# How to calculate the conditional variance for AR(2) model?

I need to find the conditional variance of $$Y_t$$ given information up to time t.

$$Y_t$$ = $$\mu$$ + $$\phi_1Y_{t-1}$$ + $$\phi_2Y_{t-2}$$ +$$\epsilon_{t}$$

Need to find the conditional variance of this equation.

I need to find three step ahead forecast.

Was able to do the first, but got stuck at second step. Var($$Y_{t+1}$$|$$\Omega_t)$$ = $$\sigma^2$$

But need to find Var($$Y_{t+2}$$|$$\Omega_t)$$?

• Is that $\mu_{t+1}$ or $\epsilon_t$?
– Stat
Dec 19, 2013 at 0:46
• should be et. sorry. Dec 19, 2013 at 1:41

## 1 Answer

Consider that all random variables at time t are constant once you condition on $\Omega_t$.

Then, you have the following:

\begin{aligned} Var(Y_{t+2}|\Omega_t) &= Var(\mu + \phi_1 Y_{t+1} + \phi_2 Y_t + e_{t+2}|\Omega_t)\\ & = Var(\phi_1 Y_{t+1}|\Omega_t) + Var(e_{t+2}|\Omega_t)\\ & = \phi_1^2 Var(Y_{t+1}|\Omega_t) + \sigma^2\\ & = \phi_1^2 \sigma^2 + \sigma^2 \qquad \text{using your result for the 1-step ahead} \end{aligned}

You can compute $Var(Y_{t+3}|\Omega_t)$ with the same procedure using the results for the 1-step and 2-steps ahead conditional variance.

• How do you calculate a Covariance in the 3 step ahead?
– user294054
Mar 27, 2021 at 21:46
• Welcome @Germania, please open another question for the covariance. In any case, try to use the equation of $Y_{t+3}$ as I did for $Y_{t+2}$.
– Ale
Mar 28, 2021 at 11:37
• – user294054
Mar 28, 2021 at 13:37