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）被广泛用于刻画风险资产收益率的条件均值与条件方差动态特征。已有实证结果表明，此类模型的残差服从具有正极值指数（extreme value index）的厚尾创新分布。因此，可借助极值理论（extreme value theory）对残差的极端分位数进行估计。本文基于残差的加权序贯尾部经验过程的弱收敛性质，推导了一系列极值指数估计量对应的极端条件风险价值（CVaR）与条件期望短缺（CES）估计量的极限分布。为构建置信区间，本文提出采用自正则化方法；相较于正态近似法，该方法可提升区间覆盖率，尽管置信区间宽度略有增加。本文还给出了估计中上端次序统计量数量的数据驱动选取准则，并通过仿真实验验证了该准则的有效性。最后，本文将所提方法应用于股票指数收益率数据，实证结果表明该方法对CVaR与CES的预测性能有所提升。