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8.5 Estimation For State-Space Models 279
stepwise procedure. For fixed Q we find ˆµ(Q) and σ 2 w (Q) that maximize the likelihood
L µ,Q,σ2
2
w . We then maximize the “reduced likelihood” L ˆµ(Q), Q, ˆσ w (Q) with
respect to Q.
To achieve this we define the mean-corrected state vectors, X∗ t Xt − F t−1 µ,
and apply the Kalman prediction recursions to {X∗ t } with initial condition X∗ 1 0.
This gives, from (8.4.1),
ˆX ∗
t+1 F ˆX ∗
t
+ t −1
t
Yt − G ˆX ∗
t
, t 1, 2,..., (8.5.5)
with ˆX ∗ 1 0. Since ˆXt also satisfies (8.5.5), but with initial condition ˆXt µ, it
follows that
ˆXt ˆX ∗
t + Ctµ (8.5.6)
for some v × v matrices Ct. (Note that although ˆXt P(Xt|Y0,Y1,...,Yt), the quantity
ˆX ∗ t is not the corresponding predictor of X∗ t .) The matrices Ct can be determined
recursively from (8.5.5), (8.5.6), and (8.4.1). Substituting (8.5.6) into (8.5.5) and
using (8.4.1), we have
ˆX ∗
t+1 F ˆXt − Ctµ + t −1
t
Yt − G ˆXt − Ctµ
F ˆXt + t −1
t
Yt − G ˆXt − F − t −1
t G Ctµ
so that
ˆXt+1 − F − t −1
t G Ctµ,
Ct+1 F − t −1
t G Ct (8.5.7)
with C1 equal to the identity matrix. The quadratic form in the likelihood (8.5.2) is
therefore
S(µ,Q,σ 2
w )
n
t1
n
t1
2 Yt − G ˆXt
t
Yt − G ˆX ∗ t
t
(8.5.8)
2 − GCtµ
. (8.5.9)
Now let Q∗ : σ −2
w Q and define L∗ to be the likelihood function with this new
parameterization, i.e., L∗ µ,Q∗ ,σ2
2
w L µ,σwQ∗ ,σ2
∗
w . Writing t σ −2
w t and
∗ −2
t σw t, we see that the predictors ˆX ∗ t and the matrices Ct in (8.5.7) depend on
the parameters only through Q∗ . Thus,
S µ,Q,σ 2
w
σ −2
w S (µ,Q∗ , 1) ,