Bir VAR(p)nin genel matris gösterimi

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

## Var(''p'')

Her y_{i} bir *k x 1* vektör ve her A_{i} *k x k* matris olmak üzere:

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

## Geniş matris gösterimi

:\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:

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}\,

## Kısa matris gösterim

*k* değişkenli bir VAR(*p*) genel bir biçimde yeniden yazılabibilir (*T* 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)

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