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X-13ARIMA-SEATS Seasonal Adjustment Program

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Frequency Domain Analyses of SEATS and X-11/12-ARIMA Seasonal Adjustment Filters for Short and Moderate-Length Time Series

David F. Findley (1) and Donald E. K. Martin (2)


We investigate frequency domain properties, revealed by the squared gain and phase delay functions, of short and moderate-length linear seasonal adjustment filters of the ARIMAmodel- based signal extraction method of SEATS. X-11/12-ARIMA filters are also considered to a limited extent. The focus is on the one-sided (concurrent) and symmetric (central) filters associated with the Box and Jenkinsís airline model for monthly time series of lengths 49 and 109. A Digital Signal Processing perspective on filter properties, favoring interpretability of the filter output, is presented. We show that important features of the finite filters actually used are often not visible in the diagnostics of the infinite filters, such as the Wiener-Kolmogorov symmetric filter gain functions provided by SEATS. For comparing competing adjustments, properties of concurrent filters, especially their phase delays, can be more important than properties of symmetric filters. Our phase delay results illustrate that adjusters who favor smoother seasonal adjustments, or who favor trends for their greater smoothness, must usually reckon with greater delay of turning point and business cycle information for the most recent months. Trend filters are considered briefly. New analytical results are obtained for transfer functions and phase delays.


Gain function; phase delay function; seasonal adjustment; trend estimation; ARIMA-model-based signal extraction; X-11-ARIMA; X-12-ARIMA.

(1) David F. Findley was the Senior Mathematical Statistican for Time Series and is now a consultant, U. S. Census Bureau, 4600 Silver Hill Road, Washington, DC 20233. email :

(2) Donald E. K. Martin is currently a professor at North Carolina State University.

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