On the Asymptotics of ADF Tests for Unit Roots1

In this paper, we derive the asymptotic distributions of AugmentedDickey-Fuller (ADF) tests under very mild conditions. The tests were originally proposed and investigated by Said and Dickey (1984) for testing unit roots in ...nite-order ARMA models with iid innovations, and are based on a ...nite AR process of order increasing with the sample size. Our conditions are signi...cantly weaker than theirs. In particular, we allow for general linear processes with martingale di¤erence innovations, possibly having conditional heteroskedasticities. The linear processes driven by ARCH type innovations are thus permitted. The range for the permissible increasing rates for the AR approximation order is also much wider. For the usual t-type test, we only require that it increase at order o(n1=2) while they assume that it is of order o(n·) for some · satisfying 0 < · · 1=3. This version: November, 2001