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Figure 8-2
Sample ACF of the series
obtained by differencing
the data in Figure 8.1.
8.2 The Basic Structural Model 265
the differenced series {Dt} satisfy θ/(1 + θ 2 ) −.4and θσ2 −8. Solving these
equations for θ and σ 2 , we find that θ −.5and σ 2 16 (or θ −2and σ 2 4).
The sample ACF of the observed differences Dt of the realization of {Yt} in Figure
8.1 is shown in Figure 8.2.
The local level model is often used to represent a measured characteristic of the
output of an industrial process for which the unobserved process level {Mt} is intended
to be within specified limits (to meet the design specifications of the manufactured
product). To decide whether or not the process requires corrective attention, it is
important to be able to test the hypothesis that the process level {Mt} is constant.
From the state equation, we see that {Mt} is constant (and equal to m1) when Vt 0
or equivalently when σ 2 v
0. This in turn is equivalent to the moving-average model
(8.2.3) for {Dt} being noninvertible with θ −1 (see Problem 8.2). Tests of the unit
root hypothesis θ −1 were discussed in Section 6.3.2.
The local level model can easily be extended to incorporate a locally linear trend
with slope βt at time t. Equation (8.2.2) is replaced by
Mt Mt−1 + Bt−1 + Vt−1, (8.2.4)
where Bt−1 βt−1. Now if we introduce randomness into the slope by replacing it
with the random walk
Bt Bt−1 + Ut−1, where {Ut} ∼WN 0,σ 2
u , (8.2.5)
we obtain the “local linear trend” model.
ACF
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
0 10 20
Lag
30 40