Questions

The ACF for the ARMA(p; q) process tails off (approximately equal to zero) after lag q in a manner similar to the AR(p) process.

AR:

There are correlations between $x_T$ and $x_t$, for AR process, the current x_i is a linear combination of its past results plus random error. Then for ACF, it will be tailed off.

For PACF, once we do the regression controlling the medium x_j s, then we can cut the relationship between x_T and x_t, meaning that PACF will cut off at certain threshold.

MA:

对于MA过程来说，只要两个变量之间超过距离超过q，其自相关函数就因为没有共同的随机项归0了，那么偏自相关系数PACF为什么拖尾呢？我们可以这样理解，对于moving average 过程来说，如果不控制中间的变量，其实两个遥远的变量之间的自相关是严格为0，但是一旦你控制了中间的变量，那么相关性就会传导到两边，如下面这个例子：

$$ x_2 = ae_1+be_2, x_3 = ae_2 +be_3, x_4 = ae_3+be_4 $$

当我们控制了 �3 ,那么 �2 , �3 之间就会有线性关系，那么就会传导到 �2,�4 ,形成相关。