Introduction

The LM principle deserves special consideration when discussing tests for misspecification because, unlike the asymptotically equivalent W and LR methods, it does not require the estimation of the more complex alternative in which the original model of interest has been embedded. The purpose of this chapter is to provide a detailed analysis of LM tests in the context of detecting specification errors and deciding how to respond to significant evidence of model inadequacy.

In Section 3.2, it is shown that several alternatives can lead to the same value of the LM statistic for a given null specification. Consequently, only a class of alternative hypotheses need be selected in order to determine the form of the LM statistic. The members of such a class will, however, correspond to quite different types of specification. It is, therefore, natural to ask whether the LM test based upon the selection of the correct class of alternative hypotheses is inferior to the LR and W tests when the latter tests are derived using the correct member of this class. This issue is examined, and some Monte Carlo evidence on the relative performance of the LM test is summarized.