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Normal Mixture Quasi‐maximum Likelihood Estimator for GARCH Models

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  • TAEWOOK LEE
  • SANGYEOL LEE

Abstract

. The generalized autoregressive conditional heteroscedastic (GARCH) model has been popular in the analysis of financial time series data with high volatility. Conventionally, the parameter estimation in GARCH models has been performed based on the Gaussian quasi‐maximum likelihood. However, when the innovation terms have either heavy‐tailed or skewed distributions, the quasi‐maximum likelihood estimator (QMLE) does not function well. In order to remedy this defect, we propose the normal mixture QMLE (NM‐QMLE), which is obtained from the normal mixture quasi‐likelihood, and demonstrate that the NM‐QMLE is consistent and asymptotically normal. Finally, we present simulation results and a real data analysis in order to illustrate our findings.

Suggested Citation

  • Taewook Lee & Sangyeol Lee, 2009. "Normal Mixture Quasi‐maximum Likelihood Estimator for GARCH Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 36(1), pages 157-170, March.
    • Handle: RePEc:bla:scjsta:v:36:y:2009:i:1:p:157-170
    DOI: 10.1111/j.1467-9469.2008.00624.x
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    1. Vlaar, Peter J G & Palm, Franz C, 1993. "The Message in Weekly Exchange Rates in the European Monetary System: Mean Reversion, Conditional Heteroscedasticity, and Jumps," Journal of Business & Economic Statistics, American Statistical Association, vol. 11(3), pages 351-360, July.
    2. repec:bla:jfinan:v:58:y:2003:i:2:p:753-778 is not listed on IDEAS
    3. Hall, Peter & Yao, Qiwei, 2003. "Inference in ARCH and GARCH models with heavy-tailed errors," LSE Research Online Documents on Economics 5875, London School of Economics and Political Science, LSE Library.
    4. Markus Haas, 2004. "Mixed Normal Conditional Heteroskedasticity," Journal of Financial Econometrics, Oxford University Press, vol. 2(2), pages 211-250.
    5. 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.
    6. Bollerslev, Tim, 1986. "Generalized autoregressive conditional heteroskedasticity," Journal of Econometrics, Elsevier, vol. 31(3), pages 307-327, April.
    7. Jensen, Søren Tolver & Rahbek, Anders, 2004. "Asymptotic Inference For Nonstationary Garch," Econometric Theory, Cambridge University Press, vol. 20(6), pages 1203-1226, December.
    8. 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.
    9. Bollerslev, Tim, 1987. "A Conditionally Heteroskedastic Time Series Model for Speculative Prices and Rates of Return," The Review of Economics and Statistics, MIT Press, vol. 69(3), pages 542-547, August.
    10. Lumsdaine, Robin L, 1996. "Consistency and Asymptotic Normality of the Quasi-maximum Likelihood Estimator in IGARCH(1,1) and Covariance Stationary GARCH(1,1) Models," Econometrica, Econometric Society, vol. 64(3), pages 575-596, May.
    11. Weiss, Andrew A., 1986. "Asymptotic Theory for ARCH Models: Estimation and Testing," Econometric Theory, Cambridge University Press, vol. 2(1), pages 107-131, April.
    12. Peter Hall & Qiwei Yao, 2003. "Inference in Arch and Garch Models with Heavy--Tailed Errors," Econometrica, Econometric Society, vol. 71(1), pages 285-317, January.
    13. Bai, Xuezheng & Russell, Jeffrey R. & Tiao, George C., 2003. "Kurtosis of GARCH and stochastic volatility models with non-normal innovations," Journal of Econometrics, Elsevier, vol. 114(2), pages 349-360, June.
    14. Berkes, István & Horváth, Lajos, 2003. "The rate of consistency of the quasi-maximum likelihood estimator," Statistics & Probability Letters, Elsevier, vol. 61(2), pages 133-143, January.
    15. A. Azzalini & A. Capitanio, 1999. "Statistical applications of the multivariate skew normal distribution," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(3), pages 579-602.
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    Cited by:

    1. Wang, Hui & Pan, Jiazhu, 2014. "Normal mixture quasi maximum likelihood estimation for non-stationary TGARCH(1,1) models," Statistics & Probability Letters, Elsevier, vol. 91(C), pages 117-123.
    2. Lee, Sangyeol & Tim Ng, Chi, 2010. "Trimmed portmanteau test for linear processes with infinite variance," Journal of Multivariate Analysis, Elsevier, vol. 101(4), pages 984-998, April.
    3. Hwang, S.Y. & Baek, J.S. & Park, J.A. & Choi, M.S., 2010. "Explosive volatilities for threshold-GARCH processes generated by asymmetric innovations," Statistics & Probability Letters, Elsevier, vol. 80(1), pages 26-33, January.
    4. Caporin, Massimiliano & Rossi, Eduardo & Santucci de Magistris, Paolo, 2017. "Chasing volatility," Journal of Econometrics, Elsevier, vol. 198(1), pages 122-145.
    5. Fiorentini, Gabriele & Sentana, Enrique, 2023. "Discrete mixtures of normals pseudo maximum likelihood estimators of structural vector autoregressions," Journal of Econometrics, Elsevier, vol. 235(2), pages 643-665.
    6. Lee, Sangyeol & Ng, Chi Tim, 2011. "Normality test for multivariate conditional heteroskedastic dynamic regression models," Economics Letters, Elsevier, vol. 111(1), pages 75-77, April.
    7. Ha, Jeongcheol & Lee, Taewook, 2011. "NM-QELE for ARMA-GARCH models with non-Gaussian innovations," Statistics & Probability Letters, Elsevier, vol. 81(6), pages 694-703, June.

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