Nonparametric Expected Shortfall Forecasting Incorporating Weighted Quantiles

Abstract

A new semi-parametric Expected Shortfall (ES) estimation and forecasting framework is proposed. The proposed approach is based on a two-step estimation procedure. The first step involves the estimation of Value-at-Risk (VaR) at different quantile levels through a set of quantile time series regressions. Then, the ES is computed as a weighted average of the estimated quantiles. The quantiles weighting structure is parsimoniously parameterized by means of a Beta weight function whose coefficients are optimized by minimizing a joint VaR and ES loss function of the Fissler-Ziegel class. The properties of the proposed approach are first evaluated with an extensive simulation study using two data generating processes. Two forecasting studies with different out-of-sample sizes are then conducted, one of which focuses on the 2008 Global Financial Crisis (GFC) period. The proposed models are applied to 7 stock market indices and their forecasting performances are compared to those of a range of parametric, non-parametric and semi-parametric models, including GARCH, Conditional AutoRegressive Expectile (CARE), joint VaR and ES quantile regression models and simple average of quantiles. The results of the forecasting experiments provide clear evidence in support of proposed models.