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.
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Paper provided by University Library of Munich, Germany in its series MPRA Paper with number
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References listed on IDEAS Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
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.
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Other versions:
Lawrence J. Christiano & Terry J. Fitzgerald, 1999.
"The Band Pass Filter,"
NBER Working Papers
7257, National Bureau of Economic Research, Inc.
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Lawrence J. Christiano & Terry J. Fitzgerald, 1999.
"The Band pass filter,"
Working Paper
9906, Federal Reserve Bank of Cleveland.
[Downloadable!]