Unconditional mean and variance of a stationary VAR(1) model I am confused while trying to find a general expression for the mean and variance of a stationary VAR model.  I am trying to do it for VAR(1).  I also can't find it in the literature.  Can anyone help me?
 A: Taking the variance of both sides of the equation
$$
y_t = \nu + A_1 y_{t-1} + u_t
$$
leads to
$$
\operatorname{Var}y_t = A_1\operatorname{Var}y_{t-1}A_1^T+\Sigma_u.
$$
Stationary implies that $\operatorname{Var}y_t =\operatorname{Var}y_{t-1}=\Gamma_0$ so you need to solve the matrix equation
$$
\Gamma_0 = A_1\Gamma_0 A_1^T+\Sigma_u.
$$
Applying the vec-function, this can be rewritten (see wikipedia) as
$$
\operatorname{vec}\Gamma_0 = (A_1\otimes A_1) \operatorname{vec}\Gamma_0 + \operatorname{vec}\Sigma_u
$$
and solved using standard methods for the unknown covariances given by
$$
\operatorname{vec}\Gamma_0 = (I-A_1\otimes A_1)^{-1} \operatorname{vec}\Sigma_u.
$$
So you don't need to work out the infinite sum from the MA$(\infty)$-representation.
A: According to Lütkepohl (2005), p. 14-15, if we have a $K$-variate VAR(1) process of the form
$$
y_t = \nu + A_1 y_{t-1} + u_t,
$$
then the unconditional mean is 
$$
(I_K-A_1)^{-1}\nu
$$
(where $I_K$ is an identity matrix of dimension $K\times K$) and the unconditional covariance for lag $h$ (i.e. $\text{Cov}(y_t,y_{t-h})$) is
$$
\sum_{i=0}^\infty A_1^{h+i}\Sigma_u {A_1^i}'
$$
where $\Sigma_u$ is the covariance matrix of the error term $u_t$. Then the unconditional variance can be obtained by taking $h=0$ in the above expression.
The same applies to VAR($p$) after having expressed the process in its alternative $Kp$-dimensional VAR(1) representation.
These results are obtained using the vector moving-average (VMA) representation of the VAR(1) process. 
References


*

*Lütkepohl, Helmut. New Introduction to Multiple Time Series Analysis. Springer Science & Business Media, 2005.

