2. Consider a series $x_t$ generated by the moving average process as:
$x_t = \mu + \epsilon_t + \theta_1 \epsilon_{t-1}$,
(3)
where $\epsilon_t$ are independently identically distributed random variables with $E(\epsilon_t) =$
0, and $Var(\epsilon_t) = \sigma^2$.
(a) [10%] Calculate the unconditional mean and the unconditional variance of
$x_t$.
(b) [5%] What is meant by saying that a process like $x_t$ is invertible? What
condition would assure that $x_t$ is invertible? If $\theta = 0.75$, does $x_t$ satisfy the
invertibility condition?
(c) [10%] What shapes of the ACF and PACF functions do you expect for $x_t$?
Derive the first 4 autocorrelations for this process ($\tau_1$ up to $\tau_4$).
(d) [10%] Carefully write the equations for the 1, 2, 3 and 4 step ahead forecasts
for $x_t$.
