Testing for Unit Roots in Seasonal Time Series with Long Period.

Abstract

Testing for seasonal unit roots has been discussed extensively in the literature. However, the test will be difficult if the time series has a long period, where the critical values for the test statistics are not available. We modify the seasonal unit roots test of Dickey, Hasza, and Fuller (1984) to investigate results for less typical, long period cases, and present some asymptotic normality properties. We also suggest an empirical adjustment to improve the normal approximation when the seasonal period is not sufficiently long.

The basic idea is to use a double-index form for the seasonal time series with a long period, where d denotes the large lag number, so that the d "channels" will be independent for each i. By applying the Classical Central Limit Theorem for iid random variables, we can obtain the asymptotic result. The convergence is proved to be order independent with respect to m and d. An advantage of this technique is that one can make the adjustment and use a standard normal as a reference distribution instead of looking into the seasonal percentile tables when doing the seasonal unit roots test, no matter what kind of deterministic terms are included in the model as long as the number of the regressors is fixed. We also show that for an AR(p) model we still obtain the asymptotic normality of the unit root statistics.