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This paper generalizes multivariate multi-quantile CAViaR models (White et al., 2015) by incorporating CoVaR specification (see Adrian and Brunnermeier, 2011) into the model specification. The proposed model presents a vector-autoregression (VAR) of financial institutions' value-at-risk (VaR) as well as their CoVaR. This model generalization is able to capture contemporaneous tail dependence of financial institutions and market indexes so that we can interpret the systemic risks of the institutions more timely. We provide consistency and asymptotic normality of the general model estimator as well as some relevant inference tests. For tracing the transmission of a single shock to a financial institution in the financial system, we also construct quantile impulse response functions (QIRF) accordingly using the local projection idea (Jorda, 2005) and score vectors. Applications to real data show strong evidence of contemporaneous effects of big banks on the market index S&P500, and supports this methodology.
