Testing for normality in linear regression models using regression and scale equivariant estimators

•Many robust-regression estimators are regression and scale equivariant.•Normality tests in linear regressions using these estimators control size exactly.•In a simulation study, we show that tests based on least squares can be biased.

In this paper we provide a general solution to the problem of controlling the probability of a type I error in normality tests for the disturbances in linear regressions when using robust-regression residuals. We show that many classes of well-known robust regression estimators belong to the class of regression and scale equivariant estimators. It is these equivariance properties that are used to reduce the nuisance parameter space under the null, from which we develop Monte Carlo and Maximized Monte Carlo tests for the null of disturbance normality. Finally, we illustrate in a simulation experiment the potential power gains from using robust-regression residuals in testing this null hypothesis.