The popular systemic risk measure CoVaR (conditional Value-at-Risk) is widely used in economics and finance. Formally, it is defined as an (extreme) quantile of one variable (e.g., losses in the financial system) conditional on some other variable (e.g., losses in a bank's shares) being in distress and, hence, measures the spillover of risks. In this article, we propose joint dynamic and semiparametric models for VaR and CoVaR together with a two-step M-estimator for the model parameters drawing on recently proposed bivariate scoring functions for the pair (VaR, CoVaR). Among others, this allows for the estimation of joint dynamic forecasting models for (VaR, CoVaR). We prove consistency and asymptotic normality of the proposed estimator and analyze its finite-sample properties in simulations. We apply our dynamic models to generate CoVaR forecasts for real financial data, which are shown to be superior to existing methods.

翻译：流行的系统性风险度量指标CoVaR（条件风险价值）广泛应用于经济学与金融学领域。其正式定义为：一个变量（如金融体系的损失）在另一个变量（如银行股价的损失）处于困境条件下的（极端）分位数，因而测度了风险的溢出效应。本文针对VaR与CoVaR提出联合动态半参数模型，并基于近期提出的(VaR, CoVaR)二元评分函数，构建模型参数的两步M估计量。该框架尤其支持对(VaR, CoVaR)的联合动态预测模型进行估计。我们证明了所提估计量的一致性与渐近正态性，并通过模拟分析其有限样本性质。将本文提出的动态模型应用于真实金融数据生成CoVaR预测值，结果表明其优于现有方法。