Determining the order of differencing in seasonal time series processes

In this paper we propose a sequential testing procedure to determine the order of differencing in seasonally observed time series processes, which builds upon existing approaches developed for nonseasonal series. We allow for the possible presence of multiple unit roots at both the zero and seasonal frequencies of the data. Multiple seasonal unit roots can be useful in circumstances where one does not wish to take logarithms of the data-set at hand. Multiple unit roots at the seasonal frequencies also appear in commonly applied seasonal adjustment filters such as Census X-11. The testing procedure developed in this paper can therefore be used to obtain some a priori insight into the likely properties of seasonally adjusted time series. In particular it may be used to investigate whether or not the seasonally adjusted series can be expected to be strictly noninvertible at the seasonal frequencies of the data. The proposed testing procedure is shown be asymptotically consistent. The unit root statistics arising at each stage of the testing procedure are shown to have familiar limiting null distributions so that, at least to an approximation, existing critical values may be used. Empirical applications are provided to illustrate the practical usefulness of our approach.

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