The presence of nuisance parameters causes unexpected complications in econometric inference procedures. A number of modified likelihood and message length functions have been developed for better handling of nuisance parameters but all of them are not equally efficient. In this paper, we empirically compare different modified likelihood and message length functions in the context of estimation and testing of parameters from linear regression disturbances that follow either a first-order moving average or first-order autoregressive error processes. The results show that estimators based on the conditional profile likelihood and tests based on the marginal likelihood are best. If there is a minor identification problem, the sizes of the likelihood ratio and Wald tests based on simple message length functions are best. The true sizes of the Lagrange multiplier tests based on message length functions are rather poor because the score functions of message length functions are biased.