The Power of Bootstrap Tests

Bootstrap tests are tests for which the significance level is calculated using some variant of the bootstrap, which may be parametric or nonparametric. We show that the power of a bootstrap test will generally be very close to the power of the asymptotic test on which it is based, provided that both tests are properly adjusted to have the correct size. We also discuss the loss of power that can occur when the number of bootstrap samples is relatively small. Some Monte Carlo results for two forms of omitted variable test in logit models are presented. These illustrate the theoretical results of the paper and demonstrate that the size-adjusted power of asymptotic tests can vary greatly depending on the method used for size adjustment. This research was supported, in part, by grants from the Social Sciences and Humanities Research Council of Canada. We are grateful to Joel Horowitz and numerous seminar participants for comments on earlier work. Much of the paper was written while the second author was visiting GREQAM.