The Lagrange multiplier test and its applications to model specification in econometrics

Abstract

This paper has two aims. The first is to exposit the various forms of the LM statistic and to collect together some of the relevant research reported in the mathematical statistics literature. The second is to illustrate the construction of LM tests by considering a number of particular econometric specifications as examples. It will be found that in many instances the LM statistic can be computed by a regression using the residuals of the fitted model which, because of its simplicity, is itself estimated by OLS. The paper contains five sections. In Section 2, the LM statistic is outlined and some alternative versions of it are discussed. Section 3 gives the derivation of the statistic for several econometric specifications. Applications in this section are the testing for a liquidity trap, autocorrelation, the error components model, diagonality of a covariance matrix in seemingly unrelated equation systems and choice between models generated by separate families of distributions. Section 4 considers the construction of a pseudo-LM statistic when estimation is difficult even under the null hypothesis and discusses the derivation of the exact distribution of the statistic. A concluding summary is given as Section