On The Asymptotic Properties Of Some Seasonal Unit Root Tests

This paper analyzes the large sample behavior of the seasonal unit root tests of Dickey, Hasza, and Fuller (1984, Journal of the American Statistical Association 79, 355–367) when applied to a series that admits a unit root at the zero but not seasonal spectral frequencies. We show that in such cases the Dickey et al. statistics have nondegenerate limiting distributions. Consequently, there is a nonzero probability that, taken in isolation, they will lead the applied researcher to accept the seasonal unit root null hypothesis and hence, incorrectly, take seasonal differences of the series, even asymptotically. The same conclusion holds if the process displays unit root behavior at any of the zero and/or seasonal frequencies. Our results therefore prove a conjecture made on the basis of Monte Carlo simulation evidence, in Ghysels, Lee, and Noh (1994, Journal of Econometrics 62, 415–442) that the tests of Dickey et al., unlike those of Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44, 215–238), are unable to separate between unit roots at the zero and seasonal frequencies.I thank Peter Burridge, Bruce Hansen, and three anonymous referees for helpful comments on earlier drafts of this paper.