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Optimal Linear Filtering, Smoothing and Trend Extraction for Processes with Unit Roots and Cointegration

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  • Dimitrios Thomakos

Abstract

In this paper I propose a novel optimal linear filter for smoothing, trend and signal extraction for time series with a unit root. The filter is based on the Singular Spectrum Analysis (SSA) methodology, takes the form of a particular moving average and is different from other linear filters that have been used in the existing literature. To best of my knowledge this is the first time that moving average smoothing is given an optimality justification for use with unit root processes. The frequency response function of the filter is examined and a new method for selecting the degree of smoothing is suggested. I also show that the filter can be used for successfully extracting a unit root signal from stationary noise. The proposed methodology can be extended to also deal with two cointegrated series and I show how to estimate the cointegrating coefficient using SSA and how to extract the common stochastic trend component. A simulation study explores some of the characteristics of the filter for signal extraction, trend prediction and cointegration estimation for univariate and bivariate series. The practical usefulness of the method is illustrated using data for the US real GDP and two financial time series.

Suggested Citation

  • Dimitrios Thomakos, 2008. "Optimal Linear Filtering, Smoothing and Trend Extraction for Processes with Unit Roots and Cointegration," Working Papers 0024, University of Peloponnese, Department of Economics.
  • Handle: RePEc:uop:wpaper:0024
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    File URL: http://econ.uop.gr/~econ/RePEc/pdf/optimal_smoothing_updated.pdf
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    References listed on IDEAS

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    1. Marianne Baxter & Robert G. King, 1999. "Measuring Business Cycles: Approximate Band-Pass Filters For Economic Time Series," The Review of Economics and Statistics, MIT Press, vol. 81(4), pages 575-593, November.
    2. Durbin, James & Koopman, Siem Jan, 2012. "Time Series Analysis by State Space Methods," OUP Catalogue, Oxford University Press, edition 2, number 9780199641178.
    3. Schlicht, Ekkehart, 2004. "Estimating the Smoothing Parameter in the So-Called Hodrick-Prescott Filter," Discussion Papers in Economics 304, University of Munich, Department of Economics.
    4. 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.
    5. Peter C.B. Phillips, 1996. "Spurious Regression Unmasked," Cowles Foundation Discussion Papers 1135, Cowles Foundation for Research in Economics, Yale University.
    6. Christoph Schleicher, 2003. "Kolmogorov-Wiener Filters for Finite Time Series," Computing in Economics and Finance 2003 109, Society for Computational Economics.
    7. 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.
    8. 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.
    9. Peter C.B. Phillips, 1998. "Econometric Analysis of Fisher's Equation," Cowles Foundation Discussion Papers 1180, Cowles Foundation for Research in Economics, Yale University.
    10. McElroy, Tucker, 2008. "Matrix Formulas For Nonstationary Arima Signal Extraction," Econometric Theory, Cambridge University Press, vol. 24(4), pages 988-1009, August.
    11. Thomakos, Dimitrios D. & Wang, Tao & Wille, Luc T., 2002. "Modeling daily realized futures volatility with singular spectrum analysis," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 312(3), pages 505-519.
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    Cited by:

    1. Theodore Alexandrov & Silvia Bianconcini & Estela Bee Dagum & Peter Maass & Tucker S. McElroy, 2012. "A Review of Some Modern Approaches to the Problem of Trend Extraction," Econometric Reviews, Taylor & Francis Journals, vol. 31(6), pages 593-624, November.
    2. D.S. Poskitt, 2016. "Singular Spectrum Analysis of Grenander Processes and Sequential Time Series Reconstruction," Monash Econometrics and Business Statistics Working Papers 15/16, Monash University, Department of Econometrics and Business Statistics.

    More about this item

    Keywords

    cointegration; forecasting; linear filtering; singular spectrum analysis; smoothing; trend extraction and prediction; unit root.;

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