Consistent Estimation of the Value at Risk When the Error Distribution of the Volatility Model is Misspecified

A two-step approach for conditional value at risk estimation is considered. First, a generalized quasi-maximum likelihood estimator is employed to estimate the volatility parameter, then the empirical quantile of the residuals serves to estimate the theoretical quantile of the innovations. When the instrumental density h of the generalized quasi-maximum likelihood estimator is not the Gaussian density, both the estimations of the volatility and of the quantile are generally asymptotically biased. However, the two errors counterbalance and lead to a consistent estimator of the value at risk. We obtain the asymptotic behavior of this estimator and show how to choose optimally h.