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Non-negativity Conditions for the Hyperbolic GARCH Model

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  • Christian Conrad

Abstract

In this article we derive conditions which ensure the non-negativity of the conditional variance in the Hyperbolic GARCH(p,d,q) (HYGARCH) model of Davidson (2004). The conditions are necessary and sufficient for p 2 and emerge as natural extensions of the inequality constraints derived in Nelson and Cao (1992) for the GARCH model and in Conrad and Haag (2006) for the FIGARCH model. As a by-product we obtain a representation of the ARCH(∞) coefficients which allows computationally efficient multi-step-ahead forecasting of the conditional variance of a HYGARCH process. We also relate the necessary and sufficient parameter set of the HYGARCH to the necessary and sufficient parameter sets of its GARCH and FIGARCH components. Finally, we analyze the effects of erroneously fitting a FIGARCH model to a data sample which was truly generated by a HYGARCH process. An empirical application of the HYGARCH(1,d,1) model to daily NYSE data illustrates the importance of our results.

Suggested Citation

  • Christian Conrad, 2007. "Non-negativity Conditions for the Hyperbolic GARCH Model," KOF Working papers 07-162, KOF Swiss Economic Institute, ETH Zurich.
  • Handle: RePEc:kof:wpskof:07-162
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    File URL: http://dx.doi.org/10.3929/ethz-a-005390226
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    Cited by:

    1. Antonio Rubia & Trino-Manuel Ñíguez, 2006. "Forecasting the conditional covariance matrix of a portfolio under long-run temporal dependence," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 25(6), pages 439-458.
    2. Charles, Amélie & Darné, Olivier, 2014. "Volatility persistence in crude oil markets," Energy Policy, Elsevier, vol. 65(C), pages 729-742.
    3. Mohamed Chikhi & Anne Péguin-Feissolle & Michel Terraza, 2013. "SEMIFARMA-HYGARCH Modeling of Dow Jones Return Persistence," Computational Economics, Springer;Society for Computational Economics, vol. 41(2), pages 249-265, February.
    4. Rafik Nazarian & Esmaeil Naderi & Nadiya G. Alikhani & Ashkan Amiri, 2014. "Long Memory Analysis: An Empirical Investigation," International Journal of Economics and Financial Issues, Econjournals, vol. 4(1), pages 16-26.
    5. Christian Conrad & Karin Loch, 2015. "Anticipating Long‐Term Stock Market Volatility," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 30(7), pages 1090-1114, November.
    6. Li, Muyi & Li, Wai Keung & Li, Guodong, 2015. "A new hyperbolic GARCH model," Journal of Econometrics, Elsevier, vol. 189(2), pages 428-436.
    7. Karanasos, Menelaos & Xu, Yongdeng, 2017. "Matrix Inequality Constraints for Vector (Asymmetric Power) GARCH/HEAVY Models and MEM with spillovers: some New (Mixture) Formulations," Cardiff Economics Working Papers E2017/14, Cardiff University, Cardiff Business School, Economics Section.
    8. Conrad, Christian & Rittler, Daniel & Rotfuß, Waldemar, 2012. "Modeling and explaining the dynamics of European Union Allowance prices at high-frequency," Energy Economics, Elsevier, vol. 34(1), pages 316-326.
    9. Morales-Pelagio, Ricardo Cristhian & López-Herrera, Francisco & Cabrera-Llanos, Agustín Ignacio, 2013. "Eficiencia de las principales acciones de la bolsa mexicana de valores: 2001-2012," eseconomía, Escuela Superior de Economía, Instituto Politécnico Nacional, vol. 0(37), pages 55-75, primer tr.
    10. Conrad, Christian & Karanasos, Menelaos & Zeng, Ning, 2011. "Multivariate fractionally integrated APARCH modeling of stock market volatility: A multi-country study," Journal of Empirical Finance, Elsevier, vol. 18(1), pages 147-159, January.
    11. Conrad, Christian & Loch, Karin & Rittler, Daniel, 2012. "On the Macroeconomic Determinants of the Long-Term Oil-Stock Correlation," Working Papers 0525, University of Heidelberg, Department of Economics.
    12. Conrad, Christian & Karanasos, Menelaos, 2010. "Negative Volatility Spillovers In The Unrestricted Eccc-Garch Model," Econometric Theory, Cambridge University Press, vol. 26(03), pages 838-862, June.
    13. 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.
    14. Karanasos, Menelaos & Yfanti, Stavroula & Karoglou, Michail, 2016. "Multivariate FIAPARCH modelling of financial markets with dynamic correlations in times of crisis," International Review of Financial Analysis, Elsevier, vol. 45(C), pages 332-349.
    15. repec:wyi:journl:002190 is not listed on IDEAS
    16. repec:eee:phsmap:v:483:y:2017:i:c:p:266-272 is not listed on IDEAS
    17. repec:eee:ecolet:v:162:y:2018:i:c:p:107-111 is not listed on IDEAS
    18. Lux, Thomas & Segnon, Mawuli & Gupta, Rangan, 2015. "Modeling and forecasting crude oil price volatility: Evidence from historical and recent data," FinMaP-Working Papers 31, Collaborative EU Project FinMaP - Financial Distortions and Macroeconomic Performance: Expectations, Constraints and Interaction of Agents.
    19. Conrad, Christian & Loch, Karin & Rittler, Daniel, 2014. "On the macroeconomic determinants of long-term volatilities and correlations in U.S. stock and crude oil markets," Journal of Empirical Finance, Elsevier, vol. 29(C), pages 26-40.
    20. repec:eee:reveco:v:51:y:2017:i:c:p:36-59 is not listed on IDEAS
    21. Klein, Tony & Walther, Thomas, 2016. "Oil price volatility forecast with mixture memory GARCH," Energy Economics, Elsevier, vol. 58(C), pages 46-58.

    More about this item

    Keywords

    Inequality constraints; Fractional integration; Long memory GARCH processes;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods

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