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8.8 Generalized State-Space Models 305
Remark 1. The preceding analysis for the Poisson-distributed observation equation
holds, almost verbatim, for the general family of exponential densities (8.8.31).
(One only needs to take care in specifying the correct range for x and the allowable
parameter space for α and λ in (8.8.37).) The relations (8.8.43)–(8.8.44), as well
as the exponential smoothing formula (8.8.48), continue to hold even in the more
general setting, provided that the parameters αt|t−1 and λt|t−1 satisfy the relations
(8.8.41)–(8.8.42).
Remark 2. Equations (8.8.41)–(8.8.42) are equivalent to the assumption that the
prior density of Xt given y (t−1) is proportional to the δ-power of the posterior distribution
of Xt−1 given Y (t−1) , or more succinctly that
f(xt; αt|t−1,λt|t−1) f(xt; δαt−1|t−1,δλt−1|t−1)
∝ f δ (xt; αt−1|t−1,λt−1|t−1).
This power relationship is sometimes referred to as the power steady model (Grunwald,
Raftery, and Guttorp, 1993, and Smith, 1979).
Remark 3. The transformed state variables Wt e Xt have a gamma state density
given by
p wt+1|y (t) g(wt+1; αt+1|t,λt+1|t)
(see Problem 8.26). The mean and variance of this conditional density are
E Wt+1|y (t) αt+1|t and Var Wt+1|y (t) αt+1|t/λ 2
t+1|t .
Remark 4. If we regard the random walk plus noise model of Example 8.2.1 as
the prototypical state-space model, then from the calculations in Example 8.8.1 with
G F 1, we have
and
E Xt+1|Y (t) E Xt|Y (t)
Var Xt+1|Y (t) Var Xt|Y (t) + Q>Var Xt|Y (t) .
The first of these equations implies that the best estimate of the next state is the same
as the best estimate of the current state, while the second implies that the variance
increases. Under the conditions (8.8.41), and (8.8.42), the same is also true for the
state variables in the above model (see Problem 8.26). This was, in part, the rationale
behind these conditions given in Harvey and Fernandes (1989).
Remark 5. While the calculations work out neatly for the power steady model,
Grunwald, Hyndman, and Hamza (1994) have shown that such processes have degenerate
sample paths for large t. In the Poisson example above, they argue that the
observations Yt convergeto0ast →∞(see Figure 8.12). Although such models