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Asymptotic theory for QMLE for the real‐time GARCH(1,1) model

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  • Ekaterina Smetanina
  • Wei Biao Wu

Abstract

We investigate the asymptotic properties of the Gaussian quasi‐maximum‐likelihood estimator (QMLE) for the Real‐time GARCH(1,1) model of Smetanina (2017, Journal of Financial Econometrics, 15(4), 561–601). The developed theory relies on the functional dependence measure and recently developed theory for derivative processes in Dahlhaus etal. (2019, Bernoulli, 25(2), 1013–1044). We prove stationarity and ergodicity of the underlying processes and consistency for the QMLE estimator under mild conditions. Furthermore, under normality of the error term, we also establish asymptotic normality for QMLE, which then becomes MLE, at the usual T rate. Finally, in our simulations we show that consistency and asymptotic normality holds for typical sample sizes.

Suggested Citation

  • Ekaterina Smetanina & Wei Biao Wu, 2021. "Asymptotic theory for QMLE for the real‐time GARCH(1,1) model," Journal of Time Series Analysis, Wiley Blackwell, vol. 42(5-6), pages 752-776, September.
    • Handle: RePEc:bla:jtsera:v:42:y:2021:i:5-6:p:752-776
    DOI: 10.1111/jtsa.12578
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    References listed on IDEAS

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    1. Babsiri, Mohamed El & Zakoian, Jean-Michel, 2001. "Contemporaneous asymmetry in GARCH processes," Journal of Econometrics, Elsevier, vol. 101(2), pages 257-294, April.
    2. Torben G. Andersen & Tim Bollerslev & Francis X. Diebold & Paul Labys, 2003. "Modeling and Forecasting Realized Volatility," Econometrica, Econometric Society, vol. 71(2), pages 579-625, March.
    3. Heejoon Han & Dennis Kristensen, 2014. "Asymptotic Theory for the QMLE in GARCH-X Models With Stationary and Nonstationary Covariates," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(3), pages 416-429, July.
    4. Bali, Turan G. & Mo, Hengyong & Tang, Yi, 2008. "The role of autoregressive conditional skewness and kurtosis in the estimation of conditional VaR," Journal of Banking & Finance, Elsevier, vol. 32(2), pages 269-282, February.
    5. Chris Brooks, 2005. "Autoregressive Conditional Kurtosis," Journal of Financial Econometrics, Oxford University Press, vol. 3(3), pages 399-421.
    6. 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.
    7. Francq, Christian & Zakoian, Jean-Michel, 2007. "Quasi-maximum likelihood estimation in GARCH processes when some coefficients are equal to zero," Stochastic Processes and their Applications, Elsevier, vol. 117(9), pages 1265-1284, September.
    8. Arnold Polanski & Evarist Stoja, 2010. "Incorporating higher moments into value-at-risk forecasting," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 29(6), pages 523-535.
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    Cited by:

    1. Ding, Yashuang (Dexter), 2023. "A simple joint model for returns, volatility and volatility of volatility," Journal of Econometrics, Elsevier, vol. 232(2), pages 521-543.

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