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Hypothesis Testing in Linear Models when the Error Covariance Matrix is Nonscalar

Thomas J. Rothenberg

Abstract

Stochastic expansions are developed for the Lagrange multiplier, likelihood ratio, and Wald statistics for testing regression coefficients in the normal linear model with unknown error covariance matrix. Under suitable regularity conditions, the likelihood ratio statistic is found to be approximately the average of the other two. Critical values are calculated so that the three tests have approximately the same size. The second-order approximate local power functions indicate that, when the null hypothesis is one dimensional, all three tests are equally powerful. When the hypothesis is multidimensional, the power functions differ; no one of the tests is uniformly more powerful than the others.