CoVaR (conditional value-at-risk) is a crucial measure for assessing financial systemic risk, which is defined as a conditional quantile of a random variable, conditioned on other random variables reaching specific quantiles. It enables the measurement of risk associated with a particular node in financial networks, taking into account the simultaneous influence of risks from multiple correlated nodes. However, estimating CoVaR presents challenges due to the unobservability of the multivariate-quantiles condition. To address the challenges, we propose a two-step nonparametric estimation approach based on Monte-Carlo simulation data. In the first step, we estimate the unobservable multivariate-quantiles using order statistics. In the second step, we employ a kernel method to estimate the conditional quantile conditional on the order statistics. We establish the consistency and asymptotic normality of the two-step estimator, along with a bandwidth selection method. The results demonstrate that, under a mild restriction on the bandwidth, the estimation error arising from the first step can be ignored. Consequently, the asymptotic results depend solely on the estimation error of the second step, as if the multivariate-quantiles in the condition were observable. Numerical experiments demonstrate the favorable performance of the two-step estimator.

翻译：CoVaR（条件风险价值）是评估金融系统性风险的关键指标，定义为随机变量在其他随机变量达到特定分位数条件下的条件分位数。它能够衡量金融网络中特定节点的风险，同时考虑多个相关节点风险的同步影响。然而，由于多变量分位数条件的不可观测性，CoVaR的估计面临挑战。为解决这一问题，我们提出一种基于蒙特卡洛模拟数据的两步非参数估计方法。第一步，利用顺序统计量估计不可观测的多变量分位数；第二步，采用核方法估计以顺序统计量为条件的分位数。我们证明了两步估计量的一致性和渐近正态性，并给出了带宽选择方法。结果表明，在带宽的温和约束下，第一步产生的估计误差可忽略不计。因此，渐近结果仅取决于第二步的估计误差，如同条件中的多变量分位数是可观测的。数值实验验证了两步估计量的优异性能。