Robustness of marginal likelihood based tests for random regression coefficients

Over the last few decades there has been a growing literature on diagnostic tests of regression disturbances. Tests that have been constructed based on marginal likelihood methods have been found to do well when the disturbances are normally distributed. This paper investigates the small-sample size and power properties of marginal likelihood based tests when testing for random regression coefficients in the presence of first-order autoregressive disturbances. We find test sizes are less robust to non-normality as the sample size increases and the relative power performance of the various tests hardly changes as non-normality is introduced. Consequently marginal likelihood score based tests typically have the best power properties with a particular approximate point optimal invariant test providing some exceptions.