and Stiglitz to the behavioural economics of Khaneman and the experimental economics of Vernon Smith, have involved a willingness to step outside the strict confines of the world of fully rational, maximising agents. Claims to a general, universal theory of how agents behave are being abandoned. As Akerlof stated in his Nobel lecture: ‘in this new style [of economics], the economic model is

20

Paul Ormerod

customized to describe the salient features of reality that describe the special problem under consideration’. In one sense this is very liberating for economics, enabling it to become more realistic and to tackle more problems in a more convincing way. But it is also a source of confusion and doubt within the discipline. Essentially, economics is a theory of individual behaviour, of the cognitive abilities of agents to gather information and the rules which they use to process it. The world of the fully rational, maximising agent provided a bedrock of security, for it lay claim to be a general theory of behaviour. All agents at all times could be presumed to follow these rules. Many people in economics still cling to this, like a comfort blanket. But its validity is being undermined, not least within economics itself by the empirical findings of experimental and behavioural economics. In common with everyone else at this conference, I believe that the complexity approach is increasing quite dramatically our ability to understand the social and economic world. But it can seem alien to many economists, no matter what doubts they may harbour individually about the validity of the core paradigm of their discipline. So the purpose of the paper is three-fold. First, and least important, to show purely for intellectual curiosity that there is indeed a tradition of complexity within economics. Second, to reassure economist readers that it is perfectly respectable to adopt a complex systems approach. This is what some of the most famous economists of the previous century were doing, even though they did not describe it as such at the time. Third, to hope that others may be inspired by the insights of these great economists to develop empirically successful models using the modern tools of the complex systems approach. In Section 2, I outline very briefly some of the key features of a complex system, and in Section 3 I consider examples of this within conventional economic theory. In Section 4, I provide examples from the works of Keynes and Hayek which bear unmistakeably the footprints of complexity.

2 Some key features of complex systems This section is not intended in any way to contribute to the literature of what actually are the features of complex systems. It is simply to set out some the key features so that these can be subsequently identified in economic theory. Perhaps the single most important feature is that the macroscopically observable properties of a complex system emerge from the inter-

Keynes, Hayek and Complexity

21

actions of its constituent parts. In the context of economics, this implies that there is a need in any theoretical model for micro-foundations. In other words, a need for rules which describe the behaviour of the individual agents in the system. The specification of such rules does not of course by itself guarantee that the agent interaction will be such as to generate the emergence of complexity. But in modelling a complex system, such micro-behaviour of agents must be specified. A further feature is a low (or even zero) ability to predict the state of the system at any given point in the future. There may very well be stable statistical distributions from which describe the range of behaviours of the macroscopic factors, so that we can reasonably estimate the proportion of time which the system spends in any particular state. But we cannot predict consistently at particular points in time with any reasonable accuracy. An important implication of this is that the understanding which individual agents have of the world is inevitably imperfect. They cannot be ascribed the cognitive powers of gathering and processing information which exist in conventional economic theory A final feature is that complex systems will typically exhibit multiple possible histories. By definition there can only ever be one actual history, but at any point in time the system has the potential to move in a variety of different ways. So the features to look for are: • • • •

Emergence Low level of predictability at a point in time Limited cognition of individual agents Multiple possible histories

3 Complex systems and conventional theory The foundations of conventional theory were developed around 1870 by Jevons and Walras, at the heyday of classical, equilibrium physics. The theory reflects the principles of the physics of that era. The full implications took a century to work out. The late 1960s and 1970s saw a number of very important papers which brought the programme of research on general equilibrium to an end. But the approach developed by Jevons and Walras remains at the core of conventional theory. General equilibrium theory remains the jewel in the crown of conventional economics, and it is indeed a formidable intellectual accomplishment. General equilibrium theory, as its name might imply, is concerned with the behaviour of all markets in an economy. A key aspect of this has been to establish the least restrictive set of conditions that must hold

22

Paul Ormerod

for the existence of equilibrium to be guaranteed. In other words, the conditions under which it can be proved there exists a set of prices such that demand and supply will be in balance in every single market. It is not the purpose of this paper to enter into a critique of general equilibrium theory. A point which might usefully be noted is that its scientific status is questionable, since it contains no testable propositions. The results of the 1970s established that market demand and supply curves can in principle take any shape, even if the individual demand curves of each agent are well-behaved by the precepts of economic theory. Further, factors of production are not necessarily paid their marginal product. It might be thought that the fact that the theory is concerned with the establishment of an equilibrium means that almost by definition it cannot be regarded as a complex system. However, starkly outside the approach of general equilibrium is the work of scholars such as Brian Arthur1 , yet nevertheless equilibrium solutions obtain in such models. Arthur’s model of path dependency and lock-in in the process of the adoption of new technologies is based upon the non-linear probability theory of Polya urns. And, for a variety of replacement rules, fixed points exist in Polya urn processes. In other words, equilibrium configurations exist. With the simplest possible replacement rule, in which a ball drawn at random from the urn is replaced along with another of the same colour, the equilibrium approached in the limit is that the urn is made up entirely of balls of one of the two colours, although we cannot tell in advance which it will be. For other replacement rules, however, different equilibria exist with different divisions of the colours. Arthur, in his classic paper, used this model in the context of the adoption of competing new technologies. An important feature of the model is that, once an agent adopts one of the technologies, he or she is not permitted subsequently to switch to the other. This reflects the fact that in the urn process, individual balls are not permitted to change colour. More balls are introduced into the urn, so that then proportions of the two colours changes. All theories are of course approximations to reality, and for the adoption of technologies, it seems reasonable to assume that, at least in the relatively short-run and in some markets, it is not easy for agents to switch once they have made an initial choice. There is, of course, a different tradition in agent-based modelling in economics which derives from the work of F¨ ollmer [4] and Kirman [5], in which agents choose between different states of the world but are subsequently allowed to change their minds. This approach is probably 1

For example, W.B. Arthur [2]

Keynes, Hayek and Complexity

23

more realistic in many situations. And of course in such models the system never settles down to an equilibrium. However, Arthur’s model certainly satisfies all four of the criteria set out above to define a complex system, even though it contains equilibrium solutions. So the fact that general equilibrium theory is concerned with equilibrium does not of itself means that it cannot be regarded as a complex system. Indeed, in general equilibrium the system has emergent properties. Each individual agent is simply maximising utility given his or her (fixed) preferences and given the vector of prices. No agent intends all markets to clear, but this can in principle emerge from the reactions of agents to prices. In a one period world, this solution is also a Pareto optimum2 . No agent can be made better off without making at least one other agent worse off. Again, agents do not intend to bring this about. It emerges from their individual actions. Further, it has become recognised that there will usually be multiple solutions in general equilibrium, e.g. [38]. We cannot say, for given tastes and preferences of agents, which of these will actually obtain. There are formidable problems for the theory in specifying the mechanism by which an equilibrium, even it can be shown to exist, actually comes into being. But in principle general equilibrium has emergent properties and multiple possible solutions. The fundamental feature of general equilibrium which means that it cannot be classified as a complex system is the level of cognition ascribed to agents. Agents are in possession of full information. The implications of this can be illustrated in Kenneth Arrow and Gerard Debreu’s original existence proof [1] in general equilibrium once the future is permitted to exist. All previous existence proofs had been for timeless, single period economies. Arrow and Debreu showed that a set of prices which cleared all markets could be guaranteed to exist if each agent prepared a complete list of all future states of the world which might obtain, and everyone must hold identical and correct beliefs regarding the prices which would exist in each potential state of the world at every point in the future. Subsequently, there has been a very partial relaxation of the enormous stringency of these conditions, but clearly we are a very long way from not just the world of complexity and limited cognition, but from anywhere at all.

2

in a multi-period world, this is in general not true: Newbery and Stiglitz (1982)

24

Paul Ormerod

4 Keynes and Hayek The features of complexity appear much more strongly in the works of these two great economists than they do in orthodox theory. In standard theory, the traces of complexity we have described are intellectual curiosities detectable in a system of thought which is in essence antithetical to the principles of complexity. In Keynes and Hayek, this is not the case at all. Instead, features of complexity are an integral part of their individual views of how the economy operates in reality. Agents are interacting in a dynamic environment with many dimensions. Their ability to comprehend the environment is limited. In so far as classical equilibrium is part of their theoretical models, most of the time the economy is operating out of equilibrium, often a long way out. It must be stressed straightaway that even today, well over half a century since Keynes and Hayek were at the peak of their powers, there is considerable disagreement amongst economists on the interpretation of much of their work. The debate is confused rather than enlightened by the fact that both of them adopted different arguments on key aspects of economics at different stages of their careers. I am offering here my own interpretation, that the underlying vision which they each offer in their various ways fits in with the modern concept of the economy as a complex system. I should also say that I am not a believer in what might termed the exegetical tendency in social science, whereby the truth is held to be revealed by deep study of the works of some past master, be it Marx, Keynes, Hayek or whoever3 . Neither Keynes nor Hayek offered a satisfactory theory of the cycle by modern scientific standards, though with the benefit of hindsight Keynes’ work in this particular area is distinctly the better of the two. But the key point is that their works on the business cycle contain many aspects which we would now regard as key features of complex systems. I illustrate this point by considering their views on the business cycle. The two features of capitalist economies which distinguish them from all other societies are, first, a trend of slow but steady positive economic growth and, second, persistent and often large fluctuations from year to year around the trend. The latter fluctuations are referred to as the business cycle. Both Keynes and Hayek believed that the business cycle is an endogenous phenomenon. In other words, it arises from 3

As a student at Cambridge I attended lectures by Richard Kahn and Joan Robinson, almost the last survivors of the group which had worked personally with Keynes. We imagined them holding spiritualist sances in Keynes’ old rooms in King’s College, knocking under the table and intoning “Maynard, are you there? Speak to us Maynard!”.

Keynes, Hayek and Complexity

25

features which are internal to the workings of the economy. Modern believers in rational agents are obliged, as in real business cycle theory, to postulate the existence of exogenous shocks as the cause of the cycle. A modern interpretation of their theories is that they regarded imperfect cognition by individual agents as the source of the cycle. Their works were a product of their times, and both were unable to escape completely from the grip of the conventional concept of equilibrium. Both believed they were working within a framework in which a stable full employment equilibrium exists in principle. But they observed that many aspects of the economic world did not seem to correspond to such a view. In particular, both Keynes and Hayek observed the catastrophic falls in output which had taken place in many Western economies in the early 1930s. They attempted to explain why the economy appears to spend much of its time away from such an equilibrium. We can see this clearly with Keynes. His major work, The General Theory of Employment Interest and Money, was published in 1936 and represents his attempt to account for the high unemployment of the time. In inter-war Britain for example, the unemployment rate averaged 9.4 per cent, peaking at 15.3 per cent in 1932. In the United States, the average was 11.1 per cent, with a high of 24.7 per cent in 19334 . Prima facie, the labour market was not in equilibrium, with supply vastly exceeding demand. His concern was to attempt to understand why this was the case, why it was that labour markets in the capitalist economies experienced prolonged periods of substantial departure from equilibrium. Keynes was at pains to retain as many of the postulates of conventional economics as possible. For example, in chapter 2 of the General Theory, he stresses that he is retaining a fundamental postulate of the classical theory of employment, namely that the wage is equal to the marginal product of labour. In chapter 1, which consists of a single paragraph only, he states that ‘the postulates of the classical theory are applicable to a special case only and not to the general case, the situation which it assumes being a limiting point of the possible positions of equilibrium [my italics]’. In other words, Keynes believed that his theory established the existence of multiple equilibria in labour markets. The classical theory, in which labour markets cleared, was simply a special case of his more general theory, just one of the numerous equilibria which could exist. Keynes chose as the focus for his theoretical attack the work of his Cambridge 4

estimates of unemployment rates in the inter-war period do vary slightly between sources. This data is taken from A Maddison, Dynamic Forces in Capitalist Development, Oxford University Press, 1991

26

Paul Ormerod

colleague Pigou5 , the designated heir of Marshall who had founded the Cambridge economics faculty. Pigou had written the Theory of Unemployment in 1933. Subsequent to Keynes’ book, Pigou argued that a unique full employment equilibrium could still be shown to exist in Keynes’ theoretical model because of the real balance effect [11]. According to this, consumption does not depend simply upon income as Keynes argued, but also on real net wealth. If high unemployment arises for whatever reason, there will be downward pressure on wages, and hence on costs and so on prices. The real value of assets denominated in money terms will rise. Consumption will therefore increase, and full employment be restored. There are many refinements of the real balance effect in economic theory subsequent to the Pigou/Keynes interchange, which need not concern us here. The important point for this paper is not whether Keynes actually succeeded in constructing a theoretical model in which multiple equilibria for unemployment could exist. It is that it was his definite intention to do so. There are further aspects to the General Theory which are even more obviously in keeping with the complex systems approach. A great deal of conventional theory is timeless, and is concerned to describe the properties of whatever equilibrium can be shown to exist. If a change in a model parameter is postulated which creates a different equilibrium, the two can be compared. Nothing at all is said about the transition between the two, in terms of either the shape of or the speed along the path. For Keynes, economies most definitely existed in time. He was concerned to show that equilibria existed in his model. But much of the time not only were economies not in equilibrium, but the extent of their deviation away from it varied in time. For example, in chapter 5 (Expectation and Employment) he wrote “An uninterrupted process of transition to a new long-period position can be complicated in detail. But the actual course of events is more complicated still. For the state of expectation is liable to constant change, a new expectation being superimposed long before the previous change has fully worked itself out”. In other words, although in principle economies may be moving at any point in time from one equilibrium to another, they will spend almost all of their time out of equilibrium. Keynes regarded expectations 5

Pigou introduced into economics the very important concept of the distinction between private and social marginal costs and benefits, and the idea that governments can use taxes and subsidies to correct for any such externalities. This still forms the basis of a great deal of policy analysis throughout the world. Perhaps even more interestingly, he has been suggested as the ‘Fifth Man’, the recruiter of the Soviet spy network at Cambridge in the inter-war years which did so much to damage British and Western intelligence during the Cold War

Keynes, Hayek and Complexity

27

as the key to the fluctuations in output which are observed in all the market-oriented developed economies. Individual agents - firms - take decisions on output in the light of their individual expectations. These are based on very limited knowledge of their environment. Further, individual expectations are formed not in isolation, but with regard to what others are thought to believe. In modern terms, Keynes’ firms are connected on a network across which different expectations either percolate or are contained. Keynes distinguished very clearly between short-run (chapter 5 of the General Theory) and long-run expectations (chapter 12 The State of Long Term Expectation). The former relate to the decision about how much to produce in the immediate future with a given level of capital stock. The second relate to decisions on whether or not to change the amount of capital stock, in other words investment. In terms of short-run expectations, Keynes postulated that these were formed by an individual agents using a very simple rule of thumb: ‘it is sensible for producers to base their expectations on the assumption that the most recently realised results will continue, except in so far as there are definite reasons for expecting a change’. In other words, Keynes’ firms are not assumed to perform complicated optimising decisions when setting their level of output. Instead, they operate with limited knowledge of the environment, and use a simple rule of thumb. The limits to agent cognition are made even more apparent in Keynes’ view of long-run expectations. In chapter 12, he writes that ‘the outstanding fact is the extreme precariousness of the basis of knowledge on which our estimates of prospective yield [of a new investment] have to be made If we speak frankly, we have to admit that our basis of knowledge for estimating the yield ten years hence of a railway, a copper mine, a textile factory, the goodwill of a patent medicine, an Atlantic liner, a building in the City of London amounts to little and sometimes to nothing; or even five years hence’. In other words it is as if - a favourite phrase of economists - firms have very low or zero cognition when taking decisions on whether or not to vary the size of the capital stock. He re-emphasises this view in chapter 22 (Notes on the Trade Cycle), where he writes that ‘the basis for such expectations is very precarious. Being based on shifting and unreliable evidence, they are subject to sudden and violent changes’, and refers to the ‘uncontrollable and disobedient psychology of the business world’. For Keynes, the long-run expectations of firms were the most important determinant of the business cycle through their impact on in-

28

Paul Ormerod

vestment6 . The long-run expectation of a firm at any point in time is not the result of a rational calculation of the amount of profit which an investment is expected to yield. Rather it is a sentiment, the degree of optimism or pessimism which the agent holds about the future. Keynes did not specify a formal model of how such expectations are generated, but there appear to be two components. Most importantly, sentiment is altered across the network of firms as a whole by ‘waves of irrational psychology’. Keynes also writes of changes in sentiment being generated as the ‘outcome of mass psychology of a large number of ignorant individuals’. This is the key feature of long run expectations. In addition, an agent seems to have the ability to change its optimism/pessimism spontaneously without regard to external factors, including the sentiments of other agents. Keynes writes of ‘spontaneous optimism’and a ‘spontaneous urge to action rather than inaction’. This is the context in which his famous phrase ‘animal spirits’appears. In modern terminology, we have agents on a network which at any point in time are in one of k states of the world, where k is the degree of optimism/pessimism. There is some kind of threshold rule by which individual agents alter their state of the world according to the state of the world of their neighbours. This could certainly generate the ‘waves’to which Keynes refers. In addition, each agent has an individual probability of changing his or her mind spontaneously, as for example in the Kirman ants model. So the interactions between agents are of decisive importance for the actual macroscopic outcome which emerges. Hayek’s major general contribution to social science was to emphasise the limits to knowledge in social and economic systems. This was the source of a profound disagreement between him and Keynes. Keynes believed not only that he had discovered why labour markets did not clear, but that he, and like-minded others, could solve the problem. So, for example, towards the end of the General Theory, he writes ‘I conclude that the duty of ordering the current volume of investment cannot safely be left in private hands’. Hayek, in contrast, believed that there are inherent limits to knowledge which no amount of intellect can overcome. His 1974 Nobel lecture, for example, is entitled The Pretence of Knowledge. In it he writes, along with much else, that ‘the social sciences, like much of biology but unlike most fields of the physical sciences, have to deal with structures of essential complexity, i.e. 6

We now know, thanks to over half a century of national accounts, that in practice it is indeed the fluctuations in investment which in general make the major quantitative contribution to movements in total output over the course of the business cycle.

Keynes, Hayek and Complexity

29

with structures whose characteristic properties can be exhibited only by models made up of relatively large numbers of variables’. Empirical evidence which supports Hayek’s view in the context of the business cycle is the track record on economic forecasts of output growth, even just one year ahead. We now have a track record of over 30 years of such forecasts, carried out by both public and private bodies. In general, the forecasting record exhibits a certain degree of accuracy in that the average error over time is smaller than the size of the variable being predicted. But the error is still large compared to the actual data, and most of the accurate forecasts are made when economic conditions are relatively stable. Exactly when they are most needed, at turning points in the economy, forecasts are at their least accurate. And the forecasting record shows no sign of getting better over time, despite the incentives for policymakers to get it right7 . 4.1 Hayek Time Hayek wrote a great deal on the business cycle and he changed his views over time as he realised weaknesses in his previous hypotheses. It has to be said that his theories here were never articulated as clearly as Keynes’ in the General Theory, but they should be considered in the context of his general views on the limits to knowledge and the importance of the emergence of spontaneous order. A potential problem in thinking about Hayek and the business cycle is that he was cited approvingly by Robert Lucas [8] , the seminal thinker behind modern (and failed) attempts to produce a rational expectations account of the cycle. Hayek essentially believed that the starting point of business cycle theory should be the framework of general equilibrium. It was not sufficient, important though this might be, to point out empirical evidence on why this theory fails to explain the cycle. It was necessary to extend the theory in order to achieve an explanation. This ex post guilt by association of Hayek with the rational expectations school needs to be set aside when considering the complexity features of his business cycle theories. In essence his view as to why the cycle exists is first, that agents have different expectations about the future. Second, and partly in consequence of this, agents find that outcomes differ from their expectations, and revise their actions as a result. So in his 1937 article Economics and Knowledge [23], for example, Hayek writes that ‘It appears that the concept of equilibrium merely means that the foresight of the different 7

for a discussion of this, see, for example, Ormerod and Mounfield [10]

30

Paul Ormerod

members of the society is in a special sense correct. It must be correct in the sense that every person’s plan is based on the expectation of just those actions of other people which those other people intend to perform and that all these plans are based on the expectation of the same set of external facts, so that under certain conditions nobody will have any reason to change his plans’. He argued that such individual plans might indeed have to be revised by external shocks. But, more importantly, the individual plans may not have been, indeed are unlikely to have been, compatible from the outset, so that revisions are inevitable. Heterogeneous agents in this view of the world are operating with limited knowledge of their environment and their foresight is imperfect. Again, it is the interactions between agents, this time in terms of the incompatibility of their individual plans, which brings about the particular macroscopic outcome for the system as a whole. Ultimately, Hayek regarded the business cycle as a monetary phenomenon, and attempted several times to articulate his views. Perhaps the best known of these is his 1929 Monetary Theory and the Trade Cycle, not least because it was famously attacked by Sraffa, a Cambridge colleague of Keynes, in the Economic Journal in 1932 [13]. An excellent modern discussion of this debate is given by Cottrell [3]. Hayek saw the source of the cycle in a divergence between the money rate of interest and what he termed the ‘natural’rate, the rate which would ensure that savings and investment would equilibrate. This divergence could obviously arise by mistakes of bankers, but he believed there was a much deeper reason. Namely, that firms at some stage become overly optimistic about the future, and revise upwards their expectations of profitability. This is remarkably similar to the concept of the ‘marginal efficiency of capital’ which Keynes put forward in the General Theory in 1936. The key points are that, first, firms’ expectations prove to be wrong and the unintended consequence is the emergence of an economic expansion which ultimately proves unsustainable. Individual firms may have complete information on their local circumstances, but they are unable to appreciate the collective consequences of their individual decisions. Second, firms do not learn from previous experience to avoid similar mistakes in the future, these false expectations are an inherent feature of the economic system. Third, the central bank does not learn how to offset these expectations in order to smooth out the cycle and restore equilibrium.

Keynes, Hayek and Complexity

31

5 Closing remarks Complex systems exhibit a range of distinguishing characteristics. For example, a low level of predictability of the system at any particular point in time. This in turn implies that the individual agents which comprise the system have limited knowledge of their environment. The macroscopic properties of complex systems emerge from the interactions of the agents, and it may not, indeed usually is not possible to deduce them even from a complete knowledge of the behaviour of the individual agents. Further, there will typically be multiple possible histories in such systems. Conventional economic theory is in essence antithetical to a complex systems view of the world. It is populated by rational agents each with high cognitive powers, and is mainly concerned to discuss the properties of equilibrium solutions to the system. Nevertheless, there are traces of features of complex systems scattered here and there. Most strikingly, general equilibrium theory exhibits multiple equilibria and an emergent phenomenon: no individual agent in general equilibrium intends all markets to clear. In contrast, the works of Keynes and Hayek have much more in common with the modern complex systems approach. Their theoretical models describe worlds in which agents have limited knowledge and foresight regarding their environment and in which the interactions between agents are of decisive importance for the macroscopic outcome. This outcome emerges from the interactions and cannot be deduced from a description of how an individual agent behaves. This dynamic view of the world, fundamental to Keynes’ General Theory, was edited out by mainstream economics after his death in the so-called IS-LM synthesis. And Hayek’s work was for many years to all intents and purposes ignored by economics. The closing decades of the 20th century and the opening ones of the present one have seen a resurgence of interest in Hayek, as the empirical evidence from experimental and behavioural economics builds. Agent behaviour in the real world appears to be much closer to Hayek’s view, with agents having limited cognition and facing inherent limits to their knowledge, than it does to the rational agent postulate of conventional economics. So the complex systems approach is part of a fine tradition within economic theory, embracing as it does two of the greatest economists of the 20th century.

32

Paul Ormerod

References 1. Arrow, K.J. and Debreu, G. (1954). Existence of a Competitive Equilibrium for a Competitive Economy. Econometrica 22, 265-90. 2. Arthur, W.B. (1989). Competing Technologies, Increasing Returns, and Lock-In by Historical Events. Economic Journal 99, 116-131. 3. Cottrell, A. (1994). Hayeks Early Cycle Theory Re-Examined. Cambridge Journal of Economics 18, 197212. 4. Follmer, H. (1974). Random Economies with Many Interacting Agents’. Journal of Mathematical Economics 1, 51-62. 5. Kirman, A. (1993). Ants, Rationality and Recruitment. Quarterly Journal of Economics 108, 137-56. 6. Newbery, D.M. and J.E. Stiglitz (1982). Optimal Commodity Stock-Piling Rules. Oxford Economic Papers 34(3), November, 403-27. 7. Hayek, F.A. (1937), Economics and Knowledge. Economica 4, 33-54. 8. Lucas, R. E. (1977). Understanding Business Cycles, in Brunner, K. and Meltzer, A. (eds.) Stabilization of the Domestic and International Economy, Amsterdam, North-Holland. 9. Maddison, A. (1991). Dynamic Forces in Capitalist Development, Oxford University Press. 10. Ormerod, P. and Mounfield, C. (2000). Random Matrix Theory and the Failure of Macro-economic Forecasting. Physica A 280, 497-504. 11. Pigou, A.C. (1943). The Classical Stationary State. Economic Journal 53, 343-351. 12. Silvestre, J. (1993). The Market-Power Foundations of Macroeconomic Policy. Journal of Economic Literature 31, 105-141. 13. Sraffa, P. (1932). Dr Hayek on Money and Capital’. Economic Journal 42, 42-53.

A look at the relationship between industrial dynamics and aggregate fluctuations Domenico Delli Gatti1 , Edoardo Gaffeo2 , and Mauro Gallegati3∗ 1 2 3

a Cattolica di Milano - Italia Universit` Universit` a degli Studi di Trento - Italia Politecnico delle Marche - Italia

1 Introduction Starting with the pioneering work of Gibrat [23], the study of the determinants and the shape of the steady-state distribution of firms’ size has long fascinated economists. While the conventional view received from the seminal work of e.g. Hart and Prais [29], Hart [27] and Mansfield [42] holds that the firms’ size distribution is significantly right-skewed and approximately log-normal, recent empirical research has lent support to the view suggested by H. Simon and his co-author (Ijiri and Simon [32]), according to whom a Pareto-Levy (or power law) distribution seems to return a better fit to the data for the whole distribution (Axtell [3]), or at least for its upper tail (Ramsden and Kiss-Haypal, 2000; Gaffeo et al. [18])4 . Regardless of the different outcomes obtained from distributionfitting exercises, the most popular explanation for right-skewness emerged so far in the literature rests on stochastic growth processes, basically because of their satisfactory performance in empirical modelling5 . From a theoretical point of view, however, random growth mod∗

4

5

We would like to thank an anonymous referee, Richard Day, Corrado Di Guilmi, Nicol` as Garrido, Sorin Solomon and seminar participants at the Universities of Leiden and Salerno for useful comments on earlier drafts. Emiliano Santoro provided excellent research assistance. Responsibility for remaining errors remains with us. Another challenging stylized fact on the drivers of corporate growth and the resulting industrial structure is the ubiquitous exponential shape of the growth rates density (Bottazzi and Secchi [6]; Bottazzi et al. [5] For recent evidence, see Geroski et al. [21], who point towards a pure random walk model for firms’ growth, and Hart and Oulton [28], who instead suggest a Galtonian, reversion-to-the-mean growth process.

34

Domenico Delli Gatti et al.

els of firms’ dynamics have been generally seen as far less satisfactory. On the one hand, several authors have simply discarded purely stochastic models as ad-hoc and uninformative, given that a proper theory of firms’ growth should be grounded on richer economic contents and maximizing rational behaviour (Sutton [51])6 . Alas, the introduction of stochastic elements in standard maximizing, game-theoretic models7 has shown that their implications as regards the steady-state firms’ size distribution are highly dependent on initial assumptions and modelling choices, to the point that we cannot find “[. . . ] any reason to expect the size distribution of firms to take any particular form for the general run of industries” (Sutton [51]). On the other hand, the original Gibrat’s random multiplicative model (also known as the Law of proportionate effect) and its numerous extensions8 all share the unpleasant property of possessing either an implosive or an explosive behaviour under rather general conditions, so that their cross-section dynamics tend alternatively towards a degenerate firms’ size distribution with zero mean and variance or a degenerate distribution with infinite mean and variance (Richiardi [47])9 . In this chapter, we propose a model which embeds idiosyncratic stochastic influences in a simple imperfect information, rational expec6

7 8

9

In spite of being very popular among economists, Sutton’s critique is not properly established. In fact, the Gibrat’s Law may be perfectly consistent with the behaviour of rational, profit-maximizing firms. In a nutshell, consider a model in which firms, ` a la Penrose [45], are constrained in their growth opportunities only by their internal resources. Since firms’ size, at the optimum, depends on current expectations on future conditions, if firms form rational expectations changes in expectations will be unpredictable. This implies that growth rates are realizations of pure random processes. See e.g. Klette and Kortum [37]. See, for example, Jovanovic [34] and Ericson and Pakes [15]. Among the many variations of the Gibrat’s model, one can cite the preferential attachment mechanism introduced by Simon and Bonini [48], the multiplicative plus additive random process due to Kesten [36], or the lower reflection barrier by Levy and Solomon [39]. To grasp the argument, consider this simple example reported by Richiardi (2004). Let us presume that the size of a firm increases or decreases, with equal probability, by 10% in each period. Now suppose that, starting from a size of 1, the firm first shrinks and then bounces back. In the first period the firm’s size is 0.9, while in the second one it is only 0.99. If we let the dynamics be inverted, so that the firm first grows and then shrinks, then the firms’ size is 1.1 in the first period, and 0.99 in the second one. Clearly, this effect is stronger the bigger the variance of the stochastic growth process, and can be contrasted only by increasing the average growth rate. Richiardi [47], resorting to simulations, shows that a degenerate long-run distribution can be prevented only by a limited number of mean/variance pairs.

Industrial Dynamics and Fluctuations

35

tations framework. We show that a right-skewed firms’ size distribution emerges as a natural feature of the endogenous cross-section dynamics, and that the upper tail is Pareto distributed. This results from the interplay of the cross-sectional dispersion (i.e., heterogeneity) of firms and the feedback exerted on it by the competitive pressure that individual actions determine through the labour and the equity markets10 . Furthermore, certain properties of the steady-state distribution are in some sense universal, i.e. they are independent of some of the model’s parameters. In other terms, our approach does not possess any tendency towards a long-run degenerate behaviour, typical of Gibrat’s processes, and it grants the modeller more degrees of freedom than the stochastic-game-theoretic models referred to above. From an analytical point of view, our results are based on the possibility of describing the economy as a Generalized Lotka-Volterra system, that is a multi-agent extension of the prey-predator framework formally introduced to economists by Richard M. Goodwin (1913-1996) back in the 1960s (Goodwin [25]). While inside the Econophysics community such a formalism has been already successfully employed to explain puzzling statistical regularities regarding stock market returns and the personal wealth distribution (Solomon and Richmond [50]), the application of this Generalized Lotka-Volterra approach to the relation between the firms’ size distribution and aggregate fluctuations is new. The organization of this chapter is as follows. In Section 2 a simple macroeconomic model resting on financial market imperfections, maximizing behaviour, rational expectations and heterogeneity is presented. Section 3 contains the characterization of the associated firms’ size cross-section dynamics and its long-run attractor or, in other terms, the equilibrium firms’ size distribution. Section 4 concludes.

2 The model The purpose of this section is to create a simple macroeconomic model that, besides explaining aggregate fluctuations, replicates the main features of industrial dynamics and the firm size distribution. At the centre of our approach there are three basic ideas: 1. The firms’ financial position matters. Firms display heterogeneous unobservable characteristics, so that lenders may not be perfectly informed on firms’ ability or willingness to pay back. As 10

Another model in which financing constraints help to explain the skewness of the firms’ size distribution is the one recently proposed by Cabral and Mata [8].

36

Domenico Delli Gatti et al.

a result, the various financial instruments whereby firms can raise means of payment are not perfect substitutes. Decisions about employment and production are conditional on the cost and contractual terms of the financial instruments available to firms (Gertler [22]). From an empirical viewpoint, it appears that firms tend to be rationed on the market for equity due to both adverse selection and moral hazard effects (Fazzari et al. [16]). 2. Agents are heterogeneous as regards how they perceive risk associated to economic decisions. Numerous experimental studies indicate that heterogeneity in risk perception is pervasive. Such a finding holds both for consumers and insurees (Hammar and Johansson-Stenman [30]; Lundborg and Lindgren [41]; Filkestein and Poterba [17]), and for entrepreneurs (Pennings and Garcia [44]; van Garderen et al.). While popular in mathematical psychology, the exploitation of these findings is far less common in theoretical economics, the random utility model discussed in Anderson et al. [2] being a relevant exception. 3. Firms interact through the labour and equity markets. While the feedbacks occurring in the labour market between hiring firms are clear enough not to deserve particular emphasis, a few words are in order to discuss interactions in the equity market. In particular, we are referring to the evidence suggesting that the number of initial public offerings (IPO), of additional new stock issues, as well as the proportion of external financing accounted for by private equity, all tend to increase as the aggregate activity expands and the equity market is bullish, and vice-versa (Choe et al. [9]; Brailsford et al. [7]). All other aspects of the model are kept as simple as possible11 . We follow the literature on information imperfections in financial markets, in particular Greenwald and Stiglitz [26], in assuming that finitely many competitive firms indexed by i = 1,. . . , I operate in an uncertain environment, in which futures markets do not exist12 . Each firm has a constant returns to scale technology which uses only labour as an input, yit = φnit , where yit is the time t output of firm i, nit its employment, and φ the labour productivity, constant and common to all. The production cycle takes one period regardless of the 11

12

For a more complete, agent-based model where aggregate (mean-field) interactions occur through the credit market, see Delli Gatti et al. [12]. The analytical details of the framework used in this paper are discussed at length in Delli Gatti [11].

