El Ghourabi, Mohamed and Francq, Christian and Telmoudi, Fedya (2013): Consistent estimation of the ValueatRisk when the error distribution of the volatility model is misspecified.

PDF
MPRA_paper_51150.pdf Download (352kB)  Preview 
Abstract
A twostep approach for conditional Value at Risk (VaR) estimation is considered. In the first step, a generalizedquasimaximum 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 heavytailed. We extended our approach to Distortion Risk Measure parameter estimation, where consistency of the gQMLEbased method is also proved. Numerical illustrations are provided, through simulation experiments and an application to financial stock indexes.
Item Type:  MPRA Paper 

Original Title:  Consistent estimation of the ValueatRisk when the error distribution of the volatility model is misspecified 
Language:  English 
Keywords:  APARCH, Conditional VaR, Distortion Risk Measures, GARCH, Generalized Quasi Maximum Likelihood Estimation, Instrumental density. 
Subjects:  C  Mathematical and Quantitative Methods > C2  Single Equation Models ; Single Variables > C22  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes C  Mathematical and Quantitative Methods > C5  Econometric Modeling > C58  Financial Econometrics 
Item ID:  51150 
Depositing User:  Christian Francq 
Date Deposited:  06 Nov 2013 03:47 
Last Modified:  30 Sep 2019 19:51 
References:  Artzner, P., Delbaen, F., Eber, JM. and D. Heath (1999) Coherent measures of risk. Mathematical Finance 9, 203228. Bardet, JM. and O. Wintenberger (2009) Asymptotic normality of the Quasimaximum likelihood estimator for multidimensional causal processes. The Annals of Statistics 37, 27302759. Bassett, GW. and RW. Koenker (1986) Strong Consistency of Regression Quantiles and Related Empirical Processes. Econometric Theory 2, 191201. Berkes, I. and L. Horv\'ath (2004) The efficiency of the estimators of the parameters in GARCH processes. The Annals of Statistics 32, 633655. Berkes, I., Horv\'ath, L. and P. Kokoszka (2003) GARCH processes: structure and estimation. Bernoulli 9, 201227. Billingsley, P. (1961) Statistical inference for Markov processes. Chicago: University of Chicago Press. Billingsley, P. (1995) Probability and Measure. Third edition John Wiley and Sons Bollerslev, T. (1986) Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics 31, 307327. Davis, R.A., Knight, K. and J. Liu (1992) Mestimation for autoregressions with infinite variance. Stochastic Processes and their Applications 40, 145180. Ding, Z., Granger C. et R.F. Engle (1993) A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83106. Engle, RF. (1982) Autoregressive conditional heteroscedasticity with estimates of the variance of UK inflation. Econometrica 50, 9871008. Fan, J., Qi, L. and D. Xiu (2013) Quasi maximum likelihood estimation of GARCH models with heavytailed likelihoods. Available at SSRN: http://ssrn.com/abstract=1540363 or http://dx.doi.org/10.2139/ssrn.1540363 Francq, C., Lepage, G. and JM. Zakoïan (2011) Twostage non Gaussian QML estimation of GARCH Models and testing the efficiency of the Gaussian QMLE. Journal of Econometrics 165, 246257. Francq, C. and JM. Zakoïan (2004) Maximum likelihood estimation of pure GARCH and ARMAGARCH processes. Bernoulli 10, 605637. Francq, C. and JM. Zakoïan (2012) Riskparameter estimation in volatility models. MPRA Preprint No. 41713. Francq, C. and JM. Zakoïan (2013) Optimal predictions of powers of conditionally heteroskedastic processes. Journal of the Royal Statistical Society  Series B 75, 345367. Glosten, L., Jagannathan, R. and D. Runkle (1993) Relationship between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance 48, 17791801. Hamadeh, T., and JM. Zakoïan (2011)Asymptotic properties of LS and QML estimators for a class of nonlinear GARCH Processes. Journal of Statistical Planning and Inference 141, 488–507. Knight, K. (1998) Limiting distributions for $L_1$ regression estimators under general conditions. The Annals of Statistics 26, 755770. Koenker, R. (2005) Quantile Regression. Cambridge: Cambridge University Press. Koenker, R. and Z. Xiao (2006) Quantile autoregression. Journal of the American Statistical Association 101, 980990. Kuester, K., Mittnik, S. and M.S. Paolella (2006) ValueatRisk predictions: A comparison of alternative strategies. Journal of Financial Econometrics 4, 5389. Lee, S.W. and B.E. Hansen (1994) Asymptotic theory for the GARCH(1,1) quasimaximum likelihood estimator, Econometric Theory 10, 2952. Lumsdaine, R.L. (1996) Consistency and asymptotic normality of the quasimaximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models. Econometrica 64, 575596. McNeil, A. J., Frey, R. and P. Embrechts (2005) Quantitative Risk Management. Princeton University Press. Mikosch, T. and D. Straumann (2006) Stable limits of martingale transforms with application to the estimation of GARCH parameters. The Annals of Statistics 34, 493522. Pollard, D. (1991) Asymptotics for Least Absolute Deviation Regression Estimators Econometric Theory 7, 186199. Straumann, D. and T. Mikosch (2006) Quasimaximum likelihood estimation in conditionally heteroscedastic Time Series: a stochastic recurrence equations approach. The Annals of Statistics 5, 24492495. Wang, S. (2000) A class of distortion operators for pricing financial and insurance risks. Journal of Risk and Insurance 67, 1536. White, H. (1982) Maximum likelihood estimation of misspecified models. Econometrica 50, 125. Wirch, J. L. and M. R. Hardy (1999) A Synthesis of Risk Measures for Capital Adequacy. Insurance: Mathematics and Economics 25, 337347. Xiao, Z. and R. Koenker (2009) Conditional quantile estimation for generalized autoregressive conditional heteroscedasticity models. Journal of the American Statistical Association 104, 16961712. Xiao, Z. and C. Wan (2010) A robust estimator of conditional volatility. Unpublished document. Zakoïan J.M. (1994) Threshold Heteroskedastic Models. Journal of Economic Dynamics and Control 18, 931955. Zhu, K. and S. Ling (2011) Global selfweighted and local quasimaximum exponential likelihood estimators for ARMAGARCH/IGARCH models. The Annals of Statistics 32, 21312163. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/51150 