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We propose a two-step semi-parametric estimation approach for dynamic Conditional VaR (CoVaR), from which other important systemic risk measures such as the Delta-CoVaR can be derived. The CoVaR allows to define reserves for a given financial entity, in order to limit exceeding losses when a system is in distress. We assume that all financial returns in the system follow semi-parametric GARCH-type models. Our estimation method relies on the fact that the dynamic CoVaR of one return conditional to a second return series is the product of the volatility of the financial entity's return and a conditional quantile term involving the innovations of the different returns. We show that the latter quantity can be easily estimated from residuals of the GARCH-type models estimated by Quasi-Maximum Likelihood (QML). The study of the asymptotic behaviour of the corresponding estimator and the derivation of asymptotic confidence intervals for the dymanic CoVaR are the main purposes of the paper. Our theoretical results are illustrated via Monte-Carlo experiments and real financial time series.
