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.

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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:  06. Nov 2013 04:07 
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URI:  https://mpra.ub.unimuenchen.de/id/eprint/51150 