Separate Misspecified Regressions and the U.S. Long-Run Demand for Money Function

Abstract

T HE purpose of this paper is to extend the traditional analysis of specification error to models that are separate (i.e., non-nested), in the sense that one may not be obtained from the other by the imposition of restrictions or as a limiting form of a suitable approximation. The general theoretical approach to testing separate families of hypotheses was pioneered by Cox (1961, 1962), and subsequent work has largely been based on the many suggestions in these two papers. In an important extension, Atkinson (1970) derived a test statistic that is asymptotically equivalent to the Cox test under the null hypothesis. However, Pereira (1977) showed that while the Cox procedure always provides a consistent test of the null, Atkinson's variation does not. This is an important result since the power of an inconsistent test does not tend to unity in samples, where power is taken to be the probability of rejecting a false null model. The general results obtained by Cox for separate distributions were specialized to regression analysis by Pesaran (1974) and Pesaran and Deaton (1978). Dastoor (1981a) and Fisher and McAleer (1981) modified the Cox test using Atkinson's procedure and demonstrated Pereira's caveat regarding test inconsistency to be unwarranted for separate linear and non-linear regressions, respectively, with normal disturbances. Several algorithms for calculating approximations to the Cox statistic via artificial regression models have recently been developed by Davidson and MacKinnon (1981), and Fisher and McAleer (1981). These test statistics, which are asymptotically equivalent under the null as well as under local alternatives (see Pesaran, 1982b), will also be analyzed with regard to the effects of misspecification. In spite of the potential usefulness of the Cox test in applied econometric research, it is surprising that there have been so few published empirical papers employing it (Quandt, 1974; Deaton, 1978). It is also notable that theoretical work involving considerations of power has concentrated on local, I rather than fixed, alternatives. Since serious misspecification arises only when the difference between the null and the alternative model does not disappear in samples, it would seem more helpful from the practical viewpoint to analyze specification error when the null is tested against a fixed alternative, and this is one of the primary objectives of the present paper. The other objective is to provide an empirical example illustrating the consequences of misspecifying separate models. The plan of the paper is as follows. Alternative procedures for testing separate regression models are outlined in section II, and the consistency of the tests is investigated in the light of specification error in section III. It is demonstrated that (i) both the Cox and Atkinson procedures provide consistent tests of the null when variables are incorrectly included in the alternative; and (ii) it is possible that neither procedure will provide a consistent test when variables are incorrectly excluded from the alternative. It is argued that specification error in the null does not affect the consistency of the tests. Some consequences of the results are discussed in section IV. An illustrative empirical example related to alternative specifications of the U.S. long-run demand for money function is presented in section V. Some concluding remarks are given in section VI. Received for publication May 18, 1981. Revision accepted for publication February 25, 1982. * Australian National University, Queen's University, and Australian National University, respectively. The authors wish to thank Noxy Dastoor, Adrian Pagan, Deane Terrell, and especially Anil Bera, for helpful comments, Les Godfrey for permission to refer to his unpublished paper, and Eva Klug for efficient research assistance. Responsibility for the final product rests with the authors alone. The present paper is a substantially revised and extended version of McAleer and Fisher (1981). I When the concept of local alternatives is used, the sample size is taken to be sufficiently large so that the asymptotic results are valid, but not so that the sequence of local alternatives has converged to the null (see Cox, 1962, p. 407). Pesaran (1982a) examines the local power of alternative tests of non-nested models, but makes the somewhat restrictive assumption that the dimension of the null model is not exceeded by that of the alternative.