A Correction Factor for Unit Root Test Statistics

Despite the fact that it is not correct to speak of Bartlett corrections in the case of
nonstationary time series, this paper shows that a Bartlett-type correction to the
likelihood ratio test for a unit root can be an effective tool to control size distortions.
Using well-known formulae, we obtain second-order (numerical) approximations
to the moments and cumulants of the likelihood ratio, which makes it possible to
calculate a Bartlett-type factor. It turns out that the cumulants of the corrected statistic
are closer to their asymptotic value than the original one. A simulation study
is then carried out to assess the quality of these approximations for the first four
moments; the size and the power of the original and the corrected statistic are also
simulated. Our results suggest that the proposed correction reduces the size distortion
without affecting the power too much.
1. INTRODUCTION
Asymptotic theory for an unstable first-order stochastic difference equation is
based upon the well-known property that a simple random walk model can be
approximated by a Wiener process (Brownian motion) and is more generally
embedded in a diffusion process (usually Ornstein-Uhlenbeck ) when the root is
local to unity (i.e., near integrated) (Phillips, 1987). The limiting distributions of
a number of test statistics and estimators can therefore be expressed in terms of
functionals of Brownian motion (for a recent survey on the topic, see Stock,
1994) and tabulated using intensive Monte Carlo simulation, usually on a case by
case basis. Unfortunately this approach does not provide the researcher with any
analytical formulae that can be used when analyzing the finite-sample properties
of the size and power of the test statistic under investigation. This fact is particularly
serious in the unit root case as, since the work of Phillips and Perron (1988)
and Schwert (1989), there has been growing evidence that tests based on functional
limiting distributions are characterized by size distortion in finite samples.