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130 Chapter 4 Spectral Analysis
Since {Xt} has spectral density fX(λ), wehave
γX(h + k − j)
π
which, by substituting (4.3.3) into (4.3.2), gives
γY (h)
∞
π
e i(h−j+k)λ fX(λ) dλ
j,k−∞
π
−π
−π
ψjψk
∞
j−∞
π
e
−π
ihλ
∞
j−∞
e i(h−j+k)λ fX(λ) dλ, (4.3.3)
−π
−ij λ
ψje
ψje
∞
−ij λ
k−∞
ψke ikλ
2
fX(λ) dλ.
e ihλ fX(λ) dλ
The last expression immediately identifies the spectral density function of {Yt} as
fY (λ) e −iλ 2 fX(λ) e −iλ e iλ fX(λ).
Remark 1. Proposition 4.3.1 allows us to analyze the net effect of applying one or
more filters in succession. For example, if the input process {Xt} with spectral density
fX is operated on sequentially by two absolutely summable TLFs 1
and 2, then
−iλ the net effect is the same as that of a TLF with transfer function 1 e
−iλ
2 e
and the spectral density of the output process
Wt 1(B)2(B)Xt
is
−iλ
1 e
−iλ
2 e 2 fX(λ). (See also Remark 2 of Section 2.2.)
As we saw in Section 1.5, differencing at lag s is one method for removing a
seasonal component with period s from a time series. The transfer function for this
filter is 1 − e −isλ , which is zero for all frequencies that are integer multiples of 2π/s
radians per unit time. Consequently, this filter has the desired effect of removing all
components with period s.
The simple moving-average filter in Example 4.3.2 has transfer function
e −iλ Dq(λ),
where Dq(λ) is the Dirichlet kernel
−1
Dq(λ) (2q + 1)
|j|≤q
e −ij λ
⎧
⎨ sin[(q + .5)λ]
, if λ 0,
(2q + 1) sin(λ/2)
⎩
1, if λ 0.
A graph of Dq is given in Figure 4.12. Notice that |Dq(λ)| is near 1 in a neighborhood
of 0 and tapers off to 0 for large frequencies. This is an example of a low-pass filter.