Testing Non-Nested Nonlinear Regression Models

In Pesaran [9], the test developed by Cox for comparing separate families of hypotheses was applied to the choice between two non-nested linear single-equation econometric models. In this paper, the analysis is extended to cover multivariate nonlinear models whenever full information maximum likelihood estimation is possible. This allows formal comparisons not only of competing explanatory variables but also of alternative functional forms. The largest part of the paper derives the results and shows that they are recognizable as generalizations of the single-equation case. It is also shown that the calculation of the test statistic involves very little computation beyond that necessary to estimate the models in the first place. The paper concludes with a practical application of the test to the analysis of the U.S. consumption function and it is demonstrated that formal tests can give quite different results to conventional informal selection procedures. Indeed, in the case examined, five alternative hypotheses, some of which appear to perform quite satisfactorily, can all be rejected using the test. THE NEED FOR STATISTICAL PROCEDURES for testing separate families of hypotheses has become more acute with the increased use of econometric techniques in practice. The usual F tests can only be applied to test nested hypotheses, i.e. those which are members of the same family. However, in practice, one is frequently faced with the problem of testing non-nested hypotheses. In an earlier article, Pesaran [9] applied the test developed by Cox [3, 4], for separate families of hypotheses to single-equation linear regression models both with autocorrelated and nonautocorrelated disturbances. In that paper, the question was confined to the selection of appropriate explanators for a given dependent variable. However, in much applied work, the investigator is required not merely to select variables but simultaneously to find an appropriate functional form. This problem can be especially acute since in many areas of research, economic theory can guide us in the choice of variables, but helps very little in the choice of functional form. As computing capacity has increased, and nonlinear estimation has become routine, the use of linearity has become more a matter of choice than of necessity; the criteria for such a choice are thus of considerable practical importance. In this paper, we extend the earlier analysis to cover these problems by deriving the comparable statistics without assuming linearity of the models. This allows formal comparisons of different explanatory variables, of different functional forms, and of the interactions between the two. We also extend the results to cover competing systems of nonlinear equations whenever full-information maximum-likelihood estimation is possible. This allows the test to be applied to non-nested simultaneous equation models as well