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8.8 Generalized State-Space Models 301
Example 8.8.4 An AR(1) process
An AR(1) process with iid noise can be expressed as an observation driven model.
Suppose {Yt} is the AR(1) process
Yt φYt−1 + Zt,
where {Zt} is an iid sequence of random variables with mean 0 and some probability
density function f(x). Then with Xt : Yt−1 we have
and
p(yt|xt) f(yt − φxt)
p xt+1|y (t)
1, if xt+1 yt,
0, otherwise.
Example 8.8.5 Suppose the observation-equation density is given by
p(yt|xt) xyt t e−xt , yt0, 1,..., (8.8.29)
yt!
and the state equation (8.8.26) is
p xt+1|y (t) g(xt; αt,λt), (8.8.30)
where αt α + y1 +···+yt and λt λ + t. It is possible to give a parameterdriven
specification that gives rise to the same state equation (8.8.30). Let {X∗ t } be the
parameter-driven state variables, where X∗ t X∗ t−1 and X∗ 1 has a gamma distribution
with parameters α and λ. (This corresponds to the model in Example 8.8.2 with
π a 1.) Then from (8.8.19) we see that p x∗ t |y(t) g(x∗ t ; αt,λt), which
coincides with the state equation (8.8.30). If {Xt} are the state variables whose joint
distribution is specified through (8.8.28), then {Xt} and {X∗ t } cannot have the same
joint distributions. To see this, note that
p x ∗
t+1 |x∗
∗
1, if xt+1 x∗
t
t
,
0, otherwise,
while
p xt+1|x (t) , y (t) p xt+1|y (t) g(xt; αt,λt).
If the two sequences had the same joint distribution, then the latter density could take
only the values 0 and 1, which contradicts the continuity (as a function of xt) of this
density.
Exponential Family Models
The exponential family of distributions provides a large and flexible class of distributions
for use in the observation equation. The density in the observation equation