Combining Value-at-Risk and Expected Shortfall forecasts via the Model Confidence Set

Abstract

To comply with increasingly stringent international standards in risk management and regulation, several approaches have been developed in the literature for forecasting tail-risk measures such as Value-at-Risk (VaR) and Expected Shortfall (ES). However, the accuracy of these measures can be significantly affected by multiple sources of uncertainty, including model misspecification, data limitations and estimation procedures. To address these challenges and enhance the predictive performance of individual models, this study introduces novel forecast combination strategies based on the Model Confidence Set (MCS) methodology. Specifically, a strictly consistent joint VaR-ES loss function is employed to identify the best-performing models, which constitute the Set of Superior Models (SSM). Subsequently, the VaR and ES forecasts of the models included in the SSM are combined using various weighting schemes. An empirical analysis based on nine stock market indices at the 2.5\% and 1\% risk levels provides evidence that the proposed combined predictors are a robust alternative for forecasting tail-risk measures, successfully passing standard backtests and consistently entering the SSM of the MCS.