In this paper we derive representations for the limiting
distributions of the regression-based seasonal unit root test
statistics of Hylleberg, Engle, Granger, and Yoo (1990, Journal of Econometrics 44,
215–238) and Beaulieu and Miron (1993,
Journal of Econometrics 55, 305–328), inter alia, when
the underlying process displays near seasonal integration. Our results
generalize those presented in previous studies by allowing for an
arbitrary seasonal periodicity (including the nonseasonal case), a wide
range of possible assumptions on the initial conditions, a range of
(seasonal) deterministic mean effects, and finite autoregressive
behavior in the driving shocks. We use these representations to
simulate the asymptotic local power functions of the seasonal unit root
tests, demonstrating a significant dependence on serial correlation
nuisance parameters in the case of the pairs of t-statistics,
but not the associated F-statistic, for unit roots at the
seasonal harmonic frequencies. Monte Carlo simulation results are
presented that suggest that the local limiting distribution theory
provides a good approximation to the finite-sample behavior of the
statistics. Our results lend further weight to the advice of previous
authors that inference on the unit root hypothesis at the seasonal
harmonic frequencies should be based on the F-statistic,
rather than on the associated pairs of t-ratios.We are grateful to Bruce Hansen and two anonymous referees for
their helpful comments and suggestions on earlier versions of this
paper.