I am using a BEKK model in the following form,
$$H_t=C^\ast{C^\ast}^\prime+\sum_{i=1}^{m}{A_i\varepsilon_{t-i}\varepsilon_{t-i}A_i^\prime+\sum_{j=1}^{s}{B_jH_{t-j}B_j^\prime}}$$
I first start with a BEKK(1,0)
$$H_t=C^\ast{C^\ast}^\prime+A_{11}\varepsilon_{t-1}\varepsilon_{t-1}^\prime A_{11}^\prime$$
I am looking at a bivariate model, $C^\ast=\left[\begin{matrix}c_{11}&0\\c_{12}&c_{12}\\\end{matrix}\right]$, $A_{11}=\left[\begin{matrix}\alpha_{11}&\alpha_{12}\\\alpha_{21}&\alpha_{22}\\\end{matrix}\right]$,
My question is, what would be suitable C's and A's to indicate causaility in variance.
I understand that multiplying this model out yields,
$$h_{11t}=c_{11}^2+\alpha_{11}^2\varepsilon_{1,t-1}^2+2\alpha_{11}\alpha_{12}\varepsilon_{1,t-1}+\alpha_{12}^2\varepsilon_{2,t-1}^2$$ $$\ h_{22t}=c_{12}^2+c_{22}^2+\alpha_{21}^2\varepsilon_{1,t-1}^2+2\alpha_{21}\alpha_{22}\varepsilon_{1,t-1}\varepsilon_{2,t-1}+\alpha_{22}^2\varepsilon_{2,t-1}^2 $$ $$h_{12t}=c_{11}c_{12}+\alpha_{21}\alpha_{11}\varepsilon_{1,t-1}^2+\alpha_{21}\alpha_{12}\varepsilon_{1,t-1}\varepsilon_{2,t-1}+\alpha_{22}\alpha_{11}\varepsilon_{1,t-1}\varepsilon_{2,t-1}+\alpha_{22}\alpha_{12}\varepsilon_{2,t-1}^2$$
