ASPECTS OF MODELLING NONLINEAR TIME SERIES

Abstract

It is common practice for economic theories to postulate non-linear relationships between economic variables, production functions being an example. If a theory suggests a specific functional form, econometricians can propose estimation techniques for the parameters, and asymptotic results, about normality and consistency, under given conditions are known for these estimates, see e.g. Judge et. a1. (1985) and White (1984) and Gallant (1987, chapter 7). However, in many cases the theory does not provide a single specification or specifications are incomplete and may not capture the major features of the actual data, such as trends, seasonality or the dynamics. When this occurs, econometricians can try to propose mort: general specifications and tests of them. There are clearly an immense number of possible parametric nonlinear models and there are also many nonparametric techniques for approximating them. Given the limited amount of data that is usually available in economics it would not be appropriate to consider many alternative models or to use many techniques. Because of the wide possibilities the methods and models available to analyze non-linearities are usually very flexible so that they can provide good approximations to many different generating mechanisms. A consequence is that with fairly small samples the methods arc inclined to over-fit, so that if the true mechanism is linear, say, with residual variance the fitted model may appear to find nonlinearity and the estimated residual variance is less than . The estimated model will then be inclined to forecast badly in the post-sample period. It is therefore necessary to have a specific research strategy for modelling non-linear relationships between time series. In this chapter the modelling process concentrates on a particular situation, where there is a single dependent variable Yt to be explained and:.!..t is a vector of exogenous variables.