Robustness to nonnormality of regression F-tests

Abstract

This study investigates the robustness to nonnormality of the null distribution of the standard F-tests for regression coefficients in linear regression models. Assuming the errors to be nonnormal with finite moments, the null distribution of the F-statistic is derived. This differs from its normal theory F-distribution. Besides the sample size and the degrees of freedom of error sum of squares, the major determinant of the sensitivity to nonnormality is the extent of the ‘nonnormality’ of the regressors or the extent of presence of ‘leveraged’ (influential) observations. The small effect in one direction when all observations are equally influential and the much larger effect in the opposite direction when the observations are extremely heterogeneous in their influences provide extremes of sensitivity within which the sensitivity of the tests will be found. We have identified several specific functions of regressors that can be used to judge the extent of nonnormality of the regressors or the extent of presence of leveraged observations.