Introduction

Three general principles employed for hypothesis testing in econometrics are the likelihood-ration (LR), Wald (W), and Lagrange-multiplier (LM) criteria. The W test was introduced by Wald (1943) and the LM test by Aitchison and Silvey (1958) and Silvey (1959). The LM test, which is also the same as the score test of Rao (1947), has been the subject of recent reports by Breusch (1979), Godfrey (1978a, 1978b, 1978c), and Breusch and Pagan (1979, 1980).

Savin (1976) and Berndt and Savin (1977) showed that a systematic numerical inequality exists for the test statistics for testing linear restrictions on the coefficients of certain linear models. The inequality relationship for the values of the test statistics is W≥LR≥LM. Breusch (1979) established that the inequality relationship holds for a general linear model with normal disturbances, provided that the unknown elements of the covariance matrix can be estimated by maximum likelihood (ML) and that the ML estimates of the coefficient parameters are asymptotically uncorrelated with those of the covariance matrix parameters.

The exact sampling distributions of the three test statistics can be complicated, so that in practice the critical regions of the tests commonly are based on asymptotic approximations. In regular problems the test statistics have the same asymptotic chi-square distribution under the null hypothesis. When the tests use the asymptotic chi-square value, they have the same critical region, and they are asymptotically equivalent. The inequality relationship among the test statistics implies that there are samples for which the large-sample tests will produce conflicting inferences.