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Long Memory and Long Run Variation

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Abstract

May 2008 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.

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File URL: http://cowles.econ.yale.edu/P/cd/d16b/d1656.pdf
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Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1656.

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Length: 23 pages
Date of creation: May 2008
Date of revision:
Handle: RePEc:cwl:cwldpp:1656

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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

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Keywords: Asymptotic expansion; Autocovariance function; Fractional pole; Fourier integral; Generalized function; Long memory; Long range dependence; Singularity;

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  1. Peter C.B. Phillips & Chang Sik Kim, 2007. "Long Run Covariance Matrices for Fractionally Integrated Processes," Cowles Foundation Discussion Papers 1611, Cowles Foundation for Research in Economics, Yale University.
  2. Katsumi Shimotsu, 2003. "Gaussian semiparametric estimation of multivariate fractionally integrated processes," Economics Discussion Papers 571, University of Essex, Department of Economics.
  3. Robinson, P.M., 2008. "Diagnostic testing for cointegration," Journal of Econometrics, Elsevier, vol. 143(1), pages 206-225, March.
  4. Granger, C. W. J., 1980. "Long memory relationships and the aggregation of dynamic models," Journal of Econometrics, Elsevier, vol. 14(2), pages 227-238, October.
  5. Offer Lieberman & Peter C.B. Phillips, 2006. "A Complete Asymptotic Series for the Autocovariance Function of a Long Memory Process," Cowles Foundation Discussion Papers 1586, Cowles Foundation for Research in Economics, Yale University.
  6. Peter M Robinson, 2007. "Multiple Local Whittle Estimation in StationarySystems," STICERD - Econometrics Paper Series /2007/525, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
  7. Peter Robinson, 2007. "Diagnostic Testing For Cointegration," STICERD - Econometrics Paper Series /2007/522, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
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