# Algebra for two period forecasting in AR (3) Model

I wondered if some folks could help fill in a knowledge gap for me with some time-series algebra please regarding the following AR (3):

$$x_t = \phi x_{t-1} + \phi_2 x_{t-2} + \phi x_{t-3} + \epsilon_t\qquad \epsilon_t \sim(0,\sigma^2)$$

In particular, could someone point out why the substitution (highlighted by substitution of the first underbrace into the second underbrace) is legitimate?

The $$t+1$$ forecast is given:

$$$$\begin{split} \underbrace{E(x_{t+1}|x_t,x_{t-1},...)} & = E(\phi_1x_t + \phi_2 x_{t-1} + \phi_3 x_{t-2} + \epsilon_{t+1}|x_t, x_{t-1},...)\\ & =\phi_1x_t + \phi_2 x_{t-1} + \phi_3 x_{t-2} \end{split}$$$$

And the $$t+2$$ forecast

$$$$\begin{split} E(x_{t+2}|x_t,x_{t-1},...) & = \underbrace{E(\phi_1 x_{t+1}} + \phi_2 x_t + \phi_3 x_{t-1} + \epsilon_{t+2}|x_t, x_{t-1},...) \\ & = \underbrace{\phi_1E(x_{t+1}|x_t,x_{t-1},...)} + \phi_2x_t +\phi_3 x_{t-1},\\ & = \underbrace{\phi_1(\phi_1x_t +\phi_2 x_{t-1} + \phi_3 x_{t-2})} + \phi_2x_t + \phi_3x_{t-1}\\ & = (\phi_1^2 + \phi_2)x_t + (\phi_1 \phi_2 + \phi_3)x_{t-1} + \phi_1\phi_3x_{t-2}. \end{split}$$$$

Would be appreciated.

There are three things going on (slightly rewriting the expectation operator):

1. Backward recursion

$$$$\begin{split} x_{t+2} &= \phi_1 x_{t+1} + \phi_2 x_{t} + \phi x_{t-1} + \epsilon_{t+2}\\ &= \phi_1(\phi_1 x_{t} + \phi_2 x_{t-1} + \phi x_{t-1} + \epsilon_{t+1}) + \phi_2 x_{t} + \phi x_{t-1} + \epsilon_{t+2}\\ &= \cdots \end{split}$$$$

1. Application / Rules of the expectation operator: $$E[a + bX] = a + bE[X]$$ for constants $$a,b$$ and a random variable $$X$$.

2. $$E_t[\epsilon_{t+h}] = 0 \quad \textrm{for all} \quad h>0$$

Applying these and collecting coefficients on $$x_t, x_{t-1}, x_{t-2}$$ gives you the result.

• Thanks @ BrsG, I appreciate your help. I edited the post a little. I appreciate your feedback on the rule. It makes sense now, you've said it. Initially I was stuck on why the first and second underbraces were equivalent. Aug 7, 2021 at 9:08
• Edited my response accordingly.
– BrsG
Aug 7, 2021 at 10:05