Forecast error for AR and MA process

Question

AR(p) process is denoted by: $X_t=\mu+\alpha_1(X_{t-1}-\mu)+\alpha_2(X_{t-2}-\mu)+...\alpha_p(X_{t-p}-\mu)+Z_t$

I don't understand forecast error.
Let $\epsilon_{t+l}$ be the forecast error at $l$ step ahead forecast .
Then it says that it can be shown that
$\epsilon _{t+l}=Z_{t+l}+\alpha_1Z_{t+l-1}+\alpha_1^2Z_{t+l-2}+....+\alpha_1^{l-1}Z_{t+1}$
How is this obtained?

My second question is for a MA(1) process it is obtained that
$V(\hat \epsilon_{t+1})=\hat\sigma^2 $ ; if $l=1$
$V(\hat \epsilon_{t+1})=\hat\sigma^2 (1+\beta_1^2)$ ; if $l>=2$

Can someone please tell me the method as to how these errors were obtained.

Answer 1

It could be useful despite I can't understand the different notation $Z_t$ and $X_t$:

We can suppose to simplify: $\mu_t=0$ and $l=0$. You can add and substitute these values by the way. We can obtain similar expression by a recursive form. Imagine the AR(1) $$X_t=\alpha_1 X_{t-1}+\epsilon_t (1) $$ and then, how we obtained the $X_{t-1}$?

$$X_{t-1}=\alpha_1 X_{t-2}+\epsilon_{t-1} (2) $$ So, you can substitute the expression (2) in the expression (1): $$X_{t}=\alpha_1^2X_{t-2}+\alpha_1\epsilon_{t-1}+\epsilon_t $$ If you repeat this process for $X_{t-2}$, $X_{t-3}$,$X_{t-4}$...you'll get something like your expression: $$X_t=\epsilon_t+ \alpha^1\epsilon_{t-1}+\alpha^2\epsilon_{t-2}+...+\alpha^i\epsilon_{t-i}$$ It demonstrates the MA model is a AR model with reiterative replacements. So, AR models is used for long memory structures and MA for short memory ones.