Bootstrapping Autoregressive Processes with Possible Unit Roots

An important question in applied work is how to bootstrap autoregressive processes involving highly persistent time series of unknown order of integration. In this paper, we show that in many cases of interest in applied work the standard bootstrap algorithm for unrestricted autoregressions remains valid for processes with exact unit roots; no pre-tests are required, at least asymptotically, and applied researchers may proceed as in the stationary case. Specifically, we prove the first-order asymptotic validity of bootstrapping any linear combination of the slope parameters in autoregressive models with drift. We also establish the bootstrap validity for the marginal distribution of slope parameters and for most linear combinations of slope parameters in higher-order autoregressions without drift. The latter result is in sharp contrast to the well-known bootstrap invalidity result for the random walk without drift. A simulation study examines the finite-sample accuracy of the bootstrap approximation both for integrated and for near-integrated processes. We find that in many, but not all circumstances, the bootstrap distribution closely approximates the exact finite- sample distribution.