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On the Spectral Properties of Matrices Associated with Trend Filters

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  • Luati, Alessandra
  • Proietti, Tommaso

Abstract

This note is concerned with the spectral properties of matrices associated with linear smoothers. We derive analytical results on the eigenvalues and eigenvectors of smoothing matrices by interpreting the latter as perturbations of matrices belonging to algebras with known spectral properties, such as the Circulant and the generalised Tau. These results are used to characterise the properties of a smoother in terms of an approximate eigen-decomposition of the associated smoothing matrix.

Suggested Citation

  • Luati, Alessandra & Proietti, Tommaso, 2008. "On the Spectral Properties of Matrices Associated with Trend Filters," MPRA Paper 11502, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:11502
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506, September.
    4. Estela Bee Dagum & Alessandra Luati, 2002. "A linear transformation and its properties with special applications in time series filtering," Quaderni di Dipartimento 2, Department of Statistics, University of Bologna.
    5. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521785167, September.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Signal extraction; Smoothing; Boundary conditions; Matrix algebras;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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