Bootstrapping autoregressions with infinite order

We consider a stationary autoregressive process of infinite order and discuss bootstrap procedures for the autocovariance function and the spectral density. After fitting an autoregressive process of increasing order to the observed data and estimating the parameter of the process we obtain estimated residuals. Now we construct an autoregressive bootstrap process with innovations which are distributed according to the empirical distribution of the estimated residuals. Based on the bootstrap sample we define the bootstrap-analogons of the empirical autocovariance function and the lag-window estimator of the spectral density. Some theoretical results concerning consistency of the bootstrap procedure are given. A short simulation study shows the practical relevance of the bootstrap procedure for the spectral density