Financial Forecasting, Sensitive Dependence

Abstract

Empirical studies show that there are at least some components in future asset returns that are predictable using information that is currently available. When the linear time series models are employed in prediction, the accuracy of the forecast does not depend on the current return or the initial condition. In contrast, with nonlinear time series models, properties of the forecast error depend on the initial value or the history. The effect of the difference in initial values in a stable nonlinear model, however, usually dies out quickly as the forecast horizon increases. For both deterministic and stochastic cases, the dynamic system is chaos if a small difference in the initial value is amplified at an exponential rate. In a chaotic nonlinear model, the reliability of the forecast can decrease dramatically even for a moderate forecast horizon. Thus, the knowledge of the sensitive dependence on initial conditions in a particular financial time series offers practically useful information on its forecastability. The most frequently used measure of initial value sensitivity is the largest Lyapunov exponent, defined as the long‐run average growth rate of the difference between two nearby trajectories. It is a global initial value sensitivity measure in the sense that it contains the information on the global dynamic property of the whole system. The dynamic properties around a single point in the system can be also described using other local measures. Both global and local measures of the sensitive dependence on initial conditions can be estimated nonparametrically from data without specifying the functional form of the nonlinear autoregressive model.