Seasonal unit root tests and the role of initial conditions

In the context of regression-based (quarterly) seasonal unit root tests, we examine the impact of initial conditions (one for each quarter) of the process on test power. We investigate the behaviour of the OLS detrended HEGY seasonal unit root tests of Hylleberg et al. (1990) and the corresponding quasi-differenced (QD) detrended tests of Rodrigues and Taylor (2007), when the initial conditions are not asymptotically negligible. We show that the asymptotic local power of a test at a given frequency depends on the value of particular linear (frequencyspecific) combinations of the initial conditions. Consistent with previous findings in the non-seasonal case (see, inter alia, Harvey et al., 2008, Elliott and Müller, 2006), the QD detrended test at a given spectral frequency dominates on power for relatively small values of this combination, while the OLS detrended test dominates for larger values. Since, in practice, the seasonal initial conditions are not observed, in order to maintain good power across both small and large initial conditions, we extend the idea of Harvey et al. (2008) to the seasonal case, forming tests based on a union of rejections decision rule; rejecting the unit root null at a given frequency (or group of frequencies) if either of the relevant QD and OLS detrended HEGY tests rejects. This procedure is shown to perform well in practice, simultaneously exploiting the superior power of the QD (OLS) detrended HEGY test for small (large) combinations of the initial conditions. Moreover, our procedure is particularly adept in the seasonal context since, by design, it exploits the power advantage of the QD (OLS) detrended HEGY tests at a particular frequency when the relevant initial condition is small (large) without imposing that same method of detrending on tests at other frequencies.