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 $\text{Var}(Z_t)$? Here's where I've gotten:
$$\text{Var}(Z_t) = \mathbb{E}[Z_tZ_t^T] = \mathbb{E}[(AZ_{t-1} + BZ_{t-2} + Ce_t)Z_t^T] = A\mathbb{E}[Z_{t-1}Z_t^T] + B\mathbb{E}[Z_{t-2}Z_t^T]$$
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:
$$\text{Var}(Z_t) = A\mathbb{E}[Z_{t-1}Z_{t-1}^T]A^T+ A\mathbb{E}[Z_{t-1}Z_{t-2}^T]B^T + B\mathbb{E}[Z_{t-2}Z_{t-1}^T]A^T + B\mathbb{E}[Z_{t-2}Z_{t-2}^T]B^T$$
But from here I can't really see a simple recursion that would lead me to a closed-form solution.
Any help?
self-study? Please add the tag then. Also, VAR(p) can be represented as VAR(1) for a higher-dimensional vector process, maybe that could help. Or perhaps an MA($\infty$) representation could be useful. – Richard Hardy Dec 30 '16 at 20:51