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Unit Root Properties of Seasonal Adjustment and Related Filters (Updated: 9/1/2011)

William R. Bell


time series, ARIMA model, X-11 seasonal adjustment, trend estimation


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 ) fixed seasonal effects, and (ii ) polynomial functions of time. This paper 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.


William R. Bell. (2010). Unit Root Properties of Seasonal Adjustment and Related Filters. (Updated: 9/1/2011 Statistical Research Division Research Report Series (Statistics #2010-08). U.S. Census Bureau. Available online at <>.

Source: U.S. Census Bureau, Statistical Research Division

Published online: January 05, 2010
Last revised: January 05, 2010


Source: U.S. Census Bureau | Statistical Research Division | (301) 763-3215 (or |   Last Revised: September 11, 2013