Bu sayfa, k değişkenli bir VAR(p) sürecinin farklı matris gösterimlerinin ayrıntılarıdır.
Her
bir k x 1 vektör ve her
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},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da84149886d490fba4c4d70a81a6a5361565387e)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ce3e0bc80793bad348d0cc6926585037f090b87)
y değişkenleri birebir yeniden yazılırsa:
k değişkenli bir VAR(p) genel bir biçimde yeniden yazılabibilir (T observations
through
![{\displaystyle Y=BZ+U\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e71bf99707f86271d8be7d271fdecf284a6b6d1f)
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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4949cb080930dac32879f8fba0599537e56c50e)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4953e2a7d925b77b6da4eba2cea808dc6039f9d6)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/be49f7c79d55456e11e91f2a7b0a6beb53872a80)
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}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92fa7abf8f3fc8ff5d9dff7a691798534ce58a4f)
One can then solve for the coefficient matrix B (e.g. using an ordinary least squares estimation of
)
- Helmut Lütkepohl. New Introduction to Multiple Time Series Analysis. Springer. 2005.