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308 Chapter 8 State-Space Models
Table 8.3 Transition probabilities for the
number of goals scored by England
against Scotland.
yt+1
p(yt+1|yt ) 0 1 2 ≥ 3
0 .214 .500 .214 .072
yt 1 .409 .272 .136 .182
2 .250 .375 .125 .250
≥ 3 0 .857 .143 0
for t 1, 2,..., where f is given by (8.8.36) and α1|0 0,λ1|0 0. The power
steady conditions (8.8.41)–(8.8.42) are assumed to hold for αt|t−1 and λt|t−1. The only
unknown parameter in the model is δ. The log-likelihood function for δ based on the
conditional distribution of y1,...,y52 given y1 is given by (see (8.8.27))
ℓ δ,y (n) n−1
ln p yt+1|y (t) , (8.8.49)
t1
where p yt+1|y (t)) is the negative binomial density (see Problem 8.25(c))
p yt+1|y (t) nb yt+1; αt+1|t,(1 + λt+1|t) −1 ,
with αt+1|t and λt+1|t as defined in (8.8.44) and (8.8.43). (For the goal data, y1 0,
which implies α2|1 0 and hence that p y2|y (1) is a degenerate density with unit
mass at y2 0. Harvey and Fernandes avoid this complication by conditioning the
likelihood on y (τ) , where τ is the time of the first nonzero data value.)
Maximizing this likelihood with respect to δ, we obtain ˆδ .844. (Starting the
equations (8.8.43)–(8.8.44) with α1|0 0 and λ1|0 δ/(1 − δ), we obtain ˆδ .732.)
With .844 as our estimate of δ, the prediction density of the next observation Y53 given
y (52) is nb(y53; α53|52,(1+λ53|52) −1 . The first five values of this distribution are given in
Table 8.4. Under this model, the probability that England will be held scoreless in the
next match is .471. The one-step predictors, ˆY1 0, ˆY2,...,Y52 are graphed in Figure
8.10. (This graph can be obtained by using the ITSM option Smooth>Exponential
with α 0.154.)
Figures 8.11 and 8.12 contain two realizations from the fitted model for the goal
data. The general appearance of the first realization is somewhat compatible with the
goal data, while the second realization illustrates the convergence of the sample path
to 0 in accordance with the result of Grunwald et al. (1994).