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A Complete Asymptotic Series for the Autocovariance Function of a Long Memory Process

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Abstract

An infinite-order asymptotic expansion is given for the autocovariance function of a general stationary long-memory process with memory parameter d in (-1/2,1/2). The class of spectral densities considered includes as a special case the stationary and invertible ARFIMA(p,d,q) model. The leading term of the expansion is of the order O(1/k^{1-2d}), where k is the autocovariance order, consistent with the well known power law decay for such processes, and is shown to be accurate to an error of O(1/k^{3-2d}). The derivation uses Erdélyi's (1956) expansion for Fourier-type integrals when there are critical points at the boundaries of the range of integration - here the frequencies {0,2}. Numerical evaluations show that the expansion is accurate even for small k in cases where the autocovariance sequence decays monotonically, and in other cases for moderate to large k. The approximations are easy to compute across a variety of parameter values and models.

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File URL: http://cowles.econ.yale.edu/P/cd/d15b/d1586.pdf
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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 1586.

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Length: 22 pages
Date of creation: Oct 2006
Date of revision:
Publication status: Published in Journal of Econometrics (2008), 147: 99-103
Handle: RePEc:cwl:cwldpp:1586

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Keywords: Autocovariance; Asymptotic expansion; Critical point; Fourier integral; Long memory;

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  1. Giraitis, Liudas & Kokoszka, Piotr & Leipus, Remigijus & Teyssiere, Gilles, 2003. "Rescaled variance and related tests for long memory in volatility and levels," Journal of Econometrics, Elsevier, vol. 112(2), pages 265-294, February.
  2. Peter M. Robinson & M. Henry, 1999. "Long and short memory conditional heteroskedasticity in estimating the memory parameter of levels," LSE Research Online Documents on Economics 304, London School of Economics and Political Science, LSE Library.
  3. Sowell, Fallaw, 1992. "Maximum likelihood estimation of stationary univariate fractionally integrated time series models," Journal of Econometrics, Elsevier, vol. 53(1-3), pages 165-188.
  4. Lieberman, Offer & Phillips, Peter C.B., 2004. "Expansions For The Distribution Of The Maximum Likelihood Estimator Of The Fractional Difference Parameter," Econometric Theory, Cambridge University Press, vol. 20(03), pages 464-484, June.
  5. Offer Lieberman & Peter C. B. Phillips, 2006. "Refined Inference on Long Memory in Realized Volatility," Cowles Foundation Discussion Papers 1549, Cowles Foundation for Research in Economics, Yale University.
  6. Offer Lieberman & Peter C. B. Phillips, 2005. "Expansions for approximate maximum likelihood estimators of the fractional difference parameter," Econometrics Journal, Royal Economic Society, vol. 8(3), pages 367-379, December.
  7. Baillie, Richard T., 1996. "Long memory processes and fractional integration in econometrics," Journal of Econometrics, Elsevier, vol. 73(1), pages 5-59, July.
  8. Robinson, P.M. & Henry, M., 1999. "Long And Short Memory Conditional Heteroskedasticity In Estimating The Memory Parameter Of Levels," Econometric Theory, Cambridge University Press, vol. 15(03), pages 299-336, June.
  9. Peter C.B. Phillips, 1998. "Econometric Analysis of Fisher's Equation," Cowles Foundation Discussion Papers 1180, Cowles Foundation for Research in Economics, Yale University.
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Cited by:
  1. Hurvich, Clifford & Wang, Yi, 2009. "A Pure-Jump Transaction-Level Price Model Yielding Cointegration, Leverage, and Nonsynchronous Trading Effects," MPRA Paper 12575, University Library of Munich, Germany.
  2. Guillaume Chevillon & Sophocles Mavroeidis, 2013. "Learning generates Long Memory," Post-Print hal-00661012, HAL.
  3. Javier Contreras-Reyes & Wilfredo Palma, 2013. "Statistical analysis of autoregressive fractionally integrated moving average models in R," Computational Statistics, Springer, vol. 28(5), pages 2309-2331, October.
  4. Phillips, Peter C.B., 2009. "Long memory and long run variation," Journal of Econometrics, Elsevier, vol. 151(2), pages 150-158, August.

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