On Nonlinear, Stochastic Dynamics in Economic and Financial Time Series

Abstract

The search for deterministic chaos in economic and financial time series has attracted much interest over the past decade. However, clear evidence of chaotic structures is usually prevented by large random components in the time series. In the first part of this paper we show that even if a sophisticated algorithm estimating and testing the positivity of the largest Lyapunov exponent is applied to time series generated by a stochastic dynamical system or a return series of a stock index, the results are difficult to interpret. We conclude that the notion of sensitive dependence on initial conditions as it has been developed for deterministic dynamics, can hardly be transfered into a stochastic context. Therefore, in the second part of the paper our starting point for measuring dependencies for stochastic dynamics is a distributional characterization of the dynamics, e.g. by heteroskedastic models for economic and financial time series. We adopt a sensitivity measure proposed in the literature which is an information-theoretic measure of the distance between probability density functions. This sensitivity measure is well defined for stochastic dynamics, and it can be calculated analytically for the classes of stochastic dynamics with conditional normal distributions of constant and state-dependent variance. In particular, heteroskedastic return series models such as ARCH and GARCH models are investigated. (author's abstract)