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

$Y_t$ = $\mu $ + $\phi_1$$Y_{t-1}$ + $\phi_2$$Y_{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($Y_{t+2}$|$\Omega_t)$?

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

1 Answer 1


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.

  • $\begingroup$ How do you calculate a Covariance in the 3 step ahead? $\endgroup$
    – user294054
    Mar 27, 2021 at 21:46
  • $\begingroup$ 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 $Y_{t+2}$. $\endgroup$
    – Ale
    Mar 28, 2021 at 11:37
  • $\begingroup$ math.stackexchange.com/q/4079759/817548 @Ale $\endgroup$
    – user294054
    Mar 28, 2021 at 13:37

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