Simultaneity and non‐linear variability in financial markets: simulation and forecasting

Non‐linear variability in financial markets can emerge from several mechanisms, including simultaneity and time‐varying coefficients. In simultaneous equation systems, the reduced‐form coefficients that determine the behaviour of jointly dependent variables are products and ratios of the original structural coefficients. If the coefficients are stochastic, the resulting multiplicative interactions will result in high degrees of non‐linearity. Processes generated in this way will scale as fractals: they will exhibit intermittent outliers and scaling symmetries, i.e. proportionality relationships between fluctuations at different separation distances. A model is specified in which both the exchange rate itself and the exchange rate residual exhibit simultaneity. The exchange rate depends on other exchange rates, while the residual depends on the other residuals. The model is then simulated using embedding noise from a t‐distribution. The simulations replicate the observed properties of exchange rates, heavy‐tailed distributions and long memory in the variance. A forecasting algorithm is specified in two stages. The first stage is a model for the actual process. In the second stage the residuals are modelled as a function of the predicted rate of change. The first and second stage models are then combined. This algorithm exploits the scaling symmetry: the residual is proportional to the predicted rate of change at separation distances corresponding to the forecast horizon. The procedure is tested empirically on three exchange rates. At a daily frequency and a 1‐day forecast horizon, two‐stage models reduce the forecast error by one fourth. At a 5‐day horizon, the improvement is 10–15 percent. At a weekly frequency, the improvement at the 1‐week horizon is on the order of 30–40 percent. Copyright © 2006 John Wiley & Sons, Ltd.