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Signal Extraction in Continuous Time and the Generalized Hodrick- Prescott Filter


  • Roberto Iannaccone


  • Edoardo Otranto

    (DEIR-Università di Sassari)


A widely used filter to extract a signal in a time series, in particular in the business cycle analysis, is the Hodrick-Prescott filter. The model that underlies the filter considers the data series as the sum of two unobserved component (signal and non signal) and a smoothing parameter which for quarterly series is set to a specified value. This paper proposes a generalization of the Hodrick-Prescott filter to a continuous time support, using the well-established relationship between cubic splines and state-space models. The spline formulation of the filter leads to a state space model with several practical advantages: first, the smoothing parameter can be either pre-specified or estimated as the other parameters in the model; second, the unobserved components can be modelled by the addition of particular ARIMA structures; lastly the model is capable of working in the presence of missing values or for irregular surveys. Monte Carlo experiments support these considerations.

Suggested Citation

  • Roberto Iannaccone & Edoardo Otranto, 2003. "Signal Extraction in Continuous Time and the Generalized Hodrick- Prescott Filter," Econometrics 0311002, EconWPA.
  • Handle: RePEc:wpa:wuwpem:0311002
    Note: Type of Document - PDF; pages: 20. PDF document submitted via ftp

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    References listed on IDEAS

    1. Juan J. Dolado & Miguel Sebastián & Javier Vallés, 1993. "Ciclical patterns of the spanish economy," Investigaciones Economicas, Fundación SEPI, vol. 17(3), pages 445-473, September.
    2. Hodrick, Robert J & Prescott, Edward C, 1997. "Postwar U.S. Business Cycles: An Empirical Investigation," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 29(1), pages 1-16, February.
    3. Koopman, Siem Jan & Harvey, Andrew, 2003. "Computing observation weights for signal extraction and filtering," Journal of Economic Dynamics and Control, Elsevier, vol. 27(7), pages 1317-1333, May.
    4. Harvey, A C & Jaeger, A, 1993. "Detrending, Stylized Facts and the Business Cycle," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 8(3), pages 231-247, July-Sept.
    5. Carter, C.K. & Kohn, R., "undated". "Semiparametric Bayesian inference for time series with mixed spectra," Statistics Working Paper _005, Australian Graduate School of Management.
    6. Morten O. Ravn & Harald Uhlig, 2002. "On adjusting the Hodrick-Prescott filter for the frequency of observations," The Review of Economics and Statistics, MIT Press, vol. 84(2), pages 371-375.
    7. Siem Jan Koopman & Neil Shephard & Jurgen A. Doornik, 1999. "Statistical algorithms for models in state space using SsfPack 2.2," Econometrics Journal, Royal Economic Society, vol. 2(1), pages 107-160.
    8. Cogley, Timothy, 2001. "Alternative definitions of the business cycle and their implications for business cycle models: A reply to Torben Mark Pederson," Journal of Economic Dynamics and Control, Elsevier, vol. 25(8), pages 1103-1107, August.
    9. Víctor Gómez & Agustín Maravall, 1998. "Seasonal Adjustment and Signal Extraction in Economic Time Series," Working Papers 9809, Banco de España;Working Papers Homepage.
    10. King, Robert G. & Rebelo, Sergio T., 1993. "Low frequency filtering and real business cycles," Journal of Economic Dynamics and Control, Elsevier, vol. 17(1-2), pages 207-231.
    11. Cogley, Timothy & Nason, James M., 1995. "Effects of the Hodrick-Prescott filter on trend and difference stationary time series Implications for business cycle research," Journal of Economic Dynamics and Control, Elsevier, vol. 19(1-2), pages 253-278.
    12. Wynne, Mark A. & Koo, Jahyeong, 1997. "Business cycles under monetary union: EU and US business cycles compared," Working Papers 9707, Federal Reserve Bank of Dallas.
    13. Nelson, Charles R. & Plosser, Charles I., 1982. "Trends and random walks in macroeconmic time series : Some evidence and implications," Journal of Monetary Economics, Elsevier, vol. 10(2), pages 139-162.
    14. Agustín Maravall & Ana del Río, 2001. "Time Aggregation and the Hodrick-Prescott Filter," Working Papers 0108, Banco de España;Working Papers Homepage.
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    More about this item


    smoothing parameter; cubic spline; state-space model; irregular surveys.;

    JEL classification:

    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C3 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables
    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling

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