General matrix notation of a VAR(p)

This page just shows the details for different matrix notations of a VAR("p") process with "k" variables.

Var("p")

:$y\_\{t\}=c\; +\; A\_\{1\}y\_\{t-1\}\; +\; A\_\{2\}y\_\{t-2\}\; +\; cdots\; +\; A\_\{p\}y\_\{t-p\}\; +\; e\_\{t\},$Where each $y\_\{i\}$ is a "k x 1" vector and each $A\_\{i\}$ is a "k x k" matrix.

Large matrix notation

:

+ egin{bmatrix}e_{1,t} \ e_{2,t}\ vdots \ e_{k,t}end{bmatrix}

Equation by equation notation

Rewriting the "y" variables one to one gives:

$y\_\{1,t\}\; =\; c\_\{1\}\; +\; a\_\{1,1\}^1y\_\{1,t-1\}\; +\; a\_\{1,2\}^1y\_\{2,t-1\}\; +cdots\; +\; a\_\{1,k\}^1y\_\{k,t-1\}+cdots+a\_\{1,1\}^py\_\{1,t-p\}+a\_\{1,2\}^py\_\{2,t-p\}+\; cdots\; +a\_\{1,k\}^py\_\{k,t-p\}\; +\; e\_\{1,t\},$$y\_\{2,t\}\; =\; c\_\{2\}\; +\; a\_\{2,1\}^1y\_\{1,t-1\}\; +\; a\_\{2,2\}^1y\_\{2,t-1\}\; +cdots\; +\; a\_\{2,k\}^1y\_\{k,t-1\}+cdots+a\_\{2,1\}^py\_\{1,t-p\}+a\_\{2,2\}^py\_\{2,t-p\}+\; cdots\; +a\_\{2,k\}^py\_\{k,t-p\}\; +\; e\_\{2,t\},$

$y\_\{k,t\}\; =\; c\_\{k\}\; +\; a\_\{k,1\}^1y\_\{1,t-1\}\; +\; a\_\{k,2\}^1y\_\{2,t-1\}\; +cdots\; +\; a\_\{k,k\}^1y\_\{k,t-1\}+cdots+a\_\{k,1\}^py\_\{1,t-p\}+a\_\{k,2\}^py\_\{2,t-p\}+\; cdots\; +a\_\{k,k\}^py\_\{k,t-p\}\; +\; e\_\{k,t\},$

Concise matrix notation

One can rewrite a VAR("p") with "k" variables in a general way which includes "T" observations $y\_\{0\}$ through $y\_\{T\}$

:$Y=BZ\; +U\; ,$

Where: :

:

:

and

:

One can then solve for the coefficient matrix "B" (e.g. using an ordinary least squares estimation of $Y\; approx\; BZ$)

References

* Helmut Lütkepohl. "New Introduction to Multiple Time Series Analysis". Springer. 2005.