Financial Risk Forecasting with Non-Stationarity

One important feature in all financial markets is that they are nonlinear dynamical systems (Brock 1986; Scheinkman and LeBaron 1989; Hsieh 1989; Brock et al. 1996). Under this framework, it is assumed that asset price changes may not be solely due to new information, as simply described by the random walk model, but are also governed by some underlying dynamics. Such nonlinearities can only be described in higher dimensions, for which the observed financial time series is considered to be a projection onto one-dimensional space that renders a random-looking structure. Clearly, we have no clue as to what relevant components or even dimensions are for the hidden dynamics. A crucial theorem due to Takens (1981) shows that the dynamical behavior of “histories” in higher dimensions constructed from the observed time series typically mimics the behavior of the underlying dynamics. The idea is first to construct the trajectory of histories and then to apply straightforward regression techniques to build ad-hoc models for the underlying dynamics. In this respect, the method of local approximation (Farmer and Sidorowich 1987; Casdagli 1989, 1992) is an effective approach using only nearby points or neighbors in the fitting. Such a model presumably captures the local structure of the underlying dynamics capable of making short-term predictions.