Time Series (M30085) 2002
Exercises 3
In these questions,
{
t
}
is a discrete, purely random process, such that
E
(
t
) = 0,
V AR
(
t
) =
σ
2
,
COV
(
t
,
t
+
τ
) = 0 for
τ
= 0.
1. Find the ACF of the second order MA process given by
X
t
=
t
+ 0
.
7
t

1

0
.
2
t

2
2. Show that the ACF of the m
th
order MA process is given by
X
t
=
∑
m
k
=0
t

k
/
(
m
+ 1)
is
ρ
τ
=
(
m
+ 1

k
)
/
(
m
+ 1)
k
= 0
,
1
, ..., m
0
k > m
3. Show that the infinite MA process
{ }
defined by
X
t
=
t
+
C
(
t

1
+
t

2
+
...
) where
C is a constant is nonstationary.
Also show that the series of first differences
{
Y
t
}
defined by
Y
t
=
X
t

X
t

1
is a first order MA process and is stationary. Find the ACF
of
{
Y
t
}
4. Find the ACF of the first order AR process defined by
X
t

μ
= 0
.
7(
X
t

1

μ
) =
t
.
Plot
ρ
τ
for
k
=

6
,

5
, ...,

1
,
0
,
+1
, ...,
+6
5. If
X
t
=
μ
+
t
+
β
t

1
, where
μ
is a constant, show that the ACF does not depend on
μ
.
6. Find the values of
λ
1
and
λ
2
such that the second order AR process defined by
X
t
=
λ
1
X
t

1
+
λ
t
X
t

2
+
t
is stationary. If
λ
1
= 1
/
3 and
λ
2
= 2
/
9, show that the ACF of
X
t
is given by
ρ
τ
= 16
/
21(2
/
3)

τ

+ 5
/
12(

1
/
3)

k

, k
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 Spring '09
 jsdkasj
 spectral density, Autocorrelation, Stationary process, ACF, Xt, order MA process

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