Estimating the predictability of economic and financial time series

Abstract: The predictability of a time series is determined by the sensitivity to initial conditions of its data generating process. In this paper our goal is to characterize this sensitivity from a finite sample by assuming few hypotheses on the data generating model structure. In order to measure the distance between two trajectories induced by a same noisy chaotic dynamic from two close initial conditions, a symmetric Kullback-Leiber divergence measure is used. Our approach allows to take into account the dependence of the residual variance on initial conditions. We show it is linked to a Fisher information matrix and we investigated its expressions in the cases of covariance-stationary processes and ARCH($\infty$) processes. Moreover, we propose a consistent non-parametric estimator of this sensitivity matrix in the case of conditionally heteroscedastic autoregressive nonlinear processes. Various statistical hypotheses can so be tested as for instance the hypothesis that the data generating process is "almost" independently distributed at a given moment. Applications to simulated data and to the stock market index S&P500 illustrate our findings. More particularly, we highlight a significant relationship between the sensitivity to initial conditions of the daily returns of the S&P 500 and their volatility.