IDEAS home Printed from https://ideas.repec.org/a/eee/econom/v189y2015i2p428-436.html
   My bibliography  Save this article

A new hyperbolic GARCH model

Author

Listed:
  • Li, Muyi
  • Li, Wai Keung
  • Li, Guodong

Abstract

There are two commonly used hyperbolic GARCH processes, the FIGARCH and HYGARCH processes, in modeling the long-range dependence in volatility. However, the FIGARCH process always has infinite variance, and the HYGARCH model has a more complicated form. This paper builds a simple bridge between a common GARCH model and an integrated GARCH model, and hence a new hyperbolic GARCH model along the lines of FIGARCH models. The new model remedies the drawback of FIGARCH processes by allowing the existence of finite variance as in HYGARCH models, while it has a form nearly as simple as the FIGARCH model. Two inference tools, including the Gaussian QMLE and a portmanteau test for the adequacy of the fitted model, are derived, and an easily implemented test for hyperbolic memory is also constructed. Their finite sample performances are evaluated by simulation experiments, and an empirical example gives further support to our new model.

Suggested Citation

  • Li, Muyi & Li, Wai Keung & Li, Guodong, 2015. "A new hyperbolic GARCH model," Journal of Econometrics, Elsevier, vol. 189(2), pages 428-436.
    • Handle: RePEc:eee:econom:v:189:y:2015:i:2:p:428-436
    DOI: 10.1016/j.jeconom.2015.03.034
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304407615001128
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jeconom.2015.03.034?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Conrad, Christian, 2010. "Non-negativity conditions for the hyperbolic GARCH model," Journal of Econometrics, Elsevier, vol. 157(2), pages 441-457, August.
    2. Marco J. Lombardi & Giampiero M. Gallo, 2002. "Analytic Hessian matrices and the computation of FIGARCH estimates," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 11(2), pages 247-264, June.
    3. Christian Conrad & Berthold R. Haag, 2006. "Inequality Constraints in the Fractionally Integrated GARCH Model," Journal of Financial Econometrics, Oxford University Press, vol. 4(3), pages 413-449.
    4. Whitney K. Newey & Douglas G. Steigerwald, 1997. "Asymptotic Bias for Quasi-Maximum-Likelihood Estimators in Conditional Heteroskedasticity Models," Econometrica, Econometric Society, vol. 65(3), pages 587-600, May.
    5. 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.
    6. 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.
    7. Paul H. Kupiec, 1995. "Techniques for verifying the accuracy of risk measurement models," Finance and Economics Discussion Series 95-24, Board of Governors of the Federal Reserve System (U.S.).
    8. Douc, Randal & Roueff, François & Soulier, Philippe, 2008. "On the existence of some processes," Stochastic Processes and their Applications, Elsevier, vol. 118(5), pages 755-761, May.
    9. 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.
    10. 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.
    11. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    12. Zaffaroni, Paolo, 2004. "Stationarity And Memory Of Arch(∞) Models," Econometric Theory, Cambridge University Press, vol. 20(1), pages 147-160, February.
    13. W. K. Li & T. K. Mak, 1994. "On The Squared Residual Autocorrelations In Non‐Linear Time Series With Conditional Heteroskedasticity," Journal of Time Series Analysis, Wiley Blackwell, vol. 15(6), pages 627-636, November.
    14. Giraitis, Liudas & Surgailis, Donatas, 0. "ARCH-type bilinear models with double long memory," Stochastic Processes and their Applications, Elsevier, vol. 100(1-2), pages 275-300, July.
    15. Robinson, P. M., 1991. "Testing for strong serial correlation and dynamic conditional heteroskedasticity in multiple regression," Journal of Econometrics, Elsevier, vol. 47(1), pages 67-84, January.
    16. Davidson, James, 2004. "Moment and Memory Properties of Linear Conditional Heteroscedasticity Models, and a New Model," Journal of Business & Economic Statistics, American Statistical Association, vol. 22(1), pages 16-29, January.
