Quasi Maximum Likelihood Estimation of Value at Risk and Expected Shortfall

Quasi maximum likelihood estimation of Value at Risk (VaR) and Expected Shortfall (ES) is discussed. The reference likelihood is that of a location-scale asymmetric Laplace distribution, related to a family of loss functions that lead to strictly consistent scoring functions for joint estimation of VaR and ES. The case of zero mean processes is considered, where quasi maximum likelihood estimators (QMLE) are consistent and asymptotically normal, as well as the case of non-zero mean processes, where quasi maximum likelihood estimators lead to inconsistent estimates due to lack of identification. In the latter situation, the asymptotic properties of two stage quasi maximum likelihood estimators (2SQMLE) are derived. QMLE and 2SQMLE are related with sample and M-estimators and compared in terms of asymptotic efficiency. A simulation study investigates the finite sample properties of QMLE, 2SQMLE, sample and M-estimators of expected shortfall.