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X-13ARIMA-SEATS Seasonal Adjustment Program

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Some Consideration of Seasonal Adjustment Variances

William R. Bell(1)


We consider alternative definitions of seasonal adjustment variances for both model-based and X-11 type seasonal adjustments. In particular, when we account for sampling error components of time series we can consider either the variance of the estimation error for the nonseasonal component of the true series, or the variance of the estimation error for the seasonally adjusted series (observed series minus estimated seasonal component). The magnitudes of these variances can be quite different. We examine the relative contributions of various error components to seasonal adjustment variances, and the effects on model-based seasonal adjustment variances of uncertainty about model parameters. The considerations are illustrated with several examples.


sampling error, canonical decomposition, forecast extension

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

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Source: U.S. Census Bureau | Center for Statistical Research and Methodology | (301) 763-1649 (or |  Last Revised: November 19, 2012