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## Looking for efficient QML estimation of conditional VaRs at multiple risk levels

Christian Francq and Jean-Michel Zakoïan
Keywords: Asymmetric Power GARCH, Distortion Risk Measures, Estimation risk, Non-Gaussian Quasi-Maximum Likelihood, Value-at-Risk.
Abstract: We consider joint estimation of conditional Value-at-Risk (VaR) at several levels, in the framework of general GARCH-type models. The conditional VaR at level $\alpha$ is expressed as the product of the volatility and the opposite of the $\alpha$-quantile of the innovation. A standard method is to estimate the volatility parameters by Gaussian Quasi-Maximum Likelihood (QML) in a first step, and to use the residuals for estimating the innovations quantiles in a second step. We argue that the Gaussian QML may be inefficient with respect to more general QML and can even be in failure for heavy tailed conditional distributions. We therefore study, for a vector of risk levels, a two-step procedure based on a generalized QML. For a portfolio of VaRs at different levels, confidence intervals accounting for both market and estimation risks are deduced. An empirical study based on stock indices illustrates the theoretical results.
Current version of the paper
Archive with the R codes and data sets for the numerical illustrations of the paper (extract the zip file and see the file readme.txt for an explanation of the R codes)