Several authors have investigated the asymptotic properties of the standard residual-based bootstrap method for unrestricted autoregressions in the random walk model (see Basawa et al. (1991a), Datta (1996)). In contrast, there are no theoretical results for the properties of the bootstrap when the true model is a higher-order autoregressive process that is integrated of order one. The main contribution of this paper is to show that in the latter case the bootstrap achieves the correct first-order asymptotic distribution for the non-unit root parameters in the augmented Dickey-Fuller (ADF) representation, but not for the estimated unit root parameter (nor for the deterministic regressors). This result is new because the presence of the estimated unit root parameter invalidates conventional arguments for the asymptotic validity of the bootstrap approach such as the sufficiency conditions presented by Beran and Ducharme (1991, Proposition 1.3).
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