# 向量自回归模型

## 定义

VAR模型描述在同一样本期间内的n变量（内生变量）可以作为它们过去值的线性函数。

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

1. ${\displaystyle \mathrm {E} (e_{t})=0\,}$  —误差项的均值为0
2. ${\displaystyle \mathrm {E} (e_{t}e_{t}')=\Omega \,}$  —误差项的协方差矩阵为Ω（一个n × 'n正定矩阵）
3. ${\displaystyle \mathrm {E} (e_{t}e_{t-k}')=0\,}$  （对于所有不为0的k都满足）—误差项不存在自相关

## 例子

${\displaystyle {\begin{bmatrix}y_{1,t}\\y_{2,t}\end{bmatrix}}={\begin{bmatrix}c_{1}\\c_{2}\end{bmatrix}}+{\begin{bmatrix}A_{1,1}&A_{1,2}\\A_{2,1}&A_{2,2}\end{bmatrix}}{\begin{bmatrix}y_{1,t-1}\\y_{2,t-1}\end{bmatrix}}+{\begin{bmatrix}e_{1,t}\\e_{2,t}\end{bmatrix}},}$

${\displaystyle y_{1,t}=c_{1}+A_{1,1}y_{1,t-1}+A_{1,2}y_{2,t-1}+e_{1,t}\,}$
${\displaystyle y_{2,t}=c_{2}+A_{2,1}y_{1,t-1}+A_{2,2}y_{2,t-1}+e_{2,t}.\,}$

## 转换VAR(p)为VAR(1)

VAR(p)模型常常可以被改写为VAR(1)模型。 比如VAR(2)模型：

${\displaystyle y_{t}=c+A_{1}y_{t-1}+A_{2}y_{t-2}+e_{t}}$

${\displaystyle {\begin{bmatrix}y_{t}\\y_{t-1}\end{bmatrix}}={\begin{bmatrix}c\\0\end{bmatrix}}+{\begin{bmatrix}A_{1}&A_{2}\\I&0\end{bmatrix}}{\begin{bmatrix}y_{t-1}\\y_{t-2}\end{bmatrix}}+{\begin{bmatrix}e_{t}\\0\end{bmatrix}},}$

## 结构与简化形式

### 结构向量自迴归

${\displaystyle B_{0}y_{t}=c_{0}+B_{1}y_{t-1}+B_{2}y_{t-2}+\cdots +B_{p}y_{t-p}+\epsilon _{t},}$

${\displaystyle {\begin{bmatrix}1&B_{0;1,2}\\B_{0;2,1}&1\end{bmatrix}}{\begin{bmatrix}y_{1,t}\\y_{2,t}\end{bmatrix}}={\begin{bmatrix}c_{0;1}\\c_{0;2}\end{bmatrix}}+{\begin{bmatrix}B_{1;1,1}&B_{1;1,2}\\B_{1;2,1}&B_{1;2,2}\end{bmatrix}}{\begin{bmatrix}y_{1,t-1}\\y_{2,t-1}\end{bmatrix}}+{\begin{bmatrix}\epsilon _{1,t}\\\epsilon _{2,t}\end{bmatrix}},}$

${\displaystyle \Sigma =\mathrm {E} (\epsilon _{t}\epsilon _{t}')={\begin{bmatrix}\sigma _{1}&0\\0&\sigma _{2}\end{bmatrix}};}$

### 简化向量自迴歸

${\displaystyle y_{t}=B_{0}^{-1}c_{0}+B_{0}^{-1}B_{1}y_{t-1}+B_{0}^{-1}B_{2}y_{t-2}+\cdots +B_{0}^{-1}B_{p}y_{t-p}+B_{0}^{-1}\epsilon _{t},}$

${\displaystyle B_{0}^{-1}c_{0}=c,}$  ${\displaystyle B_{0}^{-1}B_{i}=A_{i}}$  对于 ${\displaystyle i=1,\cdots ,p\,}$ ${\displaystyle B_{0}^{-1}\epsilon _{t}=e_{t}}$

${\displaystyle y_{t}=c+A_{1}y_{t-1}+A_{2}y_{t-2}+\cdots +A_{p}y_{t-p}+e_{t}}$