# Difference between VAR model and simple vectorial regression

So I am aware of VAR models. Specifically for the VAR(1) case:

$X = A_1~LX +\epsilon_t$ where $L$ is the lag operator.

A simple regression between vectors would be:

$Z = A_2~Y+\epsilon$ where $Z$ and $Y$ are both vectors.

My question is: Assume that we set $Z=X$ and $Y=LX$ and then apply the second model. How would be the matrices $A_1$ and $A_2$ different?

Bluntly put; how is this simple vectorial regression different from a VAR(1)?

From the point of estimation there is no difference, in both VAR(1) and multiple vector regression the matrix $A_1$ and $A_2$ are estimated by OLS and are identical.
1. Multiple vector regression usually assumes that the $Y$ are fixed, in VAR(1) $LX$ are random variables.
2. In order for $A_2$ to be well defined you need that the the matrix $(\sum YY^T)$ is invertible ($EYY^T$ and $E\varepsilon Y^T=0$ if $Y$ is random variable). For VAR(1) the roots of characteristic equation $|I-\lambda A_1|$ must satisfy certain conditions.