ON THE ASYMPTOTICS OF ADF TESTS FOR UNIT ROOTS

ABSTRACT

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(n ^1/2) while they assume that it is of order o(n ^κ) for some κ satisfying 0 < κ ≤ 1/3.

Keywords:

• ADF tests
• Unit roots
• Asymptotics
• Linear process
• Autoregressive approximation

ACKNOWLEDGMENTS

We wish to thank the Editor, an Associate Editor and two anonymous referees for useful comments, which have greatly improved the exposition of an earlier version of this paper. Park thanks the Department of Economics at Rice University, where he is an Adjunct Professor, for its continuing hospitality and secretarial support. This research was supported by Korea Research Foundation.

Notes

Pantula3-4 extended the tests to AR(1) models with martingale difference errors and to AR( p) models with ARCH errors.

Some statistical packages like SPLUS set by default the maximum lag length to be 10log₁₀(n) for the order selection criteria such as AIC and BIC.