X-13ARIMA-SEATS Seasonal Adjustment Program

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Optimal Signal Extraction with Correlated Components

Tucker S. McElroy (1) and Agustin Maravall (2)


While it is typical in the econometric signal extraction literature to assume that the unobserved signal and noise components are uncorrelated, there is nevertheless an interest among econometricians in the hypothesis of hysteresis, i.e., that major movements in the economy are fundamentally linked. While specific models involving correlated signal and noise innovation sequences have been developed and applied using state space methods, there is no systematic treatment of optimal signal extraction with correlated components. This paper provides the Mean Square Error optimal formulas for both finite samples and bi-infinite samples, and furthermore relates these filters to the more well-known Wiener-Kolmogorov (WK) and Beveridge-Nelson (BN) signal extraction formulas in the case of ARIMA component models. Then we obtain the result that the optimal filter for correlated components can be viewed as a weighted linear combination of the WK and BN filters. The gain and phase functions of the resulting filters are plotted for some standard cases. Some discussion of estimation of hysteresis models is presented, along with empirical results on several economic time series. Comparisons are made between signal extraction estimates from traditional WK filters and those arising from the hysteresis models.


ARIMA, Nonstationary, Seasonality, Time Series

(1) Tucker S. McElroy is Mathematical Statistican, Center for Statistical Research and Methodology U. S. Census Bureau, 4600 Silver Hill Road, Washington, DC 20233. email : Tucker.S.McElroy@census.gov

(2) Agustin Maravall was Chief Economist, Bank of Spain, and is now retired.

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Source: U.S. Census Bureau | Statistical Research Division | (301) 763-1649 (or x12@census.gov) |  Last Revised: August 07, 2015