This preview shows pages 1–3. Sign up to view the full content.
NOTES 3
Trends of a time series
:
Deterministic versus Stochastic Trends
Examples
:
y
t
=
a
t

0
.
5
a
t

1
where
a
t
∼
N
(0
,σ
2
a
);
Purely Stochastic.
y
t
=
β
0
+
β
1
t
+
β
2
t
2
;
Purely Deterministic
y
t
=
β
0
+
β
1
t
+
β
2
t
2
+
a
t
where
a
t
∼
N
(0
,σ
2
a
);
Stochastic + Deterministic
Estimation of a constant mean
Model:
Z
t
=
μ
+
X
t
, E
(
X
t
) = 0 for all
t.
First, we wish to estimate
μ
with observed time series
Z
1
,Z
2
,...,Z
n
.
The most common estimate of
μ
is the sample mean
¯
Z
=
1
n
n
X
i
=1
Z
t
.
Since
E
(
¯
Z
) =
μ
,
¯
Z
is an unbiased estimate of
μ
.
To investigate the precision of
¯
Z
as an estimate of
μ
, we need to make further assump
tions concerning
X
t
.
Theorem
Suppose
{
X
t
}
is a stationary time series, then
V ar
(
¯
Z
) =
γ
0
n
[1 + 2
n

1
X
k
=1
(1

k
n
)
ρ
k
]
=
γ
0
n
n

1
X
k
=

n
+1
(1


k

n
)
ρ
k
Note that the ﬁrst factor
γ
0
n
is the population variance assuming the observations are
independent. If the
{
X
t
}
series is in fact just white noise, then
ρ
k
= 0 for
k
≥
1 and
V ar
(
¯
Z
) simply reduce to
γ
0
n
.
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentProof:
V ar
(
¯
Z
) =
V ar
(
1
n
n
X
t
=1
Z
t
)
=
1
n
2
V ar
(
n
X
t
=1
Z
t
)
=
1
n
2
V ar
(
n
X
t
=1
X
t
)
=
1
n
2
[
n
X
t
=1
V ar
(
X
t
) + 2
n

1
X
t
=1
n
X
s
=
t
+1
Cov
(
X
t
,X
s
)]
=
1
n
2
[
nγ
0
+ 2
n

1
X
t
=1
n
X
s
=
t
+1
γ
t,s
]
=
1
n
2
[
nγ
0
+ 2((
n

1)
γ
1
+ (
n
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '08
 ?

Click to edit the document details