How to find the conditional variance?

Question

Suppose I have the following time series process with $a_t$ being an independent white noise with mean $0$ and variance $\sigma^2$:

$$ z_t = \frac{1}{3}a_t + \frac{1}{3}a_{t-1} + \frac{1}{3}a_{t-2} $$

How do I compute the variance conditional on $a_{t-2} = 5$

Also, the unconditional variance in this problem is $\frac{\sigma^2}{3}$ correct? Thank you!

Answer 1

• $\frac{1}{3}a_t$ has variance $\frac19\sigma^2$

$a_t,a_{t-1},a_{t-2}$ are mutually independent so

• $\mathrm{Var}(z_t)=\frac39\sigma^2=\frac13\sigma^2$
• $\mathrm{Var}(z_t\mid a_{t-2}=5)=\frac29\sigma^2$