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272 Chapter 8 State-Space Models
Definition 8.4.1 For the random vector X (X1,...,Xv) ′ ,
Pt(X) : (Pt(X1),...,Pt(Xv)) ′ ,
where Pt(Xi) : P(Xi|Y0, Y1,...,Yt), is the best linear predictor of Xi in terms
of all components of Y0, Y1,...,Yt.
Remark 1. By the definition of the best predictor of each component Xi of X,
Pt(X) is the unique random vector of the form
Pt(X) A0Y0 +···+AtYt
with v × w matrices A0,...,At such that
[X − Pt(X)] ⊥ Ys, s 0,...,t
(cf. (7.5.2) and (7.5.3)). Recall that two random vectors X and Y are orthogonal
(written X ⊥ Y) ifE(XY ′ ) is a matrix of zeros.
Remark 2. If all the components of X, Y1,...,Yt are jointly normally distributed
and Y0 (1,...,1) ′ , then
Pt(X) E(X|Y1,...,Yt), t ≥ 1.
Remark 3. Pt is linear in the sense that if A is any k × v matrix and X, V are two
v-variate random vectors with finite second moments, then (Problem 8.10)
and
Pt(AX) APt(X)
Pt(X + V) Pt(X) + Pt(V).
Remark 4. If X and Y are random vectors with v and w components, respectively,
each with finite second moments, then
P(X|Y) MY,
where M is a v×w matrix, M E(XY ′ )[E(YY ′ )] −1 with [E(YY ′ )] −1 any generalized
inverse of E(YY ′ ). (A generalized inverse of a matrix S is a matrix S −1 such that
SS −1 S S. Every matrix has at least one. See Problem 8.11.)
In the notation just developed, the prediction, filtering, and smoothing problems
(a), (b), and (c) formulated above reduce to the determination of Pt−1(Xt), Pt(Xt),
and Pn(Xt) (n >t), respectively. We deal first with the prediction problem.