Assume we have the following VAR(2) system, where $Z_t$ is an n-dimensional vector and $e_t\sim\mathcal{N}(0,I)$ are IID:
$$Z_t = AZ_{t-1} + BZ_{t-2} + Ce_t$$
Is there a nice, closed-form solution for $Var(Z_t)$? Here's where I've gotten:
$$Var(Z_t) = \mathbb{E}[Z_tZ_t^T] = \mathbb{E}[(AZ_{t-1} + BZ_{t-2} + Ce_t)Z_t^T] = A\mathbb{E}[Z_{t-1}Z_t^T] + B\mathbb{E}[Z_{t-2}Z_t^T]$$
since $e_t$ are mean zero. For the two cross-terms we have:
$$\mathbb{E}[Z_{t-1}Z_t^T] = \mathbb{E}[Z_{t-1}Z_{t-1}^T]A^T + \mathbb{E}[Z_{t-1}Z_{t-2}^T]B^T$$
$$\mathbb{E}[Z_{t-2}Z_t^T] = \mathbb{E}[Z_{t-2}Z_{t-1}^T]A^T + \mathbb{E}[Z_{t-2}Z_{t-2}^T]B^T$$ and plugging in yields:
$$Var(Z_t) = A\mathbb{E}[Z_{t-1}Z_{t-1}^T]A^T+ A\mathbb{E}[Z_{t-1}Z_{t-2}^T]B^T + B\mathbb{E}[Z_{t-2}Z_{t-1}^T]A^T + B\mathbb{E}[Z_{t-2}Z_{t-2}^T]B^T$$
But from here I can't really see a simple recursion that would lead me to a closed-form solution.
Any help?
I am not completely sure about this
$Z_t=ALZ_T+BL^2Z_t+CE_t$
$(1-AL-BL^2)Z_t=CE_t$
$Z_t=(1-(AL+BL^2)^{-1}CE_t$
Write $(1-(AL+BL^2)^{-1}$ as infinite geometric progression
$Z_t=CE_t+(AL+BL^2)CE_t+(AL+BL^2)^2CE_t+...$
Now take variance from both sides. Lag operators dont matter because $Var(E_t)=Var(E_{t-j})$ so you can just get rid of them.
$Var(Z_t)=CC^T+(A+B)CC^T(A+B)^T+...$
Wrap it back
$Var(Z_t)=(I-(A+B)(A+B)^T)^{-1}CC^T$