A permutation approach to goodness-of-fit testing in regression models

Abstract

Model checking plays an important role in parametric regression as model misspecification seriously affects the validity and efficiency of regression analysis. Model checks can be performed by constructing an empirical process from the model's fitted values and residuals. Due to a complex covariance function of the process obtaining the exact distribution of the test statistic is, however, intractable. Several solutions to overcome this have been proposed. It was shown that the simulation and bootstrap-based approaches are asymptotically valid, however, we show by using simulations that the rate of convergence can be slow. We, therefore, propose to estimate the null distribution by using a novel permutation-based procedure. We prove, under some mild assumptions, that this yields consistent tests under the null and some alternative hypotheses. Small sample properties of the proposed approach are studied in an extensive Monte Carlo simulation study and real data illustration is also provided.