Long memory and long run variation
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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- 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.
- Phillips, Peter C.B. & Kim, Chang Sik, 2007.
"Long-Run Covariance Matrices For Fractionally Integrated Processes,"
Cambridge University Press, vol. 23(06), pages 1233-1247, December.
- 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.
- 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.
- Lieberman, Offer & Phillips, Peter C.B., 2008. "A complete asymptotic series for the autocovariance function of a long memory process," Journal of Econometrics, Elsevier, vol. 147(1), pages 99-103, November.
- Katsumi Shimotsu, 2006.
"Gaussian Semiparametric Estimation of Multivariate Fractionally Integrated Processes,"
1062, Queen's University, Department of Economics.
- Shimotsu, Katsumi, 2007. "Gaussian semiparametric estimation of multivariate fractionally integrated processes," Journal of Econometrics, Elsevier, vol. 137(2), pages 277-310, April.
- Robinson, P.M., 2008. "Diagnostic testing for cointegration," Journal of Econometrics, Elsevier, vol. 143(1), pages 206-225, March.
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