Unit root inference for non-stationary linear processes driven by infinite variance innovations

Abstract

The contribution of this paper is two-fold. First, we derive the asymptotic null distribution of the familiar augmented Dickey-Fuller [ADF] statistics in the case where the shocks follow a linear process driven by in…nite variance innovations. We show that these distributions are free of serial correlation nuisance parameters but depend on the tail index of the in…nite variance process. These distributions are shown to coincide with the corresponding results for the case where the shocks follow a …nite autoregression, provided the lag length in the ADF regression satis…es the same o(T1=3) rate condition as is required in the …nite variance case. In addition, we establish the rates of consistency and (where they exist) the asymptotic distributions of the ordinary least squares sieve estimates from the ADF regression. Given the dependence of their null distributions on the unknown tail index, our second contribution is to explore sieve wild bootstrap implementations of the ADF tests. Under the assumption of symmetry, we demonstrate the asymptotic validity (bootstrap consistency) of the wild bootstrap ADF tests. This is done by establishing that (conditional on the data) the wild bootstrap ADF statistics attain the same limiting distribution as that of the original ADF statistics taken conditional on the magnitude of the innovations.