Common Cycles in Seasonal Non-stationary Time Series

This paper extends the notion of common cycles to quarterly time series having unit roots both at the zero and seasonal frequencies. It is shown that common cycles are present in the Hylleberg-Engle-Granger-Yoo decomposition of these series when there exists a linear combination of their seasonal differences which follows an MA process of order, at most, three. The pitfalls of seasonal adjustment for common cycles analysis are also documented. Inference on common cycles in seasonally cointegrated series is derived from existing statistical methods for codependence. Concepts and methods are illustrated with an empirical analysis of the comovements between consumption and output using Italian data.