ON THE ASYMPTOTICS OF ADF TESTS FOR UNIT ROOTS

In this paper, we derive the asymptotic distributions of Augmented-Dickey-Fuller (ADF) tests under very mild conditions. The tests were originally proposed and investigated by Said and Dickey (1984) for testing unit roots in finite-order ARMA models with i.i.d. innovations, and are based on a finite AR process of order increasing with the sample size. Our conditions are significantly weaker than theirs. In particular, we allow for general linear processes with martingale difference 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(nκ) for some κ satisfying 0 < κ ≤ 1/3.