Some Non-Nested Hypothesis Tests and the Relations Among Them

In this paper we discuss several statistical techniques which may be used to test the validity of a possibly nonlinear and multivariate regression model, using the information provided by estimating one or more alternative models on the same set of data. The first such techniques in econometrics were proposed by Pesaran (1974) and Pesaran and Deaton (1978) based on the work of Cox (1961, 1962). In Davidson and MacKinnon (1980), we recently proposed, for the univariate case, some new techniques which are conceptually and computationally simpler. The first major result of this paper is that the techniques we have proposed can be regarded as alternative implementations of Cox's basic idea for non-nested hypothesis testing; under the null hypothesis all of the test statistics are asymptotically the same random variable. A second major result is that, for the univariate linear regression case, our tests and Pesaran's test have asymptotic relative efficiency of unity for local alternatives. We then propose several generalizations of our procedures to the case of multivariate regression models, and show that one of these generalizations is asymptotically equivalent under the null hypothesis to the test proposed by Pesaran and Deaton. Finally, we present the results of a sampling experiment for univariate linear models which shows that the small-sample performance of our 3-test and Pesaran's test can be quite different.