A Note on Bootstrapping Autoregression Under Nonstationary Volatility

In a recent article, Xu (2008) developed the asymptotic theory for autoregressions around a polynomial trend, under nonstationary volatility. In the same article, Xu proposed a set of t-tests for the regression coefficients and claimed that these tests are asymptotically standard normal. A drawback concerning the applicability of these tests is that they are feasible only under the exact knowledge of the asymptotic order of the nonstationary volatility. In this paper it is first shown that by incorporating Eicker-White type covariance matrix estimators, asymptotically standard normal t-statistics can be obtained, that do not depend either on the asymptotic order of the nonstationary volatility, or on the degree of the polynomial trend. It is then observed that the test statistics proposed by Xu are not properly standardized and are not, in general, asymptotically standard normal. Finally, it is shown that the residual-based recursive-design wild bootstrap can be applied reducing significantly the size distortions in small samples. Simulation results offer strong evidence for the robustness of the bootstrap procedure for various volatility specifications, including the cases of a variance break and of nonstationary nonlinear heteroskedasticity