Unit Root Properties of Seasonal Adjustment and Related Filters

Linear filters used in seasonal adjustment (model-based or from the X-11 method) contain unit root factors in the form of differencing operators and seasonal summation operators. The extent to which the various filters (seasonal, seasonal adjustment, trend, and irregular) contain these unit root factors determines whether the filters reproduce or annihilate (i) polynomial functions of time, and (ii) fixed seasonal effects. This article catalogs which unit root factors are contained by the various filters for the most common approaches to model-based seasonal adjustment, and for X-11 seasonal adjustment with or without forecast extension. Both symmetric and asymmetric filters are considered.