This paper is concerned with the problem of testing a subset of the parameters which characterize the error variance-covariance matrix in the general linear regression model. Formulae for likelihood ratio, Wald, Lagrange multiplier and asymptotically locally most mean powerful test statistics based on the likelihood of a maximal invariant statistic or an equivalent marginal likelihood are given. Specific applications discussed are the problems of testing against AR(4) disturbances in the presence of AR(1) disturbances and testing for a Hildreth-Houck (1968) random coefficient against the alternative of a Rosenberg (1973) random coefficient. Monte Carlo size and power calculations for these two testing problems are reported. These results provide further evidence that supports the proposed approach to test construction. It also suggests that better handling of nuisance parameters is likely to improve the small-sample properties of asymptotically based inference procedures.