Multi-level conditional VaR estimation in dynamic models

We consider joint estimation of conditional Value-at-Risk (VaR) at several levels, in the framework of general conditional heteroskedastic models. The volatility is estimated by Quasi-Maximum Likelihood (QML) in a first step, and the residuals are used to estimate the innovations quantiles in a second step. The joint limiting distribution of the volatility parameter and a vector of residual quantiles is derived. We deduce confidence intervals for general Distortion Risk Measures (DRM) which can be approximated by a finite number of VaR's. We also propose an alternative approach based on non Gaussian QML which, although numerically more cumbersome, has interest when the innovations distribution is fat tailed. An empirical study based on stock indices illustrates the theoretical findings.