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.

自回归移动平均-广义自回归条件异方差（ARMA–GARCH）模型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特征。实证研究表明，此类模型的厚尾新息具有正的极值指数（extreme value index）。据此，可借助极值理论（extreme value theory）估计残差（residuals）的极端分位数。本文基于残差的加权序贯尾部经验过程（weighted sequential tail empirical process）的弱收敛（weak convergence）性质，推导了多类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（limiting distribution）。为构建置信区间（confidence intervals），本文提出采用自归一化（self-normalization）方法：相较于正态近似（normal approximation）方法，该方法可提升置信区间的覆盖率，尽管其生成的置信区间宽度略有扩大。本文还给出了估计过程中上阶统计量（upper order statistics）个数的数据驱动选取准则，并通过模拟实验（simulations）验证了该准则的有效性。最后，本文通过股指收益率（stock index returns）的实证应用，证实了CVaR与CES预测的提升效果。