Goodness-of-fit Testing in Linear Regression Models

Abstract: Model checking plays an important role in linear regression as model misspecification seriously affects the validity and efficiency of regression analysis. In practice, model checking is often performed by subjectively evaluating the plot of the model's residuals. This approach is objectified by constructing a random process from the model's residuals, however due to a very complex covariance function obtaining the exact distribution of the test statistic is intractable. Several solutions to overcome this have been proposed, however the simulation and bootstrap based approaches are only asymptotically valid and can, with a limited sample size, yield tests which have inappropriate size. We therefore propose to estimate the null distribution by using permutations. We show, under some mild assumptions, that with homoscedastic random errors this yields consistent tests under the null and the alternative hypotheses. Small sample properties of the proposed tests are studied in an extensive Monte Carlo simulation study, where it is demonstrated that the proposed tests attain correct size, even with strongly non-normal random errors and a very small sample size, while being as powerful as the other available alternatives. The results are also illustrated on some real data examples.