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Continuous Time Extraction of a Nonstationary Signal with Illustrations in Continuous Low-pass and Band-pass Filtering

Tucker S. McElroy(1) and Thomas Trimbur(2)


This paper sets out the theoretical foundations for continuous-time signal extraction in econometrics. Continuous-time modeling gives an effective strategy for treating stock and flow data, irregularly spaced data, and changing frequency of observation.

We rigorously derive the optimal continuous-lag filter when the signal component is nonstationary, and provide several illustrations, including a new class of continuous-lag Butterworth filters for trend and cycle estimation.


signal extraction, continuous time processes, linear filtering, Hodrick-Prescott Filter, Band-Pass Filter

(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) Thomas Trimbur is currently at the U. S. Federal Reserve Board email : Thomas.M.Trimbur@frb.gov

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Source: U.S. Census Bureau | Center for Statistical Research and Methodology | (301) 763-1649 (or x12@census.gov) |  Last Revised: November 19, 2012