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The sample autocorrelation function and the detection of long-memory processes


  • Hassani, Hossein
  • Leonenko, Nikolai
  • Patterson, Kerry


The detection of long-range dependence in time series analysis is an important task to which this paper contributes by showing that whilst the theoretical definition of a long-memory (or long-range dependent) process is based on the autocorrelation function, it is not possible for long memory to be identified using the sum of the sample autocorrelations, as usually defined. The reason for this is that the sample sum is a predetermined constant for any stationary time series; a result that is independent of the sample size. Diagnostic or estimation procedures, such as those in the frequency domain, that embed this sum are equally open to this criticism. We develop this result in the context of long memory, extending it to the implications for the spectral density function and the variance of partial sums of a stationary stochastic process. The results are further extended to higher order sample autocorrelations and the bispectral density. The corresponding result is that the sum of the third order sample (auto) bicorrelations at lags h,k≥1, is also a predetermined constant, different from that in the second order case, for any stationary time series of arbitrary length.

Suggested Citation

  • Hassani, Hossein & Leonenko, Nikolai & Patterson, Kerry, 2012. "The sample autocorrelation function and the detection of long-memory processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(24), pages 6367-6379.
  • Handle: RePEc:eee:phsmap:v:391:y:2012:i:24:p:6367-6379 DOI: 10.1016/j.physa.2012.07.062

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    References listed on IDEAS

    1. Hassani, Hossein, 2010. "A note on the sum of the sample autocorrelation function," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(8), pages 1601-1606.
    2. Lobato, Ignacio N & Savin, N E, 1998. "Real and Spurious Long-Memory Properties of Stock-Market Data," Journal of Business & Economic Statistics, American Statistical Association, vol. 16(3), pages 261-268, July.
    3. Ding, Zhuanxin & Granger, Clive W. J. & Engle, Robert F., 1993. "A long memory property of stock market returns and a new model," Journal of Empirical Finance, Elsevier, vol. 1(1), pages 83-106, June.
    4. Lo, Andrew W. & MacKinlay, A. Craig, 1989. "The size and power of the variance ratio test in finite samples : A Monte Carlo investigation," Journal of Econometrics, Elsevier, vol. 40(2), pages 203-238, February.
    5. Cochrane, John H, 1988. "How Big Is the Random Walk in GNP?," Journal of Political Economy, University of Chicago Press, vol. 96(5), pages 893-920, October.
    6. Serinaldi, Francesco, 2010. "Use and misuse of some Hurst parameter estimators applied to stationary and non-stationary financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(14), pages 2770-2781.
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    Cited by:

    1. Hossein Hassani & Emmanuel Sirimal Silva, 2015. "A Kolmogorov-Smirnov Based Test for Comparing the Predictive Accuracy of Two Sets of Forecasts," Econometrics, MDPI, Open Access Journal, vol. 3(3), pages 1-20, August.


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