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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)?

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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.

However proving that the resulting estimates have nice properties of consistency and asymptotic normality is different in both cases, since the assumptions are different for multiple vector regression and VAR(1). Here are some of the difference in assumptions:

  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.

I've listed few of the differences. Any textbook on VAR(1) and multiple vector regression has a detailed list, which you can compare by yourself.

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  • $\begingroup$ This answer is to the point, thank you very much. Could you also include a reference to one of these lists that you mention? $\endgroup$ Jun 21, 2016 at 10:07
  • $\begingroup$ For VAR(1) you could use J.D. Hamilton Time Series Analysis and for multiple regression you can adapt J. Wooldridge Cross section and panel data analysis. $\endgroup$
    – mpiktas
    Jun 21, 2016 at 10:55

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