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LONG MEMORY AND LONG RUN VARIATION

By Peter C. B. Phillips

Abstract

A commonly used defining property of long memory time series is the power law decay of the autocovariance function. Some alternative methods of deriving this property are considered working from the alternate definition in terms of a fractional pole in the spectrum at the origin. The methods considered involve the use of (i) Fourier transforms of generalized functions, (ii) asymptotic expansions of Fourier integrals with singularities, (iii) direct evaluation using hypergeometric function algebra, and (iv) conversion to a simple gamma integral. The paper is largely pedagogical but some novel methods and results involving complete asymptotic series representations are presented. The formulae are useful in many ways including the calculation of long run variation matrices for multivariate time series with long memory and the econometric estimation of such models

Topics: singularity. JEL Classification, C22, C32
Year: 2008
OAI identifier: oai:CiteSeerX.psu:10.1.1.193.8779
Provided by: CiteSeerX
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