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4.1 Spectral Densities 119
where {Zt} ∼WN 0,σ2 , then from (4.1.1), {Xt} has spectral density
σ
f(λ)
2
2π 1 − φ2
∞
1 + φ
h1
h e −ihλ + e ihλ
σ
2
2π 1 − φ2
1 + φeiλ φe−iλ
+
1 − φeiλ 1 − φe−iλ
2 σ 2
1 − 2φ cos λ + φ
2π
−1 .
Graphs of f(λ),0≤ λ ≤ π, are displayed in Figures 4.3 and 4.4 for φ .7 and
φ −.7. Observe that for φ .7 the density is large for low frequencies and small
for high frequencies. This is not unexpected, since when φ .7 the process has a
positive ACF with a large value at lag one (see Figure 4.5), making the series smooth
with relatively few high-frequency components. On the other hand, for φ −.7 the
ACF has a large negative value at lag one (see Figure 4.6), producing a series that
fluctuates rapidly about its mean value. In this case the series has a large contribution
from high-frequency components as reflected by the size of the spectral density near
frequency π.
Example 4.1.5 Spectral density of an MA(1) process
Xt .7Xt−1 + Zt ,
where
{Zt }∼WN 0,σ2
If
f
0.0 0.5 1.0 1.5
Xt Zt + θZt−1,
Figure 4-3
The spectral density
f(λ), 0≤ λ ≤ π, of
0.0 0.5 1.0 1.5 2.0 2.5 3.0
. Frequency