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8.5 Estimation For State-Space Models 281
with initial values µ M1 0, σ 2 w 8, and σ 2 v 4. The maximum likelihood
estimates of the parameters are found by first minimizing (8.5.12) with ˆµ given by
(8.5.10). Substituting these values into (8.5.11) gives ˆσ 2 w . The resulting estimates are
ˆµ .906, ˆσ 2 v 5.351, and ˆσ 2 w 8.233, which are in reasonably close agreement
with the true values.
Example 8.5.2 International airline passengers, 1949–1960; AIRPASS.TSM
The monthly totals of international airline passengers from January 1949 to December
1960 (Box and Jenkins, 1976) are displayed in Figure 8.3. The data exhibit both a
strong seasonal pattern and a nearly linear trend. Since the variability of the data
Y1,...,Y144 increases for larger values of Yt, it may be appropriate to consider a
logarithmic transformation of the data. For the purpose of this illustration, however,
we will fit a structural model incorporating a randomly varying trend and seasonal
and noise components (see Example 8.2.3) to the raw data. This model has the form
Yt GXt + Wt, {Wt} ∼WN 0,σ 2
w
Xt+1 F Xt + Vt, {Vt} ∼WN(0, Q),
where Xt is a 13-dimensional state-vector,
⎡
⎤
1
⎢ 0
⎢ 0
⎢
F ⎢ 0
⎢ 0
⎢
⎣ .
1
1
0
0
0
.
0
0
−1
1
0
.
0
0
−1
0
1
.
···
···
···
···
···
. ..
0
0
−1
0
0
.
0
⎥
0 ⎥
−1 ⎥
0 ⎥
⎥,
0 ⎥
. ⎦
0 0 0 0 ··· 1 0
and
G [ 1 0 1 0 ··· 0 ] ,
⎡
σ
⎢
Q ⎢
⎣
2
1
0
0
σ
0 0 ··· 0
2
0
2
0
0
σ
0 ··· 0
2
⎤
0
.
0
.
3
0
.
0
0
.
···
···
. ..
⎥
0 ⎥
0 ⎥.
⎥
. ⎦
0 0 0 0 ··· 0
The parameters of the model are µ,σ 2 1 ,σ2 2 ,σ2 3 , and σ 2 w , where µ X1. Minimizing
(8.5.12) with respect to Q ∗ we find from (8.5.11) and (8.5.12) that
ˆσ 2
1
, ˆσ 2
2
, ˆσ 2
3
, ˆσ 2
w
,
(170.63,.00000, 11.338,.014179)