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)

Main article: Vector autoregression

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

where each ${\displaystyle y_{i}}$ is a vector of length k and each ${\displaystyle A_{i}}$ is a k × k matrix.

What are the assumptions on the noise?

Large matrix notation

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

Equation by regression notation

Rewriting the y variables one to one gives:

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

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

${\displaystyle \qquad \vdots }$

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

Concise matrix notation

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

${\displaystyle Y=BZ+U\,}$

where:

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

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

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

and

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

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

References

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