Unit Root Tests Based on Adaptive Maximum Likelihood Estimation

Adaptive maximum likelihood estimators of unit roots in autoregressive processes
with possibly non-Gaussian innovations are considered. Unit root tests based on the
adaptive estimators are constructed. Limiting distributions of the test statistics are
derived, which are linear combinations of two functionals of Brownian motions. A
Monte Carlo simulation reveals that the proposed tests have improved powers over
the classical Dickey-Fuller tests when the distribution of the innovation is not close
to normal. We also compare the proposed tests with those of Lucas (1995, Econometric
Theory 1 1, 33 1-346) based on M-estimators.
1, INTRODUCTION
Testing for autoregressive unit roots has attracted a great deal of attention since
the influential works of Fuller (1976,1996) and Dickey and Fuller (1979). Various
extensions have been and are currently being made in several directions. One
research direction is to widen the applicability of the unit root tests under broader
correlation structures for the innovations. Phillips (1987) and Phillips and Perron
(1988) developed tests that are applicable under weakly dependent innovations.
Said and Dickey (1984, 1985), Pantula and Hall (1991), and others constructed
tests that work for the innovations of stationary autoregressive moving averages
(ARMA). Another direction is to improve powers of tests as studied by Pantula
et al. (1994), Park and Fuller (1995), Shin and So (1997), and others. The power
improvements are achieved by a proper handling of the initial observation or by
considering symmetric estimation of the unit root parameter.