Pitfalls of testing non-nested hypotheses by the lagrange multiplier method

Abstract

In applied research in econometrics a general model determined from the current knowledge of economic theory often establishes a ‘natural’ method of embedding a number of otherwise non-nested hypotheses. Under these circumstances, significant tests of various hypotheses can be carried out within the classical framework, and tests of non-nested or separate families of hypotheses do not require development of new statistical methods. The application of some suitable variant of likelihood ratio testing procedure will be quite appropriate. There are, however, many ocassions in applied econometrics where the hypotheses under consideration are intended to provide genuine rival explanations of the same given phenomenon and the state of economic theory is not such as to furnish us with a general model that contains both of the rival hypotheses in a ‘natural’ and theoretically consistent manner. A number of investigators have advocated that even when a ‘natural’ comprehensive model containing both of the hypotheses under consideration cannot be obtained from theoretical considerations, it is still appropriate to base significant tests of non-nested hypotheses upon a combined model ‘artificially’ constructed from the rival alternatives. Moreover, in a recent paper on the application of Lagrange Multiplier (LM) tests to model specification, T.S. Breusch and A.R. Pagan (1980) have claimed that Cox's test statistic is connected to an LM or ‘score’ statistic derived from the application of the LM method to an exponentially combined model earlier employed by A.C. Atkinson (1970). Although the use of ‘artificially’ constructed comprehensive models fortesting separate families of hypotheses is analytically tempting, nevertheless it is subject to two major difficulties. Firstly, in many cases of interest in econometrics, the structural parameters under the combined hypothesis are not identified. Secondly, the log likelihood function of the artificially constructed model has singularities under both the null and alternative hypotheses. The paper firstly examines the derivation of LM statistics in the case of non-nested hypotheses and shows that Atkinson's general test statistic, or Breusch and Pagan's result, can be regarded as an LM test if the parameters of the alternative hypothesis are known. The paper also shows that unless all the parameters of the combined models are identified, no meaningful test of the separate families of the hypotheses by the artificial embedding procedure is possible, and in the identified case an expression for the LM statistic which avoids the problem of the singularity of the information matrix under the null and the alternative hypotheses is obtained. The paper concludes that none of the artificially embedding procedures are satisfactory for testing non-nested models and should be abandoned. It, however, emphasizes that despite these difficulties associated with the use of artificial embedding procedures, Cox's original statistic (which is not derived as an LM statistic and does not depend on any arbitrary synthetic combination of hypotheses) can still be employed as a useful procedure for testing the rival hypotheses often encountered in applied econometrics.