General matrix notation of a VAR(p)

This page shows the details for different matrix notations of a vector autoregression 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 vector of length k and each $A_{i}$ is a k × k matrix.

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_{{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}}^{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_{{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}}^{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}}

Equation by regression notation

Rewriting the y variables one to one gives:

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

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

$\qquad \vdots$

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

Concise matrix notation

One can rewrite a VAR(p) with k variables in a general way which includes T+1 observations $y_{0}$ through $y_{T}$

Y=BZ+U\,

where:

Y={\begin{bmatrix}y_{{p}}&y_{{p+1}}&\cdots &y_{{T}}\end{bmatrix}}={\begin{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={\begin{bmatrix}c&A_{{1}}&A_{{2}}&\cdots &A_{{p}}\end{bmatrix}}={\begin{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={\begin{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}}={\begin{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={\begin{bmatrix}e_{{p}}&e_{{p+1}}&\cdots &e_{{T}}\end{bmatrix}}={\begin{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

Lütkepohl, Helmut (2005). New Introduction to Multiple Time Series Analysis. Berlin: Springer. ISBN 3540401725.

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