Stochastic Modelling
Exercises on Time Series
*
Dr. Iqbal Owadally
†
March 3, 2003
Elementary Problems
Q1.
Rewrite the following time series models using the backward shift notation. Classify
each of them as an ARIMA(
p
,
d
,
q
) process (that is, determine
p
,
d
and
q
in each case).
State whether each is (i) stationary, (ii) invertible.
(i)
X
t
= 0
.
5
X
t

1
+
Z
t
(ii)
X
t

0
.
5
X
t

1
=
Z
t

1
.
3
Z
t

1
+ 0
.
4
Z
t

2
(iii)
X
t

1
.
5
X
t

1
+ 0
.
6
X
t

2
=
Z
t
(iv) (
X
t

0
.
2)

1
.
2(
X
t

1

0
.
2) + 0
.
2(
X
t

2

0
.
2) =
Z
t

0
.
5
Z
t

1
Q2.
The following autoregressive processes are stationary. Calculate
ρ
1
,
ρ
2
and
ρ
3
.
(i)
X
t
+ 0
.
5
X
t

1

0
.
1
X
t

2
=
Z
t
;
(ii)
X
t
=

0
.
6
X
t

2
+
Z
t
;
(iii) (1

1
.
1
B
+ 0
.
18
B
2
)
X
t
=
Z
t
;
(iv)
X
t
=

αX
t

1

α
2
X
t

2

α
3
X
t

3
+
Z
t
.
Q3.
Calculate
ρ
1
and
ρ
2
for the following MA processes:
(i)
Y
t
=
Z
t

βZ
t

1
(ii)
Y
t
= (1 + 2
.
4
B
+ 0
.
8
B
2
)
Z
t
Q4.
(i) Describe the key diﬀerence between the correlograms of a stationary AR process
and an MA process of the same order.
(ii) Derive the autocorrelation function for the stationary ARMA(1, 1) process:
(
X
t

μ
)

α
(
X
t

1

μ
) =
Z
t

βZ
t

1
.
(iii) Comment on the correlogram of the ARMA(1, 1) process above.
*
Elementary problems should be attempted ﬁrst. Past exam questions are included for exam practice and
could be attempted later. They are adapted from papers set by the Exam Board of the Institute and Faculty
of Actuaries. Papers set by the Exam Board, Faculty of Actuarial Science and Statistics, Cass Business School,
City University, are separately available.
†
Contact details: Cass Building Room 5071, extension 8478, [email protected]
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentQ5.
The AR(1) process
X
t

0
.
4
X
t

1
=
Z
t
is started arbitrarily at
t
= 0 with initial condition
X
0
=
x
0
∈
R
. The sequence
{
Z
t
}
is a set of independent and identically distributed
random variables with zero mean and variance
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '09
 YuliaGel
 Variance, Autocorrelation, Stationary process, Time series analysis, Xt, exam board

Click to edit the document details