Filtering permanent cycles with complex unit roots

Separating cyclical movement from trend growth at seasonal and business cycle frequencies is important to macroeconomic research. At business cycle frequencies, time trends, first differences and the more recent Hodrick-Prescott (HP) filter are used to separate trends from cycles. At seasonal frequencies, ad-hoc methods like the Census Bureau's X-11 seasonal filter are applied. This paper reviews the criteria for permanent cycles in systems characterized by difference equations and looks at the effect of filtering data which exhibit permanent cyclicality. Second order moving averages with complex unit roots at appropriate frequencies are used to filter data at seasonal and business cycle frequencies; and spectral analysis of the filtered data is used to illustrate the effect. The X-11 seasonal filter and the HP filter are also discussed in this framework. As with any filter that is applied to data where the data generating process is unknown, filtering for specific frequencies can induce cycles at harmonics of the fundamental frequency.