Bootstrap Confidence Intervals Based on Inverting Hypothesis Tests

Most confidence intervals, whether based on asymptotic theory or the bootstrap, are implicitly based on inverting a Wald test. Since Wald test statistics are not invariant under nonlinear reparametrizations of the restrictions they test, confidence intervals based on them are not invariant either. This fact explains the well-known non invariance of bootstrap confidence intervals obtained by Hall’s percentile-t method. Davidson and MacKinnon (1999) show that bootstrap inference can be improved if the bootstrapped test statistic is asymptotically independent of the bootstrap data-generating process. In this note, it is shown for a simple AR(1) model that greatly improved coverage accuracy of confidence intervals can be obtained by explicitly inverting a set of bootstrap hypothesis tests for each of which the bootstrap data-generating process is asymptotically independent of the bootstrapped statistic. This research was supported, in part, by grants from the Social Sciences and Humanities Research Council of Canada. This note is based on a comment on a paper, Recent Developments on Bootstrapping Time Series, by Berkowitz and Kilian. The paper and the comment were published in 2000 in Econometric Reviews.