Censored Posterior and Predictive Likelihood in Bayesian Left-Tail Prediction for Accurate Value at Risk Estimation

Accurate prediction of risk measures such as Value at Risk (VaR) and Expected Shortfall (ES) requires precise estimation of the tail of the predictive distribution. Two novel concepts are introduced that offer a specific focus on this part of the predictive density: the censored posterior, a density for the model parameters that results if we replace the likelihood by a so-called censored likelihood in the posterior; and the censored predictive likelihood, which is used for Model Averaging for forecast combination. We perform extensive experiments involving simulated and empirical data. Our results show the ability of these new approaches to outperform the standard posterior and traditional Bayesian Model Averaging techniques in applications of Value-at-Risk prediction in GARCH models