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On the Model-Based Interpretation of Filters and the Reliability of Trend-Cycle Estimates


  • Tommaso Proietti

    (University of Udine)


The paper is concerned with a class of trend cycle filters, encompassing popular ones, such as the Hodrick-Prescott filter, that are derived using the Wiener-Kolmogorov signal extraction theory under maintained models that prove unrealistic in applied time series analysis. As the maintained model is misspecified, inference about the unobserved components, and in particular their first two conditional moments, given the observations, are not delivered by the Kalman filter and smoother or the Wiener-Kolmogorov filter for the maintained model. The paper proposes a model based framework according to which the same class of filters is adapted to the particular time series under investigation; via a suitable decomposition of the innovation process, it is shown that any linear time series with ARIMA representation can be broken down into orthogonal trend and cycle components, for which the class of filters is optimal. Finite sample inferences are provided by the Kalman filter and smoother for the relevant state space representation of the decomposition. In this framework it is possible to discuss two aspects of the reliability of the signals’ estimates: the mean square error of the final estimates and the extent of the revisions. The paper discusses and illustrates how the uncertainty is related to features of the series and the design parameters of the filter, the role of smoothness priors, and the fundamental trade-off between the uncertainty and the magnitude of the revisions as new observations become available.

Suggested Citation

  • Tommaso Proietti, 2004. "On the Model-Based Interpretation of Filters and the Reliability of Trend-Cycle Estimates," Econometrics 0403007, EconWPA.
  • Handle: RePEc:wpa:wuwpem:0403007
    Note: Type of Document - pdf; pages: 25

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    References listed on IDEAS

    1. Athanasios Orphanides & Simon van Norden, 2002. "The Unreliability of Output-Gap Estimates in Real Time," The Review of Economics and Statistics, MIT Press, vol. 84(4), pages 569-583, November.
    2. Michael ARTIS & Massimiliano MARCELLINO & Tommaso PROIETTI, 2002. "Dating the Euro Area Business Cycle," Economics Working Papers ECO2002/24, European University Institute.
    3. James H. Stock & Mark W. Watson, 2003. "Has the Business Cycle Changed and Why?," NBER Chapters,in: NBER Macroeconomics Annual 2002, Volume 17, pages 159-230 National Bureau of Economic Research, Inc.
    4. 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.
    5. 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.
    6. Lawrence J. Christiano & Terry J. Fitzgerald, 2003. "The Band Pass Filter," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 44(2), pages 435-465, May.
    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. Charles Nelson & Eric Zivot, 2000. "Why are Beveridge-Nelson and Unobserved-Component Decompositions of GDP so Different?," Econometric Society World Congress 2000 Contributed Papers 0692, Econometric Society.
    9. Pollock, D. S. G., 2000. "Trend estimation and de-trending via rational square-wave filters," Journal of Econometrics, Elsevier, vol. 99(2), pages 317-334, December.
    10. Michael Artis & Massimiliano Marcellino & Tommaso Proietti, 2004. "Dating Business Cycles: A Methodological Contribution with an Application to the Euro Area," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 66(4), pages 537-565, September.
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    Cited by:

    1. Tommaso Proietti & Alberto Musso, 2012. "Growth accounting for the euro area," Empirical Economics, Springer, vol. 43(1), pages 219-244, August.
    2. Proietti, Tommaso, 2008. "Structural Time Series Models for Business Cycle Analysis," MPRA Paper 6854, University Library of Munich, Germany.
    3. Proietti, Tommaso, 2007. "Signal extraction and filtering by linear semiparametric methods," Computational Statistics & Data Analysis, Elsevier, vol. 52(2), pages 935-958, October.
    4. Terence C. Mills, 2013. "Trends, cycles and structural breaks," Chapters,in: Handbook of Research Methods and Applications in Empirical Macroeconomics, chapter 3, pages 45-60 Edward Elgar Publishing.
    5. Galimberti, Jaqueson K. & Moura, Marcelo L., 2016. "Improving the reliability of real-time output gap estimates using survey forecasts," International Journal of Forecasting, Elsevier, vol. 32(2), pages 358-373.
    6. Jaqueson K. Galimberti & Marcelo L. Moura, 2011. "Improving the reliability of real-time Hodrick-Prescott filtering using survey forecasts," Centre for Growth and Business Cycle Research Discussion Paper Series 159, Economics, The Univeristy of Manchester.
    7. Siliverstovs Boriss, 2013. "Dating Business Cycles in Historical Perspective: Evidence for Switzerland," Journal of Economics and Statistics (Jahrbuecher fuer Nationaloekonomie und Statistik), De Gruyter, vol. 233(5-6), pages 661-679, October.
    8. Michal Andrle, 2013. "What Is in Your Output Gap? Unified Framework & Decomposition into Observables," IMF Working Papers 13/105, International Monetary Fund.

    More about this item


    Signal Extraction; Revisions; Kalman filter and Smoother.;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • E30 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - General (includes Measurement and Data)
    • E32 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Business Fluctuations; Cycles

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