# 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?

• I was going through my old answers and noticed this one was not accepted. Do you perhaps need further clarification? Feb 26, 2017 at 12:17

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.

• If I understand correctly, the result is the same but your representation is more convenient as it readily yields an empirically feasible solution (unlike the infinite sum). Dec 15, 2016 at 10:19
• Yes, it's just two different ways of expressing the same covariance matrix. Dec 15, 2016 at 10:21
• Thanks for the answer. Quite what I was looking for. Not sure why the answer is not accepted. It should be.
– Xbel
Jul 27, 2021 at 7:17

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.
• Is there a way to derive the unconditional variance? Jan 28, 2017 at 9:22
• @Bonsaibubble, the answer by Jarle Tufto does that. Jan 28, 2017 at 9:51
• But how does this correspond to what Lütkepohl defines? Jan 28, 2017 at 10:07
• @Bonsaibubble, both answers agree on the substance (they do not imply different things), they just use different approach and notation. Jan 28, 2017 at 10:09
• Ok, I dont understand Tufto's approach then Jan 28, 2017 at 10:13