The contribution of this paper is threefold. First, a characterization theorem of the subhypotheses comprising the seasonal unit root hypothesis is presented that provides a precise formulation of the alternative hypotheses associated with regression- based seasonal unit root tests. Second, it proposes regression-based tests for the seasonal unit root hypothesis that allow a general seasonal aspect for the data and are similar both exactly and asymptotically with respect to initial values and seasonal drift parameters. Third, limiting distribution theory is given for these statistics where, in contrast to previous papers in the literature, in doing so it is not assumed that unit roots hold at all of the zero and seasonal frequencies. This is shown to alter the large-sample null distribution theory for regression <italic>t</italic>-statistics for unit roots at the complex frequencies, but interestingly to not affect the limiting null distributions of the regression <italic>t</italic>-statistics for unit roots at the zero and Nyquist frequencies and regression <italic>F</italic>-statistics for unit roots at the complex frequencies. Our results therefore have important implications for how tests of the seasonal unit root hypothesis should be conducted in practice. Associated simulation evidence on the size and power properties of the statistics presented in this paper is given that is consonant with the predictions from the large-sample theory.
