Improvements in bootstrap inference

The fast double bootstrap can improve considerably on the single bootstrap when the bootstrapped statistic is approximately independent of the bootstrap DGP. This is because, among the approximations that underlie the fast double bootstrap (FDB), is the assumption of such independence. In this paper, use is made of a discrete formulation of bootstrapping in order to develop a conditional version of the FDB, which makes use of the joint distribution of a statistic and its bootstrap counterpart, rather than the joint distribution of the statistic and the full distribution of its bootstrap counterpart, which is available only by means of a simulation as costly as the full double bootstrap. Simulation evidence shows that the conditional FDB can greatly improve on the performance of the FDB when the statistic and the bootstrap DGP are far from independent, while giving similar results in cases of near independence.