## Description

Let Zt ∼ W N(0, σ2

) be white noise. Given a data set

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).

1) Draw a time series plot, ACF and PACF plot for the data.

2) Find the moment estimates of θ, σ

2

for fitting an MA(1) model Yt =

Zt + θZt−1 to the data.

3) Find the least squares estimates of φ1, φ2, σ2

for fitting an AR(2) model

Yt − φ1Yt−1 − φ2Yt−2 = Zt to the data. Find a 95% confidence interval for

each of φ1 and φ2.

4) Find the Yule-Walker estimates of φ1, φ2 for fitting an AR(2) model Yt −

φ1Yt−1 − φ2Yt−2 = Zt to the data.

5) Find the conditional least squares estimates of φ, θ, σ

2

for fitting an

ARMA(1,1) model Yt − φYt−1 = Zt + θZt−1 to the data.

6) Find the maximum likelihood estimates of φ, θ, σ

2

for fitting an ARMA(1,1)

model Yt − φYt−1 = Zt + θZt−1 to the data. What is the maximized value

of the log-likelihood?

7) Among AR(p) with p = 1, 2, 3, 4, 5, which model is the best in terms of

FPE?

8) Among MA(q) with q = 1, 2, 3, 4, 5, which model is the best in terms of

AICC?

9) Fit an MA(1) model to the data. Find the residuals. Find the Ljung-Box

test statistic Q(10). State H0 and H1 of the Ljung-Box test, and then

draw the conclusion.

10) Which model will you select to describe the data?

1