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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.

Suggested Citation

  • Peter C.B. Phillips, 2008. "Long Memory and Long Run Variation," Cowles Foundation Discussion Papers 1656, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1656
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    File URL: http://cowles.yale.edu/sites/default/files/files/pub/d16/d1656.pdf
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Robinson, P.M., 2008. "Diagnostic testing for cointegration," Journal of Econometrics, Elsevier, vol. 143(1), pages 206-225, March.
    4. Shimotsu, Katsumi, 2007. "Gaussian semiparametric estimation of multivariate fractionally integrated processes," Journal of Econometrics, Elsevier, vol. 137(2), pages 277-310, April.
    5. Phillips, Peter C.B. & Kim, Chang Sik, 2007. "Long-Run Covariance Matrices For Fractionally Integrated Processes," Econometric Theory, Cambridge University Press, vol. 23(06), pages 1233-1247, December.
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    Cited by:

    1. Simos Theodore, 2012. "On the Exact Discretization of a Continuous Time AR(1) Model driven by either Long Memory or Antipersistent Innovations: A Fractional Algebra Approach," Journal of Time Series Econometrics, De Gruyter, vol. 4(2), pages 1-26, November.

    More about this item

    Keywords

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

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C32 - Mathematical and Quantitative Methods - - Multiple or Simultaneous Equation Models; Multiple Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes; State Space Models

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