Industrial Dynamics and Fluctuations

37

scale of output, implying that firms have to pay for inputs before being able to sell output. Since capital markets are characterized by informational imperfections, firms’ ability to raise risk capital on external stock markets is sub-optimal, and restricted to depend on the average capitalization in the economy. As a result, firms must generally rely upon bank loans to pay for production costs (i.e., the wage bill). In real terms the demand for credit of the i-th firm is dit = wt nit − ait , where wt is the real wage, determined on an aggregate labour market, while ait is the firm i’s real equity position. For simplicity we assume that firms can borrow from banks as much as they want at the market expected real return r, and debt is totally repaid in one period. The individual demand faced by firms is affected by idiosyncratic real shocks. The individual selling price of the i-th firm is the random outcome of a market process around the average market price of output Pt , according to the law Pit = uit Pt , with expected value E(uit ) = 1 and finite variance. Let the I random variables {uit } be uniformly, but non-identically, distributed. In particular, support of the i-th firm relative price shock uit is given by [zit , 2 − zit ], with {uit } ∈ (0, 1) being a iid random variable with finite mean and variance. It follows that individual price shocks are characterized by a common and constant 2 expected value equal to 1, but the variance, V (uit ) = (1−z3it ) , evolves stochastically. This aims to capture the idea that people usually perceive the same signal in different ways or, alternatively, that people differ as regards the degree of risk they attach to random events to be forecasted. If a firm cannot meet its debt obligations (in real terms, πit+1 < 0, where πit+1 are real profits at the beginning of period t+1), it goes bankrupt. From the assumptions above, it follows that this event happens as the relative price is below a threshold given by:   wt ait (1) uit = R − φ yit where R = (1 + r). Going bankrupt is costly, not only because of direct legal and administrative costs (Altman [1]; White [54]), but also because of the indirect costs of bankruptcy-induced disruptions, like asset disappearance, loss of key employees and investment opportunities, and managerial stigma and loss of reputation (Gilson [24]; Kaplan and Reishus [35]). Estimates suggests that the costs associated to bankruptcy may be very large (White [54]; Weiss [53]). For simplicity, we assume that the real

38

Domenico Delli Gatti et al.

bankruptcy cost Cit is an increasing quadratic function of output, so that the expected bankruptcy cost becomes: 2 Pr(uit < uit ) = E (Cit ) = cyit

=

2 (u − z ) cyit it it 2 (1 − zit )   2 R wφt − cyit

ait yit



− zit

2 (1 − zit )



(2)

where c > 0 is a measure of the aversion to bankruptcy on the part of the firm’s owners and managers. As recalled above, the magnitude of the parameter measuring aversion to bankruptcy depends on a number of factors, and it can vary with the institutional framework and the economic conditions faced by firms. Clearly, the higher is c, the lower is the incentive to recur to debt in financing production and — as we will see presently — the lower is the level of output maximizing real profits. While the model could be expanded to more general cases with similar qualitative results - although at the cost of additional remarkable analytical complications – for the sake of tractability we limit the dynamics of the system in a region in which the two following conditions holds true: C1: The individual demand for credit is always positive, i.e. di > 0, ∀ i. C2: Expected profits net of expected bankruptcy costs are always t positive, so that we are assuming that Rw φ < 1. The problem of firm i consists in maximizing the expected value of real profits net of real bankruptcy costs:  wt yit max E (πit − Cit ) = yit − R − ait − yit φ     2 − zit R wφt − ayitit cyit − . 2 (1 − zit ) 

(3)

From the first order condition it follows that as c grows large13 the individual supply can be approximated to the linear function: 13

Such that the term

  Rw (1−zit ) 1− φ t   Rwt −zit c φ

tends to approach 0. It seem worthwhile to

note that this could happen well before c goes to infinity, since both terms at the numerator are lower than 1. In particular, for c ≤ 10, 000 the expression above becomes lower than 0.01 but for a negligible number of cases.

Industrial Dynamics and Fluctuations

yit ≃

R 2



wt yit φ

− zit

 ait = hit ait

39

(4)

where hit , being a function of the random variable zit , is a random variable as well, which we assume follows a distribution Π (hi ) with finite mean and variance. Individual supply is an increasing linear function of net worth, as a higher net worth reduces the marginal bankruptcy costs. In turn, for any given level of net worth higher uncertainty (a realization of the random variable zit close to 0) on the relative price makes production more risky. As firms trying to maximize the concave profit function (3) behaves in a risk-averse manner, higher uncertainty means lower production. The dynamics of production can be tracked by the evolution of the equity base, which in real terms reads as: ait+1 = uit yit − R (wt nit − ait ) + γi at

(5)

where at is the average capitalization of firms operating at time t, and γi > 0. The last term measures the amount of new equity firm i can raise on the stock market, assumed to be proportional to the average capitalization of the economy at time t. The value of γi depends on the level of development of the financial system, that is on the presence of financial instruments, markets and institutions aimed at mitigating the effects of information and transaction costs. Firms operating in economies with larger, more active and more liquid stock markets with a large number of private equity funds are characterized by a higher γ than economies with poorly developed financial systems. From the point of view of each individual firm, however, the hot market effect represents an externality arising from a mean-field interaction. Finally, we assume that at any time t the real wage is determined on the labour market according to an aggregate wage setting function, which for simplicity we assume to be linear: wt = bnt

(6)

where nt is total employment, and b > 0. Such a positive relationship between aggregate employment (at given labour supply) and real wage can be alternatively derived from union models, insider-outsider models or efficiency wage models (Lindbeck [40]).

40

Domenico Delli Gatti et al.

The parameters γi and b then capture the interdependence between aggregate outcomes on the financial and the labour markets, respectively, and individual decisions. Their values vary with the institutional context governing both market transactions and non-market interactions. Such an institutionally-constrained microeconomic analysis calls for the search of appropriate macrofoundations of microeconomics (Colander [10]). The importance of this issue will be further discussed below. Due to the constant returns technology and the supply function (4), knowledge of a it immediately translates into knowledge, at least in expectational terms, of other traditional measures of firms’ size, that is total sales (yit ) and employment (nit ). By inserting (4) and (6) into (5), and assuming rational expectations so that uit = 1 for any i and t, we obtain: ait+1 = (hit + R) ait + γi at − uit yit − Rbnt nit

(7)

Heterogeneity enters the model along two margins. First, because of cumulative differences in individual equity bases, which are influenced by past idiosyncratic shocks to relative prices. Second, because of differences in the way agents perceive risk associated to future profits. The dynamical system (7) for i ∈ (1, I) and its predictions for the firms’ size distribution will be analyzed in the next Section. For the time being we just want to highlight that this economy can display aggregate — i.e., per capita — endogenous fluctuations under rather mild conditions. To do this, let us take the cross-sectional average in both members of (7) to get: 

2  Ibht a2t + γat at = ht + R at − R (8) φ2

where to simplify calculations we have assumed that the real shock to individual aggregate demand is always equal to its average value, so that hi = h, ∀ i. Simple algebra and a suitable change of variable, i.e. 2

xt = R Ibhφt2 at , allows us to show that equation (8) can be reduced to a logistic map (Iooss, 1979): xt+1 = Γt xt (1 − xt )

(9)

where Γt = ht + R + γ is the control parameter. It is well known that such a first-order nonlinear map can display deterministic cycles if 3 < Γ < 3.57, and non-periodic — i.e., chaotic — behaviour if 3.57 <

Industrial Dynamics and Fluctuations

41

Γ < 4 (Baumol and Benhabib [4]). In this latter case, the time series generated by the one-dimensional deterministic difference equation (9) are characterized by irregular fluctuations which can mimic actual data. In particular, the dynamic path of the economy enter a chaotic regions for particular combinations of the average slope of the individual offer functions (itself parameterized by the interest rate and the real product wage), of the interest rate on loans, and of the parameter measuring spillovers in the equity market. The literature modelling endogenous business fluctuations in terms of non-linear dynamic systems and chaotic attractors is large. See Day (1996) for a nice introduction to methods and applications. In frameworks strictly related to the one we employ here, Gallegati [20] shows how to derive sudden shifts between the periodic and non-periodic regimes due to stochastic influences on the tuning parameter, while Delli Gatti and Gallegati [13] introduce technological progress to obtain fluctuating growth.

3 The firms’ size distribution This Section is devoted to analyzing the cross-section dynamics associated to system (7), which amounts to studying the firms’ size distribution and its evolution. A natural question is whether such a distribution converges towards a long-run stable (i.e., invariant) distribution, or if it is bound to fluctuate in a random-like manner. It must be stressed from the start that the aggregate dynamics generated by the aggregate model (8)-(9) resembles the Richard Goodwin’s prey-predator growth-cycle model (Goodwin [25]), which represents the first attempt to adapt the mathematical description of biological evolution due to Alfred Lotka and Vito Volterra to an investigation of the way a modern economy works. In this masterpiece of non-linear economic dynamics, Goodwin re-cast in a new, analytically elegant guise14 the Marxian analysis of capitalism and its inherent instability. The two classes of capitalist and workers compete for the national income, which at any time period is divided between profits and wages according to their relative strength. During an expansion phase, growing investment opportunities call for an increasing demand for labour. Rising wages and rising employment, in turn, determine an increase in the share of 14

The Nobel Laureate Robert Solow used the following words to comment the 1967 Goodwin’s paper: “[It] is five pages long. It does its business clearly and forcefully and stops. It contains no empty calories” (Veluppillai [52]).

42

Domenico Delli Gatti et al.

national income going to workers. As the profit share declines, however, gross investments - and therefore total output - fall. As a result, new job creation shrinks and the unemployed labour force (the socalled reserve army of the unemployed) is enlarged. The growth in the unemployed puts downward competitive pressure on wages, and thus provides greater profit opportunities. Larger profits stimulates additional capital accumulation, and economic activity and labour demand increase. The business cycle can begin again. The story we are telling is basically the same as the Goodwin’s one. During an upswing, the increase of output induces higher profits and more private equity funds. Higher production means also rising employment and higher wages, however. The increased wage bill calls for more bank loans which, when repaid, will depress profits and, through equations (4) and (5), the production and the equity level as well. The labour requirement thus decreases, along with the real wage, while profits rise. This restores profitability and the cycle can start again. Moving from an aggregate to a cross sectional perspective, it seems worthwhile to stress that the predator-prey analogy can be extended to the non-linear dynamical system (7), as it represents a Generalized Lotka-Volterra (GLV) system (Solomon [49]), whose solution describes the limit (long-run) behaviour of the firms’ size distribution as measured by their equity, employment or sales. Such a system is completely defined by: i) a stochastic autocatalytic term representing production and how it impacts on equity; ii) a drift term representing the influence played via the hot market effect by aggregate capitalization on the financial position of each firm; and iii) a time dependent saturation term capturing the competitive pressure exerted by the labour market. In what follows we will borrow from the work of Solomon and Richmond [50], to show that the cross sectional predictions of the GLV system (7) are consistent with the available empirical evidence on how firms are distributed. In particular, the right tail of the firms’ size distribution turns out to exhibit a Pareto distribution of the form: P (a) ∼ a−1−α

(10)

over several orders of magnitude. Interesting enough, the GLV model ensures a stable exponent α even in the presence of large fluctuations of the terms parameterizing the economy, namely the random equity productivity term h and the aggregate employment level n. To see how this result can be obtained, we transform the system (7) from discrete to continuous time to write the time evolution of the equity base of firm i as:

Industrial Dynamics and Fluctuations

dai (t) = ai (t + τ ) − ai (t) = [βi (t) − 1] ai (t) + γi a (t) − δ (a, t) ai (t)

43

(11) 2

where τ is a (continuos) time interval, βi = hi +R, and δ (a, t) = R bhφt2at . To ensure the limit τ → ∞ to be meaningful, let γi (t), δ (a, t) and the time averages s = βi (t) − 1 - equal for all i’s- to be of order τ . let us define the variance of the random terms β.   2   2 σ 2 = βi2 − β ∼ βi − β (12) = [εi (t)]2 where εi (t) = βi − β represents the stochastic fluctuations of the autocatalytic term. Without any loss of generality, we assume εi (t) = 0. Note that βi (t) − 1 = εi (t) + s, so that we can write:

dai (t) = ai (t + τ ) − ai (t) = [εi (t) + s] ai (t) + γi a (t) − δ (a, t) ai (t) (13) ai (t) If we introduce the change of variable ϕi (t) = a(t) , which represents the relative equity of the i-th firm, and apply the chain rule for differentials, we obtain: dϕi (t) = [εi (t) − γi ] ϕi + γi.

(14)

The system (14) shows that the stochastic dynamics of the relative equity base and, due to the linearity assumptions, of the relative sales and employment as well, reduces to a set of independent, linear equations. If we assume that the variance of the uncertainty associated to the relative price is common to all, so that σi2 = σ 2 , and the same is true for the hot market influence on individual equity, γi = γ, we obtain (Solomon and Richmond, 2001):   −2γ −1−α P (ϕ) ∼ ϕ exp 2 σ ϕ with α = 1 + 1 2 1+ σγ

2γ . σ2

distribution P (ϕ) is unimodal, as it peaks at ϕ0 =

. Above ϕ0 that is on its upper tail, it behaves like a power law

with scaling exponent α, while below ϕ0 it vanishes very fast. The theoretical value of the scaling exponent α is bounded from below to 115 , it increases with γ and it decreases with σ 2 . In other 15

The case α = 1 is also known as the Zipf’s law.

44

Domenico Delli Gatti et al.

terms, α depends exclusively on: i) how much firms are rationed in issuing new risk capital or, in other terms, on how much capital markets are affected by adverse selection and moral hazard phenomena; ii) how much individuals are heterogeneous as regards the perceived risk associated to their final demand. Given that both the degree with which these two factors bites and the array of institutions set up to confront them differ widely among countries, besides predicting that the firms’ size distribution scales down as a power law, our model suggests that the degree of industrial concentration should be country-specific16 . Both the shape of the distribution P (ϕ) – and therefore of P (a), P (y) and P (n) – and the heterogeneity of scaling among countries are consistent with the empirical evidence. Axtell [3] reports estimates for the whole universe of the U.S. firms as derived from the Census of Manufactures, suggesting that the firms’ size distribution is Pareto distributed with α = 1. Other studies concentrate on large firms in international samples, largely confirming Axtell’s results. Gaffeo et al. [18], for instance, show that the upper tail of the firms’ size distribution as derived for a pool of quoted large companies from the G7 countries is Pareto, and that the scaling exponent is comprised between 1 (total sales) and 1.15 (total assets) as one changes the proxy used to measure the firms’ size. Ramsden and Kiss-Hayp´ al [46], furthermore, provide evidence on the larger firms for a sample of 20 countries: for 16 of them, the firms’ size distribution scales as a power law with α comprised between 1 and 1.2517 . Interestingly enough, α does not directly depend on how firms interact in the labour market, which can thus be modelled recurring to alternative assumptions without affecting the shape of the long run firms’ size distribution. To put it in another way, our model suggests that a necessary condition for the firms’ size to evolve towards a power law distribution is to let firms’ decisions be affected by competitive pressures from the labour market, but also that the shape of the distribution is invariant to details. In particular, it is invariant to the precise form of the relation between wages and unemployment, to the way the productivity dynamics or fiscal variables affect it, and consequently to the level of the natural rate of unemployment. 16

17

Naldi [43] derives analytical relationships between the scaling exponent of a power law distribution and several major concentration indices, like the HirschmanHerfindahl and the Gini indices. The exceptions are China (α = 0.83; data referred to 1993), Hungary (α= 0.71; data referred to 1992), South-Africa (α= 2.27; data referred to 1994) and U.S.A. (α= 0.8; data referred to 1994).

Industrial Dynamics and Fluctuations

45

Furthermore, our model does not possess the degenerative asymptotic behaviour affecting both the basic Gibrat’s model and its numerous extensions, that is a long run dynamics which implodes or explodes but for a tiny (and empirically implausible) set of values of the average stochastic multiplicative shock. In particular, the industrial dynamics evolves towards a stable distribution with a Pareto-Levy upper tail even if the dynamics of the equity productivity term h is nonstationary, or the parameter b summarizing how the labour market works is subject to sudden, large shifts. Finally, it is worthwhile to note that the parameter γ — a proxy for the level of development of capital markets — tunes at the same time the dynamic features of aggregate fluctuations and the longitudinal characteristics of microeconomic units. Suppose γ is initially at a low value. The design and implementation of new financial regulations abating moral hazard and adverse selection problems, or the emergence of market-based financial innovations aimed at easing risk diversification, translate into an easier access to new equity financing: γ increases for all firms, regardless of their leverage position. Ceteris paribus, this results in a more egalitarian distribution of firms’ size (a higher α), due to higher positive externalities in external finance. The increase of γ, in turn, causes an increase of the control parameter of the logistic map Γ , which may move the system’s aggregate (i.e., per-capita) dynamics from, say, a periodic oscillation (limit cycle) regime to a chaotic one. In our model, reducing the inequality of access to investment opportunities may be a necessary condition for an increase in macroeconomic volatility. The intuition for this result is straightforward. A higher γimplies more powerful spillovers stemming from average capitalization towards individual laws of motion for equity. Higher individual equity bases, however, cause firms to demand more labour and, due to condition C1, to more debt. A higher labour demand, in turn, implies a higher real wage which drives towards lower profits. From an analytical viewpoint, as spillovers raise the curvature of the logistic map (9) increases, and the per-capita dynamics crosses several bifurcations migrating from a stable regime to a chaotic one. Incidentally, our findings lend some theoretical support to the argument put forth by authoritative commentators (see e.g. ECB [14]), according to whom the spurt of volatility occurred in international markets during the summer of 2007 has been partly due to the rising leverage of the corporate sector associated to the massive increase of private equity-sponsored leveraged buyout deals occurred since 2003.

46

Domenico Delli Gatti et al.

4 Conclusions The relation among the firms’ size and growth rates, their longitudinal distribution and macroeconomic activity represents an issue of major importance for the economic profession as a whole. A proper understanding of why firms grow (or do not grow) and how they distribute in the long-run may help in understanding whether microeconomic constraints are important for macroeconomic growth (Kumar at al. [38]). Furthermore, realizing that firms respond differently and, in particular, assessing how different their responses are to the business cycle may help in designing policies to reduce firms’ vulnerability (Higson et al. [31]) or to foster job creation at an aggregate level (Hart and Oulton [28]). In this chapter we show how heterogeneity in the firms’ financial position and the competitive pressure exerted by their indirect interactions on the labour and the equity markets can combine to explain both macroeconomic fluctuations and the cross-sectional distribution of microeconomic variables. In particular, a key stylized fact established by empirical studies on longitudinal firm data the firms’ size distribution is right-skewed, and its upper tail scales as a power law can be interpreted in terms of a stable, steady-state statistical equilibrium associated to microeconomic dynamics in a maximizing, rational expectations model. Our findings suggest that the degree of long-run heterogeneity, measured by the scaling exponent characterizing the right tail of the firms’ size distribution, depends on institutional factors, i.e. how financial markets work, and the degree of uncertainty affecting economic agents.

Industrial Dynamics and Fluctuations

47

References 1. Altman, E., (1984). A further empirical investigation of the bankruptcy cost question. Journal of Finance 39, 1067-1089. 2. Anderson, S., De Palma, A., Thisse, J.-F., (1992). Discrete Choice Theory and Product Differentiation. MIT Press, Cambridge. 3. Axtell, R., (2001). Zipf distribution of U.S. firm size. Nature 293, 18181820. 4. Baumol, W., Benhabib, J., (1989). Chaos: significance, mechanism and economic applications. Journal of Economic Perspectives 3, 77-105. 5. Botazzi, G., Cefis, E., Dosi, G., Secchi, A. (2007). Invariances and diversities in the evolution of Italian manifacturing industry. Small Business Economics 29, 137-159. 6. Bottazzi, G., Secchi, A. (2003). Common properties and sectoral specificities in the dynamics of U.S. manufacturing companies. Review of Industrial Organization 23, 217-232. 7. Brailsford, T., Heany, R., Powell, J., Shi, J., (2000). Hot and cold IPO markets: identification using a regime switching model. Multinational Finance Journal 4, 35-68. 8. Cabral, L., Mata, J., (2003). On the evolution of the firm size distribution: facts and theory. American Economic Review 93, 1075-1090. 9. Choe, H., Masulis, R., Nanda, V., (1993). Common stock offerings across the business cycle: theory and evidence. Journal of Empirical Finance 1, 3-31. 10. Colander, D., (1996). The macrofoundations of micro. In: Colander, D. (Ed.), Beyond Microfoundations. Cambridge University Press, Cambridge. 11. Delli Gatti, D., (1999). Firms’ size and monetary policy: sone New Keynesian reflections. In: Gallegati, M., Kirman, A. (Eds), Beyond the Representative Agent. Edward Elgar, Cheltenham. 12. Delli Gatti, D., Di Guilmi, C., Gaffeo, E., Gallegati, M., Giulioni, G., Palestrini A., (2005). A new approach to business fluctuations: heterogeneous interacting agents, scaling laws and financial fragility. Journal of Economic Behavior and Organization 56, 489-512. 13. Delli Gatti, D., Gallegati, M., (1996). Financial market imperfections and irregular growth cycles. Scottish Journal of Political Economy 43, 146-158. 14. European Central Bank, (2007). Financial Stability Review – June 2007. ECB, Frankfurt. 15. Ericson, R., Pakes, A., (1995). Markov-perfect industry dynamics: a framework for empirical work. Review of Economic Studies 62, 53-82. 16. Fazzari, S., Hubbard, G., Petersen, G., (1988). Financing constraints and corporate investment. Brookings Chapters on Economic Activity 1, 141206.

48

Domenico Delli Gatti et al.

17. Filkestein, A., Poterba, J., (2004). Adverse selection in insurance markets: policyholder evidence from the UK annuity market, Journal of Political Economy. 18. Gaffeo, E., Gallegati, M., Palestrini, A., (2003). On the size distribution of firms: additional evidence from the G7 countries. Physica A 324, 117-123. 19. van Galderen, M., Thurik, R., Bosma, N., (2005). Success and risk factors in the pre-startup phase. Small Business Economics 24, 365-380. 20. Gallegati, M., (1994). Irregular business cycles. Structural Change and Economic Dynamics 5, 73-79. 21. Geroski, P., Lazarova, S., Urga, G., Walters C., (2003). Are differences in firm size transitory or permanent. Journal of Applied Econometrics 18, 47-59. 22. Gertler, M., (1988). Financial structure and aggregate activity: an overview. Journal of Money, Credit and Banking 20, 559-588. 23. Gibrat, R., (1931). Les Ingalit´es Economiques. Sirey, Paris. 24. Gilson, S., (1990). Bankruptcy, boards, banks and blockholders: evidence on changes in corporate ownership and control when firms default. Journal of Financial Economics 27, 355-388. 25. Goodwin, R., (1967). A growth cycle. In: Fenstein, C. (Ed.), Socialism, Capitalism and Economic Growth: Essays Presented to Maurice Dobb. Cambridge University Press, Cambridge. 26. Greenwald, B., Stiglitz, J., (1993). Financial market imperfections and business cycles. Quarterly Journal of Economics 108, 77-114. 27. Hart, P., (1962). The size and growth of firms. Economica 29, 29-39. 28. Hart, P., Oulton, N., (2001). Galtonian regression, company age and job generation 1986-95. Scottish Journal of Political Economy 48, 82-98. 29. Hart, P., Prais, S., (1956). The analysis of business concentration: a statistical approach. Journal of the Royal Statistical Association 119, 150-191. 30. Hammar, O., Johansson-Stenman, H., (2004). The value of risk-free cigarettes- Do smokers underestimate risks? Health Economics 13, 53-71. 31. Higson, C., Holly, S., Kattuman, P., Platis, S., (2004). The business cycle, macroeconomic shocks and the cross section: the growth of UK quoted companies. Economica 71, 299-318. 32. Ijiri, Y., Simon, H., (1977). Skew Distributions and the Sizes of Firms. North-Holland, Amsterdam. 33. Iooss, G., (1979). Bifurcation of Maps and Applications. North Holland, Amsterdam. 34. Jovanovic, B., (1982). Selection and the evolution of industry. Econometrica 50, 649-670. 35. Kaplan, S., Reishus, D., (1990). Outside directorship and corporate performance. Journal of Financial Economics 27, 389-410. 36. Kesten, H., (1973). Random difference equations and renewal theory for products of random matrices. Acta Mathematica 131, 207-248. 37. Klette, J., Kortum, S., (2004). Innovating firms and aggregate innovation. Journal of Political Economy 112, 986-1018.

Industrial Dynamics and Fluctuations

49

38. Kumar, K., Rajan, R., Zingales, L., (2001). What determines firm size? The Center for Research in Security Prices Working Chapter No. 496, University of Chicago. 39. Levy, M., Solomon, S., (1996). Spontaneous scaling emergence in generic stochastic systems. International Journal of Modern Physics C 7, 745-751. 40. Lindbeck, A., (1992). Macroeconomic theory and the labor market. European Economic Review 36, 209-235. 41. Lundborg, P., Lindgren, B., (2002). Risk perceptions and alcohol consumption among young people. Journal of Risk and Uncertainty 25, 16583. 42. Mansfield, E., (1962). Entry, Gibrat’s Law, innovation and the growth of firms. American Economic Review 52, 1023-1051. 43. Naldi, M., (2003). Concentration indices and Zipf’s law. Economics Letters 79, 329-334. 44. Pennings, J., Garcia, P., (2004). Hedging behaviour in small and mediumsized enterprises: the role of unobserved heterogeneity. Journal of Banking and Finance 28, 951-978. 45. Penrose, E., (1959). The Theory of the Growth of the Firm. Basil Blackwell, Oxford. 46. Ramsden, J., Kiss-Haypal, G., (2000). Company size distribution in different countries. Physica A 277, 220-227. 47. Richiardi, M., (2004). Generalizing Gibrat: reasonable multiplicative models of firm dynamics with entry and exit. Journal of Artificial Societies and Social Simulation 7(1). 48. Simon, H., Bonini, C., (1958). The size distribution of business firms. American Economic Review 48, 607-617. 49. Solomon, S., (2000). Generalized Lotka-Volterra (GLV) models and generic emergence of scaling laws in stock markets. In: Ballot, G., Weisbuch, G. (Eds), Applications of Simulation to Social Sciences. Hermes, Paris. 50. Solomon, S., Richmond, P., (2001). Stability of Pareto-Zipf Law in nonstationary economies. In: Kirman, A., Zimmermann, J.-B. (Eds.), Economics with Heterogeneous Interacting Agents. Springer, Berlin. 51. Sutton, J., (1997). Gibrat’s legacy. Journal of Economic Literature 35, 40-59. 52. Velupillai, K., (ed.) (1990). Nonlinear and Multisectoral Macrodynamics: Essays in Honour of Richard Goodwin. New York University Press, New York. 53. Weiss, L., (1990). Bankruptcy resolution: direct costs and the violation of priority of claims. Journal of Financial Economics 27, 285-314. 54. White, M., (1983). Bankruptcy costs and the New Bankruptcy Code. Journal of Finance 38, 477-488. 55. White, M., (1989). The corporate bankruptcy decision. Journal of Economic Perspectives 3, 129-151.

Part II

Quantum Physics and Financial Markets

Quantum-like Viewpoint on the Complexity and Randomness of the Financial Market Olga Choustova V¨axj¨ o University - Sweden

1 Introduction 1.1 Efficient-market Hypothesis In economics and financial theory, analysts use random walk and more general martingale techniques to model behavior of asset prices, in particular share prices on stock markets, currency exchange rates and commodity prices. This practice has its basis in the presumption that investors act rationally and without bias, and that at any moment they estimate the value of an asset based on future expectations. Under these conditions, all existing information affects the price, which changes only when new information comes out. By definition, new information appears randomly and influences the asset price randomly. Corresponding continuous time models are based on stochastic processes (this approach was initiated in the thesis of L. Bachelier [4] in 1890), see, e.g., the books of R. N. Mantegna and H. E. Stanley [33] and A. Shiryaev [37] for historical and mathematical details. This practice was formalized through the efficient market hypothesis which was formulated in sixties, see P.A. Samuelson [35] and E. F. Fama [17] for details: A market is said to be efficient in the determination of the most rational price if all the available information is instantly processed when it reaches the market and it is immediately reflected in a new value of prices of the assets traded. Mathematically the efficient market hypothesis was supported by investigations of Samuelson [35]. Using the hypothesis of rational behavior and market efficiency he was able to demonstrate how qt+1 , the

54

Olga Choustova

expected value of price of a given asset at time t + 1, is related to the previous values of prices q0 , q1 , ..., qt through the relation E(qt+1 |q0 , q1 , ..., qt ) = qt .

(1)

Typically there is introduced the σ-algebra Ft generated by random variables q0 , q1 , ..., qt . The condition (1) is written in the form: E(qt+1 |Ft ) = qt .

(2)

Stochastic processes of such a type are called martingales [37]. Alternatively, the martingale model for the financial market implies that the (qt+1 − qt ) is a “fair game” (a game which is neither in your favor nor your opponent’s): E(qt+1 − qt |Ft ) = 0. (3) On the basis of information, Ft , which is available at the moment t, one cannot expect either E(qt+1 − qt |Ft ) > 0 or E(qt+1 − qt |Ft ) < 0. 1.2 Deterministic Chaos? First we remark that empirical studies have demonstrated that prices do not completely follow random walk. Low serial correlations (around 0.05) exist in the short term; and slightly stronger correlations over the longer term. Their sign and the strength depend on a variety of factors, but transaction costs and bid-ask spreads generally make it impossible to earn excess returns. Interestingly, researchers have found that some of the biggest prices deviations from a random walk result from seasonal and temporal patterns, see the book [33]. There are also a variety of arguments, both theoretical and obtained on the basis of statistical analysis of data, which question the general martingale model (and hence the efficient market hypothesis), see, e.g., [3]–[27]. It is important to note that efficient markets imply there are no exploitable profit opportunities. If this is true then trading on the stock market is a game of chance and not of any skill, but traders buy assets they think are underevaluated at the hope of selling them at their true price for a profit. If market prices already reflect all information available, then where does the trader draw this privileged information from? Since there are thousands of very well informed, well educated asset traders, backed by many data researchers, buying and selling securities quickly, logically assets markets should be very efficient and profit opportunities should be minimal. On the other hand, we see that there are many traders whom successfully use their opportunities and

Quantum-like viewpoint

55

perform continuously very successful financial operations, see the book of G. Soros [38] for discussions1 . There were also performed intensive investigations on testing that the real financial data can be really described by the martingale model, see [5]–[27]. Roughly speaking people try to understand on the basis of available financial data: Do financial asset returns behave randomly (and hence they are unpredictable) or deterministically (and in the latter case one may hope to predict them and even to construct a deterministic dynamical system which would at least mimic dynamics of the financial market)? Predictability of financial asset returns is a broad and very active research topic and a complete survey of the vast literature is beyond the scope of this work. We shall note, however, that there is a rather general opinion that financial asset returns are predictable, see [5]–[27]. On the other hand, there is no general consensus on the validity of the efficient market hypothesis. As it was pointed out in [10]: “ . . . econometric advances and empirical evidence seem to suggest that financial asset returns are predictable to some degree. Thirty years ago this would have been tantamount to an outright rejection of market efficiency. However, modern financial economics teaches us that others, perfectly rational factors may account for such predictability. The fine structure of securities markets and frictions in the trading process can generate predictability. Time-varying expected returns due to changing business conditions can generate predictability. A certain degree of predictability may be necessary to reward investors for bearing certain dynamic risks.” 1.3 Behavioral Financial Models Therefore it would be natural to develop approaches which are not based on the assumption that investors act rationally and without bias and that, consequently, new information appears randomly and influences the asset price randomly. In particular, there are two well established (and closely related ) fields of research behavioral finance and behavioral economics which apply scientific research on human and social cognitive and emotional biases2 to better understand economic 1 2

It seems that G.Soros is sure he does not work at efficient markets. Cognitive bias is any of a wide range of observer effects identified in cognitive science, including very basic statistical and memory errors that are common to all human beings and drastically skew the reliability of anecdotal and legal evidence. They also significantly affect the scientific method which is deliberately designed to minimize such bias from any one observer. They were first identified by Amos Tversky and Daniel Kahneman as a foundation of behavioral economics. Bias

56

Olga Choustova

decisions and how they affect market prices, returns and the allocation of resources. The fields are primarily concerned with the rationality, or lack thereof, of economic agents. Behavioral models typically integrate insights from psychology with neo-classical economic theory. Behavioral analysis is mostly concerned with the effects of market decisions, but also those of public choice, another source of economic decisions with some similar biases. Behavioral models form a well established domain of research in finance, see, e.g., Beja and Goldman [6] or Lux [32]. In this paper we develop a new approach that is not based on the assumption that investors act rationally and without bias and that, consequently, new information appears randomly and influences the asset price randomly. Our approach can be considered as a special econophysical [33] model in the domain of behavioral finance. In our approach information about financial market (including expectations of agents of the financial market) is described by an information field ψ(q) – financial wave. This field evolves deterministically3 perturbing the dynamics of prices of stocks and options. Since psychology of agents of the financial market gives an important contribution into the financial wave ψ(q), our model can be considered as a special psycho-financial model. 1.4 Quantum-like Models outside Physics This paper can be also considered as a contribution into applications of quantum mechanics outside the microworld, see [2], [1], [28]–[18]. This paper is fundamentally based on investigations of D. Bohm, B. Hiley, and P. Pylkk¨ anen [8], [25] on the active information interpretation of Bohmian mechanics [7], [26] and its applications to cognitive sciences, see also Khrennikov [28]. In this paper we use methods of Bohmian mechanics to simulate dynamics of prices in the financial market. We start with the development of the classical Hamiltonian formalism on the price/price-change phase space to describe the classical-like evolution of prices. This classical dynamics of prices is determined by “hard” financial conditions (natural resources, industrial production, services and so on). These conditions as well as “hard” relations between traders at the financial

3

arises from various life, loyalty and local risk and attention concerns that are difficult to separate or codify. Tversky and Kahneman claim that they are at least partially the result of problem-solving using heuristics, including the availability heuristic and the representativeness heuristic. Dynamics are given by Schr¨ odinger’s equation on the space of prices of shares.

