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 quantile 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 period. The proposed models are applied to seven 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 a simple average of quantiles. The results of the forecasting experiments provide clear evidence in support of the proposed models.

提出了一种新的半参数预期短缺 (ES) 估计和预测框架。所提出的方法基于两步估计程序。第一步涉及通过一组分位数时间序列回归估计不同分位数水平的风险价值 (VaR)。然后，将 ES 计算为估计分位数的加权平均值。通过最小化 Fissler-Ziegel 类的联合 VaR 和 ES 损失函数来优化其系数，从而对分位数加权结构进行简约参数化。首先通过使用两个数据生成过程的广泛模拟研究来评估所提出方法的特性。然后进行了两个具有不同样本外规模的预测研究，其中之一侧重于 2008 年全球金融危机时期。将所提出的模型应用于七个股票市场指数，并将它们的预测性能与一系列参数、非参数和半参数模型的预测性能进行比较，包括 GARCH、条件自回归期望 (CARE)、联合 VaR 和 ES 分位数回归模型和简单的分位数平均值。预测实验的结果为支持所提出的模型提供了明确的证据。