Confidence Intervals for Conditional Tail Risk Measures in ARMA-GARCH Models

ARMA-GARCH models are widely used to model the conditional mean and conditional variance dynamics of returns on risky assets. Empirical results suggest heavy-tailed innovations with positive extreme value index for these models. Hence, one may use extreme value theory to estimate extreme quantiles of residuals. Using weak convergence of the weighted sequential tail empirical process of the residuals, we derive the limiting distribution of extreme conditional Value-at-Risk (CVaR) and conditional Expected Shortfall (CES) estimates for a wide range of extreme value index estimators. To construct confidence intervals we propose to use self-normalization. This leads to improved coverage vis-à-vis the normal approximation, while delivering slightly wider confidence intervals. A data-driven choice of the number of upper order statistics in the estimation is suggested and shown to work well in simulations. An application to stock index returns documents the improvements of CVaR and CES forecasts.

ARMA-GARCH模型（ARMA-GARCH models）已被广泛应用于刻画风险资产收益率的条件均值与条件方差动态演化规律。实证研究结果表明，此类模型的新息项服从厚尾分布，且具有正的极值指数。故此，可借助极值理论对残差的极端分位数开展估计。基于残差的加权序贯尾部经验过程的弱收敛性质，本文推导了多类极值指数估计量对应的极端条件在险价值（Conditional Value-at-Risk，CVaR）与条件期望短缺（Conditional Expected Shortfall，CES）估计量的极限分布。为构建置信区间，本文提出采用自标准化方法：相较于正态近似法，该方法可提升置信区间的覆盖性能，尽管所得区间宽度略有增加。本文还给出了估计过程中上阶统计量个数的数据驱动选择方案，并通过仿真实验验证了该方案的良好表现。对股票指数收益率的实证应用结果表明，该方法可有效改善CVaR与CES的预测效果。