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4.2 The Periodogram 121
ACF
-0.5 0.0 0.5 1.0
Figure 4-6
The ACF of the AR(1)
0 5 10 15 20
process Xt −.7Xt−1 + Zt . Lag
4.2 The Periodogram
Xt Zt + .9Zt−1 where
{Zt }∼WN 0,σ2
If {Xt} is a stationary time series {Xt} with ACVF γ(·) and spectral density f(·), then
just as the sample ACVF ˆγ(·) of the observations {x1,...,xn} can be regarded as a
sample analogue of γ(·), so also can the periodogram In(·) of the observations be
regarded as a sample analogue of 2πf (·).
f
0.0 0.1 0.2 0.3 0.4 0.5
Figure 4-7
The spectral density
f(λ), 0≤ λ ≤ π, of
0.0 0.5 1.0 1.5 2.0 2.5 3.0
. Frequency