Bu sayfa, *k* değişkenli bir VAR(*p*) sürecinin farklı matris gösterimlerinin ayrıntılarıdır.

Var(*p*)
--------

Ana madde: Vektor Otoregresyon
Her {\displaystyle y\_{i}} bir *k x 1* vektör ve her {\displaystyle A\_{i}} *k x k* matris olmak üzere:

{\displaystyle y\_{t}=c+A\_{1}y\_{t-1}+A\_{2}y\_{t-2}+\cdots +A\_{p}y\_{t-p}+e\_{t},}

Geniş matris gösterimi
----------------------

{\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}}}

Denklem Denkleme Gösterim
-------------------------

*y* değişkenleri birebir yeniden yazılırsa:

{\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 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}\,}

Kısa matris gösterim
--------------------

*k* değişkenli bir VAR(*p*) genel bir biçimde yeniden yazılabibilir (*T* observations {\displaystyle y\_{0}} 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})

Kaynakça
--------

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