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4.4 The Spectral Density of an ARMA Process 133
Since {Zt} has spectral density fZ(λ) σ 2 /(2π), the result now follows from Proposition
4.3.1.
For any specified values of the parameters φ1,...,φp,θ1,...,θq and σ 2 , the
Spectrum>Model option of ITSM can be used to plot the model spectral density.
Example 4.4.1 The spectral density of an AR(2) process
For an AR(2) process (4.4.1) becomes
fX(λ)
σ 2
2π 1 − φ1e −iλ − φ2e −2iλ 1 − φ1e iλ − φ2e 2iλ
σ 2
2π 1 + φ 2 1 + 2φ2 + φ 2 2 + 2(φ1φ2 − φ1) cos λ − 4φ2 cos 2 λ .
Figure 4.14 shows the spectral density, found from the Spectrum>Model option of
ITSM, for the model (3.2.20) fitted to the mean-corrected sunspot series. Notice the
well-defined peak in the model spectral density. The frequency at which this peak
occurs can be found by differentiating the denominator of the spectral density with
respect to cos λ and setting the derivative equal to zero. This gives
cos λ φ1φ2 − φ1
4φ2
0.849.
The corresponding frequency is λ 0.556 radians per year, or equivalently
c λ/(2π) 0.0885 cycles per year, and the corresponding period is therefore
1/0.0885 11.3 years. The model thus reflects the approximate cyclic behavior of
f
0 200 400 600 800
Figure 4-14
The spectral density
fX (λ), 0 ≤ λ ≤ π of the
AR(2) model (3.2.20) fitted
to the mean-corrected
0.0 0.5 1.0 1.5 2.0 2.5 3.0
sunspot series. Frequency