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

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Author Info
Dimitrios Thomakos

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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.

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Publisher Info
Paper provided by University of Peloponnese, Department of Economics in its series Working Papers with number 0024.

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Length: 43 pages
Date of creation: 2008
Date of revision:
Handle: RePEc:uop:wpaper:0024

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

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  1. Peter C.B. Phillips, 1998. "Econometric Analysis of Fisher's Equation," Cowles Foundation Discussion Papers 1180, Cowles Foundation, Yale University. [Downloadable!]
  2. Schlicht, Ekkehart, 2004. "Estimating the Smoothing Parameter in the So-Called Hodrick-Prescott Filter," IZA Discussion Papers 1054, Institute for the Study of Labor (IZA). [Downloadable!]
    Other versions:
  3. Christoph Schleicher, 2003. "Kolmogorov-Wiener Filters for Finite Time Series," Computing in Economics and Finance 2003 109, Society for Computational 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, 05. [Downloadable!] (restricted)
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  5. 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. [Downloadable!] (restricted)
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  6. Peter C.B. Phillips, 1996. "Spurious Regression Unmasked," Cowles Foundation Discussion Papers 1135, Cowles Foundation, Yale University. [Downloadable!]
  7. 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. [Downloadable!] (restricted)
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  1. Dimitrios Thomakos, 2008. "A Note on Optimal Linear Filtering, Smoothing and Trend Extraction for Processes with Unit Roots with Drift," Working Papers 0025, University of Peloponnese, Department of Economics. [Downloadable!]
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