Seasonal Unit Root Tests: A Comparison

Abstract

Three major regression-based seasonal unit root tests: the DHF test introduced by Dickey et al (1984), the HEGY test proposed by Hylleberg et al. (1990) and the Kunst test introduced by Kunst (1997) are compared. The regression model for the DHF test is a reduced form of that for the Kunst test. We modify the Kunst test by using the t-statistic instead of Kunst's proposed joint F-statistic to study the influence of additional variables in the Kunst model. Also, we modify the HEGY test to test the presence of all four quarterly unit roots against the presence of roots 1 and -1. Through the comparison between the DHF test and the modified HEGY test, we find that the DHF test does not have asymptotic power one when the series only have some of the seasonal unit roots but not all of them. We call this case of partial unit roots. The asymptotic distributions derived in the paper provide the explanation of this limitation for the DHF test. Using simulation, we find that the probability that the DHF test will lead researchers to accept the seasonal unit root null hypothesis increases when the series contains more partial unit roots.

For the DHF test, the test power depends on the augmented model. We derive limits of the related estimates from two augmented models for the DHF test. Both estimates are inconsistent. The test statistic obtained from the augmented model suggested by Ghysels et al. (1992) has relatively low power. For the HEGY/Kunst test, most limiting distributions for the test statistics depend on the lag augmentation but the test statistics have few problems caused by inconsistent estimates. However, the augmented models for the HEGY/Kunst test have more variables than those for the DHF test. Based on our simulation study results, the inclusion of more variables results in more loss in power when a redundant variable is included, and more sensitivity to the size distortion when the augmented lag length is less than the true lag length.