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The effect of round-off error on long memory processes

Author

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  • La Spada Gabriele

    (Princeton University – Economics, Fisher Hall, Princeton, NJ 08540, USA)

  • Lillo Fabrizio

    (Scuola Normale Superiore di Pisa – Mathematics, Pisa, Italy; and Università degli Studi di Palermo, Dipartimento di Fisicae Chimica, Palermo, Italy)

Abstract

We study how the round-off (or discretization) error changes the statistical properties of a Gaussian long memory process. We show that the autocovariance and the spectral density of the discretized process are asymptotically rescaled by a factor smaller than one, and we compute exactly this scaling factor. Consequently, we find that the discretized process is also long memory with the same Hurst exponent as the original process. We consider the properties of two estimators of the Hurst exponent, namely the local Whittle (LW) estimator and the detrended fluctuation analysis (DFA). By using analytical considerations and numerical simulations we show that, in presence of round-off error, both estimators are severely negatively biased in finite samples. Under regularity conditions we prove that the LW estimator applied to discretized processes is consistent and asymptotically normal. Moreover, we compute the asymptotic properties of the DFA for a generic (i.e., non-Gaussian) long memory process and we apply the result to discretized processes.

Suggested Citation

  • La Spada Gabriele & Lillo Fabrizio, 2014. "The effect of round-off error on long memory processes," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 18(4), pages 1-38, September.
    • Handle: RePEc:bpj:sndecm:v:18:y:2014:i:4:p:38:n:5
    DOI: 10.1515/snde-2013-0011
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    as
    1. Gabriele La Spada & J. Doyne Farmer & Fabrizio Lillo, 2010. "Tick size and price diffusion," Papers 1009.2329, arXiv.org, revised Oct 2010.
    2. Dittmann, Ingolf & Granger, Clive W. J., 2002. "Properties of nonlinear transformations of fractionally integrated processes," Journal of Econometrics, Elsevier, vol. 110(2), pages 113-133, October.
    3. Eduardo Rossi & Paolo Santucci de Magistris, 2014. "Estimation of Long Memory in Integrated Variance," Econometric Reviews, Taylor & Francis Journals, vol. 33(7), pages 785-814, October.
    4. Saint-Paul, Gilles, 2017. "A “quantized” approach to rational inattention," European Economic Review, Elsevier, vol. 100(C), pages 50-71.
    5. Clifford M. Hurvich & Rohit Deo & Julia Brodsky, 1998. "The mean squared error of Geweke and Porter‐Hudak's estimator of the memory parameter of a long‐memory time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 19(1), pages 19-46, January.
    6. Szpiro, George G., 1998. "Tick size, the compass rose and market nanostructure," Journal of Banking & Finance, Elsevier, vol. 22(12), pages 1559-1569, December.
    7. Christos Agiakloglou & Paul Newbold & Mark Wohar, 1993. "Bias In An Estimator Of The Fractional Difference Parameter," Journal of Time Series Analysis, Wiley Blackwell, vol. 14(3), pages 235-246, May.
    8. Violetta Dalla & Liudas Giraitis & Javier Hidalgo, 2006. "Consistent estimation of the memory parameter for nonlinear time series," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(2), pages 211-251, March.
    9. Bali, Rakesh & Hite, Gailen L., 1998. "Ex dividend day stock price behavior: discreteness or tax-induced clienteles?," Journal of Financial Economics, Elsevier, vol. 47(2), pages 127-159, February.
    10. Arianos, Sergio & Carbone, Anna, 2007. "Detrending moving average algorithm: A closed-form approximation of the scaling law," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 382(1), pages 9-15.
    11. Matteo, T. Di & Aste, T. & Dacorogna, Michel M., 2005. "Long-term memories of developed and emerging markets: Using the scaling analysis to characterize their stage of development," Journal of Banking & Finance, Elsevier, vol. 29(4), pages 827-851, April.
