I'm trying to estimate CoVaR using bivariate DCC GARCH in R. The concept of CoVaR is the dependence adjusted of VaR, which was first introduced by Adrian and Brunnermeier (2011). However, this original definition of CoVaR presented some limitations, which called for a modified version proposed by Girardi and Tolga Ergun (2013). For given confidence levels $\alpha$ and $\beta$,
$$Pr(X_t^i \leq CoVaR_{\beta,t}^{i|j} \,|\,X_t^j \leq VaR_{\alpha,t}^j ) = \beta.$$
Girardi et al. suggests a three-step procedure is used to estimate CoVaR.
Step 1: VaR of each institution $i$ is obtained by estimating a univariate GARCH (1,1) models for each time period. (This part I already know how to do using the package rugarch -- make forecast return and volatility and find the quantile.)
Step 2: For the return of institution $i$ and $j$, set up a bivariate GARCH model with DCC specification to estimate the pdf of $(X^i,X^j)$. (I'm also fine with this part -- I know how to solve for the parameters using the rmgarch package.)
Step 3: If we start from the definition of $CoVaR^{i|j}_{\beta,t}$ above, by definition $Pr(X_t^i \leq VaR_{\alpha,t}^i ) = \alpha$ so
$$Pr(X_t^i \leq CoVaR_{\beta,t}^{i|j} \,,\,X_t^j \leq VaR_{\alpha,t}^j )=\alpha\beta.$$
Once $VaR_{\alpha,t}^i$ and pdf of $(X^i,X^j)$ have been estimated in previous two steps, CoVaR can be obtained by numerically solving the equation,
$$\int_{-\infty}^{CoVaR_{\beta,t}^{i|j}}\int_{-\infty}^{VaR_{\alpha,t}^{i|j}}f_t(X^i , X^j )dxdy=\alpha\beta,$$
where $f_t(X^i , X^j)$ is the bivariate density of $(X^i,X^j)$.
This is the part that got me stumped. How can I obtain the bivariate density from the DCC GARCH parameters? And after I have the density figured out, how can I solve for the integral bound to get CoVaR?
Any input is appreciated, thanks!
References
1. Tobias Adrian and Markus K. Brunnermeier. CoVaR. Technical report, National Bureau of Economic Research, 2011.
2. Giulio Girardi and A. Tolga Ergun. Systemic risk measurement: Multivariate GARCH estimation of CoVaR. Journal of Banking & Finance, 37(8):3169–3180, 2013.
