Outside of the normal distribution, conditional expectations are typically nonlinear, and in general one would imagine that these infeasible optimal forecasts would be nonlinear functions of past data. The main difficulty that arises with nonlinear forecasts is choosing a feasible forecasting method that performs well with the fairly short historical time series available for macroeconomic forecasting. With many parameters, approximation error in (27.2) is reduced, but estimation error can be increased. Many nonlinear forecasting methods also pose technical problems, such as having objective functions with many local minima, having parameters that are not globally identified, and difficulties with generating internally consistent й-step ahead forecasts from one-step ahead models.
Recognition of these issues has led to the development of a vast array of methods for nonlinear forecasting, and a comprehensive survey of these methods is beyond the limited scope of this chapter. Rather, here I provide a brief introduction to only two particular nonlinear models, smooth transition autoregressions (STAR) and artificial neural networks (NN). These models are interesting methodologically because they represent, respectively, parametric and nonparametric approaches to nonlinear forecasting, and they are interesting from a practical point of view because they have been fairly widely applied to economic data.
A third class of models that has received considerable attention in economics are the Markov switching models, in which an unobserved discrete state switches stochastically between regimes in which the process evolves in an otherwise linear fashion. Markov switching models were introduced in econometrics by Hamilton (1989) and are also known as hidden Markov models. However, space limitations preclude presenting these models here; for a textbook treatment, see Hamilton (1994). Kim and Nelson (1998, 1999) provide important extensions of this framework to multivariate models with unobserved components. The reader interested in further discussions of and additional references to other nonlinear time series forecasting methods should see the recent surveys and/or textbook treatments of nonlinear models by Granger and Terasvirta (1993), Priestly (1989), and Samorodnitsky and Taqqu (1994).
An artificial neural network (NN) model relates inputs (lagged values) to outputs (future values) using an index model formulation with nonlinear transformations. There is considerable terminology and interpretation of these formulations which we will not go into here but which are addressed in a number of textbook treatments of these models; see in particular Swanson and White (1995, 1997) for discussions and applications of NN models to economic data. Here, we consider the simplest version, a feedforward NN with a single hidden layer and n hidden units. This has the form:
yt+h = P o(L)yt + X Y;g(P; (%t) + (27.6)
where p;(L), і = 0,…, n are lag polynomials, у are unknown coefficients, and g(z) is a function that maps ^ ^ [0, 1]. Possible choices of g(z) include the indicator function, sigmoids, and the logistic function. A variety of methods are available for the estimation of the unknown parameters of NNs, some specially designed for this problem; a natural estimation method is nonlinear least squares. NNs have a nonparametric interpretation when the number of hidden units (n) is increased as the sample size tends to infinity.
Smooth transition autoregressions are piecewise linear models and have the form:
yt+h = a(L)yt + dte(L)yt + uM, (27.7)
where the mean is suppressed, a(L) and P(L) are lag polynomials, and dt is a nonlinear function of past data that switches between the "regimes" a(L) and P(L). Various functions are available for dt. For example, if dt is the logistic function so dt = 1/(1 + exp[y0 + y1Z t]), then the model is referred to as the logistic smooth transition autoregression (LSTAR) model. The switching variable dt determines the "threshold" at which the series switches, and depends on the data through Zt. For example, Zt might equal y-k, where k is some lag for the switch. The parameters of the model can be estimated by nonlinear least squares. Details about formulation, estimation and forecasting for TAR and STAR models can be found in Granger and Terasvirta (1993) and in Granger, Terasvirta, and Anderson (1993). For an application of TAR (and other models) to forecasting U. S. unemployment, see Montgomery, Zarnowitz, Tsay, and Tiao (1998).