H7
Given a set of observations
1
{
,...,
}
n
X
X
from a stationary time series, the ACVF is estimated
by the
sample autocovariance function
defined as
1
1
ˆ
()
(
)
(
)
γ
−
+
=
=−
−
∑
nh
th
n
t
n
t
hX
X
X
n
X
,
where
1
1
n
nn
t
X
X
n
=
=
∑
.
This also leads to estimating the ACF by the
sample autocorrelation function
ˆ
ˆˆ
/ (
0
)
ρ
γγ
=
hh
.
Using the divisor
n
(instead of
) in the formula for
−
ˆ
h
ensures that
ˆ
h
(and therefore
ˆ
h
)
is a nonnegativedefinite function, which is the necessary and sufficient condition for a function to be
an ACVF of a stationary process:
___________________________________________________________________________________________
It may be shown that under certain general conditions,
(
ˆ
ˆ
ˆ
~ N
,
/
,
(1), .
..,
( )
Wn
k
ρρ
)
′
=
±
ρ
,
where
W
is the covariance matrix (whose elements are given by the socalled
Bartlett’s formula
).
In particular,
1)
Purely random process
If
{ }
2
~ IID(0,
)
t
X
σ
, then
( )
1
ˆ()~AN 0
,
,
0
in
−
≠
i
.
Then a 95% CI for
is
11
1.96
1.96
⎛⎞
−<
<
+
⎜⎟
⎝⎠
,
hence we are 95% confident that
ˆ
1.96
1.96
<
+
, i.e.,
if we plot
ˆ
j
,
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 Spring '09
 gUR
 Covariance, Variance, Autocorrelation, Stationary process, Autocovariance, stationary time series, certain general conditions, sample autocovariance function

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