# 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? Commented Feb 26, 2017 at 12:17

## 2 Answers

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). Commented Dec 15, 2016 at 10:19
• Yes, it's just two different ways of expressing the same covariance matrix. Commented 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
Commented 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? Commented Jan 28, 2017 at 9:22
• @Bonsaibubble, the answer by Jarle Tufto does that. Commented Jan 28, 2017 at 9:51
• But how does this correspond to what Lütkepohl defines? Commented 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. Commented Jan 28, 2017 at 10:09
• Ok, I dont understand Tufto's approach then Commented Jan 28, 2017 at 10:13