TESTING FOR SEASONAL UNIT ROOTS IN PERIODIC INTEGRATED AUTOREGRESSIVE PROCESSES

This paper examines the implications of applying the Hylleberg, Engle, Granger, and Yoo ~1990, Journal of Econometrics 44, 215–238! ~HEGY! seasonal root tests to a process that is periodically integrated+ As an important special case, the random walk process is also considered, where the zero-frequency unit root t-statistic is shown to converge to the Dickey–Fuller distribution and all seasonal unit root statistics diverge+ For periodically integrated processes and a sufficiently high order of augmentation, the HEGY t-statistics for unit roots at the zero and semiannual frequencies both converge to the same Dickey–Fuller distribution+ Further, the HEGY joint test statistic for a unit root at the annual frequency and all joint test statistics across frequencies converge to the square of this distribution+ Results are also derived for a fixed order of augmentation+ Finitesample Monte Carlo results indicate that, in practice, the zero-frequency HEGY statistic ~with augmentation! captures the single unit root of the periodic integrated process, but there may be a high probability of incorrectly concluding that the process is seasonally integrated+