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280 Chapter 8 State-Space Models
so that
−2lnL ∗ µ,Q ∗ ,σ 2
w n ln(2π)+
n
t1
n
ln t + σ −2
w S (µ,Q∗ , 1)
n ln(2π)+ ln
t1
∗
2 −2
t + n ln σw + σw S (µ,Q∗ , 1) .
For Q∗ fixed, it is easy to show (see Problem 8.18) that this function is minimized
when
and
ˆµ ˆµ (Q ∗ )
n
t1
ˆσ 2 2
w ˆσw (Q∗ ) n −1
C ′ t G′ GCt
∗ t
−1
n
t1
C ′ t G′
n Yt − G ˆX ∗ t − GCt ˆµ
t1
∗ t
Yt − G ˆX ∗ t
2
∗ t
(8.5.10)
. (8.5.11)
Replacing µ and σ 2 w by these values in −2lnL∗ and ignoring constants, the reduced
likelihood becomes
ℓ (Q ∗
) ln n −1
n
Yt − G ˆX ∗ t − GCt ˆµ 2
+ n −1
n
ln det ∗
t . (8.5.12)
t1
∗ t
If ˆQ ∗ denotes the minimizer of (8.5.12), then the maximum likelihood estimator of the
parameters µ,Q,σ2 w are ˆµ, ˆσ 2 w ˆQ ∗ , ˆσ 2 w , where ˆµ and ˆσ 2 w are computed from (8.5.10)
and (8.5.11) with Q∗ replaced by ˆQ ∗ .
We can now summarize the steps required for computing the maximum likelihood
estimators of µ, Q, and σ 2 w for the model (8.5.3)–(8.5.4).
1. For a fixed Q∗ , apply the Kalman prediction recursions with ˆX ∗ 1 0, 1 0,
Q Q∗ , and σ 2 w 1 to obtain the predictors ˆX ∗ t . Let ∗ t denote the one-step
prediction error produced by these recursions.
2. Set ˆµ ˆµ(Q∗ ) n t1 C′ tG′ −1 n GCt/t t1 C′ tG′ (Yt − G ˆX ∗ t )/∗ t .
3. Let ˆQ ∗ be the minimizer of (8.5.12).
4. The maximum likelihood estimators of µ, Q, and σ 2 w are then given by ˆµ, ˆσ 2 w ˆQ ∗ ,
and ˆσ 2 w , respectively, where ˆµ and ˆσ 2 w are found from (8.5.10) and (8.5.11) evaluated
at ˆQ ∗ .
Example 8.5.1 Random walk plus noise model
In Example 8.2.1, 100 observations were generated from the structural model
Yt Mt + Wt, {Wt} ∼WN 0,σ 2
w ,
Mt+1 Mt + Vt, {Vt} ∼WN 0,σ 2
v ,
t1