ABSTRACT
We discuss new approaches to nonlinear forecasting and their possible application to problems in economics. We first embed the time series in a state space, and then employ straightforward numerical techniques to build nonlinear dynamical models. We pick an ad hoc nonlinear representation and fit it to the data. For higher dimensional problems we find that breaking the domain into neighborhoods and using local approximation is usually better than using an arbitrary global representation. For some examples, such as data from a chaotic fluid convection experiment, our methods give results roughly fifty times better than those of standard linear models. We present scaling laws for our error estimates based on properties of the dynamics, such as the dimension and Lyapunov exponents, the number of characteristic time scales, the extrapolation time, and the order of approximation of the model. Our methods also provide strong self-consistency requirements on the identification of chaotic dynamics and a more accurate means of computing statistical quantities such as dimension. They also naturally lead to a new method for noise reduction. We compare our methods to neural networks and argue that the basic principles are similar, except that our methods are orders of magnitude faster to implement on conventional computers. At this point we have only applied these new methods 100to a few economic time series, but so far do not see improvements over conventional linear forecasting procedures.
