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Modeling Sequences of Long Memory Positive Weakly Stationary Random Variables

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  • Dmitri Koulikov

Abstract

In this paper we introduce a new class of covariance stationary long-memory models on the positive half-line. The overall structure of the models is related to that of GARCH processes of Engle (1982) and Bollerslev (1986), whereby sequence of random variables of interest have multiplicative shocks structure. Unlike FIGARCH model of Baillie, Bollerslev and Mikkelsen (1996), our models are weakly stationary with non-summable autocovariances and hence belong to the class of long-memory models according to the criteria of McLeod and Hipel (1978). In addition, we are able to ensure positivity of all underlying components of the model, thereby improving on the results of Giraitis, Robinson, and Surgailis (2000). Apart from volatility modeling, the class of models introduced in this paper will find applications in high-frequency financial data econometrics.

Suggested Citation

  • Dmitri Koulikov, 2002. "Modeling Sequences of Long Memory Positive Weakly Stationary Random Variables," William Davidson Institute Working Papers Series 493, William Davidson Institute at the University of Michigan.
    • Handle: RePEc:wdi:papers:2002-493
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    References listed on IDEAS

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    1. Bollerslev, Tim & Ole Mikkelsen, Hans, 1996. "Modeling and pricing long memory in stock market volatility," Journal of Econometrics, Elsevier, vol. 73(1), pages 151-184, July.
    2. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 1999. "The Distribution of Exchange Rate Volatility," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-059, New York University, Leonard N. Stern School of Business-.
    3. Lee, Sang-Won & Hansen, Bruce E., 1994. "Asymptotic Theory for the Garch(1,1) Quasi-Maximum Likelihood Estimator," Econometric Theory, Cambridge University Press, vol. 10(1), pages 29-52, March.
    4. Giraitis, Liudas & Kokoszka, Piotr & Leipus, Remigijus, 2000. "Stationary Arch Models: Dependence Structure And Central Limit Theorem," Econometric Theory, Cambridge University Press, vol. 16(1), pages 3-22, February.
    5. Robert F. Engle, 2000. "The Econometrics of Ultra-High Frequency Data," Econometrica, Econometric Society, vol. 68(1), pages 1-22, January.
    6. Giraitis, Liudas & Robinson, Peter & Surgailis, Donatas, 2000. "A model for long memory conditional heteroscedasticity," LSE Research Online Documents on Economics 2103, London School of Economics and Political Science, LSE Library.
    7. Tanaka, Katsuto, 1999. "The Nonstationary Fractional Unit Root," Econometric Theory, Cambridge University Press, vol. 15(4), pages 549-582, August.
    8. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    9. Robert F. Engle & Jeffrey R. Russell, 1998. "Autoregressive Conditional Duration: A New Model for Irregularly Spaced Transaction Data," Econometrica, Econometric Society, vol. 66(5), pages 1127-1162, September.
    10. Baillie, Richard T. & Bollerslev, Tim & Mikkelsen, Hans Ole, 1996. "Fractionally integrated generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 74(1), pages 3-30, September.
    11. Giraitis, Liudas & Robinson, Peter M. & Surgailis, Donatas, 2000. "A model for long memory conditional heteroscedasticity," LSE Research Online Documents on Economics 299, London School of Economics and Political Science, LSE Library.
    12. Ding, Zhuanxin & Granger, Clive W. J., 1996. "Modeling volatility persistence of speculative returns: A new approach," Journal of Econometrics, Elsevier, vol. 73(1), pages 185-215, July.
    13. Chung, Ching-Fan & Baillie, Richard T, 1993. "Small Sample Bias in Conditional Sum-of-Squares Estimators of Fractionally Integrated ARMA Models," Empirical Economics, Springer, vol. 18(4), pages 791-806.
    14. Joann Jasiak, 1996. "Persistence in Intertrade Durations," Working Papers 1999_8, York University, Department of Economics, revised Mar 1999.
    15. C. W. J. Granger & Roselyne Joyeux, 1980. "An Introduction To Long‐Memory Time Series Models And Fractional Differencing," Journal of Time Series Analysis, Wiley Blackwell, vol. 1(1), pages 15-29, January.
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    Cited by:

    1. Indeewara Perera & Javier Hidalgo & Mervyn J. Silvapulle, 2016. "A Goodness-of-Fit Test for a Class of Autoregressive Conditional Duration Models," Econometric Reviews, Taylor & Francis Journals, vol. 35(6), pages 1111-1141, June.
    2. Monteiro, André A., 2009. "The econometrics of randomly spaced financial data: a survey," DES - Working Papers. Statistics and Econometrics. WS ws097924, Universidad Carlos III de Madrid. Departamento de Estadística.

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    More about this item

    Keywords

    conditional heteroscedasticity; long-memory; weak stationarity; econometrics of high-frequency financial data;
    All these keywords.

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation

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