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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?

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  • $\begingroup$ I think there have been some similar posts before. Have you tried searching for them here on Cross Validated? See e.g. this and this. $\endgroup$ – Richard Hardy Dec 30 '16 at 20:40
  • $\begingroup$ those are for VAR(1) processes, which are pretty standard. I haven't seen anything for any p>1 yet. $\endgroup$ – measure_theory Dec 30 '16 at 20:44
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    $\begingroup$ Is this 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. $\endgroup$ – Richard Hardy Dec 30 '16 at 20:51
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    $\begingroup$ It isn't self-study, but I think representing it as a VAR(1) process might help for what I'm trying to do. $\endgroup$ – measure_theory Dec 30 '16 at 21:00
  • $\begingroup$ See chapter 16.3.2 in William W.S. Wei, Time Series Analysis: Univariate and Multivariate Methods google.no/search?q=wei+time+series+analysis $\endgroup$ – Jarle Tufto Dec 30 '16 at 21:04

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