    17. repec:wyi:journl:002190 is not listed on IDEAS
    18. Li, Muyi & Li, Guodong & Li, Wai Keung, 2011. "Score Tests for Hyperbolic GARCH Models," Journal of Business & Economic Statistics, American Statistical Association, vol. 29(4), pages 579-586.
    19. Guodong Li & Wai Keung Li, 2005. "Diagnostic checking for time series models with conditional heteroscedasticity estimated by the least absolute deviation approach," Biometrika, Biometrika Trust, vol. 92(3), pages 691-701, September.
    20. 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.
    21. Muyi Li & Guodong Li & Wai Keung Li, 2011. "Score Tests for Hyperbolic GARCH Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 29(4), pages 579-586, October.
    22. Christoffersen, Peter F, 1998. "Evaluating Interval Forecasts," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(4), pages 841-862, November.
    23. Kwan, Wilson & Li, Wai Keung & Li, Guodong, 2012. "On the estimation and diagnostic checking of the ARFIMA–HYGARCH model," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3632-3644.
    24. Muyi Li & Wai Keung Li & Guodong Li, 2013. "On Mixture Memory Garch Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(6), pages 606-624, November.
    25. Dacorogna, Michael M. & Muller, Ulrich A. & Nagler, Robert J. & Olsen, Richard B. & Pictet, Olivier V., 1993. "A geographical model for the daily and weekly seasonal volatility in the foreign exchange market," Journal of International Money and Finance, Elsevier, vol. 12(4), pages 413-438, August.
    26. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kunal Saha & Vinodh Madhavan & Chandrashekhar G. R. & David McMillan, 2020. "Pitfalls in long memory research," Cogent Economics & Finance, Taylor & Francis Journals, vol. 8(1), pages 1733280-173, January.
    2. Pan, Qunxing & Li, Peng & Du, Xiuli, 2023. "An improved FIGARCH model with the fractional differencing operator (1-νL)d," Finance Research Letters, Elsevier, vol. 55(PB).
    3. Lanciné Bamba & Ouagnina Hili & Abdou Kâ Diongue & Assi N’Guessan, 2021. "M-Estimate for the stationary hyperbolic GARCH models," METRON, Springer;Sapienza Università di Roma, vol. 79(3), pages 303-351, December.
    4. Ling, Shiqing & McAleer, Michael & Tong, Howell, 2015. "Frontiers in Time Series and Financial Econometrics: An overview," Journal of Econometrics, Elsevier, vol. 189(2), pages 245-250.
    5. Heitham Al-Hajieh, 2017. "Evaluated the Success of Fractionally Integrated-GARCH Models on Prediction Stock Market Return Volatility in Gulf Arab Stock Markets," International Journal of Economics and Finance, Canadian Center of Science and Education, vol. 9(7), pages 200-213, July.
    6. Klein, Tony & Walther, Thomas, 2017. "Fast fractional differencing in modeling long memory of conditional variance for high-frequency data," Finance Research Letters, Elsevier, vol. 22(C), pages 274-279.
    7. Yanlin Shi, 2021. "Forecasting mortality rates with the adaptive spatial temporal autoregressive model," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 40(3), pages 528-546, April.
    8. Toktam Valizadeh & Saeid Rezakhah & Ferdous Mohammadi Basatini, 2021. "On time‐varying amplitude HGARCH model," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 26(2), pages 2538-2547, April.
    9. Ling, S. & McAleer, M.J. & Tong, H., 2015. "Frontiers in Time Series and Financial Econometrics," Econometric Institute Research Papers EI 2015-07, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    10. Dima, Bogdan & Dima, Ştefana Maria, 2017. "Mutual information and persistence in the stochastic volatility of market returns: An emergent market example," International Review of Economics & Finance, Elsevier, vol. 51(C), pages 36-59.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. repec:wyi:journl:002190 is not listed on IDEAS
    2. Muyi Li & Wai Keung Li & Guodong Li, 2013. "On Mixture Memory Garch Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(6), pages 606-624, November.
    3. Kwan, Wilson & Li, Wai Keung & Li, Guodong, 2012. "On the estimation and diagnostic checking of the ARFIMA–HYGARCH model," Computational Statistics & Data Analysis, Elsevier, vol. 56(11), pages 3632-3644.
