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Low-Pass Filter Design using Locally Weighted Polynomial Regression and Discrete Prolate Spheroidal Sequences

Author

Listed:
  • Proietti, Tommaso
  • Luati, Alessandra

Abstract

The paper concerns the design of nonparametric low-pass filters that have the property of reproducing a polynomial of a given degree. Two approaches are considered. The first is locally weighted polynomial regression (LWPR), which leads to linear filters depending on three parameters: the bandwidth, the order of the fitting polynomial, and the kernel. We find a remarkable linear (hyperbolic) relationship between the cutoff period (frequency) and the bandwidth, conditional on the choices of the order and the kernel, upon which we build the design of a low-pass filter. The second hinges on a generalization of the maximum concentration approach, leading to filters related to discrete prolate spheroidal sequences (DPSS). In particular, we propose a new class of lowpass filters that maximize the concentration over a specified frequency range, subject to polynomial reproducing constraints. The design of generalized DPSS filters depends on three parameters: the bandwidth, the polynomial order, and the concentration frequency. We discuss the properties of the corresponding filters in relation to the LWPR filters, and illustrate their use for the design of low-pass filters by investigating how the three parameters are related to the cutoff frequency.

Suggested Citation

  • Proietti, Tommaso & Luati, Alessandra, 2009. "Low-Pass Filter Design using Locally Weighted Polynomial Regression and Discrete Prolate Spheroidal Sequences," MPRA Paper 15510, University Library of Munich, Germany.
    • Handle: RePEc:pra:mprapa:15510
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    References listed on IDEAS

    as
    1. Frederick R. Macaulay, 1931. "The Smoothing of Time Series," NBER Books, National Bureau of Economic Research, Inc, number maca31-1.
    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. 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.
    4. Tommaso Proietti & Alessandra Luati, 2008. "Real Time Estimation in Local Polynomial Regression, with Application to Trend-Cycle Analysis," CEIS Research Paper 112, Tor Vergata University, CEIS, revised 14 Jul 2008.
    5. Lii, K.S. & Rosenblatt, M., 2008. "Prolate spheroidal spectral estimates," Statistics & Probability Letters, Elsevier, vol. 78(11), pages 1339-1348, August.
    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Trend filters; Kernels; Concentration; Filter Design.;
    All these keywords.

    JEL classification:

    • E32 - Macroeconomics and Monetary Economics - - Prices, Business Fluctuations, and Cycles - - - Business Fluctuations; Cycles
    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • 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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