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On the Discretization of Continuous-time Filters for Nonstationary Stock and Flow Time Series

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


This paper discusses the discretization of continuous-time filters for application to discrete time series sampled at any fixed frequency. In this approach, the filter is first set up directly in continuous-time -- since the filter is expressed over a continuous range of lags, we also refer to them as continuous-lag filters. The second step is to discretize the filter itself. This approach applies to different problems in signal extraction, including trend or business cycle analysis, and the method allows for coherent design of discrete filters for observed data sampled as a stock or a flow, for nonstationary data with stochastic trend, and for different sampling frequencies. We derive explicit formulas for the Mean Squared Error optimal discretization filters. We also discuss the problem of optimal interpolation for nonstationary processes -- namely, how to estimate the values of a process and its components at arbitrary times in-between the sampling times. A number of illustrations of discrete filter coefficient calculations are provided, including the Local Level Model trend filter, the Smooth Trend Model trend filter, the Band Pass filter, and the Henderson filter. The essential methodology can be applied to other kinds of signal extraction problems.


Continuous time processes, Hodrick-Prescott filter, Interpolants, Linear filtering, signal extraction.

(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 :

(2) Thomas Trimbur is currently at the U. S. Federal Reserve Board 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