自相关(ACF)与偏自相关(PACF)（4）

§5.偏自相关系数PACF
在$AR(1)$模型中，即使$y_{t-2}$没有直接出现在模型中，但是$y_t$和$y_{t-2}$之间也相关，偏相关系数是在排除了其他变量的影响之后两个变量之间的相关系数。
平稳的$AR(1)$模型：
$$\begin{gathered} {\rho }_0=1 \\ {\rho }_s=a_1^s \end{gathered}$$

在$AR(1)$过程中，即使$y_{t-2}$没有直接出现在模型中，但$y_t$和$y_{t-2}$之间也相关。$y_t$和$y_{t-2}$之间的自相关系数(${\rho }_2$) 等于$y_t$和$y_{t-1}$之间的自相关系数(${\rho }_1$) 乘以$y_{t-1}$和$y_{t-2}$之间的相关系数(仍为${\rho }_1$)， 所以${\rho }_2={\rho }_1^2$。
$y_t$和$y_{t-s}$的偏自相关系数，排除了插入值$y_{t-1}$到$y_{t-s+1}$间的影响。
在$AR(1)$过程中$y_t$和$y_{t-2}$之间的偏自相关系数为0。

The steps to construct the partial autocorrelation function is as follows:
（序列的每一个值减去序列的均值，得到一个新的序列：）
①Demean and construct $\{y_t^{\ast }\}$ sequence with $y_t^{\ast }=y_t-\mu$
②Form the first order autocorrelation $y_t^{\ast }={\varphi }_{11}{y}_{t-1}^{\ast }+e_t$,We can see that ${\varphi }_{11}$ is both the autocorrelation and partial autocorrelation.(由于没有插入值，所以${\varphi }_{11}$既是自相关系数又是偏自相关系数)
③Control for (netting out) the effect of $y_{t-1}$ on the correlation between $y_t$ and $y_{t-2}$ by constructing the second order autocorrelation $y_t^{\ast }={\varphi }_{21}{y}_{t-1}^{\ast }+{\varphi }_{22}{y}_{t-2}^{\ast }+e_t$,${\varphi }_{22}$ the the partial autocorrelation of $y_t$ and $y_{t-2}$
④Repeating the procedure for all additional lags $s$ yield the partial autocorrelation function (PACF)
我们可以不同构造阶数的自回归模型，得到对应的偏自相关系数。
以下将归纳出偏自相关函数（PACF）的一般表达式（即通过自相关系数求出）：
$y_t^{\ast }={\varphi }_{11}{y}_{t-1}^{\ast }+e_t\Rightarrow y_t-\mu ={\varphi }_{11}(y_{t-1}-\mu )+e_t$
$\Rightarrow E(y_t-\mu )(y_t-\mu )={\varphi }_{11}E(y_{t-1}-\mu )(y_t-\mu )+E[(y_t-\mu )e_t]$
$\Rightarrow {\gamma }_0={\varphi }_{11}{\gamma }_1+{\sigma }^2$
$y_t^{\ast }={\varphi }_{11}{y}_{t-1}^{\ast }+e_t\Rightarrow y_t-\mu ={\varphi }_{11}(y_{t-1}-\mu )+e_t$
$\Rightarrow E(y_t-\mu )(y_{t-1}-\mu )={\varphi }_{11}E(y_{t-1}-\mu )(y_{t-1}-\mu )+E[(y_{t-1}-\mu )e_t]$
$\Rightarrow {\gamma }_1={\varphi }_{11}{\gamma }_0$
$\Rightarrow {\rho }_1={\varphi }_{11}$

$y_t^{\ast }={\varphi }_{21}{y}_{t-1}^{\ast }+{\varphi }_{22}{y}_{t-2}^{\ast }+e_t$
$\Rightarrow y_t-\mu ={\varphi }_{21}(y_{t-1}-\mu )+{\varphi }_{22}(y_{t-2}-\mu )+e_t$
$\Rightarrow E(y_t-\mu )(y_{t-1}-\mu )={\varphi }_{21}E(y_{t-1}-\mu )(y_{t-1}-\mu )+{\varphi }_{22}E(y_{t-2}-\mu )(y_{t-1}-\mu )+E[(y_{t-1}-\mu )e_t]$
$\Rightarrow {\gamma }_1={\varphi }_{21}{\gamma }_0+{\varphi }_{22}{\gamma }_1$
$y_t^{\ast }={\varphi }_{21}{y}_{t-1}^{\ast }+{\varphi }_{22}{y}_{t-2}^{\ast }+e_t$
$\Rightarrow y_t-\mu ={\varphi }_{21}(y_{t-1}-\mu )+{\varphi }_{22}(y_{t-2}-\mu )+e_t$
$\Rightarrow E(y_t-\mu )(y_{t-2}-\mu )={\varphi }_{21}E(y_{t-1}-\mu )(y_{t-2}-\mu )+{\varphi }_{22}E(y_{t-2}-\mu )(y_{t-2}-\mu )+E[(y_{t-2}-\mu )e_t]$
$\Rightarrow {\gamma }_2={\varphi }_{21}{\gamma }_1+{\varphi }_{22}{\gamma }_0$
Thus,we have:
$$\{\begin{array}{l}{\gamma }_1={\varphi }_{21}{\gamma }_0+{\varphi }_{22}{\gamma }_1\\ {\gamma }_2={\varphi }_{21}{\gamma }_1+{\varphi }_{22}{\gamma }_0\end{array}\Rightarrow \{\begin{array}{l}{\rho }_1={\varphi }_{21}+{\varphi }_{22}{\rho }_1\\ {\rho }_2={\varphi }_{21}{\rho }_1+{\varphi }_{22}\end{array}$$

$\Rightarrow$
$${\varphi }_{22}=\frac{{\rho }_2-{\rho }_1^2}{1-{\rho }_1^2}$$

通过归纳得：
$${\varphi }_{ss}=\frac{{\rho }_s-{\sum }_{j=1}^{s-1}{\varphi }_{s-1,j}{\rho }_{s-j}}{1-{\sum }_{j=1}^{s-1}{\varphi }_{s-1,j}{\rho }_{s-j}}s=3,4,5,...$$

其中，${\varphi }_{s,j}={\varphi }_{s-1,j}-{\varphi }_{s,s}{\varphi }_{s-1,s-j}j=1,2,...,s-1$

Remark: We know that for a $MA(q)$ model, the $ACF$ will be 0 for all $s>q$ and for $AR(p)$ model, the $PACF$ will be 0 for all $s>p$($p$阶之后就没有${\varphi }_{ss}$了). This is a useful feature to identification of $MA$ and $AR$ model respectively.