Testing strategies for model specification

Abstract

A crucial element in the development of econometric methodology during the past decade has been the concern with testing as opposed to estimating econometric models. In this paper we discuss—especially for the econometric analysis of time series—the main types of test procedures, and we also investigate the opportunities to uphold the Neyman-Pearson theory in the context of thorough model specification testing. In applied work it is quite usual to carry out several tests on the same set of sample data. We consider an extension of the Neyman-Pearson framework to the case of such repeated testing, and examine situations where the various hypotheses under test have a particular nesting structure. For the case where a sequence of superposed alternatives is tested by so-called marginal tests, we prove that the various test statistics are asymptotically independent under a common null hypothesis if the statistics are based on either the likelihood-ratio, or the Wald, or the Lagrange-multiplier approach. Testing a particular null hypothesis against a series of juxtaposed alternatives appears to lead to independent test statistics only in specific circumstances. It is shown how independence of test statistics enables the control over the overall Type I error probability, which is an essential element in the Neyman-Pearson theory. Using the notions of constructive hypotheses and auxiliary hypotheses, we can draw a clear distinction between specification tests and misspecification tests. Next an overview is given of approaches to and examples of specification and misspecification testing. With respect to the former, attention is paid to the problem determining the order of dynamics and discriminating between system dynamics and error dynamics. The misspecification testing is reviewed for specification error, nonconstancy of coefficients, heteroscedasticity, serial dependence, and nonnormality of disturbances. Also the problem of testing for several misspecifications jointly or sequentially is considered. Finally we discuss the options and associated difficulties in implementing the various tests in an overall testing strategy.