General matrix notation of a VAR(p)

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

1 Var(p)
2 Large matrix notation
3 Equation by equation notation
4 Concise matrix notation
5 References

[edit] Var(p)

$y_{t}=c + A_{1}y_{t-1} + A_{2}y_{t-2} + \cdots + A_{p}y_{t-p} + e_{t},$

Where y is a k x 1 vector of variables of length T and A is a k x p matrix.

[edit] Large matrix notation

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

[edit] Equation by equation notation

Rewriting the y variables one to one gives:

$y_{1,t} = c_{1} + a_{1,1}^1y_{1,t-1} + a_{1,2}^1y_{2,t-1} +\cdots + a_{1,k}^1y_{k,t-1}+\cdots+a_{p,1}^1y_{1,t-p}+a_{p,2}^1y_{2,t-p}+ \cdots +a_{p,k}^1y_{k,t-p} + e_{1,t}\,$ $y_{2,t} = c_{2} + a_{1,1}^2y_{1,t-1} + a_{1,2}^2y_{2,t-1} +\cdots + a_{1,k}^2y_{k,t-1}+\cdots+a_{p,1}^2y_{1,t-p}+a_{p,2}^2y_{2,t-p}+ \cdots +a_{p,k}^2y_{k,t-p} + e_{1,t}\,$ $\ \vdots \quad = \qquad \vdots$ $y_{k,t} = c_{k} + a_{1,1}^ky_{1,t-1} + a_{1,2}^ky_{2,t-1} +\cdots + a_{1,k}^ky_{k,t-1}+\cdots+a_{p,1}^ky_{1,t-p}+a_{p,2}^ky_{2,t-p}+ \cdots +a_{p,k}^ky_{k,t-p} + e_{1,t}\,$

[edit] Concise matrix notation

One can rewrite a VAR(p) with k variables in a general way

$Y=BZ +U \,$

Where:

$Y= \begin{bmatrix}y_{1,1} &y_{1,2} & \cdots & y_{1,T} \\ y_{2,1} &y_{2,2} & \cdots & y_{2,T}\\ \vdots& \vdots &\vdots &\vdots \\ y_{k,1} &y_{k,2} & \cdots & y_{k,T}\end{bmatrix}$

$=\begin{bmatrix} c_{1} & A_{1,1}^{p=1}&A_{1,2}^{p=1}&\cdots &A_{1,k}^{p=1}&\cdots & A_{1,1}^{p=p}&A_{1,2}^{p=p}&\cdots &A_{1,k}^{p=p}\\ c_{2} & A_{2,1}^{p=1}&A_{2,2}^{p=1}&\cdots &A_{2,k}^{p=1}&\cdots & A_{2,1}^{p=p}&A_{2,2}^{p=p}&\cdots &A_{2,k}^{p=p}\\ \vdots& \vdots &\vdots &\ddots &\vdots& \ddots& \vdots &\vdots &\ddots &\vdots\\ c_{k} & A_{k,1}^{p=1}&A_{k,2}^{p=1}&\cdots &A_{k,k}^{p=1} &\cdots & A_{k,1}^{p=p}&A_{k,2}^{p=p}&\cdots &A_{k,k}^{p=p} \end{bmatrix} \begin{bmatrix} 1&1&\cdots&1\\ y_{1,1}&y_{1,2}&\cdots&y_{1,T-1}\\ y_{2,1}&y_{2,2}&\cdots&y_{2,T-1}\\ \vdots & \vdots & \vdots&y\vdots\\ y_{k,1}&y_{k,2}&\cdots&y_{k,T-1}\\ \vdots & \vdots & \vdots&\vdots\\ \vdots & \vdots & \vdots&\vdots\\ y_{1,1-p}&y_{1,2-p}&\cdots&y_{1,T-p}\\ y_{2,1-p}&y_{2,2-p}&\cdots&y_{1,T-p}\\ \vdots & \vdots & \vdots&\vdots\\ y_{k,1-p}&y_{k,2-p}&\cdots& y_{k,T-p} \end{bmatrix}$

$+ U= \begin{bmatrix} u_{1,1}&u_{1,2}&\cdots&u_{1,T}\\ u_{2,1}&u_{2,2}&\cdots&u_{1,T}\\ \vdots&\vdots&\ddots&\vdots\\ u_{k,1}&u_{k,2}&\cdots&u_{k,T} \end{bmatrix}$