Improving the Reliability of Bootstrap Confidence Intervals

This paper investigates the relation between hypothesis testing and the construction of confidence intervals, with particular regard to bootstrap tests. In practice, confidence intervals are almost always based on Wald tests, and consequently are not invariant under nonlinear reparametrisations. Bootstrap percentile-t confidence intervals are an instance of this. However, the (asymptotically) pivotal functions of data and parameters on which likelihood ratio (LR) and Lagrange multiplier (LM) tests depend can be used to construct parametrisation-invariant confidence intervals. We show that, whenever an artificial regression can be used to find the restricted estimates needed for LR and LM tests, the nonlinear equations that define the limits of a confidence interval can be solved by an algorithm based on the same artificial regression. The algorithm involves roughly as much computation for each interval limit as is needed to find the restricted estimates. Bootstrap tests are often more reliable when the bootstrap DGP is based on restricted estimates. Inverting such tests to find a confidence interval is computationally intensive, since many bootstrap samples must be generated for every set of restricted estimates considered. We show how to combine artificial regression based bootstrap testing with the algorithm for finding limits of confidence intervals. This research was supported, in part, by grants from the Social Sciences and Humanities Research Council of Canada.