Quantum-like viewpoint

57

market are mathematically described by the classical financial potential. As we have already remarked, at the real financial market “hard” conditions are not the only source of price changes. The information and market psychology play important (and sometimes determining) role in price dynamics. We propose to describe this “soft” financial factors by using the pilot wave (Bohmian) model of quantum mechanics. The theory of financial mental (or psychological) waves is used to take into account market psychology. The real trajectories of prices are determined (by the financial analogue of the second Newton law) by two financial potentials: classical-like (“hard” market conditions) and quantum-like (“soft” market conditions). Our quantum-like model of financial processes was strongly motivated by consideration by J. Soros [38] of the financial market as a complex cognitive system. Such an approach he called the theory of reflexivity. In this theory there is a large difference between market that is “ruled” by only “hard” economical factors and a market where mental factors play the crucial role (even changing the evolution of the “hard” basis, see [38]). J. Soros rightly remarked that “non mental” market evolves due to classical random fluctuations. However, such fluctuations do not provide an adequate description of the mental market. He proposed to use analogy with quantum theory. However, it was noticed that directly quantum formalism could not be applied to the financial market [38]. Traders differ essentially from elementary particles. Elementary particles behave stochastically due to perturbation effects provided by measurement devices, cf. [23], [16]. According to J. Soros, traders at the financial market behave stochastically due to free will of individuals. Combinations of a huge number of free wills of traders produce additional stochasticity at the financial market that could not be reduced to classical random fluctuations (determined by non mental factors). Here J. Soros followed the conventional (Heisenberg, Bohr, Dirac, see, e.g., [23], [16]) viewpoint to the origin of quantum stochasticity. However, in the Bohmian approach (that is nonconventional one) quantum statistics is induced by the action of an additional potential, quantum potential, that changes classical trajectories of elementary particles. Such an approach gives the possibility to apply quantum formalism to the financial market. There were performed numerous investigations on applying quantum methods to financial market, see, e.g., E. Haven [19]–[21], that were not directly coupled to behavioral modeling, but based on the

58

Olga Choustova

general concept that randomness of the financial market can be better described by the quantum mechanics, see, e.g., [20]: “A natural explanation for extreme irregularities in the evolution of prices in financial markets is provided by quantum effects.” Non-Bohmian quantum model for the financial market was developed by E. W. Piotrowski, J. Sladkowski [34]. This model can be also considered as a kind of behavioral quantum-like model. 1.5 The Problem of Quadratic Variation A pure technical objection against the deterministic Bohmian model is that Bohmian trajectories (as solutions of an ordinary differential equation) have zero quadratic variation. On the other hand, there is a general consensus that the real price trajectories have nonzero quadratic variation. One of the possible replies to this critique is that the problem of quadratic variation is merely a problem of the choice of a time scale. Therefore by considering a discrete version of Bohmian dynamics we can escape such a criticism. Thus one of the possibilities is stay rigidly on deterministic grounds and use in this way the Bohmian quantum model to criticize the efficient market hypothesis. Another possibility is to try to combine Bohmian mechanics with the traditional stochastic financial mathematics. We shall explore such a possibility in this paper. We are especially lucky, because in physics such a model was created long ago. This is the well known the Bohm-Vigier model, see, e.g., [8]. Moreover, recently this model was even applied to financial market. E. Haven used this model to describe dynamics of options [22]. The Bohm-Vigier financial model is in the complete agreement with the efficient market hypothesis. Mathematical features of the pricedynamics are similar to the conventional martingale model.

2 Price Phase Space, Modeling of Financial Energy We repeat shortly the Bohmian model for the financial market proposed in [11]–[13]. Let us consider a mathematical model in that a huge number of agents of the financial market interact with one another and take into account external economic (as well as political, social and even meteorological) conditions in order to determine the price to buy or sell financial assets. We consider the trade with shares of some corporations (e.g., VOLVO, SAAB, IKEA,...).

Quantum-like viewpoint

59

We consider a price system of coordinates. We enumerate corporations which did emissions of shares at the financial market under consideration: j = 1, 2, ...., n (e.g., VOLVO:j = 1, SAAB:j = 2, IKEA:j = 3,...). There can be introduced the n-dimensional configuration space Q = Rn of prices, q = (q1 , . . . , qn ), where qj is the price of a share of the jth corporation. Here R is the real line. Dynamics of prices are described by the trajectory q(t) = (q1 (t), . . . , qn (t)) in the configuration price space Q. Another variable under the consideration is the price change variq (t+∆t)−qj (t) , see, for example, the book able: vj (t) = q˙j (t) = lim∆t→0 j ∆t [33] on the role of the price change description. In real models we consider the discrete time scale ∆t, 2∆t, . . . . Here we should use discrete price change variable δqj (t) = qj (t + ∆t) − qj (t). We now introduce an analogue m of mass as the number of items (in our case shares) that a trader emitted to the market. We call m the financial mass. Thus each trader j (e.g., VOLVO) has its own financial mass mj (the size of the emission of its shares). The total price of the emission performed by the jth trader is equal to Tj = mj qj – market capitalization. We also introduce financial energy of the market as a function H : Q × V → R. If we use the analogue with classical mechanics, then we could consider (at least for  mathematical modeling) the financial 1 energy of the form: H(q, v) = 2 nj=1 mj vj2 + V (q1 , . . . , qn ).  Here K = 21 nj=1 mj vj2 is the kinetic financial energy and V (q1 , . . . , qn ) is the potential financial energy, mj is the financial mass of jth trader. The kinetic financial energy represents efforts of agents of financial market to change prices: higher price changes induce higher kinetic financial energies. If the corporation j1 has higher financial mass than the corporation j2 , so mj1 > mj2 , then the same change of price, i.e., the same financial velocity vj1 = vj2 , is characterized by higher kinetic financial energy: Kj1 > Kj2 . We also remark that high kinetic financial energy characterizes rapid changes of the financial situation at market. However, the kinetic financial energy does not give the attitude of these changes. It could be rapid economic growth as well as recession. The potential financial energy V describes the interactions between traders j = 1, ...., n (e.g., competition between NOKIA and ERICSSON) as well as external economic conditions (e.g., the price of oil and gas) and even meteorological conditions (e.g., the weather conditions in Louisiana and Florida). For example, we can consider the simplest

60

Olga Choustova

 interaction potential: V (q1 , . . . , qn ) = ni,j=1 (qi − qj )2 . The difference |q1 − qj | between prices is the most important condition for arbitrage. As in classical mechanics for material objects, we introduce a new variable p = mv, the price momentum variable. Instead of the price change vector v = (v1 , . . . , vn ), we consider the price momentum vector ∂V p = (p1 , . . . , pn ), pj = mj vj . The quantity fj (q) = − ∂q is called the j financial force. We can postulate the financial variant of the second Newton law: mv˙ = f (4) “The product of the financial mass and the price acceleration is equal to the financial force.”

3 The Bohmian Model for Behavior of the Financial Market Our fundamental assumption is that agents in the modern financial market are not just “classical-like agents.” Their actions are ruled not only by classical-like financial potentials V (t, q1 , . . . , qn ), but also (in the same way as in the pilot wave theory for quantum systems) by an additional information (or psychological) potential induced by a financial pilot wave. Therefore we could not use the classical financial dynamics (Hamiltonian formalism) on the financial phase space to describe the real price trajectories. Information (psychological) perturbation of Hamiltonian equations for price and price change must be taken into account. To describe such a model mathematically, it is convenient to use such an object as a financial pilot wave that rules the financial market. In some sense ψ(q) describes the psychological influence of the price configuration q to behavior of agents of the financial market. In particular, the ψ(q) contains expectations of agents4 . We underline two important features of the financial pilot wave model: 1. All shares are coupled on the information level. The general formalism [7], [8], [26] of the pilot wave theory says that if the function ψ(q1 , . . . , qn ) is not factorized, i.e., ψ(q1 , . . . , qn ) = ψ1 (q1 ) . . . ψn (qn ), then any changing the price qi will automatically change behavior of all 4

The reader may be surprised that there appeared complex numbers C. However, the use of these numbers is just a mathematical trick that provides the simple mathematical description of dynamics of the financial pilot wave.

Quantum-like viewpoint

61

agents of the financial market (even those who have no direct coupling with i-shares). This will imply changing of prices of j-shares for i = j. At the same time the ”“““hard”””” economic potential V (q1 , . . . , qn ) need not contain any interaction term: for example, V (q1 , . . . , qn ) = q12 + . . . qn2 . The Hamiltonian equations in the absence of the financial pilot wave have the form: q˙j = pj , p˙j = −2qj , j = 1, 2, . . . , n. Thus the classical price trajectory qj (t), does not depend on dynamics of prices of shares for other traders i = j (for example, the price of shares of ERIKSSON does not depend on the price of shares of NOKIA and vice versa)5 . 2 2 However, if, e.g., ψ(q1 , . . . , qn ) = cei(q1 q2 +...+qn−1 qn ) e−(q1 +...+qn ) , where c ∈ C is some normalization constant, then financial behavior of agents at the financial market is nonlocal (see further considerations). 2. Reactions of the market do not depend on the amplitude of the financial pilot wave: waves ψ, 2ψ, 100000ψ will produce the same reaction. Such a behavior at the market is quite natural (if the financial pilot wave is interpreted as an information wave, the wave of financial information). The amplitude of an information signal does not play such a large role in the information exchange. The most important is the context of such a signal. The context is given by the shape of the signal, the form of the financial pilot wave function. In fact, we need not develop a new mathematical formalism. We will just apply the standard pilot wave formalism to the financial market. The fundamental postulate of the pilot wave theory is that the pilot wave (field) ψ(q1 , . . . , qn ) induces a new (quantum) potential U (q1 , . . . , qn ) which perturbs the classical equations of motion. A modified Newton equation has the form: p˙ = f + g,

(5)

∂U where f = − ∂V ∂q and g = − ∂q . We call the additional financial force g a financial mental force. This force g(q1 , . . . , qn ) determines a kind of collective consciousness of the financial market. Of course, the g depends on economic and other “““‘hard”””’ conditions given by the financial potential V (q1 , . . . , qn ). However, this is not a direct dependence. In principle, a nonzero financial mental force can be induced by the financial pilot wave ψ in the case of zero financial potential, V ≡ 0. So 5

Such a dynamics would be natural if these corporations operate on independent markets, e.g., ERIKSSON in Sweden and NOKIA in Finland. Prices of their shares would depend only on local market conditions, e.g., on capacities of markets or consuming activity.

62

Olga Choustova

V ≡ 0 does not imply that U ≡ 0. Market psychology is not totally determined by economic factors. Financial (psychological) waves of information need not be generated by some changes in a real economic situation. They are mixtures of mental and economic waves. Even in the absence of economic waves, mental financial waves can have a large influence to the market. By using the standard pilot wave formalism we obtain the following rule for computing the financial mental force. We represent the financial pilot wave ψ(q) in the form: ψ(q) = R(q)eiS(q) , where R(q) = |ψ(q)| is the amplitude of ψ(q), (the absolute value of the complex number c = ψ(q)) and S(q) is the phase of ψ(q) (the argument of the complex number c = ψ(q)). Then the financial mental potential is com 2 puted as U (q1 , ..., qn ) = − R1 ni=1 ∂∂qR2 and the financial mental force i

as gj (q1 , . . . , qn ) = −∂U ∂qj (q1 , . . . , qn ). These formulas imply that strong financial effects are produced by financial waves having essential variations of amplitudes.

4 The Bohm-Vigier model for Behavior of the Financial Market The fundamental efficient market objection as well as the quadratic variation objection motivate consideration of the Bohm-Vigier stochastic model, instead of the completely deterministic Bohmian model. We follow here [8]. We recall that in the original Bohmian model the velocity of an individual particle is given by v=

∇S(q) . m

(6)

If ψ = ReiS/h , then Schr¨ odinger’s equation implies that dv dt = −∇(V + U ), where V and U are classical and quantum potentials respectively. In principle one can work only with the basic equation (6). The basic assumption of Bohm and Vigier was that the velocity of an individual particle is given by v=

∇S(q) + η(t), m

(7)

where η(t) represents a random contribution to the velocity of that particle which fluctuates in a way that may be represented as a random process but with zero average. In Bohm-Vigier stochastic mechanics

Quantum-like viewpoint

63

quantum potential comes in through the average velocity and not the actual one. We now shall apply the Bohm-Vigier model to financial market, see also E. Haven [22]. The equation (7) is considered as the basic equation for the price velocity. Thus the real price becomes a random process (as well as in classical financial mathematics [37]). We can write the stochastic differential equation, SDE, for the price: dq(t) =

∇S(q) dt + η(t)dt. m

(8)

To give the rigorous mathematical meaning to the stochastic differential we assume that dξ(t) , (9) η(t) = dt for some stochastic process ξ(t). Thus formally: η(t)dt = dξ(t) dt dt = dξ(t), and the rigorous mathematical form of the equation (8) is dq(t) = ∇S(q) m dt + dξ(t). The expression (9) one can consider either formally or in the sense of distribution theory (we recall that for basic stochastic processes, e.g., the Wiener process, trajectories are not differentiable in the ordinary sense almost every where). Suppose, for example, that the random contribution into the price dynamics is given by white noise, ηwhite noise (t).It can be defined as the derivative (in the sense of distribution theory) of the Wiener process: ∇S(q) ηwhite noise (t) = dw(t) dt , thus: v = m + ηwhite noise (t), In this case the price dynamics is given by the SDE: dq(t) =

∇S(q) dt + dw(t). m

(10)

What is the main difference from the classical SDE-description of the financial market? This is the presence of the pilot wave ψ(t, q), mental field of the financial market, which determines the coefficient of drift ∇S(q) m . Here S ≡ Sψ . And the ψ-function is driven by a special field equation – Schr¨ odinger’s equation. The latter equation is not determined by the SDE (10). Thus, instead of one SDE, in the quantum-like model, we have the system of two equations: dq(t) = ih

∇Sψ (q) dt + dξ(t). m

∂ψ h2 ∂ 2 ψ (t, q) + V (q)ψ(t, q). (t, q) = − ∂t 2m ∂q 2

(11) (12)

64

Olga Choustova

We have only to make one remark, namely, on the role of the constant h in Schr¨ odinger’s equation, cf. E. Haven [20], [21], [22]. In quantum mechanics (which deals with microscopic objects) h is the Planck constant. This constant is assumed to play the fundamental role in all quantum considerations. In our financial model we consider h as a price scaling parameter, namely, the unit in which we would like to measure price change. However, we do not exclude the possibility that there might be found a deeper interpretation of h. Finally we come back to the problem of the quadratic variation of the price. In the Bohm-Vigier stochastic model (for, e.g., the white noise fluctuations of the price velocity) is nonzero.

References 1. Accardi, L. (1997). Urne e Camaleoni: Dialogo sulla realta, le leggi del caso e la teoria quantistica. Il Saggiatore, Rome. 2. Aerts, D. and Aerts, S. (1995). Applications of quantum statistics in psychological studies of decision-proceses. Foundations of Science 1, 1-12. 3. Arthur, W.B., Holland, J.H., LeBaron, B., Palmer, R. and Tayler, P. (1997). Asset pricing under endogenous expectations in an artificial stock market. In: Arthur WA, Lane D, and Durlauf SN (eds) The economy as evolving, complex system-2. Addison-Wesley, Redwood City. 4. Bachelier, L. (1890). Annales Scientifiques de L’Ecole Normale Suprieure de Paris 111-17, 21-121. 5. Barnett, W.A. and Serletis, A. (1998). Martingales, nonlinearity, and chaos. Dept. Economics, Washington University – St. Louis Working Papers. 6. Beja, A. and Goldman, M.B. (1980). On the dynamic behavior of prices in disequilibrium. Journal of Finances 35,235-248. 7. Bohm, D. (1951). Quantum theory. Englewood Cliffs, New-Jersey, Prentice-Hall. 8. Bohm, D. and Hiley, B. (1993). The undivided universe: an ontological interpretation of quantum mechanics. Routledge and Kegan Paul, London. 9. Brock, W.A. and Sayers, C. (1988). Is business cycle characterized by deterministic chaos? Journal of Monetary Economics 22, 71-90. 10. Campbell, J.Y., Lo, A.W. and MacKinlay, A.C. (1997). The econometrics of financial markets. Princeton University Press, Princeton. 11. Choustova, O.A. (2001). Pilot wave quantum model for the stock market. http://www.arxiv.org/abs/quant-ph/0109122. 12. Choustova, O.A. (2004). Bohmian mechanics for financial processes. Jornal of Modern Optics 51, 1111. 13. Choustova, O.A. (2007). Quantum Bohmian model for financial market. Physica A: Statistical Physics and its Applications 374 (1), 304-314.

Quantum-like viewpoint

65

14. Conte, E., Todarello, O., Federici, A., Vitiello, F., Lopane, M., Khrennikov, A.Yu and Zbilut, J.P. (2006). Some remarks on an experiment suggesting quantum-like behavior of cognitive entities and formulation of an abstract quantum mechanical formalism to describe cognitive entity and its dynamics. Chaos, Solitons and Fractals 31, 1076-1088. 15. DeCoster, G.P. and Mitchell, D.W. (1991). Journal of Business and Economic Statistics 9, 455-462. 16. Dirac, P.A.M. (1995). The Principles of Quantum Mechanics. Claredon Press, Oxford. 17. Fama, E.F. (1970). Efficient Capital Markets: A Review of Theory and Empirical Work. Journal of Finance 25, 383-401. 18. Grib, A., Khrennikov, A., Parfionov, G. and Starkov, K. (2006). Quantum equilibria for macroscopic systems. Physics A: Mathematical and General 39, 8461-8475. 19. Haven, E. (2002). A Discussion on embedding the Black-Scholes option pricing model in a quantum physics setting. Physica A 304, 507-524. 20. Haven, E. (2003). A Black-Scholes Schrdinger Option Price: bit versus qubit. Physica A 324, 201-206. 21. Haven, E. (2004). The wave-equivalent of the Black-Scholes option price: an interpretation. Physica A 344, 142-145. 22. Haven, E. (2006). Bohmian mechanics in a macroscopic quantum system. In: Khrennikov AYu (ed) Foundations of Probability and Physics-3. American Institute of Physics, Melville, New York, 810, 330-340. 23. Heisenberg, W. (1930). Physical principles of quantum theory. Chicago Univ. Press, Chicago. 24. Hiley, B. (2001). From the Heisenberg picture to Bohm: a new perspective on active information and its relation to Shannon information. In: Khrennikov AYu (ed) Quantum Theory: Reconsideration of Foundations, V¨ axj¨ o University Press, V¨ axj¨ o, ser. Mathematical Modelling 10, 234-244. 25. Hiley, B. and Pylkk¨ anen, P. (1997). Active information and cognitive science – A reply to Kiesepp¨ a. In: Pylkk¨ anen P, Pylkk¨ o P and Hautam¨ aki A. (eds) Brain, mind and physics. IOS Press, Amsterdam, 121-134. 26. Holland, P. (1993). The quantum theory of motion. Cambridge Univ. Press, Cambridge. 27. Hsieh, D.A. (1991). Chaos and Nonlinear Dynamics: Application to Financial Markets. Journal of Finance 46, 1839-1850. 28. Khrennikov, A.Yu (2004). Information dynamics in cognitive, psychological and anomalous phenomena. Kluwer Academic, Dordreht. 29. Khrennikov, A.Yu (1999). (second edition, 2004) Interpretations of Probability. VSP Int. Sc. Publishers, Utrecht/Tokyo. 30. Khrennikov, A.Yu (2005). The principle of supplementarity: A contextual probabilistic viewpoint to complementarity, the interference of probabilities, and the incompatibility of variables in quantum mechanics. Foundations of Physics 35, 1655 -1693.

66

Olga Choustova

31. Khrennikov, A.Yu (2006). Quantum-like brain: Interference of minds. BioSystems 84, 225–241. 32. Lux, T. (1998). The socio-economic dynamics of speculative markets: interacting agents, chaos, and fat tails of return distributions. Journal of Economic Behavior and Organization 33, 143-165. 33. Mantegna, R.N. and Stanley, H.E. (2000). Introduction to econophysics. Cambridge Univ. Press, Cambridge. 34. Piotrowski, E.W. and Sladkowski, J. (2001). Quantum-like approach to financial risk: quantum anthropic principle. http://www.arxiv.org/abs/quant-ph/0110046. 35. Samuelson, P.A. (1965). Rational theory of warrant pricing. Industrial Management Review 6, 41-61. 36. Segal, W. and Segal, I.E. (1998). The BlackScholes pricing formula in the quantum context. Proceedings of the National Academy of Sciences USA 95, 4072-4080. 37. Shiryaev, A.N. (1999). Essentials of Stochastic Finance: Facts, Models, Theory. World Scientific Publishing Company, Singapore. 38. Soros, J. (1987). The alchemy of finance. Reading of mind of the market. J. Wiley and Sons, Inc., New-York.

Classical and Quantum-Like Randomness and the Financial Market Andrei Khrennikov V¨axj¨ o University - Sweden

1 Introduction 1.1 Classical probabilistic description of the financial market The financial market is a complex dynamical system and, since the publication of the thesis of L. Bachelier [1], there were performed numerous studies devoted to various aspects of random description of financial processes [2]. At the first stage of investigations Brownian motion was used to describe randomness of the financial market. This model provided a rather good approximation of some financial processes. However, later it became evident that the diversity of financial stochastic processes could not be reduced to Brownian motion. The next step was consideration of functionals of Brownian motion, especially, geometric Brownian motion [2]. Later there were considered other types of stochastic processes [2], in particular, general Levy processes. 1.2 The financial market as a cognitive system But even the possibility to describe financial market by the classical probabilistic model (and, in particular, dynamics of prices by stochastic differential equations) can be questioned [3]-[13]. We emphasize that the financial market is not a kind of a mechanical system. It is not a direct analog of a huge system of interacting physical particles 1 . Expectations of traders, exchange of information, news, political and social events are not less important than “the real development of industry and trade.” It might be more natural to consider financial market as a 1

Of course, one could proceed quite deeply by using such an analogy with statistical mechanics.

68

Andrei Khrennikov

huge information system (and not mechanical system) [3], [4], [11]–[13]. Moreover, we can even speculate that an adequate description of financial market can be approached through considering it as a cognitive system [3], [4], [11]–[13]. Hence, it may be possible to use the experience of mathematical description of cognitive systems for the financial market. 1.3 Classical and quantum cognitive models We recall basic approaches to description of brain’s functioning. There is a very strong tendency to interpret brain’s work as functioning of a huge neural network. Such a description (which is based on the classical probability theory and its application to the financial market – the network approach) generates classical financial processes. However, some groups of researchers in cognitive science and psychology do not believe that brain’s work could be totally reduced to the exchange of electric impulses between neurons. For example, it seems to be impossible to embed the consciousness into the reductionist picture. There is no any idea how a neuronal network (even extremely complex) would be able to produce the consciousness. It seems that brain’s complexity could not be reduced to complexity of networks. We remark that there were numerous attempts to apply quantum mechanics to describe mental processes, see [4] for extended bibliography. The main problem of the quantum approach to the description of mental processes is the impossibility to combine the neuronal and quantum models. The conventional interpretation of quantum mechanics is based on the notion of superposition of states for an individual quantum system, e.g., an electron. An electron can have a state of superposition to be in two different places simultaneously. On the other hand, a neuron could not be at the same time in the states firing and non-firing. 1.4 Irreducible quantum randomness? The conventional quantum interpretation of superposition induces a rather special viewpoint on randomness - individual randomness (see J. von Neumann [14], see also [15], [16]). It is commonly assumed that quantum randomness (described by the complex wave function ψ(x)) could not be reduced to classical ensemble randomness. The latter is induced by a variety of properties of elements of a statistical ensemble. It is described by the classical measure-theoretical approach based on the axiomatics of Kolmogorov.

Classical and Quantum-Like Randomness

69

We remark that A. Einstein, E. Schr¨ odinger, L. De Broglie, D. Bohm strongly criticized such a viewpoint to quantum randomness. They were sure that quantum randomness could be reduced to classical ensemble randomness. 1.5 Interference of probabilities The main problem in combining quantum probability with classical ensemble probability is to find a reasonable explanation of the interference of probabilities. Instead of the ordinary addition of probabilities of alternatives: (1) P = P1 + P2 , the quantum probabilistic calculus (based on transformations of vectors in the complex Hilbert space) predicts the general rule:  (2) P = P1 + P2 + 2 cos θ P1 P2 .

This is the so called quantum interference of probabilities. The difference between he rules (1) and (2) was considered as the strongest argument against the classical ensemble randomness and in favor of individual quantum randomness. Therefore a rather common opinion in quantum community is that quantum randomness differs crucially from the classical one and, in particular, it could not be considered as ensemble randomness. 1.6 Applications of quantum formalism outside the microworld

Such a viewpoint on randomness induces huge difficulties in applications of the quantum formalism outside the quantum domain. In particular, in cognitive sciences people should go to the quantum scales of space and time (R. Penrose even tried to go to the scales of quantum gravity). Such attempts of quantum reductionism did not find so much understanding by neurophysiologists, psychologists, cognitive scientists who did not believe that cognitive phenomena could be explained only at the quantum scale. It is even more difficult to apply the conventional quantum approach to randomness for the description of the financial market and general economic processes. In principle, one might believe that the brain functioning could be reduced to processes in the microworld (interactions of quantum particles in the brain), but it would be a funny thing to

70

Andrei Khrennikov

try to reduce the functioning of the financial market to interactions of photons and electrons, protons and neutrons composing traders of the market. Recently a solution of the interference problem in the classical ensemble framework was proposed in a series of authors papers, see [4] for the general presentation. The crucial point is that all (classical) probabilities should be considered as contextual probabilities. Here by context we understood a complex of conditions: physical, biological, economic or social. By taking into account the dependence of probabilities on contexts we can reproduce all distinguishing features of the quantum formalism: interference of probabilities, Born’s rule (i.e., the possibility to represent probability as the square of the absolute value of a complex amplitude - wave function), the representation of random variables by noncommutative operators. It seems that the essence of quantum formalism is not individual quantum randomness, but contextuality of probabilities. Such a viewpoint to the quantum probabilistic calculus - the quantum-like approach - provides new possibilities for its application. There is no more need for looking in economy for mystical objects being in superposition of their states. Any context dependent system, for instance, agents of market, may exhibit the quantum probabilistic behaviour and, hence, should be (in such a case) described by the quantum probability model. The main difference between the quantum probabilistic description and the classical one is that the dynamical equation is not a stochastic differential equation, but the Schr¨ odinger equation for the complex probability amplitude - the wave function2 . Such a quantum-like approach provides an approximative probabilistic representation of some class of processes in the situation in which the complete probabilistic description is impossible (because the complete information about a system is not available). In such a case we represent our partial knowledge about a system by the complex probabilistic amplitude. Such a complex probability amplitude ψ(t, x) describes dynamics of context (of e.g., the financial market, or some branch of it, or a corporation).

2

In our approach the wave function is just the special probabilistic representation of a context.

Classical and Quantum-Like Randomness

71

1.7 Can the financial market (or more generally, economics) be described as a quantum-like probabilistic system? If the answer to this question is positive, then one should apply the mathematical formalism of quantum probability theory to describe financial and economic processes. If it is negative, then one should be satisfied by the present approach which is based on classical stochastic differential equations. The only possibility to answer this question is to perform experimental statistical tests to verify non-classicality of financial or more general economic data. One of such tests is based on comparing classical and quantum formulas, (1) and (2). An experiment of such a type has been already done [4]. It confirmed the quantum-like behaviour of cognitive systems. We found the quantum-like interference of probabilities for answers to questions which were asked to groups of students. We remind that in our quantum-like model the interference is not the interference of waves, but the interference of probabilities. The additional interference term in the formula (2) is a consequence of dependence of the probabilities on contexts and not an exhibition of some mystical wave features of e.g., traders at the financial market. The quantum-like wave function represents contextual probabilistic features and nothing more. 1.8 Bell’s type test of “quantumness”of finances and economics In this note we describe another experimental statistical test which could be used to distinguish classical and quantum-like probabilistic behaviour in finances and economy. The last decades quantum physics has been strongly engaged in research which was related to violations of Bell’s inequality [17]. In the 60th J. Bell proved some inequality which should be valid for correlations of classical quantities, but violated for some quantum correlations. There were performed experimental tests which demonstrated that Bell’s inequality is really violated for experimental data. Our idea is to use an analogue of Bell’s test in finances and economics. However, it seems to be impossible to apply directly the physical methods to economics and finances. The main problem was that in quantum physics Bell’s inequality was tested for pairs of entangled particle. Roughly speaking these are particles-copies (at least with some approximation). Direct realization of such a test with e.g. traders of the financial market is practically impossible.

72

Andrei Khrennikov

We choose another way. We move from the original Bell’s inequality for correlations (or its analogue - Wigner’s inequality - for joint probabilities) and prove an analogous inequality for conditional probabilities. Such a test based on conditional probabilities can easily be performed with economic agents. If our Bell-like inequality for conditional probabilities should be violated, then such an experimental fact can be considered as a strong evidence in favor of quantumness of financial and economic processes.

2 Bell-like Inequalities Let P = (Ω, F, P) be a Kolmogorov probability space, [18]. For any pair of random variables u(ω), v(ω), their covariation is defined by  u(ω)v(ω)dP(ω). < u, v >= cov(u, v) = Ω

Theorem 2.1. (Bell inequality for covariations) Let ξa , ξb , ξc = ±1 be random variables on P. Then Bell’s inequality | < ξa , ξb > − < ξc , ξb > | ≤ 1− < ξa , ξc >

(3)

holds. Theorem 2.2. (Wigner inequality) Let ξa , ξb , ξc = ±1 be arbitrary random variables on a Kolmogorov space P. Then the following inequality holds: P(ξa = +1, ξb = +1) + P(ξb = −1, ξc = +1) ≥ P(ξa = +1, ξc = +1)

(4)

The detailed proofs can be found e.g. in [3]. Bell’s type inequalities are applied in the following way. One prepares pairs of entangled particles s = (s1 , s2 ). There are observables related to the first and second particles. They are labeled by some parameters taking values θ = a, b, c. Denote random variables corresponding to these observables3 for the first particle by ξa (ω), ξb (ω), ξc (ω) and for the second by ηa (ω), ηb (ω), ηc (ω). Entanglement of particles implies the precise correlations (or anti-correlations - depending on statistics): ξθ (ω) = ηθ (ω) 3

(5)

We would like to test the hypothesis that such classical random variables can be introduced – so to test the possibility to apply the classical probabilistic model.

