Q1. For each of the following ARMA models:
(i) t B B at z (1 0.2 0.6 )
2 = - +
(ii) t at (1- 0.5B + 0.4B )z = 2
(iii) t B at (1 0.2B 0.6B )z (1 0.8 ) 2 - + = -
where at ~ N(0, 1)
(a) Check if the model is stationary and invertible.
(b) Find the TAC, ρk, for k = 1, 2 3.
(c) Find the TPAC, ρkk, for k = 1, 2 3.
(Hint: use MDETERM in excel to calculate determinant)
Q2. Consider the MA(3) model, t t z (1 B B B )a 3
3
2 = -θ1 -θ 2 -θ , where at ~
N(0, 2 σ a ).
(a) Show that the l-step ahead forecast ) zˆn+l(n of zn+l is
?
?
?
?
?
?
?
≥
- =
- - =
- - - =
= -
- -
+
0 4
3
2
1
ˆ ( )
3
2 3 1
1 2 1 3 2
l
a l
a a l
a a a l
z n
n
n n
n n n
n l θ
θ θ
θ θ θ .
(b) Let ) en+l(n = zn+l - ) zˆn+l(n . Show that
) en+l(n =
?
?
?
?
?
?
?
- - - ≥
- - =
- =
=
+ + - + - + -
+ + +
+ +
+
4
3
2
1
1 1 2 2 3 3
3 1 2 2 1
2 1 1
1
a a a a l
a a a l
a a l
a l
n l n l n l n l
n n n
n n
n
θ θ θ
θ θ
θ
(c) Show that
Var( ) en+l(n )=
?
?
?
?
?
?
?
?
?
+ + + ≥
+ + =
+ =
=
(1 ) 4
(1 ) 3
(1 ) 2
1
2 2
3
2
2
2
1
2 2
2
2
1
2 2
1
2
l
l
l
l
a
a
a
a
θ θ θ σ
θ θ σ
θ σ
σSTAT S802F - Multivariate and Time Series Analysis Assignment 2
And the 95% prediction interval of zn+l is
?
?
?
?
?
?
?
?
?
± + + + ≥
± + + =
± + =
± =
+
+
+
+
ˆ ( ) (1 ) 4
ˆ ( ) (1 ) 3
ˆ ( ) (1 ) 2
ˆ ( ) 1
2 2
3
2
2
2
0.025 1
2 2
2
2
3 0.025 1
2 2
2 0.025 1
1 0.025
z n z l
z n z l
z n z l
z n z l
n l a
n a
n a
n a
θ θ θ σ
θ θ σ
θ σ
σ
where z0.025 is the critical value from Normal Distribution such that
P(z ≥ z0.025)=0.025.
Q3. Assume that 100 observations from an AR(2) model t at (1- B - B )z = 2 φ1 φ2 , where at ~ N(0, 2 σ a ), give the following SAC: r1=0.8, r2=0.5 and r3=0.4. Use method of moments to estimate φ1 and φ2.
Q4. Suppose that the following Box-Jenkins model has been tentatively identified for the time series values y1, y2, ..., y120.
t t (1- φ B)(1- φ B ) y = δ + a
12
1 1,12
The RSAC and RSPAC obtained from the above model are given below (for lags 1 - 6)
Lag 1 2 3 4 5 6
RSAC 0.71 0.09 0.07 0.10 0.08 0.04
RSPAC 0.07 0.04 0.02 0.01 0.02 0.01
(a) Calculate Q*, the Ljung-Box statistic, by using the above K=6 residual autocorrelations. Then, by using an appropriate rejection point, show that we can, by setting a equal 0.05 reject
Ho: ρ1= ρ2= ....= ρ6 =0
(b) Since the model is inadequate, we will write it as t t (1- φ B)(1- φ B ) y = δ +η 12
1 1,12
Using the above RSAC and RSPAC, calculate the corresponding krs , krt , kk rs , kk rt k = 1, 2, 3, 4 5, 6, to identify a model describing ηt .
Explain your choice of model. Please also provide the new model for STAT S802F - Multivariate and Time Series Analysis Assignment 2
yt.
Q5. Given a Time Series data provided in the data file "Q5Data.xls",
(a) Assess if the time series is stationary. If not, identify the transformation to transform it into stationary. Please show the SAS programs and explain the conclusions drawn.
(b) Identify the tentative ARIMA to describe the time series data and explain your conclusion drawn.
(c) Perform estimation and diagnostic checking. Check if the estimated model parameters fulfil the stationary and/or invertibility conditions.
Also, check for normality assumptions. Please show the SAS program and explain the conclusion drawn.
Q6. Given the Time Series data of Monthly Passenger Totals (measured in thousands of passengers) in international travel from 1949 to 1959 in "Q6data.xls"
(a) Assess if the time series is stationary. If not, identify the transformation to transform it into stationary. Please show the SAS programs and explain the conclusions drawn.
(b) Identify the tentative ARIMA to describe the time series data and explain the conclusion drawn.
(c) Perform estimation and diagnostic checking. Also, check for normality assumptions. Please show the SAS program and explain the conclusion.
(d) Perform prediction for the 12 months in year 1960 and the corresponding prediction intervals. Please show the SAS program and present the results.