Gaussian tests for seasonal unit roots based on Cauchy estimation and recursive mean adjustments

We propose tests for seasonal unit roots whose limiting null distributions are always standard normal regardless of the period of seasonality and types of mean adjustments. The seasonal models of Dickey, Hasza and Fuller (1984. Journal of American Statistical Association 79, 355–367) (DHF) and Hylleberg, Engle, Granger and Yoo (1990. Journal of Econometrics 44, 215–238) (HEGY) are considered. For estimating parameters related to the seasonal unit roots, regressor signs are used as instrumental variables while recursive sample means are used for adjusting the seasonal means. In addition to normality of the limiting null distributions, in seasonal mean models, the recursive mean adjustment provides the new tests with locally higher powers than those of the existing tests of DHF and HEGY based on the ordinary least-squares estimators. If data have a strong linear time trend, the recursive mean adjustment is a source of both power gains of some tests for local alternatives and power losses of all tests for other alternatives. Limiting normality allow evaluation of p-values and testing joint significance of subsets of seasonal unit roots.