Computational algorithms for double bootstrap confidence intervals

In some cases, such as in the estimation of impulse responses, it has been found that for plausible sample sizes the coverage accuracy of single bootstrap confidence intervals can be poor. The error in the coverage probability of single bootstrap confidence intervals may be reduced by the use of double bootstrap confidence intervals. The computer resources required for double bootstrap confidence intervals are often prohibitive, especially in the context of Monte Carlo studies. Double bootstrap confidence intervals can be estimated using computational algorithms incorporating simple deterministic stopping rules that avoid unnecessary computations. These algorithms may make the use and Monte Carlo evaluation of double bootstrap confidence intervals feasible in cases where otherwise they would not be feasible. The efficiency gains due to the use of these algorithms are examined by means of a Monte Carlo study for examples of confidence intervals for a mean and for the cumulative impulse response in a second order autoregressive model.