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Consistent estimation of the Value-at-Risk when the error distribution of the volatility model is misspecified

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  • El Ghourabi, Mohamed
  • Francq, Christian
  • Telmoudi, Fedya

Abstract

A two-step approach for conditional Value at Risk (VaR) estimation is considered. In the first step, a generalized-quasi-maximum likelihood estimator (gQMLE) is employed to estimate the volatility parameter, and in the second step the empirical quantile of the residuals serves to estimate the theoretical quantile of the innovations. When the instrumental density $h$ of the gQMLE is not the Gaussian density utilized in the standard QMLE, or is not the true distribution of the innovations, both the estimations of the volatility and of the quantile are asymptotically biased. The two errors however counterbalance each other, and we finally obtain a consistent estimator of the conditional VaR. For a wide class of GARCH models, we derive the asymptotic distribution of the VaR estimation based on gQMLE. We show that the optimal instrumental density $h$ depends neither on the GARCH parameter nor on the risk level, but only on the distribution of the innovations. A simple adaptive method based on empirical moments of the residuals makes it possible to infer an optimal element within a class of potential instrumental densities. Important asymptotic efficiency gains are achieved by using gQMLE instead of the usual Gaussian QML when the innovations are heavy-tailed. We extended our approach to Distortion Risk Measure parameter estimation, where consistency of the gQMLE-based method is also proved. Numerical illustrations are provided, through simulation experiments and an application to financial stock indexes.

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Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 51150.

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Date of creation: Oct 2013
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Handle: RePEc:pra:mprapa:51150

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Keywords: APARCH; Conditional VaR; Distortion Risk Measures; GARCH; Generalized Quasi Maximum Likelihood Estimation; Instrumental density.;

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  1. Bassett, Gilbert W. & Koenker, Roger W., 1986. "Strong Consistency of Regression Quantiles and Related Empirical Processes," Econometric Theory, Cambridge University Press, vol. 2(02), pages 191-201, August.
  2. 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, 03.
  3. Tim Bollerslev, 1986. "Generalized autoregressive conditional heteroskedasticity," EERI Research Paper Series EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
  4. Koenker, Roger & Xiao, Zhijie, 2006. "Quantile Autoregression," Journal of the American Statistical Association, American Statistical Association, vol. 101, pages 980-990, September.
  5. Philippe Artzner & Freddy Delbaen & Jean-Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228.
  6. Francq, Christian & Lepage, Guillaume & Zakoïan, Jean-Michel, 2011. "Two-stage non Gaussian QML estimation of GARCH models and testing the efficiency of the Gaussian QMLE," Journal of Econometrics, Elsevier, vol. 165(2), pages 246-257.
  7. 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.
  8. Xiao, Zhijie & Koenker, Roger, 2009. "Conditional Quantile Estimation for Generalized Autoregressive Conditional Heteroscedasticity Models," Journal of the American Statistical Association, American Statistical Association, vol. 104(488), pages 1696-1712.
  9. 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(01), pages 29-52, March.
  10. Lynn Wirch, Julia & Hardy, Mary R., 1999. "A synthesis of risk measures for capital adequacy," Insurance: Mathematics and Economics, Elsevier, vol. 25(3), pages 337-347, December.
  11. 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 575-96, May.
  12. Keith Kuester & Stefan Mittnik & Marc S. Paolella, 2006. "Value-at-Risk Prediction: A Comparison of Alternative Strategies," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 4(1), pages 53-89.
  13. White, Halbert, 1982. "Maximum Likelihood Estimation of Misspecified Models," Econometrica, Econometric Society, vol. 50(1), pages 1-25, January.
  14. Francq, Christian & Zakoian, Jean-Michel, 2012. "Risk-parameter estimation in volatility models," MPRA Paper 41713, University Library of Munich, Germany.
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