Relation between Vector Auto-regressive models and correlation matrix

Question

I am generating a multivariate time series using Vector Autoregressive Models- $$X(t) = AX(t-1) + \epsilon$$ where $X \in R^{n \times 1}$, $A \in R^{n \times n}$ and $\epsilon \in R^{n \times 1}$ is a constant. Is there a relation between $A$ matrix and correlation matrix storing the correlation information between different variables of the multivariate time series?

Correlation matrix can be defined as $$\rho _{i,j} = \text{Corr}(x_i,x_j) $$ where $\text{Corr}(x_i,x_j)$ is the correlation between $i^{th}$ and $j^{th}$ variable of the time series and $\rho _{i,j}$ is the $i,j$ element in the correlation matrix. I have asked the same question here.

Answer 1

You may express this VAR(1) model as a VMA($\infty$) model: \begin{aligned} X_t &= A X_{t-1}+\epsilon_t \\ (I-AL) X_t &= \epsilon_t \\ X_t &= (I-AL)^{-1} \epsilon_t. \\ \end{aligned} Here, $L$ stands for the lag operator and $I$ represents an identity matrix of appropriate dimension.
You can further express this as $$ X_t = \sum_{q=0}^\infty \Phi_q \epsilon_{t-q} $$ with coefficient matrices $\Phi_q$ where $\Phi_0=I$. Thus the components of $X_t$ are linear combinations of $\epsilon_t$ and its infinite lags with weights that are functions of $A$.

Given that $\epsilon$ is an i.i.d. vector with a contemporary covariance matrix $\Sigma_{\epsilon}$, we get that the covariance matrix of $X$, $\Sigma_X$ (with a typical element $\sigma_{X,ij}$), is $$\Sigma_X = \text{Cov}(X_t) = \text{Cov} \left( \sum_{q=0}^\infty \Phi_q \epsilon_{t-q} \right) = \sum_{q=0}^\infty \Phi_q \Sigma \Phi_q^{\top} $$ where $A$ is hiding in the matrices $\Phi_q$. Then the correlation between elements $X_i$ and $X_j$ can be obtained as $$ \rho_{X,ij} = \frac{ \sigma_{X,ij} }{ \sqrt{\sigma_{X,ii}\sigma_{X,jj}} }. $$