Classical and Quantum-Like Randomness

73

for all values of the parameter θ. Therefore we can place ηθ (ω) to the second place (instead of ξθ (ω)) in the inequality (4). Instead of nonphysical probabilities P(ξa = +1, ξb = +1), P(ξb = −1, ξc = +1), P(ξa = +1, ξc = +1), we obtain the physical ones P(ξa = +1, ηb = +1), P(ξb = −1, ηc = +1), P(ξa = +1, ηc = +1). The main idea (belonging to Einstein, Podolsky and Rosen) is that, although we are not able to make the second measurement on the same particle s1 without to disturb it totally, nevertheless, we can measure ξθ1 on the first particle and ηθ2 on the second particle. We obtain the inequality which can be experimentally verified: P(ξa = +1, ηb = +1) + P(ξb = −1, ηc = +1) ≥ P(ξa = +1, ηc = +1). The situation with financial and economics agents is very similar to the quantum one. They are also very sensitive to questions. Let now θ = a, b, c be three different questions. Of course, they should be really disturbing for agents. By giving the answer ξa (ω) the agent creates a new memory which will play an important role in answering the next question, e.g., b or c. The main problem is that we are not able to prepare an ensemble of “entangled agents”, i.e., to satisfy the condition of precise correlations (5). Therefore we shall change the strategy and operate with conditional probabilities. To find conditional probabilities, one need not operate with pairs of “entangled agents” of the financial market. One can perform successive measurements (in the form of questions to agents). As a simple consequence of Theorem 2.2, we obtain [3] the following mathematical result: Theorem 2.3. (Wigner inequality for conditional probabilities) Let ξa , ξb , ξc = ±1 be symmetrically distributed random variables on P. Then the following inequality holds true: P(ξa = +1|ξb = +1) + P(ξc = +1|ξb = −1) ≥ P(ξa = +1|ξc = +1).

(6)

The latter equality can easily be tested experimentally. We choose three “mutually disturbing questions,” a, b, c about finances or an economic situation. The answer “yes” is encoded by +1 and the answer “no” by -1. We prepare a homogeneous ensemble of people (with the same age, education, political and national background), say S. Then we divide it into three sub-ensembles, S1 , S2 , S3 , of the same size (without violating the homogeneous structure of the ensemble S).

74

Andrei Khrennikov

By asking questions b and a in the first ensemble (the order plays a crucial role) we shall find the frequency corresponding to the conditional probability P(ξa = +1|ξb = +1) : ν(ξa = +1|ξb = +1) =

n(ξa = +1|ξb = +1) , N (ξb = +1)

where N (ξb = +1) is the number of agents in the ensemble S1 who gave the answer ”yes” to the question b and n(ξa = +1|ξb = +1) is the number of agents who gave the answer “yes” to the question a among those who have already answered ”yes” to the question b. By asking questions b and c in the second ensemble (we repeat that the order of questions plays a crucial role!) we shall find the frequency corresponding to the conditional probability P(ξc = +1|ξb = −1) : ν(ξc = +1|ξb = −1) =

n(ξc = +1|ξb = −1) , N (ξb = −1)

where N (ξb = −1) is the number of agents in the ensemble S2 who gave the answer ”no” to the question b and n(ξc = +1|ξb = −1) is the number of agents who gave the answer “yes” to the question c among those who have already answered “no” to the question b. By asking questions c and a in the third ensemble we shall find the frequency corresponding to the conditional probability P(ξa = +1|ξc = +1). Finally, we put those frequencies into the inequality (6). If this inequality should be violated, we should obtain a strong argument supporting our the hypothesis about the quantum-like behaviour of financial or economic processes. The experimental framework can even be simplified. We can split the original ensemble S into just two sub-ensembles S1 and S2 . We ask the question b to agents in the first ensemble. We create two new subensembles with respect to the answers “yes” and “no” to this question, S1,b=+1 and S1,b=−1 . Then we shall ask the a-question to agents in the ensemble S1,b=+1 and the c-question to agents in the ensemble S1,b=−1 . In this way we shall obtain the frequencies ν(ξa = +1|ξb = +1), ν(ξc = +1|ξb = −1). By using the ensemble S2 we obtain the frequency ν(ξa = +1|ξc = +1). The main preparation constraint for this experiment is that all questions a, b, c, should induce the symmetric probability distributions: P(ξa = +1) = P(ξa = −1) = 1/2, P(ξb = +1) = P(ξb = −1) = 1/2, P(ξc = +1) = P(ξc = −1) = 1/2.

Classical and Quantum-Like Randomness

75

3 Efficient Market Hypothesis If the proposed test confirms our hypothesis about quantum-like probabilistic behaviour of the financial market, then it may have interesting consequences for foundations of financial mathematics. In the quantum-like approach the fundamental assumption of the modern financial mathematics, namely, the efficient market hypothesis [19], [20], [2] would be questioned. The financial context (situation at the financial market, including expectations, prognoses, political situation, social opinion) Cfin (t) is represented by a complex probability amplitude, financial wave function ψ(t, q), where q is the vector of prices of shares (it has a huge dimension). The evolution of ψ(t, q) is described by a deterministic equation - Schr¨ odinger’s equation. Hence, the evolution of the financial context Cfin (t) could be predicted - at least in principle. Of course, at the moment one could not even dream about the possibility to solve the problem analytically or even numerically. First of all there is no idea how “financial Hamiltonian” (quantum-like operator representing the “energy of the financial market”) should be constructed. Another problem is the huge dimension of the problem. However, our quantum-like model provides the qualitative prediction that there might be developed financial technologies which induce permanently exploitable profit opportunities (in the opposition to the conventional model based on the efficient market hypothesis). We emphasize that the possibility to create such quantum-like financial technologies does not imply lower complexity of our model comparing with the conventional one. The latter implies that financial processes can be represented by a special class of classical stochastic processes, martingales. For any such process we can construct a single Kolmogorov probability space for all realizations of this process (this is the essence of the famous Kolmogorov theorem [18]). In contrast to such a single space description, in our quantum-like model one could not assume that the quantum-like process based on the evolution of the financial context could be embedded into a single Kolmogorov probability space. In the classical financial mathematics there were performed fundamental investigations to find an adequate stochastic processes matching the real financial data: Brownian, geometric Brownian, general Levy processes. From the point of view of our quantum-like approach the problem cannot even be formulated in such a way. There is not any classical stochastic process which will match with the real financial data, because there is not a single Kolmogorov space describing the

76

Andrei Khrennikov

whole financial market. The financial data can only be represented as a quantum-like financial process. Finally we remark that the efficient market hypothesis has been questioned by many authors, from other points of view; see e.g. [21].

References 1. Bachelier L. (1890) Ann. Sc. lcole Normale Superiere 111-17: 21-121. 2. Bell J.S. (1987) Speakable and unspeakable in quantum mechanics. Cambridge Univ. Press, Cambridge. 3. Choustova O. (2001) Pilot wave quantum model for the stock market. http://www.arxiv.org/abs/quant-ph/0109122. 4. Choustova O. (2004) Bohmian mechanics for financial processes. J. Modern Optics 51: 1111. 5. Choustova O. (2007) Quantum Bohmian model for financial market. Physica A 374: 304-314. 6. d’Espagnat B. (1995) Veiled Reality. An analysis of present-day quantum mechanical concepts. Addison-Wesley. 7. Fama E.F. (1970) Journal of Finance 25, 383. 8. Haven E. (2002) A discussion on embedding the Black-Scholes option pricing model in a quantum physics setting. Physica A 304: 507-524. 9. Haven E. (2003) A Black-Sholes Schr¨ odinger option price: ‘bit’ versus ‘qubit’, Physica A 324: 201-206. 10. Haven E. (2003) An ‘h-Brownian motion’ of stochastic option prices, Physica A 344: 151-155. 11. Haven E. (2006) Bohmian mechanics in a macroscopic quantum system. In: Adenier G, Khrennikov AYu, and Nieuwenhuizen T. (eds) Foundations of Probability and Physics-3 810: 330-340. American Institute of Physics, Melville, New York. 12. Khrennikov A. Yu (1999) Interpretations of Probability. VSP Int. Sc. Publishers, Utrecht/Tokyo (second edition– 2004). 13. Khrennikov A. Yu (2004) Information dynamics in cognitive, psychological, social, and anomalous phenomena. Kluwer Academic, Dordrecht. 14. Kolmogoroff A.N. (1933) Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer Verlag, Berlin; reprinted: Kolmogorov A.N. (1956) Foundations of the Probability Theory. Chelsea Publ. Comp., New York. 15. Mandelbrot B., Hudson R. (2004) The (mis)behavior of markets. Basic Books Publication. 16. von Neumann J. (1955) Mathematical foundations of quantum mechanics. Princeton Univ. Press, Princeton, N.J. 17. Piotrowski E.W., Sladkowski J. (2001) Quantum-like approach to financial risk: quantum anthropic principle. http://www.arxiv.org/abs/quantph/0110046.

Classical and Quantum-Like Randomness

77

18. Samuelson P.A. (1965) Industrial Management Review 6, 41. 19. Segal W. and Segal I.E. (1998) Proc. Nat. Acad. Sc. USA 95: 4072. 20. Segal, W. and Segal, I.E. (1998). The BlackScholes pricing formula in the quantum context. Proceedings of the National Academy of Sciences USA 95, 4072-4080. 21. Shimony A. (1993) Search for a naturalistic world view. Cambridge Univ. Press, Cambridge.

Part III

New Empirical Approaches to Complex Economic Dynamics

Examination of the Cost-of-Carry Formula for Futures Contracts on WIG20. Wavelet and Nonlinear Cointegration Analysis Joanna Bruzda Nicolaus Copernicus University of Toru - Poland

1 The cost-of-carry model The instantaneous and non-instantaneous dependences between spot and futures index prices has been subject of numerous empirical investigations. The theoretical background of these studies is the cost-ofcarry model introduced by Corrnell and French [15]. The cost-of-carry model is an arbitrage relationship based on comparison between two alternative methods of acquiring an asset at some future date. In the first method an asset is purchased now and held until this future date. In the second case a futures contract with maturity on the required date is bought. The present value of the futures contract is invested at the risk free interest rate until delivery of the underlying asset at the maturity date. Arbitrage should ensure that the difference between the current asset price and the futures contract price is the cost of carrying the asset, which involves dividend yields and interest rates. The cost-of-carry formula gives the fair price of the futures contract: Ft,T = St e(rt −dt )(T −t)

(1)

where St is the security index price at time t, Ft,T is the index futures price at time t with maturity T , rt is the risk free interest rate, dt is the dividend yield on the security index, and (T – t) is the time to maturity of the futures contract. Taking logarithms of both sides of equation (1) we get: ft,T = st + (rt − dt )(T − t)

(2)

where ft,T and st denote logarithms of Ft,T and St . If logarithms of the security index adjusted for interest rates and dividends have unit root, equation (2) implies cointegration between logarithms of spot and

82

Joanna Bruzda

futures prices with the cointegrating coefficient of unity. The mispricing series, defined as the difference between the real futures price and the theoretical (fair) price of the contract, zt = ft,T − st − (rt − dt )(T − t)

(3)

should be weakly stationary. Fundamental assumptions underlying the cost-of-carry model are: the lack of friction on both markets, perfect substitutability and efficiency in the strong form. However, many empirical studies document the presence of certain lead-lag relations between cash and futures markets (see, for example, Kawaler et al., [24], Brooks et al., [8], Robles-Fernandez et al., [35]). The sources of mispricing and the observed lead-lag relations can be: different taxation on spot and futures markets, transaction costs, dividend uncertainty, unequal borrowing and lending rates, imperfect replication of the underlying index, different liquidity on both markets, non-synchronous trading, restrictions on short sale, full automatisation on one market only and so on. Most of empirical studies on the transmission mechanism between spot and futures markets indicate that price changes on the futures market lead that on the spot market by several minutes till even one day, while the inverse relationship is much weaker and not so long-lived. The results have essential practical implications for forecasting price changes and for verification of the efficient market hypothesis. The majority of these studies were based on the hypothesis of linear temporal relationships and took advantage of dynamic simultaneous equations models (see Kawaler et al., 1987) or cointegration and Granger causality analysis (see, for example, Bruzda, Winiewska, [12], and references therein). More recent studies relax the linearity assumption and make use of nonlinear causality tests (Abhyankar, [1], Brooks et al., [8]), nonlinear cointegration examination (Dwyer et al., [18], Martens et al., [29], Tsay, [37], Robles-Fernandez et al., [35], Bruzda, [11]) and wavelet analysis (Lin, Stevenson, [28]). The last approach enables to examine relationships between price levels directly instead of price changes. What is analysed are prices reconstructed at different levels of information detail. In what follows, I apply wavelet multiresolution analysis to investigate lead-lag relations between spot and futures index prices. I suggest a joint application of nonlinear cointegration, nonlinear Granger causality tests and wavelet analysis. To my knowledge this has not been done yet in the econometric literature. Such an approach turns out to be useful in precise examination of the transmission mechanism between markets and

Wavelet and Nonlinear Cointegration Analysis

83

helps to answer the question, where actually the price is formed. In the study I examine the no-arbitrage relationship for the Warsaw Stock Exchange (WSE) FW20 futures contract – one of the most frequently traded futures contracts in the Middle-East Europe. This investigation is partially motivated by some unique characteristics of the emerging Polish capital market, like relatively high risk free interest rates, limited interest in using short sale and irregular and rare payments of dividends. That is why it might be interesting to compare these results based on intraday data with those presented in studies that were exclusively devoted to examine spot and futures index prices on developed markets (see, for example, Martens et al., 1998, for an investigation concerning futures contracts on S&P500, Robles-Fernandez et al., [35], for an analysis of intraday returns in the Eurostoxx50 cash and futures indexes). The presence of any lead-lag relations between spot and futures index prices is inconsistent with the standard cost-of-carry formula and the market efficiency hypothesis that, however, was not examined here directly. Such an examination would require including additional information about transaction costs and dividends, what would make it possible to evaluate the index arbitrage profitability (see Biakowski, Jakubowski, [5], for an example of such analysis concerning the WSE). The rest of the paper is organised as follows. Section 2 briefly describes the methodology used in this study with special emphasis on wavelet analysis. Sections 3 presents the empirical results divided into two parts: the analysis of original series and the examination of decomposed processes. Finally, Section 4 provides a short conclusion.

2 Methodology The classical Fourier analysis enables to decompose a process into constituent sinusoids of different frequencies. Another way to think of Fourier analysis is as a technique of transforming our view of the process from a time-based one to a frequency-based one. However, Fourier analysis has a serious drawback, which consists in loosing time information. When looking at the Fourier transform of a process, it is impossible to tell when a particular event took place. If characteristics of a signal do not change much over time, this is not an obstacle in precise analysis of the process. The situation changes when the signal contains a trend or some transitory characteristics. In an effort to correct deficiencies of Fourier analysis several solutions were proposed. One of them is to adapt the Fourier transform to analyse only a small part of the signal

84

Joanna Bruzda

at a time. This adaptation is called Short-Time or Windowed Fourier Transform (STFT or WFT). However, this approach has a drawback connected with the same size of a time window for all frequencies, while many economic processes may require a more flexible approach – the one where we can vary the window size as to analyse long-term movements with larger windows and short-term fluctuations with smaller windows. Wavelet analysis represents the next logical step in frequencydomain analyses and is a kind of windowing technique with variablesized windows. A wavelet is – in simplest words – a “small wave”, or – being more precise – a waveform of effectively limited duration and an average value of zero1 . Wavelet analysis consists in breaking up a signal into shifted and scaled versions of the original (or mother) wavelet. The main advantage of this kind of frequency-domain analysis is a precise description of local features of a signal. In contrast to WFT that is unable to detect events taking place in a range of a time window, wavelet analysis becomes a tool for analysing nonstationary processes or processes with transient characteristics that result from changing parameters and (or) non-linearity of underlying mechanisms2 . Thanks to their ability to adjust time windows wavelets enable to see both the forest and the trees and make it possible to escape Heisenberg’s indeterminacy principle – the law that says that one cannot be simultaneously precise in the time and the frequency domain. A basis of wavelet analysis is the wavelet transform, which – similarly to the Fourier transform – can be continuous or discrete. Let us consider a real-valued function ψ(·) satisfying two basic properties: ∞ ψ(u) du = 0, −∞ (4) ∞ 2 (u) du = 1. ψ −∞ We will refer to this function as a mother wavelet. The continuous wavelet transform (CWT) of function x(·) is:  ∞ ψλ,t (u) x(u) du (5) W (λ, t) = −∞

where 1

2

Examples of wavelets include the Haar wavelet, the Morlet wavelet, the Meyer wavelet, the families of wavelets introduced by I. Daubechies: db and symmlet, as well as many other – for a quick overview see, for example, Misiti et al. [30]. About contributions of wavelets to the analysis of economic time series see Ramsey, Lampart [34], Ramsey [33], Gencay et al. [19], Schleicher [36].

Wavelet and Nonlinear Cointegration Analysis

85

  1 u−t λ > 0. (6) ψλ,t (u) = √ ψ λ λ As a result of applying the continuous wavelet transform we obtain a set of wavelet coefficients, which depend on scale λ and time t. This set is an equivalent representation of the function x(·). Let x = (x0 , x1 , . . . , xN −1 )′ be a data vector of length N = 2n . For j = 0, 1, . . . , n − 1 and k = 0, 1, . . . , 2j − 1 we define the discrete wavelet transform (DWT) of vector x with respect to ψ(·) as: n N −1 Wj,k = (7) xn ψj,k n=0 N where ψj,k (·) are scaled and shifted versions of the mother wavelet, i.e.  ψj,k (t) = 2−j/2 ψ 2−j t − k .

(8)

. . . ⊂ V2 ⊂ V1 ⊂ V0 ⊂ V−1 . . . .

(9)

DWT operates on scales, which are powers of 2 – these are the socalled dyadic scales. Dyadic are also shifts of the form k · 2j . Although DWT can be defined without referring to CWT, we will treat it as a discretisation of the continuous wavelet transform, obtained as a result of critical sampling of CWT. The discrete wavelet transform is closely related to multiresolution analysis introduced by S. Mallat at the end of the eighties. This analysis consists in a multiple-level representation of a signal, where at each level the signal is decomposed into two components: approximation and detail. At each consecutive level the approximation from the previous decomposition is represented again as a sum of the subsequent approximation and detail. Approximations are the high-scale, low-frequency components of the signal, whereas details are the lowscale, high-frequency components. Proceeding in this way any squareintegrable function can be represented with any accuracy as a sequence of details. Technically, multiresolution analysis projects a function on a set of closed functional subspaces of consecutive approximations:

Furthermore, since the subspaces are nested, one can represent Vj−1 as the direct sum of the coarsely approximated subspace Vj and its orthogonal complement Wj : Vj−1 = Vj ⊕ Wj

(10)

86

Joanna Bruzda

where Wj represent details of the signal apparent at the level of scale 2j , which are absent in less precise scales. One can think of the subspaces Vj as different levels of magnification, revealing more and more details. Additionally, it is presumed that the subspaces Vj are self-similar, i.e. f (t) ∈ Vj ⇔ f (2j t) ∈ V0

(11)

and invariant relative to shifts of the form k · 2j . It is also assumed that there exists a function ϕ ∈ V0 such that the set {ϕj,n ; n ∈ Z} is an orthonormal basis in the subspace Vj , where ϕj,n = 2−j/2 ϕ(2−j t − n). The function ϕ is called father wavelet (scaling function). Looking for an orthonormal basis in the subspace W0 one should define: √  gk ϕ(2t − k) (12) ψ(t) = 2 k

Moreover we have also:

ϕ(t) =

√  2 hk ϕ(2t − k)

(13)

k

Equations (11) and (12) are called scaling equations or dilation equations and define sequences {gk } and {hk } that are impulse responses of two filters: the high-pass filter and the low-pass filter, respectively. The two filters are called quadrature mirror filters. If G and H denote the transfer functions of the quadrature mirror filters, a wavelet decomposition of a signal x, consisting of 2n elements, has the form: W = [Gx, GHx, GH 2 x, . . . , GH n−1 x, GH n x, H n x]

(14)

In the next step of the multiresolution analysis the initial signal is reconstructed at different levels of resolution with the help of the so-called inverse discrete wavelet transform (see for details Percival, Walden, [31], pp. 93–95). This reconstruction enables to represent the signal as a sum of the coarsest approximation and a sequence of details, i.e. x = A1 + D1 = A2 + D1 + D2 = A3 + D1 + D2 + D3 = . . .. As is well known, economic actions take place at different scales, i.e. they depend on a time horizon. Economists often emphasise the importance of discerning between a long-run and a short-run behaviour. As Ramsey and Lampart [34] notice, the classics of economics (J. Hicks, for example) saw the necessity to distinguish more time horizons and only pedagogical regards decided that these two periods have been popularised. Wavelets with multiresolution analysis offer the possibility

Wavelet and Nonlinear Cointegration Analysis

87

of going beyond this simplifying dichotomy by decomposing time series into several layers of orthogonal scales. The scales can be analysed individually and compared with across other series. It is worth stressing that further two properties of wavelets make them potentially very useful in investigating economic time series3 : (i) since the base scale includes any non-stationary component, the data need not be detrended or differenced prior to the analysis; (ii) the nonparametric nature of wavelets takes care of a potential non-nonlinear relationship without losing information. In the empirical part of the paper I take advantage of these particular properties of wavelets and analyse decomposed price levels instead of price changes making use of two methods of nonlinear dynamical econometrics: nonlinear cointegration analysis and nonlinear causality examination4 . The first concept of nonlinear cointegration was introduced by Granger and Hallman [21] and Granger [20]. The authors define processes with short and extended memory in mean that are more suitable for analysis of nonlinear phenomena than the traditional I(0) and I(1) processes. Let Yt be a constant-mean process and It = {Yt−j , j ≥ 0} an information set. Let ft,h , h > 0 be a conditional forecast of Yt h periods ahead, i.e. E (Yt+h /It ) = ft,h . Then the process Yt has short memory in mean (is an SMM process) if ft,h converges to the mean value, m, with h → ∞, i.e. E (ft,h − m)2 < ch , where ch → 0. On the other hand, if ft,h is a function of It for each h, the process Yt has extended memory in mean (is an EMM process). The generalisation of the notion of cointegration consists in considering (possibly nonlinear) functions of EMM processes that form an SMM process. For simplicity Granger and Hallman [21] analyse the relationships in the form: g(Xt ) − h(Yt ) = Wt

(15)

where Xt , Yt ∼ EMM, Wt ∼ SMM. The set A = {(x, y), g(x) = h(y)} is called an attractor. To look for nonlinear attractors Granger and Hallman [21] suggest the following procedure. As many unit root tests will treat SMM processes as stationary ones, after finding the appropriate transformations

3 4

Comp. Schleicher [36]. A similar approach was proposed by Arranz and Escribano [3], who suggested applying standard cointegration tests to trend components of original processes rather that the processes themselves. The trend components were obtained by applying certain linear and nonlinear filters. The authors recommend such an approach as an outlier-robust method of testing for cointegration.

88

Joanna Bruzda

g(·) and h(·) the residuals wt = g(xt ) − h(yt ) are checked for stationarity with, for example, the augmented Dickey-Fuller test and the test equation in the form: ∆wt = δwt−1 +

k 

βj ∆wt−j + ut

(16)

j=1

In Granger and Hallman [21] to search for the functions g(·) and h(·) the alternating conditional expectations – ACE – algorithm of Breiman and Friedman [6] is recommended. The ACE algorithm looks for transformations that maximise the correlation coefficient between transformed variables. The algorithm is implemented on finite dat our a sets by replacing conditional expectations with data smooths. In my empirical investigation, similarly to Granger and Hallman study, linear regression smoothing in windows of a fixed size was used. It is worth noticing that, in practice, before examining the residuals wt = g(xt ) − h(yt ) both – the original series and their transformations – are tested for unit roots in order to assure that the nonlinear transformations preserve the I(1) property. The ACE method has an indefinite influence on the asymptotic distribution of the ADF test statistic. To overcome this problem, Granger and Hallman suggest admitting the number of regressors equal 2 and making use of tabulated standard critical values. The solution utilised further in the paper consists of applying bootstrap techniques. The Granger-Hallman approach has a serious drawback in that it applies classical linear tools to possibly nonlinear processes. To remove this drawback Dufr´enot and Mignon [17] introduced the notion of comixing as a generalisation of the Granger and Hallman concept of nonlinear cointegration to the case of nonlinear processes. A stochastic process is strongly mixing (or α-mixing) if dependences between past and future events become negligible with the increase of the time span between these events. More formally, if Xt is a sequence of random variables and Fst = σ(Xs , . . . , Xt ) is a σ-algebra generated by the sequence, then Xt is α-mixing when αm → 0 with m → ∞, where αm are coefficients of the form: αm = sup t

sup t ∞ } {F ∈F−∞ , G∈Ft+m

|P (G ∩ F ) − P (G)P (F )| .

(17)

The generalised I(d) process is defined as such a non-mixing random sequence, which becomes strongly mixing after applying the difference operator d times. Now let consider two non-mixing processes Xt and

Wavelet and Nonlinear Cointegration Analysis

89

Yt and define a non-linear function f which is measurable with respect to an appropriate σ-field. These processes are said to be co-mixing if (1) there exists a sequence f (Xt , Yt , θ) that is α-mixing for θ = θ∗ and non-mixing for θ = θ∗ or if (2) there exists a sequence f (Xt , Yt , θ) that is non-mixing but with dependence structure weaker than the dependence inherent to both Xt and Yt (see Dufr´enot and Mignon, [17], p. 217). To investigate the co-mixing property one can use methods requiring a parametric or nonparametric estimation of the function f or, alternatively, an approach not requiring such an estimation may be applied. In the first case residuals from a long-term relationship are tested for the mixing condition with, for example, the KPSS stationarity test, the modified R/S analysis or significance tests of mutual information coefficients, higher order moments and other measures of nonlinear dependence like maximal correlation coefficients or certain entropy measures (compare Chaouachi et al., [14]). Besides, tests for short memory in information suggested by Aparicio and Escribano [2] in the context of the so-called cointegration in information might be used. In the second group of tests for co-mixing one can single out the rank test for non-linear cointegration, tests based on first differences of ranges and tests for cointegration in information (for an overview see Bruzda, 2006, and references therein). In the empirical investigation I make use of the Breitung [7] rank test for nonlinear cointegration which is indifferent to monotonic transformations of variables. Let consider two I(1) processes Y1t = f1 (X1t ) and Y2t = f2 (X2t ), where the functions f1 (·) and f2 (·) are of the same monotonicity (for example, both are increasing) but not known. Only X1t and X2t are observable. There exists nonlinear cointegration between X1t and X2t if the difference ηt = Y2t − Y1t is stationary. To test for nonlinear cointegration the initial series are replaced with ranks in the form:

= Rank(x1t among x11 , x12 , . . . , x1T ) = RT (x1t ) = Rt (y1t ), RT (x2t ) = Rank(x2t among x21 , x22 , . . . , x2T ) = = Rt (y2t ).

(18)

The Breitung test statistics are two distance measures: κ = T −1 sup |dt | t

(19)

90

Joanna Bruzda

and ξ = T −3

T 

d2t

(20)

t=1

where dt = RT (x2t ) − RT (x1t ). The null hypothesis of the lack of any cointegration is rejected if the statistics (18) and (19) are less than appropriate critical values obtained via Monte Carlo simulations. However, the statistics κ and ξ are constructed under the assumption that Y1t and Y2t are independent. Breitung (2001) suggests two kinds of correction for correlation between Y1t and Y2t . If the correlation coefficient ρ is small, the corrected test statistics are given by: κ∗ =

κ σ ˆ∆d

ξ∗ =

ξ

(21) (22)

2 σ ˆ∆d

where 2 = T −2 σ ˆ∆d

T  t=2

(dt − dt−1 )2

(23)

If the value of ρ is close to 1 or –1, the statistics κ∗ and ξ ∗ should be modified (assuming the 5% significance level) as follows: κ∗∗ =

κ∗ 1 − 0.174(ρR )2

ξ ∗∗ =

(24)

ξ∗ 1 − 0.462ρR

(25)

where ρR is the correlation coefficient for differences of ranks, i.e. ∆RT (x1t )∆RT (x2t )

t=2

R

ρ = 

T 

T 

t=2

∆RT (x1t )2

(26) 

T 

t=2

∆RT (x2t )2



The statistics (22) and (23) have the same asymptotic distribution as (20) and (21), respectively. Breitung [7] suggests also a generalisation of his cointegration test to the case of k + 1 processes. Let assume that processes Y0t = f0 (X0t ), Y1t = f1 (X1t ), . . . , Ykt = fk (Xkt ) are all integrated of order

Wavelet and Nonlinear Cointegration Analysis

91

1 and functions fj (·), j = 0, 1, . . . , k, are of the same monotonicity. Let RT (xt ) = [RT (x1t ), . . . , RT (xkt )]′ be a k × 1 vector and bT – a vector of coefficients in the regression of RT (x0t ) against RT (xt ). By ′ denoting uR t = RT (x0t ) − bT RT (xt ) the multivariate rank statistic can be written as: T −3

T  

t=1 2 σ ˆ∆u

Ξ ∗ [k] =

uR t

2

(27)

where 2 σ ˆ∆u = T −2

T  t=2

R 2 (uR t − ut−1 )

(28)

is a correction for possible serial correlation in the residuals. Nonlinear cointegration tests provide information about the presence of any long-term relationship between examined processes. If one uses an estimated adjustment process (14) to construct a multivariate error correction model based on generalised difference operators (see Dufrenot, Mignon, [17], p. 202), it is also possible to say what is the direction of information flow in the long run. To examine possibly nonlinear causal relationships in the short run several nonlinear Granger causality tests have been proposed with the two most popular: Brooks and Hinich test (see Brooks et al., [8], Brooks, Hinich, [9]) and the modified Beak and Brock [4] test. The second test has been suggested by Hiemstra and Jones [23] and proceeds as follows. Let Xt and Yt are strict stationary weakly dependent stochastic processes. By Xtm an m×1 vector of leads is denoted, i.e. Xtm = {Xt , Xt+1 , . . . , Xt+m−1 }. For given m, lx, ly > 0 the process Yt does not nonlinearly Granger-cause the process Xt if the following condition is fulfilled:     lx  lx  < e, Y ly − Y ly  < e) = − Xs−lx P (Xtm − Xsm  < e  Xt−lx  t−ly s−ly    lx  lx m m    Xt−lx − Xs−lx < e), = P (Xt − Xs  < e (29) where P (·|·) denotes conditional probability and · is the maximum norm in the ℜm space. Then (see Hiemstra, Jones, 1994) we have: √

n



C1(m + lx, ly, e, n) C3(m + lx, e, n) − C2(lx, ly, e, n) C4(lx, e, n)



∼ N (0, σ 2 (m, lx, ly, e)) a

(30)

92

Joanna Bruzda

where       ly  m+lx ly  m+lx  − Xs−lx C1(m + lx, ly, e) = P Xt−lx  < e, Yt−lx − Ys−lx  < e ,      lx − X lx  < e, Y ly − Y ly  < e , C2(lx, ly, e) = P Xt−lx   s−lx s−lx   t−lx  m+lx m+lx  C3(m + lx, e) = P Xt−lx − Xs−lx  < e ,   lx lx  < e . − Xs−lx C4(lx, e) = P Xt−lx Estimators of C1, C2, C3, C4 and the variance σ 2 (m, lx, ly, e) are given in Hiemstra, Jones [23]. As Diks and Panchenko [16] point out, the testing procedure based on (26) shows serious size distortion when GARCH effects are present. The simplest remedy for it is GARCH filtering prior to an application of the test which, however, may also influence the structure of dependences between processes.

