Chaos and the exchange rate
Daniela Federici and
Giancarlo Gandolfo ()
No 4A.1, CeNDEF Workshop Papers, January 2001 from Universiteit van Amsterdam, Center for Nonlinear Dynamics in Economics and Finance
The interest of economists in chaos theory started in the 1980s. The first to draw the attention of economists to chaos theory was, in fact, Brock (1986), who examined the quarterly US real GNP data 1947-1985 using the Grassberger-Procaccia correlation dimension and Lyapunov exponents. Subsequent studies generally found absence of evidence for chaos in macroeconomic variables (GNP, monetary aggregates) while the study of financial variables such as stock-market returns and exchange rates gave mixed evidence. Studies aimed at detecting chaos in economic variables can be roughly classified into two categories. On the one hand, there are studies that simply examine the data and apply various tests, such as the studies mentioned. These tests have been originally developed in the physics literature and typically require several thousand observations. Apart from this data problem, such an approach is not very satisfactory from our point of view, which aims at finding the dynamic model underlying the data. On the other hand, structural models are built and analysed. This analysis can in principle be carried out in several ways: a) theoretically, namely showing that plausible economic assumptions give rise to dynamic structures having one of the mathematical forms known to give rise to chaotic motion; b) empirically, namely building a theoretical model and then b1) giving plausible values to the parameters, simulating the model, and testing the resulting data series for chaos; or b2) estimating the parameters econometrically, and then proceeding as in b1. Existing chaotic exchange rate models (De Grauwe and Versanten, 1990; Reszat, 1992; De Grauwe and Embrechts, 1992, 1993a,b; De Grauwe, Dewachter, Embrechts, 1993; Ellis, 1994; Szpiro, 1994; Da Silva, 1997) follow the structural approach: they are structural models built in discrete time (difference equations). From the theoretical point of view, these models show that with orthodox assumptions (PPP, interest parity, etc.) and introducing economically plausible nonlinearities in the dynamic equations, it is possible to obtain a dynamic system capable of giving rise to chaotic motion. However, none of these models is estimated, and the conclusions are based on simulations: the empirical validity of these models is not tested. Hence they can all be classified in category b1. In this paper, a continuous time exchange rate model is built as a nonlinear set of three differential equations and its theoretical properties (steady state, stability. etc.) are analysed. The model is then econometrically estimated in continuous time with Italian data and examined for the possible presence of chaotic motion. So far as we know, this is the first (tentative) study in category b2. However, this is not the main motivation of this paper. From our point of view it is important to show that the continuous time estimation of systems of nonlinear differential equations is a very powerful tool in the hands of the profession to tackle dynamic nonlinear problems.
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