Periodic Integration and Seasonal Unit Roots

Seasonality is pervasive across a wide range of economic time series and it substantially complicates the analysis of unit root non-stationarity in such series. This paper reviews recent contributions to the literature on non-stationary seasonal processes, focussing on periodically integrated (P I) and seasonally integrated (SI) processes. Whereas an SI process captures seasonal non-stationarity essentially through an annual lag, a P I process has (a restricted form of) seasonally-varying autoregressive coefficients. The fundamental properties of both types of process are compared, noting in particular that a simple SI process observed S times a year has S unit roots, in contrast to the single unit root of a P I process. Indeed, for S > 2 and even (such as processes observed quarterly or monthly), an SI process has a pair of complex-valued unit roots at each seasonal frequency except the Nyquist frequency, where a single real root applies. Consequently, recent literature concerned with testing the unit roots implied by SI processes employs complex-valued unit root processes, and these are discussed in some detail. A key feature of the discussion is to show how the demodulator operator can be used to convert a unit root process at a seasonal frequency to a conventional zero-frequency unit root process, thereby enabling the well-known properties of the latter to be exploited. Further, circulant matrices are introduced and it is shown how they are employed in theoretical analyses to capture the repetitive nature of seasonal processes. Discriminating between SI and P I processes requires care, since testing for unit roots at seasonal frequencies may lead to a P I process (erroneously) appearing to have an SI form, while an application to monthly US industrial production series illustrates how these types of seasonal non-stationarity can be distinguished in practice. Although univariate processes are discussed, the methods considered in the paper can be used to analyze cointegration, including cointegration across different frequencies