3 Examination of the cost-of-carry model for FW20 In the empirical study I examine the cost-of-carry relationship for the Warsaw Stock Exchange (WSE) FW20 futures contract – one of the most frequently traded futures contracts in the Middle-East Europe. This study is partially motivated by some unique characteristics of the emerging Polish capital market, like relatively high risk free interest rate, limited interest in using short sale and irregular and rare payments of dividends. Futures contracts on the WSE Index WIG20 have been traded since 16th January 1998. The instrument comprises blocks of shares of 20 blue chips stocks listed on the main market of the Warsaw Stock Exchange. Two criteria are taken into account in selecting companies for this index: turnover (with weight 0,6) and market value (with weight 0,4). Changes in the WIG20 portfolio take place every quarter, according to changes in turnovers and market values. Three series of the futures contracts on WIG20 are always in trade. Their delivery dates are in the nearest three months of the March quarterly cycle, i.e. March, June, September and December. Expiry of a series is followed by introduction of the series with expiry date in 9 months, as per the March quarterly cycle. The last trading day for a contract is usually the third Friday of the expiry month. Abbreviated contract names are FW20kr, where: F – type of instrument; W20 – abbreviated name of the underlying instrument; k – delivery month code (H - March, M – June, U – October, Z – December);

Wavelet and Nonlinear Cointegration Analysis

93

r – last digit of delivery year. In November 2000 the continuous quotation system WARSET has been introduced in the WSE. Since then, futures contracts are traded continuously between 9:00 and 16:00. At 16:10 the closing price is determined. In contrast to the futures prices continuous trade of assets entering WIG20 starts at 10:00. The futures contracts on this index are settled at 10 PLN times quotations on the WIG20 index. In the empirical examination three contracts with expiry date in 2004 where analysed: FW20M4, FW20U4 and FW20Z4, adjusted to one-minute basis. The data were taken from the webpage http://www.bossa.pl. For each contract a one-month period before expiry was considered, when the contracts are most frequently traded. For each contract and the underlying instrument the period 10:01-16:00 of each day was taken into account. It gave series of lengths: 7920, 8280 and 7920, respectively. As a proxy for the risk free interest rate the daily Warsaw Inter-Bank Offered Rate (WIBOR) was used. For each contract the underlying quotations on WIG20 have been corrected with this interest rate. All data have been transformed into logarithms. As opposed to former studies taking advantage of a wavelet decomposition, on reconstructed series have been executed standard cointegration and causality tests as well as their nonlinear counterparts, i.e. the Granger-Hallman and Breitung nonlinear cointegration tests together with the Hiemstra-Jones test for nonlinear Granger causality. The study may be viewed as an attempt to join wavelet analysis and nonlinear time series tools in the form of certain nonlinear cointegration and nonlinear causality tests. Former studies utilizing wavelet analysis (see Lin, Stevenson, [28]) were based on the hypothesis of linear relationships. The original series have been decomposed with a 7-level wavelet decomposition based on the db2 wavelet that has a compact support and two vanishing moments5 . Boundary value replication was used to lengthen samples to some power of 2. Further in the text the following abbreviations are used: m, u, z stand for ln(FW20M4), ln(FW20U4) and ln(FW20Z4), respectively, the underlying corrected ln(WIG20) for each contract is denoted by mw, uw and zw, respectively, djk and djkw (j = 1, 2, ..., 7; k = m, u, z) are the reconstructed details and a7k (k = m, u, z) denote the reconstructed approximations.

5

A similar analysis was conducted with the Haar (db1) and sym8 wavelets which are frequently utilised in economic and financial applications (see Gencay et al., [19]) – the conclusions were similar to presented above.

94

Joanna Bruzda

3.1 Examining the original series All the initial series: m, u, z, mw, uw, zw, turned out to be integrated of order 16 , while the KPSS test (see Kwiatkowski et al. 1992) executed on fully modified OLS residuals from potential long-term relationships7 rejects the null hypothesis of cointegration8 at the 1% significance level – see Table 1. However, totally different results were obtained with the Breitung nonlinear cointegration test that strongly indicates the presence of certain (nonlinear) long-term relationships – see Table 2. Table 1. KPSS cointegration tests on original series Series m, mw u, uw z, zw

Cointegrating equation KPSS test statistic ηˆµ mw = 1, 0791 + 0, 8545m 1,9250 uw = 0, 9750 + 0, 8702u 2,6178 zw = 0, 8223 + 0, 8908z 1,4172

Critical values (see Harris, Inder, 1994) for α = 1%: 0,550; α = 5%: 0,320; α = 10%: 0,234; truncation in the Newey-West estimator of the long-term variance: l = [5T 0,25 ]. Table 2. Breitung cointegration tests on original series Series m, mw u, uw z, zw

κ∗ statistics 0,1233 0,1600 0,2325

ξ ∗ statistics 0,0015 0,0016 0,0025

Ξ ∗ [1] statistics 0,0013 0,0015 0,0022

Critical values were obtained by generating 1000 independent random walks with the number of observations set to 8000. For α = 1% they are as follows: κ∗ : 0,321; ξ ∗ : 0,012; Ξ ∗ [1]: 0,011. The null hypothesis of no cointegration is rejected if the value of the test statistic is lower than the appropriate critical value. To shed more light on the examined relationships the original series have been transformed with the alternating conditional expectations algorithm in which regression smoothing with window size set to 100 was 6 7 8

Detailed results are available upon request. See Phillips and Hansen (1990) for a description of the FM OLS method. The KPSS test applied to fully modified residuals from the long-term relationship (the so-called Harris-Inder test – see Harris, Inder, 1994) is treated as a test for linear co-mixing – see Section 2.

Wavelet and Nonlinear Cointegration Analysis

95

utilised. Table 3 presents the results of unit root tests on transformed series, which unambiguously indicate that the ACE transformation preserves the I(1) property9 . Table 3. Unit root tests on transformed series Series

Transformed Transformed Transformed Transformed Transformed Transformed

m u z mw uw zw

ADF statistic KPSS statistic (bootstrapped 5% (bootstrapped 5% critical value) critical value) -1,6178 (-3,51) 5,0728 (0,96) -0,1572 (-3,92) 16,2563 (1,78) -0,5104 (-3,12) 13,6256 (1,20) -1,7281 (-3,68) 4,7723 (0,98) -0,2859 (-3,55) 16,1953 (1,45) -0,7093 (-4,15) 13,6188 (0,90)

The reported results concern the test equation with an intercept and no trend. Number of bootstrap replications: 1000; maximal order of autoregression: 256; truncation in the Newey-West estimator of the long-term variance: l = [5T 0,25 ]. In the statistical inference bootstrapped p-values were utilised. In the case of the ADF test the bootstrap method applied was the sieve bootstrap of Chang and Park [13], while the KPSS test was based on the Kuo and Tsong (2005) method taking advantage of the fact that testing for stationarity of Yt is equivalent to testing for a unt root in the moving average part of ∆Yt . Subsequently, according to the Granger and Hallman procedure, the adjustment series (the difference between the two transformed series) has been examined for stationarity, again by using bootstrapped p-values – see Table 4. In the case of the contracts with expiry date in September and December both the ADF and the KPSS tests indicate cointegration, while in the case of the first contract both tests consistently point at the lack of cointegration. Figure 1 presents the nonlinear transformations of variables and the potential attractors. Number of bootstrap replications: 1000; maximal order of autoregression: 100; truncation in the Newey-West estimator of the long-term variance: l = [5T 0,25 ]. 9

Further investigation, not presented here, indicates that first differences of the transformed processes are stationary.

96

Joanna Bruzda Table 4. Cointegration tests on transformed series Series

Transformed m and mw Transformed u and uw Transformed z and zw

ADF statistic KPSS statistic (bootstrapped 5% (bootstrapped 5% critical value) critical value) -5,3459 (-3,32) 1,2266 (0,82) -6,3128 (-3,84) 0,4339 (1,05) -6,7414 (-3,45) 0,6142 (0,80)

Due to the fact that the transformation of the series are approximately linear, in the next step the adjustment series obtained by applying the Granger and Hallman method were used to construct error correction models describing the short-term dynamics of our two-variable systems. In each of the three cases error correction terms turned out to be significant in both equations of the VEC models. This observation leads to the conclusion that in the long run a causal feedback takes place. Also in the short run the causal relationships were bidirectional and stretched from 9 till 14 minutes. Additionally, especially in the case of the FW20Z4 contract, futures prices lead spot prices by longer periods than spot prices lead quotations on futures contracts. Table 5 contains the estimation output for one of the VEC models – the VEC model for the FW20U4 contract. 3.2 Examining the decomposed series The above analysis provides a kind of aggregate information about the strength and direction of information flow between the spot and futures markets, where the aggregation here is over a full range of frequencies. To perform a more accurate analysis of lead-lag relations between these markets subsequently I turned to examine dependencies between prices reconstructed at different levels of information detail. Figure 2 presents results of the multilevel decomposition of quotations on the FW20M4 contract. In a similar way quotations on the remaining two contracts and the underlying instruments were decomposed. In further analysis approximations a7 and details d1–d7 were investigated. The ADF test with bootstrapped critical values indicated that all approximations were I(1) processes, whereas all details were stationary. For all pairs of details VAR models were built and tests for Granger causality were performed. On the other hand, nonstationary approximations were examined with the help of linear and nonlinear cointegration tests and their short-run dynamics was analysed by means of Vector Error Correction (VEC) models.

Wavelet and Nonlinear Cointegration Analysis

97

Fig. 1. Transformations of variables and the resulting attractors for contracts with expiry date date in June – 1, September – 2 and December – 3; (a) scatter plots X − Y , where X are futures prices and Y are spot prices; (b) transformations g(X); (c) transformations h(Y ); (d) attractors g(X) = h(Y ).

Table 6 contains the outcome of linear Granger causality tests with numbers of lags in VAR models chosen according to the Akaike Information Criterion. It is worth noticing that wavelet filtration generally causes long lag lengths to appear in the test equations. z denotes the mispricing series; the estimation method was SUR with Newey-West HAC standard errors, but the results obtained with ordinary standard errors estimates were almost the same. The results in Table 6 indicate that for oscillations, whose periods are very short (up to 16 minutes), quotations on WIG20 lead quotations on the futures contracts. Starting with d4 an inverse relationship is observed. The linear Granger causality tests were supplemented with the Hiemstra-Jones tests for nonlinear causality. Standardised residuals from VAR models with the numbers of lags as in Table 6 were investigated10 . The results (see Table 7) indicate the presence of nonlinear dependences for details from 1 to 3. Besides, one can observe that the flow of information is much stronger from the futures market to the cash market than in the other direction. To sum up the overall results of causality tests one can notice that there are two different ‘causality patterns’: one for details d1–d3 and second for details d4–d7. In the first case there is a linear flow of information from the underlying instrument to the futures contract and a nonlinear causality in the other direction. This may show that in the very short run volatility on the futures market causes that on the spot market, while information about price levels flows from the underlyings to futures contracts. For oscillation with periods over 16 minutes till about 4 hours there is a linear 10

In order to preserve the dependence structure I decided not to perform the GARCH filtering prior to the nonlinear causality tests. Due to this the documented results are likely to be oversized and should be treated with some caution.

98

Joanna Bruzda

Table 5. VEC model for the FW20U4 contract and the underlying index

Explanatory variable

const ∆u(-1) ∆u(-2) ∆u(-3) ∆u(-4) ∆u(-5) ∆u(-6) ∆u(-7) ∆u(-8) ∆u(-9) ∆u(-10) ∆u(-11) ∆u(-12) ∆uw (-1) ∆uw (-2) ∆uw (-3) ∆uw (-4) ∆uw (-5) ∆uw (-6) ∆uw (-7) ∆uw (-8) ∆uw (-9) ∆uw (-10) ∆uw (-11) ∆uw (-12) z

Equation for futures prices (dependent variable ∆u) Coefficient Standard tstatistic p-value error 0,0000 0,0000 2,6672 0,0077 -0,2190 0,0116 -18,879 0,0000 -0,1216 0,0118 -10,305 0,0000 -0,0675 0,0120 -5,6250 0,0000 -0,0493 0,0121 -4,0744 0,0000 -0,0680 0,0121 -5,6198 0,0000 -0,0252 0,0121 -2,0826 0,0373 -0,0518 0,0121 -4,2810 0,0000 -0,0458 0,0121 -3,7851 0,0002 -0,0321 0,0121 -2,6529 0,0080 -0,0289 0,0120 -2,4083 0,0160 -0,0307 0,0118 -2,6017 0,0093 -0,0282 0,0113 -2,4956 0,0126 0,2458 0,0115 21,3739 0,0000 0,1227 0,0118 10,3983 0,0000 0,0906 0,0119 7,6134 0,0000 0,0671 0,0120 5,5917 0,0000 0,0098 0,0112 0,8750 0,3816 0,0415 0,0120 3,4583 0,0005 0,0374 0,0120 3,1167 0,0018 0,0460 0,0120 3,8333 0,0001 0,0260 0,0120 2,1667 0,0303 -0,0157 0,0118 -1,3305 0,1834 0,0033 0,0117 0,2821 0,7779 0,0164 0,0115 1,4261 0,1539 -0,0014 0,0001 -14,000 0,0000 R2 = 0,0885; Su = 0,0005; DW = 1,9511

Equation for spot prices (dependent variable ∆uw ) Coefficient Standard tstatistic p-value error 0,0000 0,0000 1,2390 0,2154 0,1083 0,0116 9,3362 0,0000 0,1200 0,0120 10,0000 0,0000 0,1008 0,0121 8,3306 0,0000 0,0951 0,0123 7,7317 0,0000 0,0785 0,0122 6,4344 0,0000 0,0765 0,0122 6,2705 0,0000 0,0477 0,0122 3,9098 0,0001 0,0466 0,0122 3,8197 0,0001 0,0325 0,0122 2,6639 0,0077 0,0127 0,0121 1,0496 0,2939 0,0174 0,0119 1,4622 0,1437 -0,0119 0,0115 -1,0348 0,3008 -0,1011 0,0115 -8,7913 0,0000 -0,0848 0,0119 -7,1261 0,0000 -0,0663 0,0120 -5,5250 0,0000 -0,0599 0,0121 -4,9504 0,0000 -0,0737 0,0121 -6,0909 0,0000 -0,0815 0,0121 -6,7355 0,0000 -0,0393 0,0121 -3,2479 0,0012 -0,0433 0,0120 -3,6083 0,0003 -0,0387 0,0120 -3,2250 0,0013 -0,0476 0,0120 -3,9667 0,0001 -0,0216 0,0118 -1,8305 0,0672 -0,0221 0,0116 -1,9052 0,0568 0,0014 0,0001 14,000 0,0000 R2 = 0,0436; Su = 0,0005; DW = 1,9551

Wavelet and Nonlinear Cointegration Analysis

99

information flow from the futures market to the spot market. Thus, it seems that in the short run prices are formed on the futures market.

Fig. 2. Multiresolution decomposition of the FW20M4 contract: the original series m, approximations a7 - a1, coefficients of the continuous wafelet transform cfs (colour range from white to black corresponds to values from low to big), and details d7 - d1. Scales 1 - 7 approximately represent oscilla-tions with periods 2-4, 4-8, 8-16, 16-32, 32-64, 64-128, 128-256.

To examine the direction of information flow in the long run relationships between approximations were studied. The approximations contain fluctuations with periods over 256 minutes. On all three pairs of approximations the KPSS test for linear co-mixing and the Granger and Hallman test for nonlinear cointegration were executed. The results are given in Tables 8–10. The estimation method of VAR models was SUR with Newey-West HAC standard errors.

100

Joanna Bruzda Table 6. Linear causality tests on details

Series

Number of lags

d1m, d1mw d1u, d1uw d1z, d1zw d2m, d2mw d2u, d2uw d2z, d2zw d3m, d3mw d3u, d3uw d3z, d3zw d4m, d4mw d4u, d4uw d4z, d4zw d5m, d5mw d5u, d5uw d5z, d5zw d6m, d6mw d6u, d6uw d6z, d6zw d7m, d7mw d7u, d7uw d7z, d7zw

70 180 70 138 138 130 210 250 202 250 250 252 246 246 246 256 256 256 256 256 256

Null hypothesis: djkw does not Granger-cause djk F statistic p-value 0,9006 0,7088 1,5960 0,0000 0,1808 0,1444 1,1263 0,1499 1,6088 0,0000 1,1809 0,0800 1,1984 0,0279 1,1966 0,0197 1,5473 0,0000 0,8250 0,9785 0,8958 0,8772 0,8140 0,9851 1,9734 0,0000 0,4845 1,000 0,9639 0,6442 0,6326 1,000 0,3936 1,000 0,5953 1,000 1,7704 0,0000 0,3988 1,000 1,0204 0,4002

Null hypothesis: djk does not Granger-cause djkw F statistic p-value 1,1852 0,1391 1,1321 0,1111 0,9681 0,5534 0,8035 0,9554 1,0198 0,4205 0,8560 0,8403 0,6920 0,9998 0,9493 0,7050 1,3973 0,0002 1,2682 0,0031 0,8314 0,9741 1,1120 0,1115 1,4142 0,0000 1,8888 0,0000 1,3897 0,0001 0,9181 0,8184 0,6875 1,000 0,4334 1,000 1,2333 0,0074 1,6135 0,0000 1,8076 0,0000

2uw →2u

2u →2uw

2mw →2m

2m →2mw

1zw →1z

1z →1zw

1uw →1u

1u →1uw

1mw →1m

1m →1mw

Direction of causal relationship 10 205 (4,75) 165 (3,77) 221 (4,02) 93 (1,72) 48 (0,90) 126 (2,41) 67 30 (3,15) (0,75) 70 46 (3,24) (1,21) -8 36 (0,69) -31 -34

5 54 (1,83) 55 (1,84) 30 (0,86) 19 (0,57) 0 (0,01) -8

Lag

20 195 (2,27) 204 (2,46) 271 (2,76) 11 (0,10) -160

22 (0,19) 10 69 (0,18) (1,00) 25 148 (0,50) (2,37) -58 161 (1,69) -108 138 (1,33)

- 85

15 128 (1,96) 110 (1,70) 210 (2,63) 48 (0,56) -122

30 305 (2,40) 181 (1,36) 402 (3,10) 23 (0,12) -130

48 (0,52) 215 (2,87) 99 (0,76) 50 (0,35)

130 (1,19) 227 (2,32) 145 (0,88) -177

- 123 -126

-136

25 169 (1,49) 199 (1,80) 359 (2,64) - 26

108 (0,78) 210 (1,68) 26 (0,11) -439

-702

35 364 (2,16) 90 (0,50) 372 (2,12) 134 (0,60) -179

307 (2,19) 298 (2,43) 354 (1,80) -1013

-884

40 410 (2,21) 170 (0,88) 386 (2,13) 127 (0,43) -66

Table 7: Nonlinear causality tests on details

50 661 (3,58) -1

325 (2,05) 295 (2,13) 182 (0,33) -295

415 (2,67) 225 (1,25) 511 (1,07) -200

637 (4,07) 72 (0,19) 84 (0,25) -1256 -937

45 451 (2,05) 149 (0,57) 417 (1,69) 249 (0,69) -208

704 (4,64) 286 (1,70) 720 (5,10) -125

718 (3,71) 302 (0,68) 259 (0,54) -1130

55 735 (3,19) -135

621 (4,24) 175 (0,95) 739 (5,10) -2

60 568 (2,40) 38 (0,09) 752 (4,52) 331 (0,75) 417 (0,76) -888

Wavelet and Nonlinear Cointegration Analysis 101

4u →4uw

4mw →4m

4m →4mw

3zw →3z

3z →3zw

3u →3uw 3uw →3u

3mw →3m

-98

-187

-107

175 213 (1,75) (1,57) -55 16 (0,12) 59 181 (0,86) (2,17) -29 -30

79 (0,37) 70 (0,38) 140 (1,23) -117

106 (0,22) 24 (0,06) -199

-107

-252

-417

536 (1,48) 69 (0,21) -227

65 (0,28) -432 -398 -594 -58 452 245 (1,60) (0,58) 722 770 589 (4,32) (3,58) (2,23) -181 -453 -1561

-74

-5

161 187 (0,69) (0,68) -329 115 (0,46) 129 -73 (0,59)

150 (0,82) 63 (0,37) -240 105 (0,38) 617 (3,34) -346

41 112 8 113 508 45 (0,15) (0,32) (0,01) (0,11) (0,45) (0,03) -117 -410 -350 -297 -857 -1301

261 144 (2,09) (1,00) -81 -88 -157 -33 147 (1,11) -19 -10 -167 -5 -14 -192 -205 -46 2 -31 -175 50 186 52 91 185 (0,09) (0,53) (1,62) (0,31) (0,45) (0,93) 319 489 460 511 419 645 715 522 (7,85) (8,63) (5,75) (6,50) (4,50) (5,95) (6,13) (3,58) -137 99 152 -108 -43 125 81 -82 (1,91) (1,87) (0,77) (0,40) -322 11 -335 21 71 -85 -52 237 (0,27) (0,28) (0,76) (1,44) -335 10 -349 -41 13 -157 -138 -163 (0,25) (0,14) -330 -18 -447 -67 21 -99 21 140 (0,22) (0,14) (0,84)

31 -50 (1,04) -69 -26

2zw →2z

3m →3mw

-18

2z →2zw

-62

Lag

Direction of causal relationship

Table 7: Nonlinear causality tests on details

102 Joanna Bruzda

Lag

-354 -336 -337 -56 -55 -65 -63 -53 -54 -5 -10 -16 -11

Direction of causal relationship

4uw →4u

4z →4zw

4zw →4z 5m →5mw 5mw →5m 5u →5uw

5uw →5u

5z →5zw

5zw →5z

6m →6mw

6mw →6m 6u →6uw

6uw →6u

-28

-24 -43

-21

-206

-205

-243

7 (0,17) 11 (0,27) -56 -201 -200 -249

-66

-54 -98

-52

-774

-773

-943

-506 -693 -692 -957

-403

-424

-128

-104 -186

-107

-62

-21

-64

-168 -54 -1 -68

-299

-217 -410

-227

-171

-114

-222

-121 -151 -74 -231 -73

-246 -68 -3 -49

17 (0,18) -552 73 (0,81) -545 9 (0,21) -518 -13 -1091 2 (0,03) -878 -8

-458

-779

-431 -418 -290 -805

-23

53 (0,49) 9 (0,17) -23 -19

-4

-183

-64 -159 -77 -172

-56

-40 -66

-6

-32

-82

-542

-322 -388 -275 -594

85 142 45 153 44 -51 (1,00) (1,52) (0,33) (1,06) (0,25) -69 -112 -440 -74 -33 -343

Table 7: Nonlinear causality tests on details

-74

-45 -121

-106 -83 -58 126 (0,88) 84 (0,68) 27 (0,19) 40 (0,26) -17

-212

-42

-106

-181

-358 -410 -345 -93

12 0,027 -557

-191

-93 -284

-543

-204 -719

14 -68 (0,07) -68 -195

-507 -179 -143 91 (0,57) 30 (0,21) -8

-650

-245

Wavelet and Nonlinear Cointegration Analysis 103

-7 -15 12 (2,73) -5 -5 9 (1,80) 1 (0,36) 1 (0,36)

6z →6zw

6zw →6z 7m →7mw

7zw →7z

7z →7zw

7mw →7m 7u →7uw 7uw →7u

Lag

Direction of causal relationship

-3

-39 8 (1,50) -9 -15 3 (0,46) -4

-20

-11

-13

-23 -32 -10

-84 -1

-57

-12

-14

-29 -31 -7

-149 -2

-103

-23

-25

-48 -51 -21

-313 -14

-238

-39

-44

-78 -84 -42

-761 -34

-625

-52

-57

-65 -58 -20

21 (0,41) -8 -33

-80

-88

-103 -90 -37

-127

-140

-164 -142 -67

8 -13 (0,14) -35 -67 -61 -104

Table 7: Nonlinear causality tests on details

-210

-233

-263 -215 -119

-131 -191

-75

-362

-408

-464 -354 -210

-251 -354

-172

-714

-840

-949 -647 -421

-606 -782

-479

104 Joanna Bruzda

Wavelet and Nonlinear Cointegration Analysis

105

The values in the table are the differences between the two conditional probabilities in the Hiemstra-Jones test multiplied by 1000 together with the appropriate test statistics in brackets. In the case of negative differences the values of test statistics are omitted. The number of lags was the same for both variables. The number of leads of the variable X, m, was set to 1. Under the null hypothesis of no (nonlinear) causality the statistics are distributed asymptotically as a N(0,1). jk →jl denotes a causal relationship from djk to djl. The critical value for the right-sided test at the 1% (5%) level of significance is 2,33 (1,65). Table 8. KPSS cointegration tests on approximations Series a7m, a7mw a7u, a7uw a7z, a7zw

Cointegrating equation KPSS test statistic ηˆµ mw = 3, 4325 + 0, 5375m 2,3926 uw = 1, 0139 + 0, 8650u 2,7112 zw = 1, 7844 + 0, 7631z 4,4372

Critical values (see Harris, Inder, [22]) for α = 1%: 0,550; α = 5%: 0,320; α = 10%: 0,234; truncation in the Newey-West estimator of the long-term variance: l = [5T 0,25 ]. As indicated by the results in Table 8, in all three cases the hypothesis of linear co-mixing is rejected. After transforming the approximations with the ACE algorithm bootstrap unit root tests were performed. Table 9 contains the outcomes of the tests executed on transformed approximations, while Table 10 – the appropriate results for differences between two approximations. It turns out that the nonlinear transformations preserve the I(1) property for all six processes. Besides, I obtained a joint confirmation for the presence of cointegration in the case of two pairs of processes corresponding to contracts with expiry date in September and December. In the case of the FW20M4 contract the results are ambiguous as both tests (the ADF test and the KPSS test) reject their null hypotheses. It is worth noticing that, to a large extent, the outcomes of cointegration tests on the approximations stay in accordance with the former results concerning the original data. In my interpretation wavelet filtering might be seen as a kind of smoothing that removes oscillations with relatively shorter periods which may obscure a general tendency in the processes under study. Due to this cointegration tests on filtered processes may provide more accurate information about the existence of any long-term relationship between the processes being more robust to the presence of outliers, structural

106

Joanna Bruzda

changes and, first of all, volatility clustering (see also Arranz, Escribano, [3]). Finally, Vector Error Correction Models for the filtered series have been constructed. The models were characterised by long lags lengths varying from 67 for FW20Z4, through 129 for FW20U4, ending up with 137 for FW20M4. An interesting feature of these models is that in all three cases an error correction term was significant in the equation for futures prices, while only once in the equation for spot prices (see Table 11). This observation leads to the conclusion that in the long run the direction of information flow is from the spot market to the futures market. Eventually, prices seem to be formed on the spot market. It is worth underlying, however, that the discovery of this empirical fact was possible after wavelet filtering was applied. Interestingly, despite some unique characteristics of the Polish emerging capital market, to a large extent, the patterns of lead-lag relations I found are similar to that observed on leading capital markets. First of all, the similarity concerns the direction of information flow in the short run which is from futures to spot prices (compare, for example, Abhyankar, 1998, and references therein). However, using the nonlinear cointegration methodology in the long run I found an inverse relationship. A different result was obtained by Robles-Fernandez et al. [35] for the Eurostoxx50 cash and futures index. Applying a specific test for linear co-mixing the authors found a bidirectional relationship in the long run. The general test for co-mixing I applied here was not able to discovery any sign of cointegration. On the other hand, assuming possibly nonlinear adjustment to certain nonlinear long-term relationships I obtained quite different results pointing at a unidirectional flow of information in the long run. Table 9. Unit root tests on transformed approximations Series

ADF statistic strapped 5% value) Transformed a7m -1,6116 (-3,44) Transformed a7u -0,0826 (-4,28) Transformed a7z -0,2201 (-3,73) Transformed a7mw -1,3288 (-4,00) Transformed a7uw -0,2997 (-3,58) Transformed a7zw -0,1638 (-4,17)

(boot- KPSS statistic critical strapped 5% value) 5,0786 (1,00) 16,5180 (1,78) 13,5860 (1,13) 4,8521 (0,94) 16,4540 (2,26) 13,5953 (1,28)

(bootcritical

Wavelet and Nonlinear Cointegration Analysis

107

The reported results concern the test equation with an intercept and no trend. Number of bootstrap replications: 1000; maximal order of autoregression: 256; truncation in the Newey-West estimator of the long-term variance: l = [5T 0,25 ]. Table 10. Cointegration tests on transformed approximations Series

Transformed a7mw Transformed a7uw Transformed a7zw

a7m

ADF statistic strapped 5% value) and -5,9411 (-4,72)

(boot- KPSS statistic critical strapped 5% value) 1,0650 (0,94)

a7u

and -7,4091 (-4,05)

0,2766 (0,80)

a7z

and -7,7141 (-5,42)

0,4472 (0,73)

(bootcritical

Number of bootstrap replications: 1000; maximal order of autoregression: 256; truncation in the Newey-West estimator of the long-term variance: l = [5T 0,25 ]. Table 11. VEC models for approximations Futures contract FW20M4 FW20U4 FW20Z4

p-value for error correction Equation for futures Equation for spot prices prices 0,0239 0,0003 0,0726 0,5895 0,0967 0,4791

4 Final conclusions Wavelet decompositions seem to be a very promising tool in the analysis of nonstationary economic processes as they make it possible to examine relationships between original series instead of their differences or detrended counterparts. As nonparametric techniques, they enable to investigate nonlinear processes without loosing information and do not

108

Joanna Bruzda

require very long time series. Wavelet multiresolution analysis enables to analyse dependences for different levels of information detail and thereby for different frequency intervals. Due to this, it goes beyond the simplifying dichotomy of analysing dependences in the short and long run only. The empirical analysis of relationships between spot and futures index prices on the emerging Polish capital market indicates that in the very short run the flow of information between the markets is bidirectional, though for oscillations with periods over 16 minutes futures prices lead spot prices. In the long run, however, quotations on the underlyings determine quotations on the futures contracts. Eventually, in the long run prices seem to be formed on the spot market. However, the discovery of this fact was only possible after wavelet analysis and nonlinear cointegration examination had been jointly applied. The wavelet filtering provided more precise information about long-term tendencies in the examined series and enabled to find the proper direction of information flow in the long run that was not apparent, when original series were analysed. On the other hand, nonlinear cointegration tests were able to find nonlinear attractors, where linear-based tests failed. Besides, nonlinear causality tests provided some additional information about the direction of information flow in the short run.

References 1. Abhyankar A. (1998) Linear and Nonlinear Granger Causality: Evidence from the UK Stock Index Futures Market. Journal of Futures Market 18:519–540 2. Aparicio F.M., Escribano A. (1998) Information-Theoretic Analysis of Serial Dependence and Cointegration. Studies in Nonlinear Dynamics and Econometrics 3:119–140 3. Arranz M.A., Escribano A. (2004) Outliers-Robust ECM Cointegration Tests Based on the Trend Components. Spanish Economic Review 6:243– 266 4. Beak E., Brock W. (1992) A Nonparametric Test for Independence of a Multivariate Time Series. Statistica Sinica 2:137–156 5. Biakowski J., Jakubowski J. (2003) The Test of Market Efficiency and Index Arbitrage Profitability on Emerging Polish Stock and Futures Index Markets. Postgraduate Research Programme: Capital Markets and Finance in the Enlarged Europe. Europa-Universit¨ at Viadrina 24:1–26 6. Breiman L., Friedman J.H. (1985) Estimating Optimal Transformations for Multiple Regression and Correlation. Journal of the American Statistical Association 80:580–598

Wavelet and Nonlinear Cointegration Analysis

109

7. Breitung J. (2001) Rank Tests for Nonlinear Cointegration. Journal of Business & Economic Statistics 19:331–340 8. Brooks C., Garrett I., Hinich M.J. (1999) An Alternative Approach to Investigating Lead-Lag Relationships between Stock and Stock Index Futures Markets. Applied Financial Economics 9:605–613. 9. Brooks C., Hinich M.J. (1999), Cross-Correlations and CrossBicorrelations in Sterling Exchange Rates. Journal of Empirical Finance 6:385–404 10. Bruzda J. (2006) Empirical Verification of Money Demand Models: NonLinear Cointegration Analysis. Dynamic Econometric Models 7:113–123 11. Bruzda J. (2007) The Cost-of-Carry Model for the FW20 Futures Contracts. Threshold Cointegration Framework in: Milo W, Wrzesiski P (eds) Forecasting Financial Markets and Economic Decision-Making. Acta Universitatis Lodzienskis, Lodz, in press 12. Bruzda J., Wisniewska E. (2002) Investigating Dependences between Futures and Spot Prices on an Example of Futures Contracts on WIG20 in: Tarczynski W (ed) Capital Market. Effective Investing (in Polish). Wydawnictwo Naukowe Uniwersytetu Szczecinskiego, Szczecin 13. Chang Y., Park J.Y. (2003) A Sieve Bootstrap for the Test of a Unit Root. Journal of Time Series Analysis 24:379–400 14. Chaouachi S., Dufrenot G., Mignon V. (2004) Modelling the Misalignments of the Dollar-Sterling Real Exchange Rate: A Nonlinear Cointegration Perspective. Economics Bulletin 3:1–11 15. Cornell B., French K. (1983) The Pricing of Stock Index Futures. Journal of Futures Markets 3:1–14 16. Diks C., Panchenko V. (2005) A Note on the Hiemstra-Jones Test for Granger Non-Causality. Studies in Nonlinear Dynamics and Econometrics 9:article 4. 17. Dufrenot G., Mignon V. (2002) Recent Developments in Nonlinear Cointegration with Applications to Macroeconomics and Finance. Kluwer Academic Publishers, Boston 18. Dwyer G.P., Locke P., Yu W. (1996) Index Arbitrage and Nonlinear Dynamics between the SP500 Futures and Cash. Review of Financial Studies 9:301–332 19. Gen¸cay R.F., Selcuk F., Whitcher B. (2002) An Introduction to Wavelets and Other Filtering Methods in Finance and Economics. Academic Press, San Diego 20. Granger C.W.J. (1995) Modelling Nonlinear Relationships Between Extendend-Memory Variables. Econometrica 63:265–279 21. Granger C.W.J., Hallman J.J. (1991) Long-Memory Processes With Attractors. Oxford Bulletin of Economics and Statistics 53:11–26 22. Harris D., Inder B. (1994) A Test of the Null of Cointegration in: Hargreaves CP (ed) Nonstationary Time Series Analysis and Cointegration. Oxford University Press, Oxford.

