TESTING FOR SEASONAL UNIT ROOTS IN PERIODIC INTEGRATED AUTOREGRESSIVE PROCESSES

Abstract

This paper examines the implications of applying the Hylleberg, Engle, Granger, and Yoo (1990,Journal of Econometrics44, 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 roott-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 HEGYt-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. Finite-sample 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.