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Generalized Fractional Processes with Long Memory and Time Dependent Volatility Revisited

Author

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  • M. Shelton Peiris

    (School of Mathematics and Statistics, The University of Sydney, Sydney 2006, Australia)

  • Manabu Asai

    (Faculty of Economics, Soka University, Tokyo 192-8577, Japan)

Abstract

In recent years, fractionally-differenced processes have received a great deal of attention due to their flexibility in financial applications with long-memory. This paper revisits the class of generalized fractionally-differenced processes generated by Gegenbauer polynomials and the ARMA structure (GARMA) with both the long-memory and time-dependent innovation variance. We establish the existence and uniqueness of second-order solutions. We also extend this family with innovations to follow GARCH and stochastic volatility (SV). Under certain regularity conditions, we give asymptotic results for the approximate maximum likelihood estimator for the GARMA-GARCH model. We discuss a Monte Carlo likelihood method for the GARMA-SV model and investigate finite sample properties via Monte Carlo experiments. Finally, we illustrate the usefulness of this approach using monthly inflation rates for France, Japan and the United States.

Suggested Citation

  • M. Shelton Peiris & Manabu Asai, 2016. "Generalized Fractional Processes with Long Memory and Time Dependent Volatility Revisited," Econometrics, MDPI, vol. 4(3), pages 1-21, September.
    • Handle: RePEc:gam:jecnmx:v:4:y:2016:i:3:p:37-:d:77417
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    1. J. Durbin & S. J. Koopman, 2000. "Time series analysis of non‐Gaussian observations based on state space models from both classical and Bayesian perspectives," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 62(1), pages 3-56.
    2. Jacquier, Eric & Polson, Nicholas G & Rossi, Peter E, 2002. "Bayesian Analysis of Stochastic Volatility Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 20(1), pages 69-87, January.
    3. Hallin, Marc, 1978. "Mixed autoregressive-moving average multivariate processes with time-dependent coefficients," Journal of Multivariate Analysis, Elsevier, vol. 8(4), pages 567-572, December.
    4. Liesenfeld, Roman & Richard, Jean-Francois, 2003. "Univariate and multivariate stochastic volatility models: estimation and diagnostics," Journal of Empirical Finance, Elsevier, vol. 10(4), pages 505-531, September.
    5. Deo, Rohit S. & Hurvich, Clifford M., 2001. "On The Log Periodogram Regression Estimator Of The Memory Parameter In Long Memory Stochastic Volatility Models," Econometric Theory, Cambridge University Press, vol. 17(4), pages 686-710, August.
    6. Maria Caporale, Guglielmo & A. Gil-Alana, Luis, 2011. "Multi-Factor Gegenbauer Processes and European Inflation Rates," Journal of Economic Integration, Center for Economic Integration, Sejong University, vol. 26, pages 386-409.
    7. Hallin, Marc & Ingenbleek, Jean-François, 1983. "Nonstationary Yule-Walker equations," Statistics & Probability Letters, Elsevier, vol. 1(4), pages 189-195, June.
    8. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    9. Bos, Charles S. & Koopman, Siem Jan & Ooms, Marius, 2014. "Long memory with stochastic variance model: A recursive analysis for US inflation," Computational Statistics & Data Analysis, Elsevier, vol. 76(C), pages 144-157.
    10. David K. Backus & Stanley E. Zin, 1993. "Long-memory inflation uncertainty: evidence from the term structure of interest rates," Proceedings, Federal Reserve Bank of Cleveland, pages 681-708.
    11. 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.
    12. Jacquier, Eric & Polson, Nicholas G & Rossi, Peter E, 1994. "Bayesian Analysis of Stochastic Volatility Models: Comments: Reply," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(4), pages 413-417, October.
    13. Ferrara, Laurent & Guegan, Dominique, 2001. "Forecasting with k-Factor Gegenbauer Processes: Theory and Applications," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 20(8), pages 581-601, December.
    14. McAleer, Michael, 2005. "Automated Inference And Learning In Modeling Financial Volatility," Econometric Theory, Cambridge University Press, vol. 21(1), pages 232-261, February.
    15. Weiss, Andrew A., 1986. "Asymptotic Theory for ARCH Models: Estimation and Testing," Econometric Theory, Cambridge University Press, vol. 2(1), pages 107-131, April.
    16. Singh, N. & Peiris, M. Shelton, 1987. "A note on the properties of some nonstationary ARMA processes," Stochastic Processes and their Applications, Elsevier, vol. 24(1), pages 151-155, February.
    17. Chesney, Marc & Scott, Louis, 1989. "Pricing European Currency Options: A Comparison of the Modified Black-Scholes Model and a Random Variance Model," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 24(3), pages 267-284, September.
    18. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
    19. Hassler, Uwe & Wolters, Jurgen, 1995. "Long Memory in Inflation Rates: International Evidence," Journal of Business & Economic Statistics, American Statistical Association, vol. 13(1), pages 37-45, January.
    20. 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.
    21. Javier Hidalgo & Philippe Soulier, 2004. "Estimation of the location and exponent of the spectral singularity of a long memory process," Journal of Time Series Analysis, Wiley Blackwell, vol. 25(1), pages 55-81, January.
    22. Breidt, F. Jay & Crato, Nuno & de Lima, Pedro, 1998. "The detection and estimation of long memory in stochastic volatility," Journal of Econometrics, Elsevier, vol. 83(1-2), pages 325-348.
    23. Bougerol, Philippe & Picard, Nico, 1992. "Stationarity of Garch processes and of some nonnegative time series," Journal of Econometrics, Elsevier, vol. 52(1-2), pages 115-127.
    24. Henry L. Gray & Nien‐Fan Zhang & Wayne A. Woodward, 1989. "On Generalized Fractional Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 10(3), pages 233-257, May.
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    Cited by:

    1. Shelton Peiris & Manabu Asai & Michael McAleer, 2017. "Estimating and Forecasting Generalized Fractional Long Memory Stochastic Volatility Models," JRFM, MDPI, vol. 10(4), pages 1-16, December.
    2. Asai Manabu & Peiris Shelton & McAleer Michael & Allen David E., 2020. "Cointegrated Dynamics for a Generalized Long Memory Process: Application to Interest Rates," Journal of Time Series Econometrics, De Gruyter, vol. 12(1), pages 1-18, January.
    3. Asai, Manabu & McAleer, Michael & Peiris, Shelton, 2020. "Realized stochastic volatility models with generalized Gegenbauer long memory," Econometrics and Statistics, Elsevier, vol. 16(C), pages 42-54.
    4. Lahmiri, Salim & Bekiros, Stelios, 2020. "Nonlinear analysis of Casablanca Stock Exchange, Dow Jones and S&P500 industrial sectors with a comparison," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 539(C).
    5. Beaumont, Paul & Smallwood, Aaron, 2019. "Inference for likelihood-based estimators of generalized long-memory processes," MPRA Paper 96313, University Library of Munich, Germany.
    6. Asai, M. & Peiris, S. & McAleer, M.J. & Allen, D.E., 2018. "Cointegrated Dynamics for A Generalized Long Memory Process," Econometric Institute Research Papers EI 2018-32, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    7. Beaumont, Paul & Smallwood, Aaron, 2019. "Conditional Sum of Squares Estimation of Multiple Frequency Long Memory Models," MPRA Paper 96314, University Library of Munich, Germany.
    8. Stéphane Goutte & David Guerreiro & Bilel Sanhaji & Sophie Saglio & Julien Chevallier, 2019. "International Financial Markets," Post-Print halshs-02183053, HAL.

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