110

Joanna Bruzda

23. Hiemstra C., Jones J.D. (1995) Testing for Linear and Nonlinear Granger Causality in the Stock Price-volume Relation. Journal of Finance 49:1639– 1664 24. Kawaller I.G., Koch P.D., Koch T.W. (1987) The Temporal Price Relationship between S&P 500 Futures and the S&P 500 Index. Journal of Finance 42:1309–1329 25. Kuo B-S, Tsong C-C (2005) Bootstrap Inference for Stationarity. Discussion Paper 50, Helsinki Center of Economic Research 26. Kwiatkowski D., Phillips P.C.B., Schmidt P., Shin Y. (1992) Testing the Null Hypothesis of Stationarity Against the Alternative of A Unit Root. Journal of Econometrics 54:159–178 27. Leybourne S.J., McCabe B.P.M. (1993) A Simple Test for Cointegration. Oxford Bulletin of Economics and Statistics 55:97–103 28. Lin S-J, Stevenson M. (2001) Wavelet Analysis of Index Prices in Futures and Cash Markets: Implication for the Cost-Of-Carry Model. Studies in Nonlinear Dynamics and Econometrics 5:87–102 29. Martens M., Kofman P., Vorst TCF (1998) A Threshold Error-Correction Model for Intraday Futures and Index Returns. Journal of Applied Econometrics 13:245–263 30. Misiti M., Misiti Y., Oppenheim G., Poggi J-M (1996) Wavelet Toolbox For Use with MATLAB. The MathWorks 31. Percival D.B., Walden A.T. (2000) Wavelet Methods for Time Series Analysis. Cambridge University Press, Cambridge 32. Phillips P.C.B., Hansen B.E. (1990) Statistical Inference in Instrumental Variables Regression with I(1) Processes. Review of Economic Studies 57:99–125 33. Ramsey J.B. (1999) The Contribution of Wavelets to the Analysis of Economic and Financial Data. Philosophical Transactions of the Royal Society of London Series A 357:2593–2606 34. Ramsey J.B., Lampart C. (1998) The Decomposition of Economic Relationships by Time Scale Using Wavelets: Expenditure and Income. Studies in Nonlinear Dynamics and Econometrics 3:23–42 35. Robles-Fernandez M.D., Nieto L., Fernandez M.A. (2004), Nonlinear Intraday Dynamics in Eurostoxx50 Index Markets. Studies in Nonlinear Dynamics and Econometrics 8:article 3 36. Schleicher C. (2002) An Introduction to Wavelets for Economists. Working Paper 2002-3, Bank of Canada 37. Tsay R.S. (1998) Testing and Modeling Multivariate Threshold Models. Journal of the American Statistical Association 93:1188–1202

A Complex Approach to Estimate Shadow Economy: The Structural Equation Modelling Roberto Dell’Anno1 and Friedrich Schneider2 1 2

Universit` a degli Studi di Foggia - Italia University Linz - Austria

1 Introduction This article develops some ideas of the application of the “complexity” approach in economics. The complexity approach criticizes the scientific method by distrusting sample reductionism and proposes a multidisciplinary approach. Hence, it abolishes old paradigms by arguing to build up another one with the endowment of greater realism. We argue that one should promote the sharing of knowledge and/or methodologies among disciplines and, for economics, limiting the “autistic”(or autarchy) process, which is critically discussed in economics already. Remembering Keynes’s (1936, p. viii) words, the problem for economics seems to be not so much to develop new ideas but to have the difficulties of “escaping from old ideas” and from “habitual modes of thought and expression”. An increasing body of literature3 , known as “complexity approach”in economics has pointed out that economic data provide little evidence of either linear or simple dynamic relations and/or of lasting convergence to stationarity or (regular) cyclical behaviour. In contrast the result in empirical economic research quite often is that the economic processes emerge from complex interactions that constitute evolutionary or complex non-linear processes inside an economy. Although, the fundamental role of these characteristics of an economic system came from its founders (e.g. Marshall, Keynes, Hayek and Simon) it has been increasingly neglected by neoclassical analysis. This approach in mainstream economics began to slide away starting in the 1940’s due to the influence of major new players in our economics pro3

See e.g. Durlauf [24], Rosser [43], Velupillai [50], Albin and Duncan [2], Arthur et al., Day [17].

112

Roberto Dell’Anno, Friedrich Schneider

fession, such as Samuelson and Hicks, who demonstrated the natural complementarity between technical virtuosity and rationality assumptions (Laibson and Zeckhauser, [38]). Among the plausible reasons of a disconnection between theoretical research and practical applied economic research is the fascination of a formalised or mathematized approach (like in natural sciences) in mainstream economics. The analytical-formal method became progressively one of the primary tools to demonstrate the scientific rigor of the economic analysis. The consequence has been to lead the economists to dodge the question about the relationship between “real world”and “economists’s planet”. Neoclassical economists inconsistently defend their scientific approach against this criticism. On one side, in fact, they defend their own models as a necessary simplification of the real world, but on the other side, they accept to pay the price for the negation of the complexity of the economic system due to an enormous mathematical complication. Reminding Rabin’s [42] sarcasm: “Economists cannot really claim (with a straight collective face) to be very “complexityaverse”. Look at our journals. [...] Economists do not shy away from complicated models nearly as much as some claim when embroiled in the midst of abstract methodological debates. It is odd on the one hand to be told during such debates that economists must forego behavioral realism for the sake of keeping our models simple – when on the other hand we are holding a copy of Econometrica”. Given the barriers to communicate across traditional disciplines in social sciences, considerable effort has been made to build a bridge between research in economics and other social sciences. There are two bridges between these disciplines, one built around experimental methods and the other around theoretical modelling known as behavioural economics. In this paper we propose an additional “bridge”between quantitative methods for social research, (e.g. psychometric, marketing, education) and economics. In the following, we suggest, that the “Structural Equation Modelling”(SEM), which is extensively used in social science, can take into account an unobserved variable, and can be considered as helpful methodology for economic research. In particular, our analysis concentrates on one special specification of the SEM approach: the Multiple Indicators Multiple Causes (MIMIC) model, which is quite often applied to estimate the Shadow Economy4 . 4

Compare Bajada and Schneider [3], Dell’Anno [19], Dell’Anno and Schneider [21,22], Giles [27], Giles and Tedds [28], and Schneider and Enste [45].

Shadow Economy

113

The great flexibility of the structural equation model are the following points: 1. it can model the relationships among observed and (unobserved) latent variables, 2. the opportunity to set up a structural theory including also nonlinear as well as feedback loops among the variables, 3. the chance to estimate cyclical or asymmetric interaction effects among the variables and, 4. finally, the dynamic specifications of the statistical model, lead us to consider SEM as an interesting statistical tool to take into account the difficulties of complexity approach in economic research. In general all estimates of shadow economy are “vulnerable”and no one can really claim to be confident of the full reliability of his/her estimates. As specialists in this field know quite well, estimating shadow economy is an exercise that quite often does not allow to apply welldefined methods and commonly accepted assumptions. Hence, our main objective of this paper is twofold: (i) to summarize MIMIC procedure for estimation of the underground economy and (ii) to review its advantages and disadvantages. For this purpose, we provide an example of the best practices in SEM modelling in applied economics. The rest of the paper is organized as follows. Section 2 deals with the basic concepts of the complexity approach relevant for our current analysis estimating the size and development of the shadow economy. Section 3 describes the SEM approach in detail as well the MIMIC method. Section 4 illustrates the limits, advantages and disadvantages of this methodology. The last section concludes.

2 Complexity and Economics The first question that comes up when dealing with the complexity approach is about its definition. As reported by Horgan [32], Seth Lloyd of MIT compiles a list of more than forty different ways in which the term is used in scientific discourse. The definition of “complexity”is problematic also for the double meaning that the term complex means. First “complex”in its common meaning is synonymous for the terms complicated, twisted, difficult, etc. Secondly, “complex”in a technical scientific sense is a deeper concept. Rather than trying to define what complexity is, Corning [12] states it would be more useful to identify the properties that are commonly associated with the term. He suggest

114

Roberto Dell’Anno, Friedrich Schneider

that “complexity often (not always) implies the following attributes: (1) a complex phenomenon consists of many parts (or items, or units, or individuals); (2) there are many relationships/interactions among the parts; and (3) the parts produce combined effects (synergies) that are not easily predicted and may often be novel, unexpected, even surprising”Corning [12]. In this sense, Complex describes something that is not decomposable, not explicable without a lose of the essence. Therefore, complex means not only a complicated structure, it means that a complex system is something that cannot be simplified by analysing separately its single components. The whole structure possesses characteristics that the single parts do not have. Up to nowadays, most of the existing research on economic complexity has been theoretical. According to Durlauf [24] this is not surprising, since a first goal of research on economic complexity has been the determination of the ways in which complex systems represent an extension/or alternative to standard economic theory. In our paper, an attempt is made to reduce this gap by promoting SEM strategies in empirical economic research to demonstrate that in this area too, the economic system is a complex one. When an economic system is recognized as “complex”, we should analyze it with an interdisciplinary approach. One way of interdisciplinarity is the use of methods from one discipline to another. It allows to develop deeper understandings of specific systems, such interdisciplinary approach should help elucidate the general structure and behaviour of complex (economic) systems. In this sense SEM, by modelling the general structure of system, makes easier to understand how a complex system works. Now the interesting point is how to link this kind of modelling of causal dynamics either in terms of reciprocal causality or of recursive causality with the processes of economic interaction at the micro- and macro- level. In our opinion, if interactions among economic variables are considered as essential aspects to understand the economic phenomenon then, SEM could become an important role for future empirical research. Using SEM to model the interaction of an economic system, in fact is able to take into account some features (e.g. nonlinear relationships, reciprocal causality) essential for understanding complex economic structures. In this sense, SEM could be one of the best methods to model complex economic interactions.

Shadow Economy

115

3 The Structural Equation Approach for Shadow Economy Research 3.1 General Remarks about the SEM The SEM procedure includes unobservable variable(s) identified as latent ones, which can be defined or described by observed variables. Cooley [11] argues that SEM allows to establish a theoretical model in order to determine the degree to which the explanatory variables are related to the unobservable variable. Hence, SEM is a generalization of many familiar techniques such as regression, path analysis, discriminant analysis, canonical correlation, confirmatory factor analysis, etc. All these methods can be treated as special cases of SEM, and several authors give the SEM approach a high value. For Stevens [48] SEM is “one of the most important advances in quantitative methodology in many years”. Also Capraro et al. [9] argue “SEM subsumes all other parametric statistical analyses it provides some interesting options for the researcher ”. It has been also termed “the single most important contribution of statistics to the social and behavioural sciences during the past twenty years”(Lomax [40]). The statistical idea behind SEM is to compare a sample covariance matrix with the parametric structure imposed on it by a hypothesized model5 . From the original application of the SEM approach nowadays this type of research is extended too already. For instance Cziraky [14] develops a generalisation of the classical LISREL model in order to come to dynamic autoregressive distributed lag (ADL) models6 . One should remember that an exploratory factor analysis is a technique used to discover underlying latent factors which can become important elements of the theory behind SEM, while SEM is an applied method to test the consistency of a “structural”theory through data. In this sense it is a largely “confirmatory”, rather than “exploratory”technique. In fact in confirmatory factor analysis the theory is tested by examining the consistency of actual data with the hypothesized relationships among all of the unobserved (latent) factors and the observed (measured) variables. 5

6

Estimation of a Structural equations model with latent variables can be done by means of a computer program for the analysis of covariance structures, such a LISREL (Linear Structural Relations). A useful overview of the LISREL software package in an economics journal is Cziraky [13]. For an overview about SEM literature is widespread, among these: Hayduk [30], Bollen [5], Hoyle [33], Maruyama, Byrne, Muthen [41], Cziraky [15].

116

Roberto Dell’Anno, Friedrich Schneider

In general, such a confirmatory factor analysis has two goals: (i) to estimate the parameters (coefficients, variances, etc.) and (ii) to assess the fit of the model. Applying this to the shadow economy research these two goals mean (i) to measure the relationships of a set of observed causes and indicators with shadow economy (latent variable), and (ii) to test if the researcher’s theory or the derived hypotheses as whole fit the used data. Investigating the size and development of the shadow economy the (DY)MIMIC analysis is often used but provides only relative estimates, hence not ones of the absolute size of shadow economy. Hence, this method requires information about an additional procedure so called benchmarking or calibration procedure in order to calculate the values of the absolute size of the shadow economy. Currently, the chosen calibration procedure is one of the most controversial issues when using and evaluating the (DY)MIMIC estimations of the shadow economy. Compared with the regression and the factor analysis, SEM is a rarely used method by the economists. In our view, this is caused by an under-evaluation of SEM capabilities with respect to its potential contribution for the economic research. In the following, we consider exclusively the SEM applications to estimate shadow economy, and in this context, a particular model specification is utilized: the Multiple Indicators Multiple Causes model7 . The applications of MIMIC (or Model) approach for statistical estimation of the size of the shadow economy are growing. One of the earliest was Frey and Weck-Hannemann [25], followed by other economists, who used this approach to estimate the size of the shadow economy: Loayza [39], for Latin American countries, Giles [27] for New Zealand, Giles and Tedds [28] for Canada, Dell’Anno [18] for Italy, Dell’Anno and Schneider [21] for OECD countries, Cziraky and Gillman [16] for Romania, Croatia and Bulgaria, Bajada and Schneider [3] for Asiapacific countries, Schneider [44] for 110 countries all over the world, Chaudhuri et al. for India, Dell’Anno et al. [20] for France, Greece and Spain, Dell’Anno [19] for Portugal, Dell’Anno and Solomon [23] for the USA. 3.2 The MIMIC Estimation Procedure How does the MIMIC procedure work? Using the standard LISREL notation of Joreskog and Sorbom [36], equation 1 is a measurement 7

Seminal studies using the MIMIC model include Zellner [52], Hauser and Goldberger, Jreskog and Goldberger [35], Aigner et al.; in the context of time series applications with MIMIC models Watson and Engle (1983).

Shadow Economy

117

equation where ηt (unobserved or latent) variable determines y ′ = ′ (y1 , y2 , . . . , yd ) column vector of indicators subject to a random error term εt . ηt is an unobserved or latent and is a scalar. Following Dell’Anno and Solomon [23], Λ is a (d × 1) column vector of parameters that relates yt to ηt : yt = ληt + εt

(1)

Equation 2 is a structural equation which shows that the unobserved variable ηt is determined by xt set of exogenous causes (x1 , x2 , . . . , xc ) and ζt a structural disturbance error term. γ is a (1 × c) vector of structural parameters: ηt = γxt + ζt

(2)

Without loss of generality, all variables are taken to have zero means. In equations (1) and (2) it is assumed that: the elements of ζt and εt are normal, independent and identically distributed8 ; the variance of the structural disturbance term ζt is Ψ and the covariance matrix of the measurement errors is a diagonal covariance matrix9 Θε . Substituting equation 1 and 2 yields a reduced form equation which expresses a relationship between the observed variables xt and yt . This is shown in equation 3: yt = Π ′ xt + zt ,

(3)

where: Π = λγ ′ is a c × d reduced form coefficients matrix and has rank one expressed in terms of c and d elements of λ and γ; zt = λζt +εt is a reduced form disturbance vector; z has an d × d reduced form covariance matrix (ω) given by: ω = λϕλ′ + Θε

(4)

where: ϕ = var(λ) and Θε =the reduced-form covariance matrix of the measurement errors. 8

9

The assumption on independence between structural disturbance ζt , and measurement errors the measurement errors εt could be considered as too restrictive, when mainly using economic dataset and, consequently, espoused to question the validity of this approach. Hayduk [30] explains it “. . . is purely a matter of arbitrary convention”for SEM analysis. Dell’Anno [18] in the context MIMIC model presents a re-parameterization of the MIMIC able to test the assumption on independence between structural disturbance ζt , and measurement errors εt . In the standard MIMIC model the measurement errors are assumed to be independent of each other, but this restriction could be relaxed (Stapleton [47]).

118

Roberto Dell’Anno, Friedrich Schneider

For the SEM nomenclature, the equations system with the relationships between the latent variable ηt (shadow economy) and the causes xt is called the “structural model” (eq. 2); the links among indicators yt and underground economy is called the “measurement model” (eq. 3). The reduced form of the equation matrix has two restrictions: (i) the c × d coefficient matrix Π which has rank one, and (ii) Θε which is the sum of a rank one matrix and a diagonal d × d matrix. The necessary condition for identification is that the number of structural parameters should be equal to the number of reduced form parameters. From equations 1 and 2 our structural model has c elements in γ, an element in the variance of ζt , c(c+1) elements in the variance of xt and d elements 2 contained in λ and in the variance of εt . From equation 3 and 4, the reduced form model contains cdelements in Π, d(d+1) elements in Θε and 2 c(c+1) elements contained in the variance of xt . An observation of the 2 reduced form parameters shows that unique solutions to the structural parameters λ and γ cannot be obtained from the reduced form model. This is because altering the scale of η yields an infinite number of solutions to λ and γ from the same reduced form solution. This inability to obtain unique solutions toλ and γ causes an identification problem which can be resolved by fixing the scale of the unobserved variable. This is the sufficient condition for identification which can be achieved by setting one of the coefficients in the column matrix λ to a constant. An alternative is to fix the variance of the unobserved variable η to 1 but the former is more convenient for economic interpretation. 3.3 Application of the MIMIC Procedure In a first step the researcher has to translate his/her theory into a structural model. In a second step it is necessary to fix one coefficient to the value one in order to give the latent variable an interpretable scale. If the researcher has set up his/her model and fixed the one coefficient to one, in a third step the estimation method has to be chosen. The Maximum Likelihood Estimation (MLE procedure) is the most widely used in SEM. It assumes multivariate normal data and a reasonable sample size10 . If the data are continuous but not normally distributed, 10

There are several rules of thumb about the sample size in the literature (Garson [26]): the sample size should be at least contain 50 observations or have more than 8 times observations then the number of independent variables in the model. Another one, based on Stevens [48], is to have at least 15 observations per measured variable or indicator. Bentler and Chou [4] recommend at least 5

Shadow Economy

119

an alternative method is an asymptotically distribution free estimation procedure, which in LISREL is known as WLS (weighted least squares). All goodness-of-fit measures are a function of sample size and degrees of freedom. Most of these take into account not only the fit of the model but also the model complexity. On the one side, if we have a very large sample, the statistical test will almost certainly be significant with respect to the degrees of freedom. On the other side, if we have small samples the model is very likely to be accepted even if the fit is poor. This is particular important in the analysis of shadow economy, since usually both the data availability is poor and the model complexity is high. When the model fit is not adequate, it has become common practice to modify the model, by deleting non-significant parameters in order to improve the fit and select the most suitable model specification.

4 Advantages and disadvantages of the MIMIC estimates of Shadow Economy It is widely accepted by most scholars, who estimate the size and development of the shadow economy, that such an empirical exercise is a “minefield”regardless which method is used. In evaluating the estimations of the shadow economy, we should keep in mind, that already Schneider and Schneider and Enste [45] warned that there is no best or commonly accepted method. Each approach has its strengths and weaknesses and can provide specific insights and results. Although from the first use the MIMIC approach has been “accompanied”by strong criticisms11 , in the last 10 years it is increasingly used for estimation of the shadow economy12 . The MIMIC approach offers several advantages in comparison with the other statistical methods to estimate shadow economy. According to Giles and Tedds [28], MIMIC provides a wider approach than most other competing methods, since it allows one to take multiple indicator variables and multiple causal variables into consideration at the same time. Moreover, it is quite flexible, allowing one to vary the choice of

11

12

observations per parameter estimate (including error terms as well as path coefficients). If possible, the researcher should go beyond these minimum sample size recommendations particularly when the data are not normally distributed or are incomplete. Compare e.g. the criticism by Helberger and Knepel [31] with respect to the pioneering work of Frey and Weck-Hannemann [25]. Compare the studies quoted on section 3.1.

120

Roberto Dell’Anno, Friedrich Schneider

causal and indicator variables according to the particular features of the economy under study, the period in question, and the availability of data. Again, following Giles and Tedds [28], the MIMIC model leads to a formal estimation and to testing procedures, such as those based on the method of MLE. These procedures are well known and are generally “optimal”, if the sample is sufficiently large. A further advantage of MIMIC approach has been stressed by Schneider and Enste [45]. They emphasize that the MIMIC approach leads to some progress in the estimation techniques of underground economy, because this methodology allows a wide flexibility in its application, therefore it is potentially superior over all other the estimation methods. Compared with other methods Cassar [10] argues that MIMIC does not need restrictive assumptions to operate (with exception of the calibrating process). Also, Thomas [49] argues that the only real constraint of this approach is not in its conceptual structure but the choice of variables. Of course this method has its disadvantages or limitations, which are identified in the literature. We will discuss the six most important ones, which deal with the model implementations, the sample used, and the reliability of MIMIC estimates: (1) When estimating the shadow economy using the MIMIC model approach the most common objection concerns the meaning of the latent variable (Helberger and Knepel [31]; Giles and Tedds [28]; Smith [46]; Hill; Dell’Anno [18], because the MIMIC approach is largely a confirmatory rather than exploratory technique. This means a researcher is more likely to determine whether a certain model is valid, rather to “find”a suitable model. Therefore, it is possible that the theoretical construct “shadow economy”could include other potential definitions, (i.e. traditional crime activities, do-it-yourself, etc.). This criticism which is probably the most common in literature, remains difficult to overcome as it goes back to the theoretical assumptions behind the choice of variables and empirical limitation on the availability of data. (2) Another objection is expressed by Helberger and Knepel [31]. They argue that the MIMIC estimations lead to an instable estimated coefficients with respect to changes the sample size and with respect to alternative model specifications. As Dell’Anno [18] shows, instability disappears asymptotically as the sample size grows large and if data is stationary and normally distributed. However, Schneider and Enste [28] argue that it is difficult to undertake this estimation procedure using a pure time-series data, and to get reliable data on variables other than taxes.

Shadow Economy

121

Dell’Anno [18] points out additional objections: (i) to calculate the confidence intervals associated with estimates of the latent variable, (ii) to test the hypothesis of independence between structural and measurement errors, (iii) to identify exhaustively the properties of the residuals, and (iv) to apply the SEM approach to small sample sizes and time series analysis. We believe that these cited weaknesses are the main limitations of this approach. (3) In the SEM analysis, the researcher has to solve or overcome at least two important trade-offs: the first one is the increasing complexity of the model and the decreasing degrees of freedom13 ; the second one is the increasing number of variables and the increasing sample size. Which combination is the best equilibrium to proceed here, it is an open question. (4) A further criticism is pointed out by Dell’Anno [18]. When using the MIMIC-approach he finds that there is a frequent possibility of the MIMIC approach to encounter an indefinite covariance matrix in the estimation procedure. According to Bollen and Long [6] this problem arises when the data contains too little information, like small sample size, to few indicator variables, small factor loadings, missing values, etc. Unfortunately, these are obstacles quite usual when one uses this kind of estimation application in economics. (5) Another criticism about the reliability of the MIMIC estimates of shadow economy is related to the benchmark method (Breusch [7,8]). This criticism has its origin in the complications researchers face, when he wants to convert the index of shadow economy (estimated by MIMIC model) into cardinal values. This is not an easy task, as the latent variable and its unit measure are not observed. The model provides just a set of estimated coefficients from which one can calculate an index which shows the dynamics of the unobserved factor. Such a calibration – regardless which methodology is used - requires experimentation, and a comparison of the calibrated values in a wide academic debate, although at this stage of research on the MIMIC approach, it is not clear which benchmark method is the best or most reliable one. In which way to proceed here is still problematically and unexplored hence every suggestion about this aspect of technique is welcome. In the following, we summarize four different benchmarking strategies which have been recently applied to convert the index of the shadow economy into absolute values. We use the following symbols: 13

A sufficient number of degrees of freedom is need to test if the proposed model specification is confirmed by data.

122



Roberto Dell’Anno, Friedrich Schneider η ordinal GDP t

is the value of the index of the shadow economy as ratio of official GDP estimated at time t by using structural equation (eq. 2); T is the subscript to indicate the base-period. The base-period T is the time in which there is the exogenous estimate of the shadow economy;  η ∗ GDP T is the shadow economy as percentage of official GDP estimated by an auxiliary method at the base-period T; ′ γˆ is the vector (of dimension q) of structural coefficients estimated by MIMIC model; xt is the vector of the q-causes observed at time t; GDP is the real official Gross Domestic Product. It is hypothesized that the GDP is the reference indicator. The first benchmarking/calibration procedure is one applied by Giles and Tedds [28]:  η ordinal η f inal =µ , (5) GDP t GDP t  η ∗   η ordinal is a constant. It is multiplied for whereµ = GDP GDP T T the index estimated by equation (2) in order to satisfy the condition that the shadow economy as percentage of official GDP at the baseperiod is equal to the value obtained from the external source. The second one is proposed by Dell’Anno [21]. They  and Schneider estimate the index of Shadow Economy ηtordinal by the following ′ structural equation: ∆ηtordinal = γˆ ∆xt . The calibration consists to apply an additive constant (τ ). It is chosen in order to satisfy the usual condition that the shadow economy as percentage of official GDP at the base-period is equal to the   

∗

η ordinal +τ

η = T¯GDP ¯ = value obtained from the external source: GDP T T¯  ordinal ′ ˆ +τ +γ (xT¯ −x0 ) η0 . By defining τ˜ = τ + η0ordinal , the absolute valGDPT¯ ues of Shadow economy are calculated with:

ηt = τ˜ + γˆ′ (xt − x0 )

(6)

The third benchmarking/calibration strategy applied by Bajada and Schneider (2005) is: gtf inal = gT∗ + γˆ′ ∆xt ,

(7)

where gT∗¯ is chosen in order to satisfy the condition gTf inal = gT∗ +

γˆ ∆xT¯ . gtf inal indicates the growth rate of shadow economy and gTf inal ′

Shadow Economy

123

is equal to the growth rate of shadow economy estimated in the baseperiod by an auxiliary method. Finally, Dell’Anno [19] proposes a fourth benchmarking procedure. It requires a transformation of the indicator chosenas reference variable (GDPt ) from absolute values to an index of GDPt GDPT . According with identification rule (λ1 = −1) the index of the shadow economy as a percentage of GDP in the base-period is linked to the chain index of real GDP as follow: Measurement equation:

ηˆt − ηˆt−1 GDPt − GDPt−1 =− . GDPT GDPT

(8)

The estimates of structural model are used to obtain an ordinal time series index for the latent variable14 . Structural Equation:

ηˆt ′ = γˆ xt . GDPT

(9)

The index is scaled to take up to a value of µ and further transformed to the shadow economy as ratio of current GDP. These operations are shown in the following benchmarking equation15 :   ηT∗ GDPT GDPT ηˆt ηt = . (10) GDPT GDPT ηˆT GDPt GDPt Where:  (a) ηˆt /GDPT is the index calculated by eq.(9); (b) ηT∗ /GDPT = µ is the exogenous estimate of shadow economy at

T; the base-period (c) ηˆT /GDPT is the value of index estimated by eq. (9) at the base period; (d) GDPT /GDPt is able to convert the index of shadow economy as changes respect to base-period in shadow economy respect to current GDP; 14

15

This procedure is valid whether observed causes are stationary or they are all differenced to same degree. In fact, to calculate the levels of the latent variable multiplying the structural coefficients for raw (unfiltered) data, it is equivalent to compute the changes in the index by multiplying coefficients for the differenced causes and then to integrate them.   The form of equation 10 is equivalent of ηt = ηˆt ηT∗ ηˆT . It is preferable the eq. 10 as benchmark formula because the external value of Shadow Economy ( µ ) is usually expressed as ratio of GDP. Then, although it seems a more complicated formula, it is easier to apply as benchmark method.

124

Roberto Dell’Anno, Friedrich Schneider

(e) (ηt /GDPt ) is the estimated shadow economy as a percentage of GDP at the time t. The multiplication of (b) and (c) fulfills the constraint that estimated shadow economy divided to GDP at the reference-period is equal to (µ) and the last part of LHS in eq. (10) converts the index of changes respect to the base-period into a ratio between shadow economy and current GDP. As Dell’anno and Schneider [22] argue this method shares the proportional properties of Giles and Tedds’s [28] benchmarking strategy. It attempts to overcome the Breusch’s [8] criticism to assign the monetary unit of measure from the reference indicator to the latent variable. Finally, we argue that standardization of any methodology requires experimentation, comparisons and a wide academic debate, although, at this stage of research on the MIMIC approach, it is not clear which benchmark method is the best or most reliable one. In which way to proceed here is still problematically and unexplored hence further research is necessary about this aspect of technique. (6) The last criticism about MIMIC estimates refers to the methodological side and, according with aims of this paper, may be the most relevant. Breusch [8] argues that the statistical properties/nature of MIMIC approach is unsuitable for economic questions/problems because this approach was designed to psychometric application and “to measuring intelligence seems far removed from estimating the underground economy”. Dealing with this critique, the main problem of the MIMIC approach lies in the strong difference between economic and psychological variables. Although, we agree that it is (still) problematical to apply this methodology to an economic dataset and to specify a macroeconomic model through the MIMIC framework, it doesn’t mean to abandon this approach. At the contrary, following an interdisciplinary approach to economics, the marked criticism should be considered as incentives for further (economic) research in this field rather than to suggest not to use this method because of the difficulties in the implementation of the MIMIC method.

5 Conclusions This paper deals with the empirical research strategies/possibilities of economic systems, especially if one assumes that the economic system is a complex one. According with Durlauf [24] the empirical side of the literature on economic complexity is growing, but it still suffers from

Shadow Economy

125

serious weaknesses, which it is not surprising given the difficulties of e.g. estimating the size and development of a shadow economy. According our view, we argue that one can overcome these problems when it one uses multidisciplinary methods. First, we realize that while economics theory is increasingly opening to the social sciences (e.g. behavioural economics, experimental economics), the methods used in empirical research seem to be stiffer to share/use methods of other social disciplines (e.g. psychometrics, marketing, educations, social research, etc.). Hence opening towards social sciences can help the economic profession to develop some adequate empirical methodology to analyse complex phenomenon of an economic system like the shadow economy. Second, we formulate two principles in line with such a methodology: The first principle is that one should not only study the object but also analyze the process; the second one is that we have to consider a method that takes into consideration the relationships, i.e. direct, indirect and cyclical effects among observed and unobserved economic variables. In our view, these two principles can be found in Structural Equation Modelling. The advice to use more methods of other disciplines is not new one. Unfortunately it seems discouraging to observe that, after more than thirty years, Goldberger’s [29] advice on “...numerous incentives for econometricians to break through those fences which still separate the social sciences.” is still largely unheard! We consider that the best way to support interdisciplinary approach in empirical economics is by showing examples of this kind of application. In this sense, we present the method of structural equation modelling. It is an appropriate econometric tool to study a complex phenomenon like the shadow economy. In particular, we review the literature and estimation procedure on one special case of structural equation modelling, the Multiple Indicators and Multiple Causes (MIMIC) model. Third, according to Popper’s thought, the growth of human knowledge gains from problems and from attempts to solve them. These attempts require the formulation of theories which must go beyond existing knowledge and therefore require creatively and imagination. In the shadow economy research, where the estimation step is particularly challenging, researchers are forced to use of “imagination”because existing estimation procedures are not convincing and complications are still numerous. We admit that only a few hypotheses are commonly accepted in this filed, one of those is that the underground economy is

126

Roberto Dell’Anno, Friedrich Schneider

a very controversial issue and extremely difficult to measure. Notwithstanding economists cannot surrender in the light of such difficulties. Fourth, we showed that several criticisms are misguided, while others effectively highlight weaknesses of MIMIC approach. The latter, in our opinion should be considered mainly due to the fact that we are still in an “initial stage of research”in this field. The main difficulties go back to the properties of SEM that is designed to use as a quantitative method in the social sciences. This means that is quite difficult to implement a time series analysis with this method and there are no specific guidelines to convert the index provided by MIMIC method into real figures. In our opinion the most attractive and important arguments of criticisms are related to the process of benchmarking (i.e. calculating the absolute values of the shadow economy) and the (potential) questionability of the economic hypotheses behind the MIMIC specification. Further attempts to adapt this procedure for econometric analysis of an economic problem are certainly necessary. Fifth, in applied economies for an evaluation of MIMIC approach if is important to remember that, to measure the shadow economy, is just one important aspect for a more general economic analysis, because economists are often more interested to understand the economic phenomenon of “shadow activities”rather then to just measure them. In fact, for policy makers and (macro-) economists it is much more attractive to be aware which are the main causes for the dynamics of the shadow economy compared with a detailed knowledge of the size of the shadow economy over time. This bearing in mind, the lack of other reliable methods, and the additional information provided by Model approach lead us to the conclusion that the (DY)MIMIC approach is a very useful tool for our economic profession, e.g. to analyse the shadow economy. Sixth, with respect to the suggestion of some authors (e.g. Breusch [8]) to abandon Model approach for shadow economy estimation, we argue this is not a good advice. In our view, not using the MIMIC model at this stage of the empirical research in economics is not only premature, but the contrary, we believe that further research and a greater opening among disciplines could provide constructive comments to overcome the difficulties of the MIMIC approach. Finally, we conclude that considering the current state of art, the MIMIC model is still one of the best approaches to this purpose, and provide a meaningful example of the advantages derived by an open-

Shadow Economy

127

minded and multidisciplinary approach desirable for the economic research.

