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?

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#### Best Answer

Taking the variance of both sides of the equation $$ y_t = nu + A_1 y_{t-1} + u_t $$ leads to $$ operatorname{Var}y_t = A_1operatorname{Var}y_{t-1}A_1^T+Sigma_u. $$ Stationary implies that $operatorname{Var}y_t =operatorname{Var}y_{t-1}=Gamma_0$ so you need to solve the matrix equation $$ Gamma_0 = A_1Gamma_0 A_1^T+Sigma_u. $$ Applying the vec-function, this can be rewritten (see wikipedia) as $$ operatorname{vec}Gamma_0 = (A_1otimes A_1) operatorname{vec}Gamma_0 + operatorname{vec}Sigma_u $$ and solved using standard methods for the unknown covariances given by $$ operatorname{vec}Gamma_0 = (I-A_1otimes A_1)^{-1} operatorname{vec}Sigma_u. $$ So you don't need to work out the infinite sum from the MA$(infty)$-representation.