Nonlinear Econometric Modelling: A Selective Review

In the discipline of economics, the vast array of economic theories which are available to economists often contain nonlinear elements (e.g. first order conditions, functional forms, etc.). In the past, however, many of the testable implications of economic theories, for example were formulated and derived with one of the main objectives being the specification of linear econometric models. This is perhaps not surprising for at least two reasons. First, up until quite recently, the statistical tools available for the examination of economic variables were better able to handle estimation and inference within a linear context. Second, and perhaps just as importantly, the computational ability of early computers (and even earlier, slide rules) was limited enough to make it infeasible to estimate either large or complex econometric models. However, as computers have become more efficient, so have the tools, algorithms, tests, and modelling strategies used by increasing numbers of practicing economists also become more complex. Indeed, one feature of the econometrics profession in recent years seems to be that we are always able to develop estimation and inferential procedures (and associated nonlinear econometric models) which are complex enough to fully tax the capability of even the most powerful computers. Of course, this is not a feature of economics alone, as physics, biology, chemistry, and other "harder" sciences have clearly done likewise. Furthermore, it should be noted that we are not merely developing new and more complex theories in order to take advantage of computational ability. Rather, we are taking advantage of the opportunity of "better" empirically modelling a system which we have always known to be complex and nonlinear. In this sense, the growth in nonlinear modelling in applied economics, for example, seems quite natural. In this chapter, we discuss two varieties of nonlinear models which may be of some interest to economists and econometricians. First we consider what are referred to as "stochastic unit root" (STUR) models. In these models, roots of AR processes are assumed to vary (according to some well defined stochastic process) with average or mean values of unity, rather than being fixed at unity. These models are not "near unit root" models, however, as the unit root in a simple AR(1) process, for example, is not assumed to approach unity as the sample size gets large. Rather, STUR models focus on a reasonable claim that very few economic series are precisely characterized as containing "pure" or constant unit roots, and attempt to examine whether some economic series can be better forecast using more general nonlinear models. Second, we discuss the class of nonlinear models called artificial neural networks (ANNs). These models are nonparametric in the sense that the number of parameters one fits to the data increases with the sample size. Estimation and modelling using ANNs is discussed, some findings are presented. For example, evidence concerning the relative forecasting ability of ANN, STUR, and a variety of simple linear models is presented. Finally, we discuss various empirical nonlinear modelling issues, with particular attention paid to modelling outliers, and the persistence of shocks.