Journal Article

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

Hoga, Yannick

2019

Overview

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.

• Publisher: Taylor & Francis,American Statistical Association (ASA),Taylor & Francis Ltd
• Subject: ARMA-GARCH models / Conditional expected shortfall / Conditional Value-at-Risk / Confidence intervals / Extreme value index / Self-normalization / Stochastic models