Permutation tests in linear regression

Abstract

Exact permutation tests are available only in rather simple linear models. The problem is that, although standard assumptions allow permuting the errors of the model, we cannot permute them in practice, because they are unobservable. Nevertheless, the residuals of the model can be permuted. A proof is given here which shows that it is possible to approximate the unobservable permutation distribution where the true errors are permuted by permuting the residuals. It is shown that approximation holds asymptotically and almost surely for certain quadratic statistics as well as for statistics which are expressible as the maximum of appropriate linear functions. The result is applied to testing the significance of predictors as well as to diagnostic checking of heteroscedasticity, autocorrelation, change-points, and changing regression function. Also a simulation experiment is made in order to evaluate the performance of the proposed tests.