Skip Main Navigation Skip To Navigation Content

X-13ARIMA-SEATS Seasonal Adjustment Program

You are here: Census.govSubjects A to ZX-13ARIMA-SEATSSeasonal Adjustment PapersPapers by Year › Abstract of Bell (2010)
Skip top of page navigation

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

William R. Bell (1)


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.


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

(1) William R. Bell is Senior Mathematical Statistician for Small Area Estimation, U.S. Census Bureau, 4600 Silver Hill Road, Washington, DC 20233. email:

[PDF] or PDF denotes a file in Adobe’s Portable Document Format. To view the file, you will need the Adobe® Reader® Off Site available free from Adobe.

This symbol Off Site indicates a link to a non-government web site. Our linking to these sites does not constitute an endorsement of any products, services or the information found on them. Once you link to another site you are subject to the policies of the new site.

Source: U.S. Census Bureau | Center for Statistical Research and Methodology | (301) 763-1649 (or |  Last Revised: April 02, 2015