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Optimal Linear Filtering, Smoothing and Trend Extraction for the m-th Differences of a Unit Root Process: A Singular Spectrum Analysis Approach

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Author Info

  • Dimitrios Thomakos

    ()
    (Department of Economics, University of Peloponnese Tripolis, Greece)

  • Hossein Hassani

    ()
    (Executive Business Centre, Bournemouth University)

  • Kerry Patterson

    ()
    (Department of Economics, University of Reading)

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    Abstract

    The problem of optimal linear filtering, smoothing and trend extraction for m-period differences of processes with a unit root is studied. Such processes arise naturally in economics and finance, in the form of rates of change (price inflation, economic growth, financial returns) and finding an appropriate smoother is thus of immediate practical interest. The filter and resulting smoother are based on the methodology of Singular Spectrum Analysis (SSA). An explicit representation for the asymptotic decomposition of the covariance matrix is obtained. The structure of the impulse and frequency response functions indicates that the optimal filter has a “permanent” and a “transitory component”, with the corresponding smoother being the sum of two such components. Moreover, a particular form for the extrapolation coefficients that can be used in out-of-sample prediction is proposed. In addition, an explicit representation for the filtering weights in the context of SSA for an arbitrary covariance matrix is derived. This result allows one to examine the specific effects of smoothing in any situation. The theoretical results are llustrated using different data sets, namely U.S. inflation and real GDP growth.

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    File URL: http://www.reading.ac.uk/web/FILES/economics/emdp2013101.pdf
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    Bibliographic Info

    Paper provided by Henley Business School, Reading University in its series Economics & Management Discussion Papers with number em-dp2013-04.

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    Length: 33 pages
    Date of creation: 11 Sep 2013
    Date of revision:
    Handle: RePEc:rdg:emxxdp:em-dp2013-04

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    Related research

    Keywords: core inflation; business cycles; differences; euro; linear filtering; singular spectrum analysis; smoothing; trend extraction and prediction; unit root;

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    References

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    1. Christina Beneki & Bruno Eeckels & Costas Leon, 2012. "Signal Extraction and Forecasting of the UK Tourism Income Time Series: A Singular Spectrum Analysis Approach," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 31(5), pages 391-400, 08.
    2. Marianne Baxter & Robert G. King, 1995. "Measuring Business Cycles Approximate Band-Pass Filters for Economic Time Series," NBER Working Papers 5022, National Bureau of Economic Research, Inc.
    3. Lawrence J. Christiano & Terry J. Fitzgerald, 1999. "The Band Pass Filter," NBER Working Papers 7257, National Bureau of Economic Research, Inc.
      • 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, 05.
    4. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-72, June.
    5. Hassani, Hossein & Heravi, Saeed & Zhigljavsky, Anatoly, 2009. "Forecasting European industrial production with singular spectrum analysis," International Journal of Forecasting, Elsevier, vol. 25(1), pages 103-118.
    6. Cochrane, John H, 1988. "How Big Is the Random Walk in GNP?," Journal of Political Economy, University of Chicago Press, vol. 96(5), pages 893-920, October.
    7. 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.
    8. Hassani, Hossein, 2007. "Singular Spectrum Analysis: Methodology and Comparison," MPRA Paper 4991, University Library of Munich, Germany.
    9. Durbin, James & Koopman, Siem Jan, 2001. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, number 9780198523543, September.
    10. Beveridge, Stephen & Nelson, Charles R., 1981. "A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the `business cycle'," Journal of Monetary Economics, Elsevier, vol. 7(2), pages 151-174.
    11. 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.
    12. Christoph Schleicher, 2003. "Kolmogorov-Wiener Filters for Finite Time Series," Computing in Economics and Finance 2003 109, Society for Computational Economics.
    13. Franses, Philip Hans & Paap, Richard, 2004. "Periodic Time Series Models," OUP Catalogue, Oxford University Press, number 9780199242030, September.
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