Bir Var(P)Nin Genel Matris Gosterimi

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

y_

bir k x 1 vektör ve her

A_

k x k matris olmak üzere: :

y_=c + A_y_ + A_y_ + \cdots + A_y_ + e_,

Geniş matris gösterimi :

\beginy_ \\ y_\\ \vdots \\ y_\end=\beginc_ \\ c_\\ \vdots \\ c_\end+ \begin a_^1&a_^1 & \cdots & a_^1\\  a_^1&a_^1 & \cdots & a_^1\\  \vdots& \vdots& \ddots& \vdots\\ a_^1&a_^1 & \cdots & a_^1 \end \beginy_ \\ y_\\ \vdots \\ y_\end + \cdots + \begin a_^p&a_^p & \cdots & a_^p\\  a_^p&a_^p & \cdots & a_^p\\  \vdots& \vdots& \ddots& \vdots\\ a_^p&a_^p & \cdots & a_^p \end \beginy_ \\ y_\\ \vdots \\ y_\end   + \begine_ \\ e_\\ \vdots \\ e_\end

Denklem Denkleme Gösterim y değişkenleri birebir yeniden yazılırsa:

y_ = c_ + a_^1y_ + a_^1y_ +\cdots + a_^1y_+\cdots+a_^py_+a_^py_+ \cdots +a_^py_ + e_\,

y_ = c_ + a_^1y_ + a_^1y_ +\cdots + a_^1y_+\cdots+a_^py_+a_^py_+ \cdots +a_^py_ + e_\,

y_ = c_ + a_^1y_ + a_^1y_ +\cdots + a_^1y_+\cdots+a_^py_+a_^py_+ \cdots +a_^py_ + e_\,

Kısa matris gösterim k değişkenli bir VAR(p) genel bir biçimde yeniden yazılabibilir ( T observations

y_

through

y_

Y=BZ +U \,

Where: :

Y= \beginy_ & y_ & \cdots & y_\end = \beginy_ & y_ & \cdots & y_ \\ y_ &y_ & \cdots & y_\\ \vdots& \vdots &\vdots &\vdots \\ y_ &y_ & \cdots & y_\end

B= \begin c & A_ & A_ & \cdots & A_ \end =  \begin c_ & a_^1&a_^1 & \cdots & a_^1 &\cdots & a_^p&a_^p & \cdots & a_^p\\  c_ & a_^1&a_^1 & \cdots & a_^1 &\cdots & a_^p&a_^p & \cdots & a_^p \\  \vdots & \vdots& \vdots& \ddots& \vdots & \cdots & \vdots& \vdots& \ddots& \vdots\\ c_ & a_^1&a_^1 & \cdots & a_^1 &\cdots & a_^p&a_^p & \cdots & a_^p  \end

Z= \begin 1 & 1 & \cdots & 1 \\ y_ & y_ & \cdots & y_\\ y_ & y_ & \cdots & y_\\ \vdots & \vdots & \ddots & \vdots\\ y_ & y_ & \cdots & y_ \end = \begin 1 & 1 & \cdots & 1 \\ y_ & y_ & \cdots & y_ \\ y_ & y_ & \cdots & y_ \\ \vdots & \vdots & \ddots & \vdots\\ y_ & y_ & \cdots & y_ \\ y_ & y_ & \cdots & y_ \\ y_ & y_ & \cdots & y_ \\ \vdots & \vdots & \ddots & \vdots\\ y_ & y_ & \cdots & y_ \\ \vdots & \vdots & \ddots & \vdots\\ y_ & y_ & \cdots & y_ \\ y_ & y_ & \cdots & y_ \\ \vdots & \vdots & \ddots & \vdots\\ y_ & y_ & \cdots & y_  \end

and :

U=  \begin e_ & e_ & \cdots & e_ \end= \begin e_ & e_ & \cdots & e_ \\ e_ & e_ & \cdots & e_ \\ \vdots & \vdots & \ddots & \vdots \\ e_ & e_ & \cdots & e_  \end.

One can then solve for the coefficient matrix B (e.g. using an ordinary least squares estimation of

Y \approx BZ

)

Referanslar

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

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