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260 Chapter 8 State-Space Models
likelihood when only a subset of the complete data set is available. The EM algorithm
is particularly well suited for estimation problems in the state-space framework. Generalized
state-space models are introduced in Section 8.8. These are Bayesian models
that can be used to represent time series of many different types, as demonstrated by
two applications to time series of count data. Throughout the chapter we shall use
the notation
{Wt} ∼WN(0, {Rt})
to indicate that the random vectors Wt have mean 0 and that
Rt, if s t,
E WsW ′
t
8.1 State-Space Representations
0, otherwise.
A state-space model for a (possibly multivariate) time series {Yt,t 1, 2,...} consists
of two equations. The first, known as the observation equation, expresses the
w-dimensional observation Yt as a linear function of a v-dimensional state variable
Xt plus noise. Thus
Yt GtXt + Wt, t 1, 2,..., (8.1.1)
where {Wt} ∼WN(0, {Rt}) and {Gt} is a sequence of w × v matrices. The second
equation, called the state equation, determines the state Xt+1 at time t + 1 in terms
of the previous state Xt and a noise term. The state equation is
Xt+1 FtXt + Vt, t 1, 2,..., (8.1.2)
where {Ft} is a sequence of v × v matrices, {Vt} ∼WN(0, {Qt}), and {Vt} is uncorrelated
with {Wt} (i.e., E(WtV ′ s ) 0 for all s and t). To complete the specification,
it is assumed that the initial state X1 is uncorrelated with all of the noise terms {Vt}
and {Wt}.
Remark 1. A more general form of the state-space model allows for correlation
between Vt and Wt (see TSTM, Chapter 12) and for the addition of a control term
Htut in the state equation. In control theory, Htut represents the effect of applying
a “control” ut at time t for the purpose of influencing Xt+1. However, the system
defined by (8.1.1) and (8.1.2) with E WtV ′
s 0 for all s and t will be adequate for
our purposes.
Remark 2. In many important special cases, the matrices Ft,Gt,Qt, and Rt will
be independent of t, in which case the subscripts will be suppressed.