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264 Chapter 8 State-Space Models
Figure 8-1
Realization from a random
walk plus noise model.
The random walk is
represented by the solid
line and the data are
represented by boxes.
and Mt mt, the deterministic “level” or “signal” at time t. We now introduce
randomness into the level by supposing that Mt is a random walk satisfying
Mt+1 Mt + Vt, and {Vt} ∼WN 0,σ 2
v , (8.2.2)
with initial value M1 m1. Equations (8.2.1) and (8.2.2) constitute the “local level”
or “random walk plus noise” model. Figure 8.1 shows a realization of length 100 of
this model with M1 0,σ2 v 4, and σ 2 w 8. (The realized values mt of Mt are
plotted as a solid line, and the observed data are plotted as square boxes.) The differenced
data
Dt : ∇Yt Yt − Yt−1 Vt−1 + Wt − Wt−1, t ≥ 2,
constitute a stationary time series with mean 0 and ACF
⎧
⎪⎨ −σ
ρD(h)
⎪⎩
2 w
2σ 2 w + σ 2 , if |h| 1,
v
0, if |h| > 1.
Since {Dt} is 1-correlated, we conclude from Proposition 2.1.1 that {Dt} is an MA(1)
process and hence that {Yt} is an ARIMA(0,1,1) process. More specifically,
Dt Zt + θZt−1, {Zt} ∼WN 0,σ 2 , (8.2.3)
where θ and σ 2 are found by solving the equations
θ
1 + θ 2 −σ 2 w
2σ 2 w + σ 2 v
and θσ 2 −σ 2
w .
For the process {Yt} generating the data in Figure 8.1, the parameters θ and σ 2 of
0 10 20 30
0 20 40 60 80 100