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. Kategori:Ekonometri