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H14
The secondorder properties of a time series are completely described by its ACVF
γ
h
,
or equivalently, under mild conditions (a sufficient condition is

∞
=−∞
< ∞
∑
h
h
),
by its Fourier transform,
which is called the
spectral density function
or the
spectrum
:
0
1
11
()
2
c
o
s
22
ih
h
hh
h
f
e
ω
ωγ
ππ
∞∞
−
=−∞
=
⎛⎞
==
+
⎜
⎝⎠
∑∑
h
⎟
.
(1)
The spectrum of a time series can be obtained as long as we know the model satisfied by the time series.
Often in practice we only have a finite set of time series data and we would like to estimate from it the
spectrum of the process.
Given a finite realization
1
{
,...,
}
N
X
X
of a stationary time series, the ACVF is estimated
by the sample autocovariance function
1
1
ˆ
(
)
(
)
Nh
Nt
h
N
t
t
hX
X
XXX
N
−
+
=
−
−
∑
N
.
(
H7
)
It would seem that the natural way to estimate
f
would be to replace
h
in (1)
by its sample estimate:
1
(1
)
1
ˆ
ˆ
c
o
s
2
N
h
hN
f
h
π
−
=−
−
=
∑
.
(2)
Note that the sum now is restricted to
<
,
since for a time series of length
N
,
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 Spring '09
 gUR

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