Consider a series x_(t) generated by the moving average process as:
x_(t)=mu +epsi _(t)+ heta _(1)epsi _(t-1),
where epsi _(t) are independently identically distributed random variables with E(epsi _(t))=
0 , and Var(epsi _(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 heta =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 up to {: au _(4)).
(d) [10%] Carefully write the equations for the 1, 2, 3 and 4 step ahead forecasts
for x_(t).
2. Consider a series xt generated by the moving average process as:
Xt=+Et+01Et-1
(3)
where &t are independently identically distributed random variables with E(et) = 0, and Var(et) = o2.
(a) [10%] Calculate the unconditional mean and the unconditional variance of Xt. (b) [5%] What is meant by saying that a process like xt is invertible? What condition would assure that xt is invertible? If 0 = 0.75, does xt satisfy the invertibility condition?
(c) [10%] What shapes of the ACF and PACF functions do you expect for xt? Derive the first 4 autocorrelations for this process (Ti up to T4).
(d) [10%] Carefully write the equations for the 1, 2, 3 and 4 step ahead forecasts for xt.