On Nonlinear, Stochastic Dynamics in Economic and Financial Time Series

The search for deterministic chaos in economic and financial time series has attracted much interest over the past decade. Evidence of chaotic structures is usually blurred, however, by large random components in the time series. In the first part of this paper, a sophisticated algorithm for estimating the largest Lyapunov exponent with confidence intervals is applied to artificially generated and real-world time series. Although the possibility of testing empirically for positivity of the estimated largest Lyapunov exponent is an advantage over other existing methods, the interpretability of the obtained results remains problematic. For instance, it is practically impossible to distinguish chaotic and periodic dynamics in the presence of dynamical noise even for simple dynamical systems. We conclude that the notion of sensitive dependence on initial conditions, as it has been developed for deterministic dynamics, can hardly be transferred into a stochastic context. Therefore, the second part of the paper aims to measure the dependencies of stochastic dynamics on the basis of a distributional characterization of the dynamics. For instance, the dynamics of financial return series are essentially captured by heteroskedastic models. We adopt a sensitivity measure proposed in literature and derive analytical expressions for the most important classes of stochastic dynamics. In practice, the sensitivity measure for the a priori unknown dynamics of a system can be calculated after estimating the conditional density of the system's state variable.