Looking for Efficient QML Estimation of Conditional VaRs at Multiple Risk Levels

Abstract

We consider joint estimation of conditional Value-at-Risk (VaR) at several levels - within the framework of general GARCH-type models. The conditional VaR at level a is expressed as the product of the volatility and the opposite of the ? -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 innovation's 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.