Bootstrapping general first order autoregression

In this paper we consider general first order autoregression, including the stationary, the explosive and the unstable cases. It is well-known in the literature that the usual bootstrap method for the least squares parameter estimator is asymptotically consistent for the stationary and the explosive cases, but does not work in the unstable case, where the parameter value is equal to +1 and or -1. We propose a modified bootstrap method, which turns out to be asymptotically consistent in all possible situations. Furthermore, we derive tests for stationarity and nonstationarity for first order autoregressions. The bootstrap method is used to obtain critical values. Some simulation results are also enclosed.