    12. Francois Schmitt & Daniel Schertzer & Shaun Lovejoy, 2000. "Multifractal Fluctuations In Finance," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 3(03), pages 361-364.
    13. Clifford M. Hurvich & Eric Moulines & Philippe Soulier, 2005. "Estimating Long Memory in Volatility," Econometrica, Econometric Society, vol. 73(4), pages 1283-1328, July.
    14. Grech, Dariusz & Mazur, Zygmunt, 2013. "On the scaling ranges of detrended fluctuation analysis for long-term memory correlated short series of data," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(10), pages 2384-2397.
    15. Yoshihiro Yajima, 1989. "A Central Limit Theorem Of Fourier Transforms Of Strongly Dependent Stationary Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 10(4), pages 375-383, July.
    16. Alfarano, Simone & Lux, Thomas, 2007. "A Noise Trader Model As A Generator Of Apparent Financial Power Laws And Long Memory," Macroeconomic Dynamics, Cambridge University Press, vol. 11(S1), pages 80-101, November.
    17. Velasco, Carlos, 2000. "Non-Gaussian Log-Periodogram Regression," Econometric Theory, Cambridge University Press, vol. 16(1), pages 44-79, February.
    18. Hansen, Peter R. & Lunde, Asger, 2014. "Estimating The Persistence And The Autocorrelation Function Of A Time Series That Is Measured With Error," Econometric Theory, Cambridge University Press, vol. 30(1), pages 60-93, February.
    19. Harris, Lawrence, 1990. "Estimation of Stock Price Variances and Serial Covariances from Discrete Observations," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 25(3), pages 291-306, September.
    20. Arteche, Josu, 2004. "Gaussian semiparametric estimation in long memory in stochastic volatility and signal plus noise models," Journal of Econometrics, Elsevier, vol. 119(1), pages 131-154, March.
    21. Dalla, Violetta & Giraitis, Liudas & Hidalgo, Javier, 2006. "Consistent estimation of the memory parameter for nonlinear time series," LSE Research Online Documents on Economics 6813, London School of Economics and Political Science, LSE Library.
    22. Clifford M. Hurvich & Bonnie K. Ray, 2003. "The Local Whittle Estimator of Long-Memory Stochastic Volatility," Journal of Financial Econometrics, Oxford University Press, vol. 1(3), pages 445-470.
    23. Shao, Xiaofeng & Wu, Wei Biao, 2007. "Local Whittle Estimation Of Fractional Integration For Nonlinear Processes," Econometric Theory, Cambridge University Press, vol. 23(5), pages 899-929, October.
    24. Ying-Hui Shao & Gao Feng Gu & Zhi-Qiang Jiang & Wei-Xing Zhou & Didier Sornette, 2012. "Comparing the performance of FA, DFA and DMA using different synthetic long-range correlated time series," Papers 1208.4158, arXiv.org.
    25. Shimotsu, Katsumi & Phillips, Peter C.B., 2006. "Local Whittle estimation of fractional integration and some of its variants," Journal of Econometrics, Elsevier, vol. 130(2), pages 209-233, February.
    26. Violetta Dalla & Liudas Giraitis & Javier Hidalgo, 2006. "Consistent estimation of the memory parameterfor nonlinear time series," STICERD - Econometrics Paper Series 497, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    27. Hosking, Jonathan R. M., 1996. "Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series," Journal of Econometrics, Elsevier, vol. 73(1), pages 261-284, July.
    28. Gottlieb, Gary & Kalay, Avner, 1985. "Implications of the Discreteness of Observed Stock Prices," Journal of Finance, American Finance Association, vol. 40(1), pages 135-153, March.
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    1. Martin D. Gould & Mason A. Porter & Stacy Williams & Mark McDonald & Daniel J. Fenn & Sam D. Howison, 2013. "Limit order books," Quantitative Finance, Taylor & Francis Journals, vol. 13(11), pages 1709-1742, November.
    2. Martin D. Gould & Mason A. Porter & Stacy Williams & Mark McDonald & Daniel J. Fenn & Sam D. Howison, 2010. "Limit Order Books," Papers 1012.0349, arXiv.org, revised Apr 2013.
    3. Gianbiagio Curato & Fabrizio Lillo, 2013. "Modeling the coupled return-spread high frequency dynamics of large tick assets," Papers 1310.4539, arXiv.org.

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