On Regression-based Tests for Seasonal Unit Roots in the Presence of Periodic Heteroscedasticity∗

In this paper we analyse the behaviour of regression-based tests for seasonal unit roots when the error process is periodically heteroscedastic. We show, using the case of quarterly data to illustrate, that the limiting null distributions of tests for unit roots at the zero and Nyquist frequencies are unaffected by the presence of periodic heteroscedastic behaviour in the error process. Tests at the harmonic seasonal frequencies are shown to be either unaffected or to display a discrete shift in their limiting distribution, depending on the specific nature of the periodic heteroscedasticity. In extreme cases certain of these limiting distributions are shown to be degenerate while others are known functions of the well-known Dickey-Fuller distributions. Monte Carlo evidence demonstrates that the asymptotic theory developed in this paper provides a very good prediction for the finite sample behaviour of the unit root test statistics.