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}y_{1,t} \ y_{2,t}\ vdots \ y_{k,t}end{bmatrix}=egin{bmatrix}c_{1} \ c_{2}\ vdots \ c_{k}end{bmatrix}+egin{bmatrix}a_{1,1}^1&a_{1,2}^1 & cdots & a_{1,k}^1\ a_{2,1}^1&a_{2,2}^1 & cdots & a_{2,k}^1\ vdots& vdots& ddots& vdots\a_{k,1}^1&a_{k,2}^1 & cdots & a_{k,k}^1end{bmatrix}egin{bmatrix}y_{1,t-1} \ y_{2,t-1}\ vdots \ y_{k,t-1}end{bmatrix}+ cdots +egin{bmatrix}a_{1,1}^p&a_{1,2}^p & cdots & a_{1,k}^p\ a_{2,1}^p&a_{2,2}^p & cdots & a_{2,k}^p\ vdots& vdots& ddots& vdots\a_{k,1}^p&a_{k,2}^p & cdots & a_{k,k}^pend{bmatrix}egin{bmatrix}y_{1,t-p} \ y_{2,t-p}\ vdots \ y_{k,t-p}end{bmatrix}

+ 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: :$$Y=egin{bmatrix}y_{p} & y_{p+1} & cdots & y_{T}end{bmatrix} =egin{bmatrix}y_{1,p} & y_{1,p+1} & cdots & y_{1,T} \ y_{2,p} &y_{2,p+1} & cdots & y_{2,T}\vdots& vdots &vdots &vdots \ y_{k,p} &y_{k,p+1} & cdots & y_{k,T}end{bmatrix}

:$$B=egin{bmatrix} c & A_{1} & A_{2} & cdots & A_{p} end{bmatrix} = egin{bmatrix}c_{1} & a_{1,1}^1&a_{1,2}^1 & cdots & a_{1,k}^1 &cdots & a_{1,1}^p&a_{1,2}^p & cdots & a_{1,k}^p\ c_{2} & a_{2,1}^1&a_{2,2}^1 & cdots & a_{2,k}^1 &cdots & a_{2,1}^p&a_{2,2}^p & cdots & a_{2,k}^p \ vdots & vdots& vdots& ddots& vdots & cdots & vdots& vdots& ddots& vdots\c_{k} & a_{k,1}^1&a_{k,2}^1 & cdots & a_{k,k}^1 &cdots & a_{k,1}^p&a_{k,2}^p & cdots & a_{k,k}^p end{bmatrix}

:$$Z=egin{bmatrix}1 & 1 & cdots & 1 \y_{p-1} & y_{p} & cdots & y_{T-1}\y_{p-2} & y_{p-1} & cdots & y_{T-2}\vdots & vdots & ddots & vdots\y_{0} & y_{1} & cdots & y_{T-p}end{bmatrix} =egin{bmatrix}1 & 1 & cdots & 1 \y_{1,p-1} & y_{1,p} & cdots & y_{1,T-1} \y_{2,p-1} & y_{2,p} & cdots & y_{2,T-1} \vdots & vdots & ddots & vdots\y_{k,p-1} & y_{k,p} & cdots & y_{k,T-1} \y_{1,p-2} & y_{1,p-1} & cdots & y_{1,T-2} \y_{2,p-2} & y_{2,p-1} & cdots & y_{2,T-2} \vdots & vdots & ddots & vdots\y_{k,p-2} & y_{k,p-1} & cdots & y_{k,T-2} \vdots & vdots & ddots & vdots\y_{1,0} & y_{1,1} & cdots & y_{1,T-p} \y_{2,0} & y_{2,1} & cdots & y_{2,T-p} \vdots & vdots & ddots & vdots\y_{k,0} & y_{k,1} & cdots & y_{k,T-p} end{bmatrix}

and

:$$U= egin{bmatrix}e_{p} & e_{p+1} & cdots & e_{T}end{bmatrix}=egin{bmatrix}e_{1,p} & e_{1,p+1} & cdots & e_{1,T} \e_{2,p} & e_{2,p+1} & cdots & e_{2,T} \vdots & vdots & ddots & vdots \e_{k,p} & e_{k,p+1} & cdots & e_{k,T} end{bmatrix}.

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.