This paper is concerned with tests of the covariance matrix of the disturbances in the linear regression model that involve nuisance parameters which cannot be eliminated by usual invariance arguments. Score-based tests, namely Lagrange multiplier (LM) and locally most mean powerful (LMMP) tests are derived from the marginal likelihood. Applications considered include (i) testing for random regression coefficients; (ii) testing for secondorder autoregressive (AR(2)) disturbances in the presence of AR(1) disturbances; and (iii) testing for ARMA(1,1) disturbances; each in the presence of AR(1) disturbances. An empirical size and power comparison shows that typically the new tests have more accurate asymPtotic critical values and slightly more power than their respective conventional counterparts.