Efficient Tests for an Autoregressive Unit Root

The asymptotic power envelope is derived for point-optimal tests of a unit root in the
autoregressive representation of a Gaussian time series under various trend specifications.
We propose a family of tests whose asymptotic power functions are tangent to the power
envelope at one point and are never far below the envelope. When the series has no
deterministic component, some previously proposed tests are shown to be asymptotically
equivalent to members of this family. When the series has an unknown mean or linear
trend, commonly used tests are found to be dominated by members of the family of
point-optimal invariant tests. We propose a modified version of the Dickey-Fuller t test
which has substantially improved power when an unknown mean or trend is present. A
Monte Carlo experiment indicates that the modified test works well in small samples.
KEYWORDPSo:wer envelope, point optimal tests, nonstationarity, Ornstein-Uhlenbeck
processes.
1. INTRODUCTION
FOLLOWINGTHE SEMINAL WORK of Fuller (1976) and Dickey and Fuller (19791,
econometricians have developed numerous alternative procedures for testing
the hypothesis that a univariate time series is integrated of order one against the
hypothesis that it is integrated of order zero. The procedures typically are based
on second-order sample moments, but employ various testing principles and a
variety of-methods to eliminate nuisance parameters. Banerjee et al. (1993) and
Stock (1994) survey many of the most popular of these tests. Although numerical
calculations (e.g., Nabeya and Tanaka (1990)) suggest that the power functions
for the tests can differ substantially, no general optimality theory has been
developed. In particular, there are few general results (even asymptotic) concerning
the relative merits of the competing testing principles and of the various
methods for eliminating trend parameters.