A modified bootstrap for autoregression without stationarity

In this paper, we consider a bootstrap approximation to the distribution of the least squares estimator <(beta)over cap> of the autoregressive parameter beta in a first-order autoregressive process which may or may not have a stationary solution. Our bootstrap procedure is a modification of the standard bootstrap and employs a data based shrinkage towards the critical values beta = +/-1. The error in estimation of the sampling distribution of <(beta)over cap> by the above procedure converges to zero as the sample size grows, irrespective of the value of the autoregressive parameter. This result is in sharp contrast with the behavior of the standard bootstrap for which a random limiting distribution emerges at the critical values beta = +/-1, resulting in a failure of the bootstrap procedure. The asymptotic validity of our procedure enables us, inter alia, to construct a confidence interval for beta which will have the correct coverage probability (asymptotically) under any real beta. The theoretical validity result for the modified bootstrap is supported by appropriate simulations. Finally, we also indicate an extension of the proposed modified bootstrap method to the pth order autorepressive processes.