Consistent estimation of the Value-at-Risk when the error distribution of the volatility model is misspecified
AbstractA 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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Bibliographic InfoPaper provided by University Library of Munich, Germany in its series MPRA Paper with number 51150.
Date of creation: Oct 2013
Date of revision:
APARCH; Conditional VaR; Distortion Risk Measures; GARCH; Generalized Quasi Maximum Likelihood Estimation; Instrumental density.;
Find related papers by JEL classification:
- C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models &bull Diffusion Processes
- C58 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Financial Econometrics
This paper has been announced in the following NEP Reports:
- NEP-ALL-2013-11-09 (All new papers)
- NEP-ECM-2013-11-09 (Econometrics)
- NEP-RMG-2013-11-09 (Risk Management)
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