Simulation-based finite sample normality tests in linear regressions

In the literature on tests of normality, much concern has been expressed over the problems associated with residual-based procedures. Indeed, the specialized tables of critical points which are needed to perform the tests have been derived for the location-scale model; hence, reliance on available significance points in the context of regression models may cause size distortions. We propose a general solution to the problem of controlling the size of normality tests for the disturbances of standard linear regressions, which is based on using the technique of Monte Carlo tests. We study procedures based on 11 well-known test statistics: the Kolmogorov-Smirnov, Anderson-Darling, Cramer-von Mises, Shapiro-Wilk, Jarque-Bera and D'Agostino criteria. Evidence from a simulation study is reported showing that the usual critical values lead to severe size problems (over-rejections or under-rejections). In contrast, we show that Monte Carlo tests achieve perfect size control for any design matrix and have good power.