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Garch models without positivity constraints: exponential or log garch?

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  • Francq, Christian
  • Wintenberger, Olivier
  • Zakoian, Jean-Michel

Abstract

This paper studies the probabilistic properties and the estimation of the asymmetric log-GARCH($p,q$) model. In this model, the log-volatility is written as a linear function of past values of the log-squared observations, with coefficients depending on the sign of the observations, and past log-volatility values. Conditions are obtained for the existence of solutions and finiteness of their log-moments. We also study the tail properties of the solution. Under mild assumptions, we show that the quasi-maximum likelihood estimation of the parameters is strongly consistent and asymptotically normal. Simulations illustrating the theoretical results and an application to real financial data are proposed.

Suggested Citation

  • Francq, Christian & Wintenberger, Olivier & Zakoian, Jean-Michel, 2012. "Garch models without positivity constraints: exponential or log garch?," MPRA Paper 41373, University Library of Munich, Germany.
  • Handle: RePEc:pra:mprapa:41373
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    References listed on IDEAS

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    Cited by:

    1. Ming Chen & Qiongxia Song, 2016. "Semi-parametric estimation and forecasting for exogenous log-GARCH models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 93-112, March.
    2. Christian Francq & Genaro Sucarrat, 2018. "An Exponential Chi-Squared QMLE for Log-GARCH Models Via the ARMA Representation," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 16(1), pages 129-154.
    3. Chen, Min & Zhu, Ke, 2015. "Sign-based portmanteau test for ARCH-type models with heavy-tailed innovations," Journal of Econometrics, Elsevier, vol. 189(2), pages 313-320.
    4. Christophe Chorro & Dominique Guegan & Florian Ielpo & Hanjarivo Lalaharison, 2014. "Testing for Leverage Effects in the Returns of US Equities," Documents de travail du Centre d'Economie de la Sorbonne 14022r, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne, revised Jan 2017.
    5. Olivier Wintenberger, 2013. "Continuous Invertibility and Stable QML Estimation of the EGARCH(1,1) Model," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(4), pages 846-867, December.
    6. Ke Zhu & Wai Keung Li, 2015. "A New Pearson-Type QMLE for Conditionally Heteroscedastic Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 33(4), pages 552-565, October.
    7. repec:spr:testjl:v:27:y:2018:i:1:d:10.1007_s11749-016-0506-2 is not listed on IDEAS
    8. Sucarrat, Genaro & Grønneberg, Steffen, 2016. "Models of Financial Return With Time-Varying Zero Probability," MPRA Paper 68931, University Library of Munich, Germany.
    9. Sucarrat, Genaro & Grønneberg, Steffen & Escribano, Alvaro, 2016. "Estimation and inference in univariate and multivariate log-GARCH-X models when the conditional density is unknown," Computational Statistics & Data Analysis, Elsevier, vol. 100(C), pages 582-594.
    10. Ming Chen & Qiongxia Song, 2016. "Semi-parametric estimation and forecasting for exogenous log-GARCH models," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(1), pages 93-112, March.
    11. Christophe Chorro & Dominique Guegan & Florian Ielpo & Hanjarivo Lalaharison, 2017. "Testing for Leverage Effects in the Returns of US Equities," Post-Print halshs-00973922, HAL.
    12. Escribano, Álvaro & Sucarrat, Genaro, 2013. "Unbiased QML Estimation of Log-GARCH Models in the Presence of Zero Returns," UC3M Working papers. Economics we1321, Universidad Carlos III de Madrid. Departamento de Economía.
    13. Zhu, Ke, 2015. "Hausman tests for the error distribution in conditionally heteroskedastic models," MPRA Paper 66991, University Library of Munich, Germany.
    14. Esmeralda Gonçalves & Joana Leite & NazarÉ Mendes-Lopes, 2016. "On the Distribution Estimation of Power Threshold Garch Processes," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(5), pages 579-602, September.
    15. Ali Ahmad & Christian Francq, 2016. "Poisson QMLE of Count Time Series Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(3), pages 291-314, May.
    16. Li, Dong & Li, Muyi & Wu, Wuqing, 2014. "On dynamics of volatilities in nonstationary GARCH models," Statistics & Probability Letters, Elsevier, vol. 94(C), pages 86-90.
    17. repec:gam:jjrfmx:v:10:y:2017:i:4:p:17-:d:113895 is not listed on IDEAS
    18. Christophe Chorro & Dominique Guegan & Florian Ielpo & Hanjarivo Lalaharison, 2014. "Testing for Leverage Effect in Financial Returns," Documents de travail du Centre d'Economie de la Sorbonne 14022, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    19. Min-Hsien Chiang & Ray Yeutien Chou & Li-Min Wang, 2016. "Outlier Detection in the Lognormal Logarithmic Conditional Autoregressive Range Model," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 78(1), pages 126-144, February.

    More about this item

    Keywords

    log-GARCH: Quasi-Maximum Likelihood: Strict stationarity: Tail index;

    JEL classification:

    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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