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X-13ARIMA-SEATS Seasonal Adjustment Program

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An Iterated Parametric Approach to Nonstationary Signal Extraction

Tucker S. McElroy (1) and Andrew Sutcliffe (2)


Consider the three-component time series model that decomposes observed data (Y) into the sum of seasonal (S), trend (T), and irregular(I) portions. Assuming that S and T are nonstationary and that I is stationary, it is demonstrated that widely-used Wiener-Kolmogorov signal extraction estimates of S and T can be obtained through an iteration scheme applied to optimal estimates derived from reduced two-component models for YS = S+YT = T+I. This "bootstrapping" signal extraction methodology is reminiscent of the iterated nonparametric approach of the U.S. Census Bureau's X-11 program. The analysis of the iteration scheme provides insight into the algebraic relationship between full model and reduced model signal extraction estimates.


ARIMA component model, nonstationary time series, seasonal adjustment, signal extraction, Wiener-Kolmogorov Filtering, X-11

(1) Tucker S. McElroy is Mathematical Statistican, U. S. Census Bureau, 4600 Silver Hill Road, Washington, DC 20233. email :

(2) Andrew Sutcliffe is currently a statistican with the Austalian Bureau of Statistics. email:

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Source: U.S. Census Bureau | Center for Statistical Research and Methodology | (301) 763-1649 (or |  Last Revised: August 07, 2015