References 1. Aigner D. J., Hsiao C., Kapteyn A., Wansbeek T., (1984). Latent variable models in econometrics. In: Griliches, Z., Intriligator, M.D., (Eds) Handbook of Econometrics, vol. 2, North-Holland, Amsterdam, pp. 1321–1393. 2. Albin P. S., Duncan K. F., (1998). Barriers and Bounds to Rationality: Essays on Economic Complexity and Dynamics in Interacting Systems. Princeton: Princeton University Press. 3. Bajada C., Schneider F., (2005). The Shadow Economies of the AsiaPacific, Pacific Economic Review 10(3), 379-401. 4. Bentler P. M., Chou C. P., (1987). Practical issues in structural modeling. Sociological Methods and Research 16(1), 78-117 5. Bollen K. A., (1989). Structural equations with latent variables, New York: John Wiley & Sons. 6. Bollen K. A., Long J. S., (1993). Testing structural equation models. Beverly Hills, CA: Sage. 7. Breusch T., (2005a). The Canadian Underground Economy: An Examination of Giles and Tedds, Canadian Tax Journal 53(2), 367-391. 8. Breusch T., (2005b). Estimating the underground economy using MIMIC models, working paper, Canberra, Australia (http://econwpa.wustl.edu/eps/em/papers/0507/0507003.pdf). 9. Capraro R. M., Kulm G., Capraro M. M., (2002). Investigating the Complexity of Middle Grade Students’ Understandings of Mathematical Constructs: An Example from Graphic Representation. Paper presented at the Annual Meeting of the American Educational Research Association (New Orleans, LA, April 1-5, 2002). Downloadable from ERIC - the Education Resources Information Center. 10. Cassar A., (2001). An index of the underground economy in Malta. Bank of Valletta Review 23, 44-62. 11. Cooley W. W., (1978). Explanatory observational studies. Educational Researcher 7, 9-15. 12. Corning P.A., (1998). Complexity Is Just a Word!, Technological Forecasting and Social Change 59, 197-200. 13. Cziraky D., (2004a). LISREL 8.54: A program for structural equation modelling with latent variables. Journal of Applied Econometrics 19, 135141. 14. Cziraky D., (2004b). Estimation of dynamic structural equation models with latent variables. Advances in Methodology and Statistics 1(1), 185204.

128

Roberto Dell’Anno, Friedrich Schneider

15. Cziraky, D. (2005). A unifying statistical framework for dynamic structural equation models with latent variables. (http://stats.lse.ac.uk/ciraki/framework.pdf). 16. Cziraky D., Gillman M., (2003). Inflation-growth causality with a latent underground economy model. Research Report, WIIW/GDNSEE, research field II Enterprise Development, Informal Economy and Labour Markets. Vienna: The Vienna Institute for International Economic Studies (www.wiiw.ac.at/balkan/files/ GDN EnterpriseLabourInformal InflationGrowthUnderground.pdf). 17. Day R. H., (1994). Complex Economic Dynamics, Volume I: An Introduction to Dynamical Systems and Market Mechanisms. Cambridge: MIT Press. 18. Dell’Anno R., (2003). Estimating the Shadow Economy in Italy: a Structural Equation Approach, Working paper 2003-7, Department of Economics, University of Aarhus, DK. 19. Dell’Anno R., (2007). The Shadow Economy in Portugal: an analysis with the MIMIC approach, Journal of Applied Economics, 10(2) 253-277. 20. Dell’Anno R., Gomez-Antonio M., Alanon Pardo A., (2007). Shadow Economy in three different Mediterranean Countries: France, Spain and Greece. A MIMIC Approach, Empirical Economics 33(1),51-84. 21. Dell’Anno R., Schneider F., (2003). The Shadow Economy of Italy and other OECD Countries: What do we know? Journal of Public Finance and Public Choice/Economia delle scelte pubbliche 21(2/3), 97-120. 22. Dell’Anno R., Schneider F., (2006). Estimating the underground economy: a response to T. Breusch’s critique, Discussion Paper, 2006-07, Johanes Kepler University of Linz, Department of Economics, Austria. 23. Dell’Anno R., Solomon H. O., (2007). Shadow Economy and unemployment rate in U.S.A. Is there a structural relationship? An empirical analysis, Forthcoming Applied Economics. 24. Durlauf S. N., (2005). Complexity and Empirical Economics. The Economic Journal 115, 225-243. 25. Frey B. S., Weck-Hannemann H., (1984). The Hidden Economy as an ‘Unobservable’ Variable. European Economic Review 26, 33-53. 26. Garson D. G., (2005). Statnotes: An online textbook. (http://www2.chass.ncsu.edu/garson/pa765 /structur.htm). 27. Giles D.E.A., (1999). Measuring the hidden Economy: Implications for Econometric Modeling. The Economic Journal 109(46), 370–380. 28. Giles D.E.A., Tedds L. M., (2002). Taxes and the Canadian Underground Economy, Canadian Tax Foundation: Toronto. 29. Goldberger A. S., (1972). Structural equation methods in the social sciences. Econometrica 40(6), 979–1001. 30. Hayduk L. A., (1987). Structural equation modelling with LISREL. Essential and advances. The Johns Hopkins University Press. London.

Shadow Economy

129

31. Helberger C., Knepel H., (1988). How Big is the Shadow Economy? A ReAnalysis of the Unobserved-Variable Approach of B.S. Frey and H.WeckHannemann. European Economic Review 32, 965-976. 32. Horgan J., (1997). The End of Science: Facing the Limits of Knowledge in the Twilight of the Scientific Age, paperback edition. New York: Broadway Books. 33. Hoyle R. H., (Eds.) (1995). Structural Equation Modeling: Concepts, Issues, and Applications. Thousand Oaks, CA: Sage. 34. Jaccard J., Wan C. K., (1996). LISREL approaches to interaction effects in multiple regression. Thousand Oaks, CA: Sage Publications. 35. J¨ oreskog K.G., Goldberger A. S., (1975). Estimation of a model with multiple indicators and multiple causes of a single latent variable. Journal of the American Statistical Association 70, 631-639. 36. J¨ oreskog K.G., Sorbom D., (1993). LISREL 8, User’s Reference Guide. Chicago: Scientific Software International. 37. Keynes J. M., (1936). General theory of employment, interest and money. London:Macmillan. 38. Laibson D. I., Zeckhauser R., (1998). Amos Tversky and the Ascent of Behavioral Economics, Journal of Risk and Uncertainty 16, 7-47. 39. Loayza N. V., (1996). The Economics of the Informal Sector: A Simple Model and Some Empirical Evidence from Latin America, CarnegieRochester Conference Series on Public Policy 45, 129–162. 40. Lomax R., (1989). Covariance structure analysis: Extensions and developments. In: Thompson, B. (Eds.), Advances in social science methodology. vol. 1. Greenwich, CT: JAI Press, pp. 170-204 41. Muthen B. O., (2002). Beyond SEM: general latent variable modeling. Behaviormetrika 29(1), 81-117. 42. Rabin M., (2002). A perspective on psychology and Economics. European Economic Review 46, 657-85. 43. Rosser J. B. Jr., (1999). On the complexities of complex economic dynamics. Journal of Economic Perspectives 12(4), 169-192. 44. Schneider F., (2005). Shadow Economies around the World: What do we really know. European Journal of Political Economy 21(3), 598-642. 45. Schneider F., Enste D. H., (2000). Shadow Economies: Size, Causes, and Consequences. Journal of Economic Literature 38(1), 77-114 46. Smith R. S., (2002). The Underground Economy: Guidance for Policy Makers?. Canadian Tax Journal 50(5), 1655-1661. 47. Stapleton D. C., (1978). Analyzing political participation data with a MIMIC Model. Sociological Methodology 15, 52-74. 48. Stevens J., (1996). Applied multivariate statistics for the social sciences (3rd ed.). Lawrence Erlbaum Associates, Inc., USA. 49. Thomas J., (1992). Informal economic activity. LSE, Handbooks in Economics, London, Harvester Wheatsheaf. 50. Velupillai K. V., (Eds.) (2005). Computability, Complexity and Constructivity in Economic Analysis. Blackwell. Victoria, Australia.

130

Roberto Dell’Anno, Friedrich Schneider

51. Watson M. W., Engle R. F., (1983). An alternative Algorithms for the estimation of Dynamic Factor, MIMIC and Varying Coefficient Regression models. Journal of Econometrics 23, 385–400. 52. Zellner A., (1970). Estimation of regression relationships containing unobservable variables. International Economic Review 11, 441–454.

Will house prices rise in 2007? A comparative assessment of house prices in London and Dublin Peter Richmond Trinity College Dublin 2 - Ireland

1 Introduction House prices are of concern to many people and in recent years they seem to have gone through one of their periodic up turns. In a previous paper [4] we offered empirical evidence suggesting that prices in both the UK and Dublin had peaked and suggested prices computed relative to inflation would fall over the next few years. Here we assess progress in the light of further data now available. During early 2006, the press headlines were saying that prices continue to rise in the UK and London in particular. However opinion expressed in recent weeks by economists and others - who most laymen would believe should know what is going on - seems to be more divided. In October, the UK Royal Institution of Chartered Surveyors reported that prices were rising at their fastest rate for four years1 . The Financial Times of November 4th 2006 contained a feature suggesting that social forces such as immigration were changing the very fabric of the house market. This seems to echo the view of commentators who at the peak of the dot com boom were suggesting that the nature of the economy had changed and PE ratios that had reached very high values were now the norm. Simultaneously in the past few days, the UK Abbey National Building Society has launched a new mortgage scheme, enabling homebuyers “who cannot save up [the deposit] quickly enough” to borrow up to five times their joint salary about which one commentator2 remarked “It’s a piece of crass opportunism with “top of the market”

1 2

http://news.bbc.co.uk/1/hi/business/6043192.stm hhttp://www.telegraph.co.uk/money/main.jhtml?view=DETAILS&grid=A1Your View&xml=/money/2006/11/03/ccjeff03.xml

132

Peter Richmond

written all over it”. It will be interesting to see to what extent this facility is taken up. Elsewhere in the same newspaper is an article indicating that the use by sellers of the sealed bid, a device used to extract the maximum price, is making a come back. Other voices including the Governor of the Bank of England have suggested that the “anticipated fall in inflation for September [2006] may not persist for long”3 . In August, the Bank raised borrowing costs to 4.75% from 4.5%, the first rate increase in two years and rates are expected to rise yet again in November 2006 as inflation, measured by the Consumer Price index (CPI), rises above the target of 2%. During November, however, the Governor said that inflation might be lower than expected over Christmas because of the plunge in oil prices. At the same time some economists close to the monetary policy committee responsible for interest rate rises offered the opinion that not just one but two more interest rate increases may be required in the coming months! A point to make at this juncture is that the UK has two measures of inflation. The CPI adopted by the UK government recently does not take account of house mortgage repayments; the Retail Price index (RPI) however does take account of these payments. It is this measure that we shall use when calculating relative house prices. At the time of writing, the CPI has fallen to 2.4% whereas the RPI has risen to 3.6%. The layman may justifiably be confused and perhaps agree with economist, Paul Ormerod who has recently criticized the modus operandi of the UK monetary policy committee [3]. Simultaneously, concern is being expressed about personal insolvency which is “at a record high”4 . A YouGov survey at the beginning of November 2006 showed that the number of people who said they were having trouble repaying their debts has risen from 21% in April to 30% and according to Credit Suisse First Boston, annual personal insolvencies are likely to hit 200,000 by 2009. The number of house repossessions is now at its highest level since the early 1990s5 . In Dublin, house prices that have risen slowly for many decades have been rising very quickly since 1997. One commentator in the first part of 2006 reported that “Frenzy in the Irish property market has intensified as house price inflation was accelerating at an annualized rate of 15%. 3 4

5

http://news.bbc.co.uk/1/hi/business/6038526.stm Financial Times, November 4/5 2006 page 4. Quote attributed to Louise Brittain, Head of personal insolvency division, Blake Tilley. http://www.telegraph.co.uk/opinion/main.jhtml?xml=/opinion/2006/11/05 /do0503.xml&sSheet=/opinion/2006/11/05/ixopinion.html

House Prices in London and Dublin

133

However at the time of writing this manuscript (early November 2006), it is clear that things are changing. The Irish Sunday Independent very recently carried a headline suggesting that the housing boom has ground to a halt. On the strength of this comment, it is easy to see that the financial climate is changing but it is difficult to decide what it means for the future. In the next sections we examine empirical past data for house prices in London, extending previously published data to cover 2006. The results remain consistent with previous observations that a price peak or bubble is in play and is at its peak with the data following those found by Roehner for the western region of the US. Similar data for Dublin suggests that the city has joined the global group of cities that exhibit similar house price dynamics and that house price dynamics are now synchronized with London.

2 House prices in the UK House price data for the UK and Northern Ireland can be obtained from both the Halifax and Nationwide Building society web sites. Both are similar and 1figure 1 below has been downloaded from the Nationwide Building Society6 .

Fig. 1. House prices (£) in the United Kingdom (1973-2006)

6

http://www.nationwide.co.uk/hpi/historical.htm

134

Peter Richmond

The peak that occurred in the UK during the late 1980s is clearly visible. The geographical variation across the UK has been studied by Roehner. Note the lower line for Northern Ireland. Uncorrelated with the frenetic activity on the UK mainland during the 1980s this is now rising steeply. The depression during the 1980s arguably reflects the difficulties faced by the province at that time; the current rapid rise is probably a catch up reflecting not only the new optimism as to the outcome of the peace process but also a spill-over of optimism from the Irish Republic. It is even more revealing to compute the house prices in constant money terms, i.e, relative to the retail price index (RPI). Data from the Nationwide Building Society database allows us to look at this from 1952 through to 2006 for average UK house prices. This representation illustrates interesting peaks and troughs that mirror those observed by Roehner who recently computed data in this way for the Western US.

Fig. 2. log10 [UK house price (£)/RPI] (1987 prices) for the UK over the period 1952-2006

The comparable graph of US data showed strong peaks at ∼1969, 1979 and 1989. Peaks in the UK data are clearly visible at 1973/4, 1979 and 1989. The US data actually showed some evidence of a weak peak or shoulder at ∼1973. This was ignored by Roehner who used smoothed data points and focussed on the almost identical character of the major peaks. However, we now see a pronounced peak in London in 1973. In passing, the author recalls that Australia experienced a similar housing boom during the period 1973-4 from which he profited as he relocated from Australia back to the UK. Looking even more closely at the UK

House Prices in London and Dublin

135

data there is a suggestion of a very weak peak around 1969 but without the supporting evidence of the strong peak in the US at this time, one would probably ignore this. This all suggests house price correlations over the past 50 years have increased not only between the UK and the US since World War 2 but also across the English speaking world. At this point we note that according to the US Office of Federal Housing Enterprise Oversight (www.ofheo.gov). US Home prices were 10.06 percent higher in the second quarter of 2006 than they were one year earlier. Appreciation for the most recent quarter was 1.17 percent, or an annualized rate of 4.68 percent. The quarterly rate reflects a sharp decline of more than one percentage point from the previous quarter and is the lowest rate of appreciation since the fourth quarter of 1999. The decline in the quarterly rate over the past year is the sharpest since the beginning of OFHEOs House Price Index (HPI) in 1975. Spokesman, James Lockhart, said “These data are a strong indication that the housing market is cooling in a very significant way. Indeed, the deceleration appears in almost every region of the country.” We also note recent work by Sornette and Zhow who have predicted, using albeit more complex arguments based on the existence of log periodic oscillations, that the US data would peak in 2006. We can now plot house prices for London in this way. These are shown in figure 3 (bold solid line) together with the average UK prices (faint continuous line).

Fig. 3. log10 Relative house prices for both the UK and London together with trend lines (see text for discussion)

136

Peter Richmond

The lower straight (light gray) line is the long term trend (best fit) for the average UK prices over the period 1952-2006. The central straight line (dashed) is the trend for London prices assuming that the gradient is equal to those for the long term UK average. The upper (dash dash dot) line is computed assuming a 40% increase over the trend. It is interesting to note that the lower (40% decrease) line essentially coincides with the UK long term average. The limit at the upper level correlates closely with the borrowing limits imposed by UK building societies. Assuming no change in earnings and 4 times an average salary is required to purchase a property at trend, roughly 6 times average salary is required to buy at the peak. Figures compiled by the Nationwide Building Society for gross price to earnings ratios for first time buyers are shown in Figure 4 where it can be seen that over the past few years, the ratio of trend to peak values is ∼1.4.

Fig. 4. Gross price to earnings ratios for first time buyers. The ratio between the trend value and the peak at 2006 is ∼1.4.

3 Stock markets and housing markets In figure 5 we compare stock indices for US (Dow Jones index) and UK (FTA index) markets since the late 19th century and see that the impact of new technology following the industrial revolution finally began to drive strongly the US economic engine after the Civil war.

House Prices in London and Dublin

137

The UK economic engine however did not really get into gear until after the Second World War. By the 1980s it is clear that there is essentially complete correlation between the two indices.

Fig. 5. Graph of UK stock index (FTA) (1800-1990) and Dow Jones index (1890-1990)

As the housing peaks have reached their current zenith it also seems that house building stocks have also reached a peak. Building stocks in the UK declined by around 20% from January to June 2006. Over the same period, US building stocks declined by a little more, ∼ 40%. Houses are much more illiquid than common stocks at least common stocks for major companies - and the housing market has not developed the sophistication of that for common stocks with its use of options, shorting and so on. We may assume two types of person buy houses. Potential home owners who want a home in which to live (and those who are selling in order to say relocate or otherwise trade up or down) and speculators who invest in so-called buy to let. Some potential home owners will always be faced with the prospect of buying at a peak in the market. They could choose to rent and wait for the market to cool. On the other hand they might have the opportunity to raise enough money to buy on a large mortgage. In November 2006, the Abbey national Building Society in the UK raised the borrowing limit for individuals to 5 times salary. A few lucky (high income earners) individuals will obviously be able to raise enough money to buy a house even at the relatively high levels prevailing in the current market. However, since the numbers of people on higher income eventually diminishes (according to a Pareto law), the number of buyers will obviously diminish as prices rise. Other options such as extending the term

138

Peter Richmond

of a mortgage to 50 years or more is also being mooted by some building societies. Whether people will then be ready to commit to such huge repayments over their lifetime remains to be seen. It is also possible for groups of people to come together and jointly finance a house purchase. Equally it is possible that as older people reach the state where their children leave and live elsewhere they will have sufficient funds to buy small properties at high prices. Some parents who feel they will have enough to fend for themselves in old age might also give money to their children to pay a deposit. The extent to which any of these options might be taken up is not clear. Demand from immigrants has also been proposed as offering scope for added demand. However many immigrants are not usually wealthy people and are generally in the same position as the average person in the community with respect to their ability to raise high mortgages. Interest and demand from speculators can certainly be expected to diminish. Speculators are interested in rental returns and whilst rental returns may be attractive (typically ∼7% or more) when buying property in a trough or even up to the level of the trend, rental income without large increases in rents become increasingly unattractive (typically ∼2-4% before any expenses on maintenance!) as the price rises above trend levels. So at some point the investor is better off keeping his money in the piggy bank or buying stocks where yields of 3-5% or possibly more are possible. Indeed one would expect that the simple house buyer seeking a house in which to live would perceive the value of weighing up the option of keeping money in the bank and renting against chasing price levels upwards. So we conclude, that without increases in rents from current levels, as prices rise to around 40-50% above the trend demand essentially goes to zero. It is clear from the 2006 data that the price is now essentially tracking this upper level. Economic pundits and real estate experts are expressing divided views, a sure sign that the market is at a point where confidence is beginning to change from being positive to negative on the future state of house prices. Quite what will tip the market is not possible to say. However from the point of view of the physicist one can say the consensus seems to be in the vicinity of a critical point where even a small shock can set the next direction of confidence. The UK is entering a period where interest rates are rising. Inflation is increasingly a concern of policy makers at this time and at the time of writing (9th November 2006) UK interest rates (see Figure 6) have been raised from 4.75% to 5%. A majority of analysts expect the Bank’s rate-setting Monetary Policy Committee to increase rates again in the

House Prices in London and Dublin

139

New Year. This will put added pressure on recent buyers with relatively high mortgages.

Fig. 6. UK Bank rates since 1976 (Source Bank of England)

Clearly from figure 6 we can see that interest rates per se are not in any way directly correlated with house price changes. One way to think about the issue is by analogy with a magnet at a critical point when only a weak magnetic field is sufficient to orient all the spins in the same direction. So any small shock that influences confidence or housing demand can tip the balance and, as a result, move prices can then move strongly in a particular direction. Interest rate increase might thus act as such a trigger but then so could other economic news of a negative nature. Furthermore, changes are likely not to be evident immediately. Houses are generally traded via estate agents who set the sale price in agreement often with vendors at levels determined by those in the recent past. Vendors often hope they can secure a sale at the highest level. It is only when it becomes clear that the house is not selling that the property is re-launched at a lower price. They do so then in yet more difficult conditions. Such a realisation can take a few months to come about. This leads to the phenomenon of prices being (as economists say) “sticky downwards” - people dont cut them unless they have to. At this point some vendors can be locked into moving to alternative housing and price reductions can be aggressive to avoid excessive payments on bridging loans. In this way prices can fall more quickly that they rose.

140

Peter Richmond

If as a result, past trends continue we can expect falls in real (inflation adjusted) house prices of typically ∼6-8% per annum over the next 4-5 years (see figure 10a). Of course whether the shape of the current peak will evolve as previous peaks did in the past we cannot say with certainty. Perhaps governments will find some hitherto unknown policies or tools that they can implement to counteract such a downturn but on the basis of past experience it seems unlikely. At this point it is worth noting that some attempts to influence the housing market have been made. The Economist7 commented on one particular attempt of UK Chancellor Gordon Brown who in 2005 expressed his desire to increase the number of home owners by a million over the next five years, taking home ownership rate towards 75% and the government subsidized loans for people who could not afford to buy homes on their own. In effect the home owner needed only to purchase between 50% and 75% of the equity. For 80,000 of the purchases according to the Economist article, the government (via housing associations) finances the remaining cost. For 20,000 of the deals the mortgage lenders and government will each own 12.5% of the equity. As the Economist also noted, by actually increasing the demand for cheap housing, the measure may hold up prices at the bottom end of the market and essentially prop up the housing market preventing a crash. It is conceivable then that this is what is actually now happening - public subsidies are holding up the housing market!

4 House prices in Ireland Figure 7 shows quarterly data for house prices both inside and outside Dublin over the period 1995 to the first quarter of 2006. Both curves are rising strongly over this period in line with both US and UK data. Can we say any more? As a first step we have converted the data from EUR into GBP using a ratio 1.46=1 and superimposed the data onto that for the UK and Northern Ireland (Figure 5). It is immediately evident that the data over this period appears to have broken away from provincial areas such as Northern Ireland. Prices for Dublin are now essentially following those for London and the data for houses outside Dublin is following that of a provincial city in England. Additional data for Irish house prices back to about 1970 are also available on www.environ.ie8 . The data are annual and there 7 8

The Economist ‘Gordon Brown’s House Sale’May 28th 2005, page 37 I am grateful to John Frain of the Department of Economics, Trinity College Dublin for drawing this data set to my attention

House Prices in London and Dublin

141

Fig. 7. House prices in the Republic of Ireland (1997-2006)

Fig. 8. House prices (quarterly data) in the Republic of Ireland converted to and superposed on similar graphs for London and Northern Ireland.

are problems about changing composition of the types and locations of houses however the picture remains unchanged as can be seen in Figure 9. The consumer price indices for Ireland and the UK are shown in figure 10. The difference between the two curves is interesting. Both countries entered the EU at the same time in 1973. Prices then began to rise more rapidly in Ireland than in the UK. It was during this period

142

Peter Richmond

Fig. 9. House prices in the UK and Dublin (1973-2006)

Fig. 10. UK Retail Price index and Irish Consumer Price index, 1949-2005

that the Thatcher government began the extensive period of privatization and sought to bring the UK out of the difficult period of industrial unrest experienced in the 1970s. During the 1990s inflationary pressures in both countries are once again comparable. However since the turn of the 21st century, prices have risen very strongly in Ireland.

House Prices in London and Dublin

143

We now convert the house price data using the consumer price index to real (adjusted to 1980) prices. The superposition of this data for Dublin onto similar prices for London is shown in figure 11.

(a) 11a

(b) 11b

144

Peter Richmond

Figs 11 (a and b) show Dublin prices (again adjusted to 1980 prices) (Log /Irish consumer price index) compared with similar ratios for London. Figure 11a shows the quarterly data for Ireland supperposed on the UK data illustrated previously. Figure 11b shows the annual data that extends back to 1973. The effect of the rapid increase in the CPI since 2000 is clear. Real (inflation adjusted) prices in Ireland have already fallen away from the London trend. The latter peak would appear to have already reached its zenith and is about to decline. At the time dealing with referees comments on this text (March 2007) the Irish press is already carrying headlines suggesting the housing market is indeed in decline and some real prices in Dublin are actually being reduced. So it seems clear that Dublin, as a result of the economic expansion during the last 20 years, has now emerged once again as a major city and global hot spot that correlates with the US that has already suffered declining house prices during 2006.

5 Conclusions House prices, it would seem, have exhibited regular, almost universal behaviour across time and geographic domain. Roehner has suggested that major cities (E.g., London, Los Angeles, New York, Sydney and Paris) are hot spots where this activity is galvanised. From the perspective of the physicist, this suggests that the house market is some kind of complex system. As a result of strong economic activity over the past two decades, Dublin seems to have joined the group of global hot spots. This increase in correlation of house prices between cities is very reminiscent of phase locking in complex systems observed recently by Holyst. The current data suggests that real (inflation adjusted) prices in both London and Dublin have peaked and are close to a tipping point. As of March 2007, in Dublin even actual prices might be beginning to decline according to recent press comment. In London prices seem to be steady with most economic pundits still predicting a rise albeit a small one during 2007. Has the UK economy changed in some way analogous to the changes seen in the stock indices as we entered the 20th century? This seems unlikely especially since house prices in both the US and now Dublin do seem to be following the traditional pattern. Could the gravity defying prices in London be due to innovative mortgages that now allow the punters to stretch their ability to secure mortgages of 5 times their salary? Or could the public subsidy outlined in the

House Prices in London and Dublin

145

Economist 2005 article and discussed above be responsible? Maybe one or both of these factors are responsible for the current position. For the moment, we can only wait and see if this essentially simple forecasting method proves to be correct. One can speculate that another peak will begin around 2011-12 and the cycle will then last through to ∼ 2016. By the time of the next house price boom it is possible that other European cities such as Warsaw, Prague and Budapest will have joined the list of hotspots.

References 1. Helbing, D., Schonhof, M., Stark, H.U. and Holyst, J.A. (2005). How individuals learn to take turns: Emergence of alternating cooperation in a congestion game and the prisoner’s dilemma. Advanced Complex Systems 8, 87. 2. Lucy, B.M., (2006). A recipe for change and growth. An analysis prepared for Homeloan Management Limited. 3. Ormerod, P. (2001). The illusion of economic control, or why membership of the MPC is a sinecure. Manchester Statistical Society. 4. Richmond, P. (2006). PhysicaA (In press) 5. Roehner, B. M. (2006). Real estate price peaks: a comparative overview. Evolutionary and Institutional Economics Review 2 (2), 167-182. 6. Roehner, B. M. (2001). Hidden Collective Factors in Speculative Trading, Springer, Berlin, ISBN 3-540-41294-8. 7. Roehner, B. M., private communication 8. Zhou, W.-X. and Sornette, D. (2005). Is There a Real Estate Bubble in the US? arXiv: physics/0506027 v1 3.

Part IV

Models of Risk Management and Decision Support

Risk Measuremant and Fair Valuation Assessment in the Life Insurance Field Mariarosaria Coppola1 , Valeria D’Amato1 , Emilia di Lorenzo1 , and Marilena Sibillo2 1 2

Universit` a degli Studi di Napoli Federico II - Italia Universita` degli Studi di Salerno - Italia

1 The regulation background in solvency tool New solvency requirements are coming into the insurance business management, in particular through Pillar 2 of Solvency 2. The guidelines lead towards an increasing importance ascribed to the particular risk profile of the single company. Regarding the mathematical provision, this statement reveals to be fundamental, being in the aim of the correct management to keep not more than the sufficient capital amount as solvency capital, and this statement strongly depends on the company risk profile. The authorities engaged for defining the new solvency rules and the reports resulting from the working activity indicate the valuation of the mathematical provision at their fair value, meaning with this expression an estimated market price; in practice, the fair value of the mathematical provision, that is the net present value of the residual debt toward the policyholder, is calculated at current interest rates and at current mortality rates. Obviously the hypotheses made on these risk components evolution in time implies monetary consequences on the final amount. These two main risk drivers constitute the basic elements for depicting the risk profile of an insurance business and, as consequence, for correctly determining the capital solvency adequacy. The tool of the quantification of the risk coming out from the two sources, financial and demographic, namely interest and mortality rates, becomes crucial for its practical implications. As analysed in several papers [6, 7, 9], the financial risk and the mortality risk are respectively due to the random movements of the interest rates and to the random deviation of the real number of deaths from their expected values, both involved in the fair valuation. Besides these two

150

Mariarosaria Coppola et al.

components a third one, the longevity risk, is due to the improvement in the mortality trend, strongly felt in all the industrialized countries. The longevity risk causes systematic deviations of the number of deaths from the forecasted ones. In general, as showed in [6, 7], the financial risk and the longevity risk produce a risk characterized by the systematic nature, and cannot be controlled by pooling operations. The mortality risk, on the contrary, is characterised by the accidental nature. All the three risk components have to be carefully considered in the fair value context: in this statement the correct debt quantification involves the constraint of the capital amount “minimum” for solvency purposes, freeing at the same time the residual financial availability. In the paper, by applying stochastic conditional calculations, we propose formulas for measuring the contribution of each risk source to the global risk connected to the fair valuation of the mathematical provision, being sure this can be an useful instrument for practitioners. In section 2 we introduce formulas for the fair valuation of the mathematical provision in the case of a generic life insurance contract. In section 3 two examples with survival and death benefits respectively are analysed. In section 4 formulas for quantifying the contribution of each risk source to the global risk are proposed and finally some consideration are collected in section 5.

2 The fair valuation of the mathematical provision of a life insurance contract The fair valuation framework within which we will work is based on the following assumptions about the stochastic context. We consider two probability spaces {Ω, ℑ′ , ℘′ }, {Ω, ℑ′′ , ℘′′ }, where ℑ′k and ℑ′′k are the sigma-algebras referred to the financial and survival information flow respectively, with the hypotheses of independence of the randomness in interest rates on the randomness in mortality rates, as commonly assumed. These two spaces generate the probability space {Ω, ℑ, ℘} , represented by the filtration ℑk ⊆ ℑ with ℑk = ℑ′k ∪ ℑ′′k and {ℑ′k } ⊆ ℑ′ ,{ℑ′′k } ⊆ ℑ′′ . The market in which the valuation is framed is frictionless, with continuous trading, no restrictions on borrowing or short sales, being the zero bonds and the stocks infinitely divisible. Even though the following formulas refer to a generic life insurance contract with a single premium payment at time 0, they can be easily generalized to the cases of periodic premium payments, as showed in the examples in section 2.

Risk Measurement in Life Insurance

151

We indicate by v(t, j) the present value at time t of one monetary unit due at time j and by Xj the stochastic cash flow at time j relating to a generic life insurance contract. Referring to a portfolio of homogeneous life contracts, we can write the fair value at time t of the stochastic stream of loss:    (1) (Nj Xj v(t, j)/ℑt ) Vt = E  j>t

In formula (1) Nj indicates the random number of claims at time j. In this general case, the claims are represented by survivors or dead persons according to the nature of the considered contract. In (1) the trading strategy replicating the portfolio flow at time j is represented and its generic element is constituted by Nj Xj units of unitary zero coupon bond [3] issued in t and maturing in j with the following price: P (t) (t, j) = Nj Xj v(t, j) In formula (1) with the symbol E we indicate the expected value calculated in a risk neutral probability measure deriving by the hypotheses of completeness of the market. This assumption, acceptable in relation to the pure financial aspect of the insurance product, is less realistic if its demographic component is taken into account. As suggested in [1], the risk neutral probability expected value can be considered using the most opportune probability measure for the demographic component, taking into account both the systematic and the unsystematic mortality risks. If c is the number of homogeneous policies at time 0 issued on policyholders aged x at issue, we can write:    Vt = E  (c1x+t,j Xj v(t, j)/ℑt ) (2) j>t

in which the variable 1x+t,j takes the value 1 if the insured aged x at issue and living at time of valuation t is dead or lives at time j, according to the contractual case, and the value 0 otherwise. On the basis of the independence of the financial variables on the demographic ones, we have:  ! Vt = (3) cXj E 1{ x + t, j}/ℑt E [v(t, j)/ℑt ] j>t

Indicating by:

152

Mariarosaria Coppola et al.

