## Description

1. Given a data set Y1, . . . , Y20.

Y = (1.33, −0.56, −1.31, −0.37, 0.05, 0.46, 2.00, −0.19, −0.25, 1.07,

−0.17, 1.14, 0.63, −0.75, 0.15, 0.71, 0.45, −0.14, 0.57, 1.43).

(a) Fit an MA(2) model to {Yt}, find the k-step ahead forecast and

the 95% prediction intervals for k = 1, 2, 3, . . ..

(b) With the MA(2) model fitted in (a), find the partial autocorrelations φ11, φ22 and φ33 using the first principle.

(c) Fit an AR(1) model to {Yt}, find the k-step ahead forecast and

the 95% prediction intervals for k = 1, 2, 3, . . ..

(d) With the AR(1) model fitted in (c), find Cov(e20(k), e20(l)), where

k 6= l are positive integers.

(e) Fit an ARMA(1,1) model to {Yt}, find the 1st and 2nd-step ahead

forecast and the 95% prediction intervals.

(f) Fit an ARIMA(1,1,0) model to {Yt}, find the 1st and 2nd-step

ahead forecast and the 95% prediction intervals.

Note: You could use the R function arima() for model fitting.

2. Consider the GARCH(1,1) model

Xt = σtt

, t

iid∼ N(0, 1),

σ

2

t = α0 + α1X

2

t−1 + β1σ

2

t−1

,

where α0, α1, β1 ≥ 0 and α1 + β1 < 1.
(a) Express Xt+1 and Xt+2 in terms of Xt
, σt
, t+1 and t+2.
(b) Given observed values of σ1, X1, X2 and X3, express the likelihood
function L(α0, α1, β1) in terms of σ1, X1, X2 and X3.
3. Prove the following result.
Theorem 1 If Xt is a GARCH(p, q) process, then X2
t
is an ARMA(m, p)
process with noise νt = σ
2
t
(
2
t − 1), where m = max(p, q).
1