Detecting seasonal unit roots: An approach based on the sample autocorrelation function

In this paper we suggest a new and consistent procedure designed to test for the presence of unit roots in quarterly time series, be they at the zero or seasonal frequencies. For each frequency the proposed statistic is based on the correlogram of a transformation (which removes unit roots that may exist at frequencies other than that of interest) of the residuals from a regression of the observed series on relevant deterministic components. Each statistic is proportional to the maximum lag length for which the residual autocorrelations at all lower lag lengths are strictly positive. Critical values for each test statistic are calculated using Monte Carlo simulation assuming a seasonal random walk data-generation process (DGP). The robustness properties of each statistic to different integrated DGPs are examined along with their respective power characteristics. These simulations indicate that the procedure suggested in this paper provides a useful complement to the set of test statistics proposed by Hylleberg et al. ('Seasonal integration and cointegration', Journal of Econometrics, Vol. 44 (1990), pp. 215-238).