Higher-order Improvements of the Parametric Bootstrap for Markov Processes

This paper provides bounds on the errors in coverage probabilities of maximum likelihood-based, percentile-t, parametric bootstrap confidence intervals for Markov time series processes. These bounds show that the parametric bootstrap for Markov time series provides higher-order improvements (over confidence intervals based on first order asymptotics) that are comparable to those obtained by the parametric and nonparametric bootstrap for iid data and are better than those obtained by the block bootstrap for time series. Additional results are given for Wald-based confidence regions. The paper also shows that k-step parametric bootstrap confidence intervals achieve the same higher-order improvements as the standard parametric bootstrap for Markov processes. The k-step bootstrap confidence intervals are computationally attractive. They circumvent the need to compute a nonlinear optimization for each simulated bootstrap sample. The latter is necessary to implement the standard parametric bootstrap when the maximum likelihood estimator solves a nonlinear optimization problem.