    4. Pan, Qunxing & Li, Peng & Du, Xiuli, 2023. "An improved FIGARCH model with the fractional differencing operator (1-νL)d," Finance Research Letters, Elsevier, vol. 55(PB).
    5. Carl Lönnbark, 2016. "Asymmetry with respect to the memory in stock market volatilities," Empirical Economics, Springer, vol. 50(4), pages 1409-1419, June.
    6. Torben G. Andersen & Tim Bollerslev & Peter F. Christoffersen & Francis X. Diebold, 2005. "Volatility Forecasting," PIER Working Paper Archive 05-011, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    7. Andersen, Torben G. & Bollerslev, Tim & Christoffersen, Peter F. & Diebold, Francis X., 2006. "Volatility and Correlation Forecasting," Handbook of Economic Forecasting, in: G. Elliott & C. Granger & A. Timmermann (ed.), Handbook of Economic Forecasting, edition 1, volume 1, chapter 15, pages 777-878, Elsevier.
    8. Klein, Tony & Walther, Thomas, 2016. "Oil price volatility forecast with mixture memory GARCH," Energy Economics, Elsevier, vol. 58(C), pages 46-58.
    9. HAFNER, Christian & PREMINGER, Arie, 2016. "On Asymptotic Theory for ARCH(infinite) Models," LIDAM Discussion Papers CORE 2016030, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    10. Lee, Oesook, 2018. "Stationarity and functional central limit theorem for ARCH(∞) models," Economics Letters, Elsevier, vol. 162(C), pages 107-111.
    11. Peter M Robinson & Paolo Zaffaroni, 2005. "Pseudo-Maximum Likelihood Estimation of ARCH(8) Models," STICERD - Econometrics Paper Series 495, Suntory and Toyota International Centres for Economics and Related Disciplines, LSE.
    12. Robinson, Peter M. & Zaffaroni, Paolo, 2005. "Pseudo-maximum likelihood estimation of ARCH(∞) models," LSE Research Online Documents on Economics 58182, London School of Economics and Political Science, LSE Library.
    13. Toktam Valizadeh & Saeid Rezakhah & Ferdous Mohammadi Basatini, 2021. "On time‐varying amplitude HGARCH model," International Journal of Finance & Economics, John Wiley & Sons, Ltd., vol. 26(2), pages 2538-2547, April.
    14. repec:awi:wpaper:0472 is not listed on IDEAS
    15. Robinson, Peter M. & Zafaroni, Paolo, 2005. "Pseudo-maximum likelihood estimation of ARCH models," LSE Research Online Documents on Economics 4544, London School of Economics and Political Science, LSE Library.
    16. Conrad, Christian, 2010. "Non-negativity conditions for the hyperbolic GARCH model," Journal of Econometrics, Elsevier, vol. 157(2), pages 441-457, August.
    17. Yanlin Shi & Yang Yang, 2018. "Modeling High Frequency Data with Long Memory and Structural Change: A-HYEGARCH Model," Risks, MDPI, vol. 6(2), pages 1-28, March.
    18. Christian Francq & Jean-Michel Zakoïan, 2013. "Optimal predictions of powers of conditionally heteroscedastic processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 75(2), pages 345-367, March.
    19. Diongue, Abdou Kâ & Guégan, Dominique, 2007. "The stationary seasonal hyperbolic asymmetric power ARCH model," Statistics & Probability Letters, Elsevier, vol. 77(11), pages 1158-1164, June.
    20. Bhaskar Bagchi & Biswajit Paul, 2023. "Effects of Crude Oil Price Shocks on Stock Markets and Currency Exchange Rates in the Context of Russia-Ukraine Conflict: Evidence from G7 Countries," JRFM, MDPI, vol. 16(2), pages 1-18, January.
    21. Lang, Korbinian & Auer, Benjamin R., 2020. "The economic and financial properties of crude oil: A review," The North American Journal of Economics and Finance, Elsevier, vol. 52(C).
    22. Liudas Giraitis & Donatas Surgailis & Andrius Škarnulis, 2015. "Integrated ARCH, FIGARCH and AR Models: Origins of Long Memory," Working Papers 766, Queen Mary University of London, School of Economics and Finance.

    More about this item

    Keywords

    ARCH(∞); Hyperbolic GARCH; Long-range dependence; QMLE;
    All these keywords.

    JEL classification:

    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:econom:v:189:y:2015:i:2:p:428-436. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/jeconom .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.