On regression-based tests for seasonal unit roots in the presence of periodic heteroscedasticity

We analyse regression-based tests for seasonal unit roots when the shocks are periodically heteroscedastic. We show, using the case of quarterly data to illustrate, that the limiting marginal null distributions of tests for unit roots at the zero and Nyquist frequencies are unaffected by the periodic heteroscedasticity. However, tests at the harmonic seasonal frequencies are shown to be either unaffected or to display a shift in their limiting distribution, depending on the specific nature of the periodic heteroscedasticity. In extreme cases certain of these limiting distributions are degenerate while others are simple functions of the well-known Dickey–Fuller distributions. Monte Carlo evidence shows that the asymptotic theory provides a very good prediction for the finite sample behaviour of the unit root test statistics. We show that tests with approximately correct size may be obtained by simulating their sampling distributions using periodic variance parameters estimated from the data in hand. Though laborious, this procedure seems to be the best available, since more conservative approaches sacrifice significant power.