  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.
