Modeling dynamic nonlinear systems

Uncertainty about the nature and significance of nonlinearities and the manner in which dynamics affect future realizations makes model specification the most difficult aspect of modeling dynamic systems. However, nonlinear dynamic systems with unknown structure can be approximated to any arbitrary level of significance by dynamic polynomial expansions with stationary, but serially correlated errors. This suggests an approach in which the data are permitted a larger role in specification of the model. The degree of the polynomial approximation supported by the data can be determined given an arbitrary specification of model dynamics. The residuals of these dynamic approximations include excluded higher degree terms and higher order dynamics, and as a result, may be nonlinear and will be characterized by complex serial correlation. Since even nonlinear processes have linear state space representations, a state space (dynamic factor or instrumental variable) approach can be used for data-based specification of the important nonlinear terms. A recently introduced multivariate time series modeling algorithm known as system theoretic time series (Aoki 1987), can be used to create a particular state space representation of the residuals of the approximate model with considerable gains to forecast accuracy. Although the mathematical development of the approximate encompassing system theoretic model is tedious and difficult in places, the persistent reader will be rewarded with a model specification procedure which consistently produces highly reliable models of nonlinear dynamic systems.; Model development is framed largely in terms of fish and cattle population dynamics, with resulting forecasts sufficiently reliable for policy application. Both production systems are characterized by structure which is known in part, but not fully observable and subject to substantial random variation. The approach developed here should be widely applicable to other unknown or unobservable dynamic systems such as the forecasting of macro- and microeconomic time series.; Chapter I develops the encompassing model, and a methodology for formally approximating the general model. Chapter II reviews system theoretic time series methods. Omitted higher order dynamics and nonlinearities embodied in the residuals of the dynamic approximation are modeled in Chapter III, using system theoretic time series techniques, to obtain improved forecasts.