Unit Root Properties of Seasonal Adjustment and Related Filters: Special Cases

Bell (2012) catalogued unit root factors contained in linear filters used in seasonal adjustment (model-based or from the X-11 method) but noted that, for model-based seasonal adjustment, special cases could arise where filters could contain more unit root factors than was indicated by the general results. This article reviews some special cases that occur with canonical ARIMA model based adjustment in which, with some commonly used ARIMA models, the symmetric seasonal filters contain two extra nonseasonal differences (i.e., they include an extra (1 - B)(1 - F)). This increases by two the degree of polynomials in time that are annihilated by the seasonal filter and reproduced by the seasonal adjustment filter. Other results for canonical ARIMA adjustment that are reported in Bell (2012), including properties of the trend and irregular filters, and properties of the asymmetric and finite filters, are unaltered in these special cases. Special cases for seasonal adjustment with structural ARIMA component models are also briefly discussed.