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8.2 The Basic Structural Model 263
State-Space Models with t ∈{0, ±1,...}
Consider the observation and state equations
Yt GXt + Wt, t 0, ±1,..., (8.1.10)
Xt+1 F Xt + Vt, t 0, ±1,..., (8.1.11)
where F and G are v × v and w × v matrices, respectively, {Vt} ∼WN(0,Q),
{Wt} ∼WN(0,R), and E(VsW ′ t ) 0 for all s, and t.
The state equation (8.1.11) is said to be stable if the matrix F has all its eigenvalues
in the interior of the unit circle, or equivalently if det(I − Fz) 0 for all z
complex such that |z| ≤1. The matrix F is then also said to be stable.
In the stable case the equations (8.1.11) have the unique stationary solution (Problem
8.1) given by
∞
Xt F j Vt−j−1.
j0
The corresponding sequence of observations
∞
Yt Wt + GF
j0
j Vt−j−1
is also stationary.
8.2 The Basic Structural Model
A structural time series model, like the classical decomposition model defined by
(1.5.1), is specified in terms of components such as trend, seasonality, and noise,
which are of direct interest in themselves. The deterministic nature of the trend
and seasonal components in the classical decomposition model, however, limits its
applicability. A natural way in which to overcome this deficiency is to permit random
variation in these components. This can be very conveniently done in the framework
of a state-space representation, and the resulting rather flexible model is called a
structural model. Estimation and forecasting with this model can be encompassed in
the general procedure for state-space models made possible by the Kalman recursions
of Section 8.4.
Example 8.2.1 The random walk plus noise model
One of the simplest structural models is obtained by adding noise to a random walk.
It is suggested by the nonseasonal classical decomposition model
Yt Mt + Wt, where {Wt} ∼WN 0,σ 2
w , (8.2.1)