Px+t,j = t px Pj the probability that the claim happens at time j to an insured aged x in 0 and living at time t and recalling that t px is the probability that an insured aged x survives at the age x + t, formula (3) can be written as follows:  [cXj t px Pj E [v(t, j)/ℑt )] (4) Vt = j>t

3 Cash flows examples On the baselines exposed in the preceding section, here we structure the cash flows of two life insurance cases exemplifying the fundamental sections of the life business: the contract in case of life and the contract in case of death. In the first case the benefit is due if the insured lives at the paying time; in the death case the benefit is payable if the insured’s death occurs in a defined time interval. To illustrate the contract in case of life we choose the deferred life annuity with periodic premiums payable during the deferment period. It constitutes the theoretical actuarial base for describing the pension cash flow. For the contract in case of death we consider the typical case of the temporary life insurance. Example 1: The deferred life annuity with periodic premiums. Let consider a life annuity issued on an insured aged x, with anticipated premiums Aj payable for the first m years and benefits Bj payable at the end of each year after m, if the insured is alive. The following is the relating cash flow scheme: X0 = −A1

Xj =

"

−Ax+j Bx+j

j<m j>m

with j = 1, 2, ... both payable in case of the insured’s life. Recalling the indicator function properties, formula (4) becomes:  (Xj j px+t E [v(t, j)/ℑt )]} (5) Vt = ct px { j>t

Example 2: The temporary life insurance with periodic premiums.

Risk Measurement in Life Insurance

153

The second example concerns a temporary life insurance with duration n, issued on an insured aged x, with premiums Aj payable at the beginning of each year during the whole duration and benefit Bj payable at the end of the year in which the insured dies, if this event happens before n. The cash flow results: X0 = −A1 Xj =

"

−Ax+j Bx+j

with j = 1, 2, ..., n − 1 " −Bn Xn = 0

if the insurer is alive in j if the insurer dies in j-1, j

if the insurer dies in n-1, n otherwise

Formula (4) can be expressed as follows:  ! Aj j px+t + Bj j−1/1 qx+t )E [v(t, j)/ℑt ) } Vt = ct px {

(6)

j>t

In formula (6) j−1/1 qx+t is the probability that the insured aged x + t dies between the ages x + t + j − 1 and x + t + j.

4 Risk measurement tools In this section we consider the impact of the risk due to the randomness in the choice of the model used for the two main risk sources involved in the fair value formula. Interest rate scenario risk The risk measurement tool we propose concerns the model risk deriving from the randomness in the choice of the scenario for the interest rates in the fair valuation. In the following formula:    V arFt′ [E(Vt )] = V arFt′ [E  (c1x+t,j Xj v(t, j))] (7) j>t

the conditioning with respect to Ft′ represents the randomness in the choice of the interest rate stochastic process. Formula (7) can be written as follows:

154

Mariarosaria Coppola et al.

V arFt′ [E(Vt )] = c2t p2x



(Xh Xk Ph Pk cov[v(t, h), v(t, k)])

(8)

h>t k>t

and represents the measure of the variability in the fair value reserve aroused by the uncertaity in the choice of the interest rate stochastic model. In this sense it can be considered a measure of the interest rate model risk. Table risk In this subsection we calculate the impact of the risk due to the randomness in the choice of the mortality table used in the fair valuation. As showed in [2] in the particular contractual case of the insured loan, the influence of the survival function used for reserve calculation in a fair valuation context originates important consequences in the mathematical provision results. The formula we propose for quantifying the table risk in the case of a generic life insurance product is the following:    (9) (1x+t,j Xj v(t, j))] V arTt [E(Vt )] = c2 V arTt [E  j>t

in which the conditioning on Tt indicates the randomness in the choice of the survival function used to determine the probabilities. By analogy with the preceding case, the index expressed in formula (9) can be consiered a measure of the table risk.

5 Conclusions The formulas proposed in this paper are characterised by two aspects particularly meaningful in the current regulation context concerning the insurance business; in this statement the requests in identifying the impact of the different risk sources are more pressing and practitioners consequently need opportune measurement tools. The first aspect is their relatively simple implementation and, not less important, the immediate interpretation of the results. The second one consists in their flexibility, meaning that the proposed measurement tools are applicable in any hypotheses of interest rate process and survival function. In particular, considering that the paper is framed in the liability valuation item, the interest rate behaviour has to be described by means of a stochastic process based on the arbitrage principle in

Risk Measurement in Life Insurance

155

the interest rate term structure scenario; for example the Cox Ingersoll and Ross (CIR) stochastic model is suitable. Methodologies for the CIR parameter estimation are well known in literature and examples of applications can be found in [4] and in [3]. Contextually, several mortality assumptions can be chosen for implementing in the table risk index model. They have to be selected on the basis of which we consider the best estimate of the future survival evolution. For example the Heligman Pollard law, with different level of projection, as well as the Lee Carter one, are possible descriptions of the human survival phenomenon. The two random aspects characterised by the financial and the demographic nature makes the insurance business not perfectly tradeable in the market, differently of the corresponding pure financial product. This circumstance implies that the mortality behaviour cannot be described together with the financial process by means of a synthetic rate capturing the valuation assessment. The longevity bond market, still in an evolution phase, isn’t able to express a structured market for the insurance products yet, both for the incompleteness of the ages at issue represented in the contracts and for the aspect of the incomplete liquidity. The stochastic framework for interest rates and survival probabilities allows to consider not only the impact of each risk component on the global risk of the fair value of the mathematical provision, but the interaction between the risk components too. This statement seems to be the right methodology for correctly managing insurance business with regard to its risk mapping.

References 1. Ballotta, L. and Haberman, S. (2006). The fair valuation problem of guaranteed annuity options: The stochastic mortality environment case. Insurance: Mathematics and Economics 38, 195–214. 2. Coppola, M., D’Amato, V., Di Lorenzo, E. and Sibillo, M. (2006). Fair value and demographic aspects of the insured loan. Proceedings of the “10th International Congress on Insurance: Mathematics and Economics”. Leuven 3. Coppola, M., Di Lorenzo, E. and Sibillo, M. (2005). Fair valuation scheme for life annuity contracts. Proceedings of the “XI International Symposium on Applied Stochastic Models and Data Analysis”(ASMDA). 4. Cocozza, R., Di Lorenzo,E. and Sibillo, M. (2004). Life insurance risk indicators: a balance sheet approach. Proceedings of the “9th International Congress on Insurance: Mathematics and Economics”. Rome

156

Mariarosaria Coppola et al.

5. Cocozza, R., Di Lorenzo, E., Orlando, A. and Sibillo, M. (2006). The VaR of the mathematical provision: critical issue, Proceedings of “28th International Congress of Actuaries, ICA 2006” - Parigi , http://papers.ica2006.com/2002.html. 6. Coppola, M., Di Lorenzo and E., Sibillo, M. (2000). Risk sources in a life annuity portfolio: decomposition and measurement tools. Journal of Actuarial Practice 8, Nos. 1 and 2, 43–72. 7. Coppola, M., Di Lorenzo, E. and Sibillo, M. (2002). Further remarks on risk sources measuring: the case of a life annuity portfolio. Journal of Actuarial Practice 10, Nos. 1 and 2, 229–242. 8. Coppola, M., Di Lorenzo, E. and Sibillo, M. (2003). Stohastic Analysis in Life Office Management: Applications to Large Portfolios. Applied Stochastic Models in Business and Industry 19, 31–42. http://www3.interscience.wiley.com/cgi-bin/jtoc?ID=66002616 9. Di Lorenzo, E. and Sibillo, M. (2002). Longevity risk: measurement and application perspective. Proceedings of “2nd Conference on Actuarial Science and Finance”, Samos http://www.stat.ucl.ac.be/Samos2002/proceedSibillo.pdf

Analysis of Qualitative and Quantitative Rankings in Multicriteria Decision Making Livia D’Apuzzo1 , Gabriella Marcarelli2 , and Massimo Squillante2 1 2

Universit`a di Napoli - Italy Universit`a del Sannio - Italy

1 Introduction The decision procedures applied in MCDM (Multicriteria Decision Making) are the most suitable in coping with problems involved by social choices, which have to satisfy a high number of criteria. In such a framework an important role is played by the Analytic Hierarchy Process (A.H.P., for short), a procedure developed by T.L. Saaty at the end of the 70s [14], [15], [16], and widely used by governments and companies in fixing their strategies [10], [16], [19]. The A.H.P. shows how to use judgement and experience to analyze a complex decision problem by combining both qualitative and quantitative aspects in a single framework and generating a set of priorities for alternatives. Once the decision problem has been defined, the fundamental steps in the procedure are: 1st. to decompose the decision problem in a hierarchy of two or several levels (see Fig. 1): at the top there is the goal of the decision problem, while at the bottom there are the alternatives among which the decision maker has to operate a choice; the goal and the elements at the intermediate levels act as criteria for the elements of the lower level; 2nd. to determinate the local weights of the elements of a level with respect to the criteria of the next higher level: the priorities are obtained by means of series of pairwise comparisons, expressing how much each element dominates the others; 3rd. to determinate the final evaluations of the alternatives by synthesizing the local evaluations of the elements of each level. The following schema represents a very simple decision problem requiring a hierarchy with only one intermediate level of criteria. The

158

Livia D’Apuzzo et al.

alternatives xi , i = 1, 2, in the lowest level, must be valued with respect to each criterion Cj and every criterion Cj must be valued with respect to the goal O. Fig. 1

oo O OOOOO O ooo

o op1

p2 p3 OO OOO oo O ooo C2 ? C1 ?TTTT C3 j j ? w1 TT  ?? jw13 jj  2 TTT jj2j w11? w3 ? w12jjTjTjTjTjTjTw 2 2  T T j j

X1

X2

If wij denotes the local evaluation of the alternative xi , with respect to the criterion Cj and pj is the weight of Cj , then the global evaluations of the alternatives are provided by w1 = p1 w11 + p2 w12 + p3 w13 w2 = p1 w21 + p2 w22 + p3 w23 For the determination of the global evaluations of the alternatives, in a more complex situation, we refer to [16], [19]. 1.1 Determination of the local weights by means of pairwise comparisons We focus on the second crucial step in a decision procedure. If X = {x1 , x2 , . . . , xn } is the set of the alternatives (or elements of a level of the hierarchy) and C is a criterion of the next higher level, then the elements of X are compared in pairs with respect to C: to each pair (xi , xj ) a positive number aij is assigned expressing how much xi is preferred to xj as regards C; obviously aij > 1 implies that xi is strictly preferred to xj , whereas aij < 1 expresses the opposite preference and aij = 1 means that xi and xj are indifferent. For the determination of the preference ratios aij , Saaty indicates the following fundamental scale translating the comparisons expressed in verbal terms into the corresponding numbers.

Multicriteria Decision Making

159

Table 1: Fundamental scale for pairwise comparisons. Verbal scale Numerical values Equally important 1 Moderately more important 3 Strongly more important 5 Very strongly more important 7 Extremely more important 9 Intermediate values to reflect compromises 2, 4, 6, 8. Alternative methods can be applied to assigne preference ratios aij For example the rating technic [21]: the expert has a budget of 100 points that he shares between the elements xi , xj , in such a way that the amount of points (rating) assigned to each one of them represents its importance. Then the preference ratio aij is calculated as ratio of the ratings of the two elements. The preference ratios generate a pairwise comparison matrix A = (aij ), that is assumed to be reciprocal, that is ′ s.

r) aji = 1/aij

∀ i, j = 1, . . . , n

(reciprocity);

so the strict preference of xi over xj , represented by aij > 1, is also expressed by the inequality aji < 1. Given the matrix A, it is easy to show that there exists a priority vector w = (w1 , w2 , . . . , wn ) verifying the desirable condition wi = aij wj

(1)

if and only if A is consistent, that is A verifies the stronger condition c)

aij ajk = aik

∀ i, j, k = 1, . . . , n

(consistency).

We say also that a positive vector verifying (1) is a consistent vector. The consistency is equivalent to each one of the following conditions: • n is the unique positive eigenvalue of A; • the columns of A are proportional vectors. Consistent vectors are: the positive eigenvector associated to n, every column of the matrix A, the vectors obtained by applying the arithmetic or the geometric mean to each row of the matrix. These vectors are proportional and, normalized to 1, provide the same dominance vector w∗ whose components supply the numerical evaluations of the elements of X.

160

Livia D’Apuzzo et al.

The request of consistency deals with the following ideal situation: if the evaluator, for instance, considers xi two times better than xj and xj four times better than xk , then he considers xi eight times better than xk . Unfortunately, in the real life the evaluator is usually inconsistent when he expresses his judgements, especially if he compares objects with respect to a qualitative attribute such that ”appealing” or ”usefulness”. The inconsistent behaviour of the evaluator leads to some problems we shall exhibit below. In the case of inconsistency it is just possible to look for vectors whose components provide ratios wi /wj that approximate the preference ratios aij ′ s: wi ≃ aij (2) wj Let λmax be the greatest eigenvalue of A. Then the Consistency Index λmax − n CI = (3) n−1 has been proposed by Saaty as a measure of the consistency of the pairwise comparison matrix A: if CI = 0 then the matrix is consistent, and the more CI is far from 0 the more the matrix is far from the consistency. Moreover Saaty indicates as priority vectors verifying (2): the positive eigenvectors associated to λmax , and each one of the vectors provided by the three following methods: arithmetic average of normalized columns (ANC), normalization of row arithmetic averages (NRA) and normalization of the geometric mean of rows (NGM) [17]. If w = (w1 , w2 , . . . , wn ) is a vector provided by the mentioned methods, then the consistent index CI represents also a measure of proximity of the ratios wi /wj to the preference ratios aij ′ s. 1.2 Coherence of quantitative rankings It is hard to compute the measure CI in the case that the order n of the matrix A is greater than 5. Moreover, if the matrix A is inconsistent, then it may happen that the right eigenvector method, and the methods (ANC), (NRA), (NGM) provide vectors that indicate different rankings [4]. So we have to deal with the following Problem A. What is the actual ranking of a set of alternatives with respect to a criterion C? Is it possible to determine the actual ranking independently of quantitative evaluations? In other situations the vectors provided by the mentioned methods indicate the same ranking, but it does not agree with the preference

Multicriteria Decision Making

161

ratios aij ′ s: it may happen that “wi > wj ” whereas “aij < 1” indicates that the alternative xj is preferred to the alternative xi [4]. Another question comes out from the following consideration: as A is a reciprocal matrix, if aij represents the priority of xi over xj , then aji represents the corresponding antipriority. So it is natural to expect that the ranking on X can be obtained by applying the usual methods both to the rows and to the columns of the matrix A. More exactly, assume that w = (w1 , w2 , . . . , wn ) is a priority vector provided by the right eigenvector method or acting on the rows of the matrix A by means of the geometric or the arithmetic mean; moreover also suppose that w1 > w2 > . . . > wn , so that the ranking indicated by w is x1 ≻ x2 ≻ . . . ≻ xn , where “ ≻ ” stands by “is better than” or ”is strictly preferred to”. Then, it would be natural to presume that by means of the left eigenvector or by acting on the columns of A by the above mean operation we would get an antipriority vector v = (v1 , v2 , . . . , vn ) stating the same ranking, that is: v1 < v2 < . . . < vn . Unfortunately, this is not the case when the matrix is inconsistent [5] [20]. Hence, once stated the actual ranking, it is necessary to deal with the following Problem B. Which conditions ensure that the above methods provide priority vectors and antipriority vectors agreeing with the actual ranking? Finally, also if w = (w1 , w2 , . . . , wn ) is a priority vector agreeing with the preference ratios aij , so that wi > wj ⇔ aij > 1, w may not represent the different intensities of the preference, that is it may happen wi /wk > wr /ws

whereas

aik < ars .

So the following Problem C. Which condition on the matrix A ensures the existence of vectors representing the intensity of the preferences? We have shown some solutions of the above problems under the following assumption for the matrix A s) aij = 1

f or

i = j

that, togheter with the condition

(no indifference),

162

Livia D’Apuzzo et al.

t) aij > 1 and ajk > 1 ⇒ aik > 1 (transitivity), ensures that the alternatives are completely ordered by means of a strict prefernce relation [4]. We say that a matrix A verifying the conditions r, t, s is a Rts matrix.

2 Notations Let X = {x1 , x2 , . . . , xn } be a set of alternatives and A = (aij ) the related pairwise comparison matrix. From now we assume that A is a Rts matrix and we denote with ˜ = {a1 , a2 , . . . , an } the set of the rows of A; - A ˇ - A = {a1 , a2 , . . . , an } the set of the columns of A; - n(ai ) the number of the components of ai greater than 1; - n0 (ai ) the number of the components of ai greater than or equal to 1. Then, because of the assumption s, n0 (ai ) = n(ai ) + 1. In order to present some results in a synthetic way, we consider the n =]0, +∞[n embodied with the relation of strict partial order ⊲ set R+ defined by u = (u1 , . . . , un ) ⊲ v = (v1 , . . . , vn ) ⇔ (uj > vj

∀j = 1, 2, . . . , n).

Then ⊲ denotes the related partial order u ⊲ v ⇔ (u ⊲ v

or

u = v);

and ⊳ and ⊳ denote the relations opposite to ⊲ and ⊲. n , then Moreover we assume that, if u, v ∈ R+   - uv indicates the vector uv11 , . . . , uvnn ; - 1 indicates the vector (1, 1,  . . . , 1);  1 1 - v denotes the vector v = v11 , v12 , . . . , v1n .

(4)

Multicriteria Decision Making

163

So the condition of reciprocity r related to a pairwise comparison matrix can be expressed as r′ ) ak =

1 ak

∀ k = 1, 2, . . . , n.

3 Qualitative actual ranking The condition s is equivalent to the completeness of the asymmetric relation ≻ on X defined by xi ≻ xj ⇔ aij > 1.

(5)

The condition t is equivalent to the transitivity of ≻. So, under the assumptions that A is Rts, ≻ is a strict preference order (i.e. a transitive, asymmetric and complete relation) and a permutation (i1 , i2 , . . . , in ) of (1, 2, . . . , n) is available so that the elements of X are ordered in the decreasing chain: (6) xi1 ≻ xi2 ≻ . . . ≻ xin . Then (6) represents the actual ranking on X [4]. The following proposition provides a tool to recognize a Rts matrix and determine the actual ranking. Proposition 1. [7] A = (aij ) is a Rts matrix if and only if there exists a permutation α =(i1 , i2 , . . . , in ) of (1, 2, . . . , n) such that n(ai1 ) = n − 1 > n(ai2 ) =n − 2 > . . . > n(aih ) = = n − h > · · · > n(ain ) = 0

(7)

or, equivalentely, n0 (ai1 ) = n > n0 (ai2 ) = n − 1 > . . . > n0 (aih ) = n − h + 1 > · · · > n0 (ain ) = 1.

(8)

Moreover xi ≻ xj ⇔ n(ai ) > n(aj ) ⇔ n0 (ai ) > no (aj ).

(9)

164

Livia D’Apuzzo et al.

Example 1. The matrix 

 1 4 7 8  1/4 1 5 6   A=  1/7 1/5 1 4  1/8 1/6 1/4 1

is Rts and the actual ranking is represented by the chain x1 ≻ x2 ≻ x3 ≻ x4 because n0 (a1 ) = 4 > n0 (a2 ) = 3 > n0 (a3 ) = 2 > n0 (a4 ) = 1. Once stated the actual ranking (6), a vector may represent it at different levels, as the following definition indicates. Definition 1. A vector w = (w1 , w2 , . . . , wn ) ∈]0, +∞[n is called: I) ordinal evaluation vector, or coherent priority vector, if and only if the equivance wi > wj ⇐⇒ ai ≻ aj holds, that is, by (5) wi > wj ⇐⇒ aij > 1;

(10)

II) value ratio vector, or intensity vector, if and only if wr wi > ⇔ aik > ars , wk ws

(11)

and as a consequence: wr wi = ⇔ aik = ars . wk ws By Proposition 1 and the equivalences (9), the same permutation α=(i1 , i2 , . . . , in ) provides the inequalities 8 and the actual ranking 6. As a consequence the vector n0 (A) = (n0 (a1 ), n0 (a2 ), . . . n0 (an )) is a coherent priority vector and a positive vector w is a coherent priority vector if and only if w = (φ(n0 (a1 )), φ(n0 (a2 )), . . . φ(n0 (an ))) , with φ strictly incresing positive function on ]0, +∞[. Anyway n0 (A) does not take the effective values of the ais ′ s into account. So we shall consider coherent priority vectors and intensity vectors obtained by aggregating the elements of the matrix by means of a suitable functional.

Multicriteria Decision Making

165

Definition 2. Let F be a functional n → F (u) ∈ ]0, +∞[ F : u = (u1 , u2 , . . . , un ) ∈ R+

(12)

and wF the vector wF = (F (a1 ), F (a2 ), . . . , F (an )).

(13)

Then - F is an ordinal evaluation functional if and only if wF is a coherent priority vector; - F is an intensity functional if and only if wF is an intensity vector. Definition 3. - A = aij is a priority matrix if and only if each column ah is a coherent priority vector; - A = aij is an intensity matrix if and only if each column ah is an intensity vector.

4 Weak consistency and priority vectors The problem of finding ordinal evaluation vectors for inconsistent matrices has been widely investigated by Basile and D’Apuzzo [1996, 1997, 2006, 2006 (2)]. They introduce the condition wc) aij > 1 and ajk > 1 ⇒ aik > aij ∨ ajk

(weak consistency),

and characterize the class F of the aggregation operators 12 that, when applied to the rows a1 , . . . , an of a weakly consistent matrix, provide wF as coherent priority vectors [Basile and D’Apuzzo 2002, 2006]. Proposition 2. A = (aij ) is a priority matrix if and only if aij > 1 ⇔ ai ⊲ aj ⇔ aj ⊲ ai .

(14)

Proof. By Definition 3 and equivalence 10 defining a coherent priority vector, A = (aij ) is a priority matrix if and only if aij > 1 ⇐⇒ (aih > ajh

∀h = 1, 2, ..., n).

So the assertion follows by 4 and Condition r′ . Theorem 1. [7] The following statements are equivalent

166

Livia D’Apuzzo et al.

(i) A = (aij ) is a weakly consistent matrix; ˜ = {a1 , a2 , . . . , an }; (ii) ⊲ is a strict simple order on A (iii) A = (aij ) is a priority matrix. Hence, by Proposition 2, Theorem 1 (ii) and equivalence 5 defining ˜ = {a1 , a2 , . . . , an } is ordered ≻, A is a priority matrix if and only if A by ⊲ in accordance with the actual ranking 6, that is: ai1 ⊲ ai2 ⊲ · · · ⊲ ain , ˇ is ordered as follows: ai1 ⊳ ai2 ⊳ · · · ⊳ ain . whereas A Theorem 2. [7] Let A = (aij ) be a priority matrix and F a functional 12. Then F is an ordinal evaluation operator if and only if its restriction ˜ is strictly increasing with respect to the relation ⊲, that is to A ai ⊲ aj ⇒ F (ai ) > F (aj ). The above theorem allows us to indicate some important examples of ordinal evaluation functionals as: - the quasilinear mean operators ([1], [2]) Fφp (a1 , a2 , . . . , an ) = φ−1

n 



pi φ(ai ) ,

i=1

(15)

n and built up starting from a strict monotonic function φ on R+ a non negative weighting vector p = (p1 , p2 , . . . , pn ) verifying the  condition pi = 1; - the ordered quasilinear means

Oφp (a1 , a2 , . . . , an ) = Fφp (a∗1 , a∗2 , . . . , a∗n ) = φ−1

n  i=1



pi φ(a∗i ) ,

(16)

obtained by applying the quasilinear mean 15 to the decreasing rearrangement (a∗1 , a∗2 , . . . , a∗n ) of a = (a1 , a2 , . . . , an ) (see also [3]) . Hence the methods to estimate priorities using arithmetic and geometric mean provide coherent priority vectors. Corollary 1. [7] Let A = (aij ) be a priority matrix. Then every right positive eigenvector associated to the maximum eigenvalue is a coherent priority vector.

Multicriteria Decision Making

167

Corollary 2. Let A be a priority matrix and F an ordinal evaluation operator. Then the vector vF = (F (a1 ), F (a2 ), . . . , F (an ))

(17)

is a coherent antipriority vector. Proof. By Proposition 2 and Theorem 2.

5 Intensity vectors and intensity matrices Proposition 3. The existence of an intensity vector implies the condition δ)

aik > ars ⇒ air > aks

(index − exchangebility).

n be an intensity vector. Then Proof. Let w = (w1 , w2 , . . . , wn ) ∈ R+

aij > ars ⇔

wj wi wr wi > ⇔ > ⇐⇒ air > ajs . wj ws wr ws

Proposition 4. δ ⇒ wc. Proof. Assume aij > 1 and ajk > 1, and suppose that δ is verified. Then, by δ, r and the inequalities ajk > 1 = aii and aij > 1 = akk we get aik > aij and aik > ajk ; so the second side of the implication wc is verified. By Propositions 3 and 4, the existence of the intensity vector requires that the matrix A is weakly consistent, but, as the following example shows, the weak consistency does not ensure the existence of an intensity vector. Example 2. The matrix in the Example 1 is weakly consistent because a1 ⊲a2 ⊲a3 ⊲a4 (see Theorem 1); nevertheless A does not verify δ because a13 > a24 , but a12 = a34 . So, because of Proposition 3, there not exists an intenisity vector.

Proposition 5. A = (aij ) is an intensity matrix if and only if qc)

aij ajk > arj ajs ⇔ aik > ars

(quasi

consistency).

168

Livia D’Apuzzo et al.

Proof. The condition qc can be written aik > ars ⇔

aij arj > ∀j = 1, . . . , n akj asj

(18)

and this proves that each column aj of A is an intensity vector. Corollary 3. qc ⇒ δ ⇒ wc. Proof. By Propositions 5, 3, 4. The condition qc is obviously sufficient for the existence of an intensity vector. Neverthless, an intensity vector may existe also if the matrix A does not verify the condition qc, as the following example shows: Example 3. The matrix 

 1 9/8 3 8  8/9 1 2 4  A   1/3 1/2 1 2  1/8 1/4 1/2 1

induces the ranking x1 ≻ x2 ≻ x3 ≻ x4 on the set of alternatives X = {x1 , x2 , x3 , x4 }. A does not verify the condition qc: indeed it is a14 a42 = 2 = a24 a43 , but a12 = 9/8 < a23 = 2. Nevertheless the column a3 is an intensity vector. Lemma 1. [9] The condition qc is equivalent to each one of the following conditions (j) (jj)

a

a

aik > ars ⇔ a i ⊲ ar ; k s wc and “(i = r, k = s) ⇒ (aik > ars > 1 ⇔

Moreover if qc, then aik = ars ⇔

ai ak

=

ai ak

⊲

ar as

⊲ 1)”.

ar as .

Theorem 3. The following statements are equivalent: (k) A is an intensity ) matrix; * a ˜ is totally ordered by ⊲; (kk) the set A÷ = a i : ai , aj ∈ A j ) a (kkk) A is a priority matrix and the set A÷ = a i ∈ A÷ : j is totally ordered by ⊲.

ai aj

⊲ 1

*

Multicriteria Decision Making

169

Proof. The equivalence (k) ⇔ (kk) follows by Proposition 5 and Lemma 3, condition j. The equivalence (k) ⇔ (kkk) follows by Proposition 3, Lemma 3 (condition jj), Corollary 3 and by Theorem 1 for which the rows of a weakly consistent matrix are completely ordered by ⊲. Final remark. Because of the inconsistency of judgements expressed bymeans of pairwise comparisons, traditional methods used to get the quantitativeranking may not agree with the expressed comparisons. Under condition s and t the actual qualitative ranking on the set of alternatives is determined. Then, the problem is to find vectors representing the ranking either as ordinal evalution vectors or as cardinal evaluation vectors (see Def. 1). So conditions weaker than consistency have been introduced and their links with the actual ranking have been analyzed. The condition wc ensures the existence of large classe of ordinal evaluation vectors and is a necessary condition for the existence of an intensity vector. The stronger condition δ is a necessary condition too, but is not sufficient. The condition of quasi consistency, qc, characterizes the intensity matrices, that are matrices which columns are intensity vectors; these matrices are weakly consistent and Theorem 3 provides a simple tool to verify if a weakly consistent matrix is an intensity matrix. Nevertheless the condition qc is not a necessary condition for the existence of an intensity vector. So a further investigation should be carried out in order to get conditions weaker than the quasi consistency and ensuring the existence of intensity vectors.

References 1. Aczel, J. (1966). Lectures on Functional Equation and their Applications. New York and London. Accademic Press. 2. Aczel, J. and Saaty, T. L. (1983). Procedures for Synthesizing Ratio Judgements. Journal math. psychology 27, 93-102. 3. Basile, L. and D’Apuzzo L. (1996). Ordering for classes of aggregation operators, International Journal of Uncertainty, Fuzziness and Knowledge Based Systems, World Scientific Publishing Co. Pte. Ltd., NJ - USA, 4.2, (1996), 145–156. 4. Basile, L. and D’Apuzzo L. (1997). Ranking and Weak Consistency in the A.H.P. Context. Rivista di matematica per le scienze economiche e sociali, 20 (1). 5. Basile, L. and D’Apuzzo L. (2002). Weak Consistency and Quasi-Linear Means Imply the Actual Ranking. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, World Scientific Publishing, 10, no. 3, 227-239.

170

Livia D’Apuzzo et al.

6. Basile, L. and D’Apuzzo L. (2006). Matrici di valutazione ordinale e matrici intensit´ a. P. Amenta, L.D’Ambra, M.Squillante, A.G.S.Ventre, Metodi, Modelli e Tecnologie dell’informazione a Supporto delle Decisioni, Franco Angeli, 143-158. 7. Basile, L. and D’Apuzzo L. (2006). Transitive Matrices, Strict Preference and Ordinal evaluation Operators. Soft Computing - A Fusion of Foundations, Methodologies and Applications, vol.10, no.10 Springer- Verlag, 933-940 ( http://dx.doi.org/10.1007/s00500-005-0020-z) 8. Basile, L. and D’Apuzzo L. (2006). Transitive Matrices, Strict Preference and Intensity Operators. Mathematical Methods in Economics and Finance, 1, 21-36. 9. D’Apuzzo L., Mrcarelli G. and Squillante M. (2006). Pairwise Comparison Matrices: Properties And Strict Preference Order. International Journal of Intelligent Systems. Accepted for the publication. 10. Fusco Girard L. and Nijkamp P. (1997). Le valutazioni per lo sviluppo sostenibile della citt e del territorio. Franco Angeli, Napoli. 11. Krovc, J. (1987). Ranking Alternatives-Comparison of Different Methods Based on Binary Comparison Matrices. European Journal of Operational Research. 32, 86-95. North Holland. 12. Narasimhan, R. (1982).A Geometric Averaging Procedure for Constructing Super Transitive Approximation to Binary Comparison Matrices. Fuzzy Sets and Systems, 8, 53-6. 13. Roberts, F.S. (1979). Measurement Theory with Applications to Decisionmaking, Utility, and Social Siences.Encyclopedia of Mathematics and its Applications, vol. 7. GIANCARLO ROTA, editor. Addison-Wesley Publishing Company. Massachusetts. 14. Saaty, T. L. (1977). A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology, vol. 15, n. 3, pp. 234-281. 15. Saaty, T. L. (1978). Exploring the interface between hierarchies, multiple objectives and fuzzy sets. Fuzzy Sets and Systems, vol.1, pp. 57-68. 16. Saaty, T. L. (1980). The Analytic Hierarchy Process. McGraw-Hill. New York. 17. Saaty T. L. and Vargs L.G. (1982). The Logic of Priorities Applications in Business, Energy, Health, and Transportation. Kluwer-Nijhoff, Boston. 18. Saaty, T. L. (1986). Axiomatic Foundation of the Analytic Hierarchy Process. Management Sciences, 32, 841-855. 19. Saaty, T. L. (1988). Decision Making for Leaders. University of Pittsburgh, Pittsburgh. 20. Saaty, T. L. (1990). Eigenvector and Logarithmic Least Squares. European journal of operational research, 48, 156-160. 21. Voogd H. (1983). Multi-criteria evaluations for urban and regional planning, London Princeton Univ. 22. Yager, R.R. (1988). On Ordered Weighted Averaging Aggregation Operators in Multicriteria Decision Making In I.E.E.E., Transactions on Systems, Man, and Cybernetics, 18, no. 1, 183-190.