[The problem to be considered in this paper is that in a linear regression model, y = Xβ + ε (where X is n × k of rank r ≤ k), the disturbance vector ε′ = ( extlesstex-math extgreater$ extbackslashvarepsilon _\1\, extbackslashvarepsilon _\2\,..., extbackslashvarepsilon _ $ extless/tex-math extgreater) is distributed according to the null hypothesis, H0, as multivariate normal with mean vector 0 and variance-covariance matrix proportional to extlesstex-math extgreater$ extbackslashSigma _\0$ extless/tex-math extgreater, against the alternative hypothesis, H_1, that it is distributed as multivariate normal with mean vector 0 and variance-covariance matrix proportional to Σ _1. Three test statistics, extlesstex-math extgreaters_\1\,s_\2\ extless/tex-math extgreater, and s_3, all functions of estimated disturbances from the fitted regression are proposed to test the hypothesis H_0. It is shown (in Section 3) that all three tests based on extlesstex-math extgreaters_\1\,s_\2\ extless/tex-math extgreater, and s_3 are unbiased and that the test T(1) based on s_1 is most powerful. In Section 4, ε is assumed to have a special covariance structure, namely, a first order stationary Markov process, uniform covariance structure, and moving average of order one, and the general results obtained in Section 3 are simplified. It is also shown that the hypothesis H_0, in general, cannot be tested and that only an implication of it can be tested. Section 5 contains three numerical illustrations comparing the results proposed in this study with the Durbin-Watson procedure.]