Empirical Likelihood Estimation of Value-at-Risk and Expected Shortfall With Moment Constraints

This paper proposes efficient estimation of risk measures by fully exploring the first and second moment information in a GARCH framework. We propose a quantile estimator based on inverting an empirical likelihood weighted distribution estimator. It is found that the new quantile estimator is uniformly more efficient than the simple empirical quantile and a quantile estimator based on normalized residuals. We show that the same conclusion applies to the estimation of conditional Expected Shortfall. We find that these proposed estimators for conditional Value-at-Risk and expected shortfall are asymptotically mixed normal. Simulation evidence provided.