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Conflict among Criteria for Testing Hypotheses in the Multivariate Linear Regression Model

Ernst R. Berndt
N. Eugene Savin

Abstract

This paper surveys alternative testing criteria in the linear multivariate regression model, and investigates the possibility of conflict among them. We consider the asymptotic Wald, likelihood ratio (LR), and Lagrange multiplier (LM) tests. These three test statistics have identical limiting chi-square distributions; thus their critical regions coincide. A strong result we obtain is that a systematic numerical inequality relationship exists; specifically, Wald @> LR @> LM. Since the equality relationship holds only if the null hypothesis is exactly true in the sample, in practice there will always exist a significance level for which the asymptotic Wald, LR, and LM tests will yield conflicting inference. However, when the null hypothesis is true, the dispersion among the test statistics will tend to decrease as the sample size increases. We illustrate relationships among the alternative testing criteria with an empirical example based on the three reduced form equations of Klein's Model I of the United States economy, 1921-1941.