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

A WELL DEVELOPED EXACT STATISTICAL THEORY exists for hypothesis testing in the normal linear regression model when the errors are independent and homoscedastic. In the more general case where the error covariance matrix is nonscalar and depends on a set of unknown parameters, exact analysis is difficult and reliance is usually placed on asymptotic approximations for large sample size n. In this paper, higher-order asymptotic expansions are developed for comparing the size and power of some common procedures for testing linear hypotheses on the regression coefficients in a class of generalized normal linear models. The class investigated is essentially the same as in Breusch 4 and Magnus 6 and includes many of the examples of heteroscedasticity and autocorrelation discussed in the literature. We assume simply that the regressors are nonrandom and that the error covariance matrix is a smooth function of a few parameters that can be efficiently estimated by maximum likelihood. Tests based on the Wald, likelihood ratio, and Lagrange multiplier principles are considered. These principles lead to three tests which, though distinct in finite samples, are locally asymptotically equivalent and share certain asymptotic optimality properties. Of course, there are infinitely many other tests that are asymptotically equivalent to the ones examined here. Although the techniques of this paper can be applied to any of them, our results concern only the tests arising from the three traditional principles. We show that, to a second order of approximation under local alternatives, the likelihood ratio test statistic is a simple average of the Wald statistic and the Lagrange multiplier statistic. When the null hypothesis is one dimensional, the three tests are, to second order, equally powerful; that is, after the critical regions are adjusted so that the tests have (to order n - i) the same size, the local power functions differ by terms of smaller order than n -. When the null hypothesis contains more than one