Sieve bootstrap for time series

We study a bootstrap method which is based on the method of sieves. A linear process is approximated by a sequence of autoregressive processes of order p = p(n), where p(n) --> infinity, p(n)= o(n) as the sample size n --> infinity. For given data, we then estimate such an AR(p(n)) model and generate a bootstrap sample by resampling from the residuals. This sieve bootstrap enjoys a nice nonparametric property, being model-free within a class of linear processes. We show its consistency for a class of nonlinear estimators and compare the procedure with the blockwise bootstrap, which has been proposed by Kunsch in 1989. In particular, the sieve bootstrap variance of the mean is shown to have a better rate of convergence if the dependence between separated values of the underlying process decreases sufficiently fast with growing separation. Finally, a simulation study helps to illustrate the advantages and disadvantages of the sieve compared to the blockwise bootstrap.