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


  • Tucker McElroy
  • Thomas M. Trimbur


This article 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 (MSE) 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 (LLM) trend filter, the smooth trend model (STM) trend filter, and the Band Pass (BP) filter. The essential methodology can be applied to other kinds of trend extraction problems. Finally, we provide an extended demonstration of the method on CPI flow data measured at monthly and annual sampling frequencies.

Suggested Citation

  • Tucker McElroy & Thomas M. Trimbur, 2011. "On the Discretization of Continuous-Time Filters for Nonstationary Stock and Flow Time Series," Econometric Reviews, Taylor & Francis Journals, vol. 30(5), pages 475-513, October.
  • Handle: RePEc:taf:emetrv:v:30:y:2011:i:5:p:475-513
    DOI: 10.1080/07474938.2011.553554

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    Cited by:

    1. Trimbur Thomas & McElroy Tucker, 2017. "Signal Extraction for Nonstationary Time Series with Diverse Sampling Rules," Journal of Time Series Econometrics, De Gruyter, vol. 9(1), pages 1-37, January.
    2. Tucker McElroy, 2013. "Forecasting continuous-time processes with applications to signal extraction," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(3), pages 439-456, June.

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