Dynamic Co-Quantile Regression

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 a dynamic "Co-Quantile Regression", which jointly models VaR and CoVaR semiparametrically. We propose a two-step M-estimator 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 the asymptotic normality of the proposed estimator and simulations illustrate its good finite-sample properties. We apply our co-quantile regression to correct the statistical inference in the existing literature on CoVaR, and to generate CoVaR forecasts for real financial data, which are shown to be superior to existing methods.