Economic Bubbles And Change Points

Testing for Bubbles and Change–Points
` Alan Kirman and Gilles Teyssiere1

Journal of Economic Dynamics and Control (2005), vol 29, 765-799. DOI:10.1016/j.jedc.2004.01.005

Abstract A model for a financial asset is constructed with two types of agents, who differ in terms of their beliefs. The proportion of the two types changes over time according to stochastic processes which model theinteraction between the agents. Agents do not persist in holding “wrong” beliefs and bubble–like phenomena in the asset price occur. We consider tests for detecting bubbles in the conditional mean and multiple changes in the conditional variance of the process. A wavelet analysis of the series generated by our models shows that the strong persistence in the volatility is likely to be the outcome of amix of changes in regimes and a moderate level of long–range dependence. These results are consistent with what has been observed by Kokoszka and Teyssi`re (2002) and Teyssi`re (2003). e e Keywords: Market interaction, bubbles, long–memory heteroskedasticity, pseudo long–memory, change–point, wavelets. JEL Classification: C52, C22, D40, G12.

Corresponding author. Laboratoire de StatistiqueAppliqu´e et MOd´lisation Stochastique (SAMOS), Unie e versit´ Paris 1 – Panth´on–Sorbonne. E–mail: stats@gillesteyssiere.net http://www.gillesteyssiere.net e e This is the outcome of a long lasting project which started with Kirman (1991), “Testing for Bubbles”, and then supersedes this former paper. This version of the paper has been largely rewritten in Spring 2002 during the visit of Teyssi`re tothe Department of Mathematics and Statistics of Utah State University, that he wishes to thank e as well as Piotr and Gudi Kokoszka for their hospitality. We are greatly indebted to Lajos Horv´th and Piotr Kokoszka, with whom joint works on the issue of change– a point tests for ARCH/GARCH processes strongly influenced the final version of this paper, to Clive Granger for his probing questions on thestatistical properties of our processes, to Murad Taqqu and Patrice Abry who suggested the use of the wavelet analysis, to Remis Leipus and Alfredas Raˇkauskas for long discussions on the c issue of bubbles testing, and Hans F¨llmer with whom one of the stochastic models was developed. We would o like to thank Raj Bhansali, George Evans, Liudas Giraitis, Cars Hommes, Davar Khoshnevisan, DannyQuah, Tom Sargent, Leigh Tesfatsion, Michael Woodford, two careful and constructive referees, and participants in the International Conference on Long-Range Dependent Stochastic Processes and their Applications, Bangalore, January 2002, in the colloquium of the Department of Mathematics and Statistics of Utah State University, and in the Stochastics seminar at the Department of Mathematics of theUniversity of Utah, May 2002, for comments and suggestions.

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I can calculate the motions of heavenly bodies but not the madness of people. Isaac Newton

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Introduction

Although the presence of “bubbles” and “herding” behaviour in financial markets is widely accepted there are still few theoretical models which generate such phenomena. Yet they are widely held to be responsible for theinstability of foreign exchange markets for example. Indeed, “price bubbles” in markets although particularly associated with the markets for financial assets, have been documented for a wide variety of markets over a considerable period of time. One of the earliest bubbles was that in the price of red mullet in the first century A.D. The red mullet fever is documented by Cicero, Horace, Juvenal andMartial. A survey of other historical bubbles, such as the Tulip, South Sea and Mississippi bubbles, may be found in Garber (2000). More recently there has been a substantial literature on the theoretical basic for and testing of bubbles, see for example Blanchard and Watson (1982), Flood and Garber (1980), Meese (1986), Tirole (1985), West (1988), Woo (1987), Stiglitz (1990), Flood and Hodrick...