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6.5 Seasonal ARIMA Models 207
ACF
-0.5 0.0 0.5 1.0
Figure 6-17
The sample ACF of the
differenced accidental
0 10 20 30
deaths {∇∇12Xt }. Lag
The values ˆρ(12) −0.333, ˆρ(24) −0.099, and ˆρ(36) 0.013 suggest a
moving-average of order 1 for the between-year model (i.e., P 0 and Q 1).
Moreover, inspection of ˆρ(1),..., ˆρ(11) suggests that ρ(1) is the only short-term
correlation different from zero, so we also choose a moving-average of order 1 for
the between-month model (i.e., p 0 and q 1). Taking into account the sample
mean (28.831) of the differences {Yt}, we therefore arrive at the model
Yt 28.831 + (1 + θ1B)(1 + 1B 12 )Zt, {Zt} ∼WN 0,σ 2 , (6.5.7)
for the series {Yt}. The maximum likelihood estimates of the parameters are obtained
from ITSM by opening the file DEATHS.TSM and proceeding as follows. After
differencing (at lags 1 and 12) and then mean-correcting the data, choose the option
Model>Specify. In the dialog box enter an MA(13) model with θ1 −0.3,
θ12 −0.3, θ13 0.09, and all other coefficients zero. (This corresponds to the
initial guess Yt (1−0.3B) 1−0.3B 12 Zt.) Then choose Model>Estimation>Max
likelihood and click on the button Constrain optimization. Specify the number
of multiplicative relations (one in this case) in the box provided and define the
relationship by entering 1, 12, 13 to indicate that θ1 × θ12 θ13. Click OK to return
to the Maximum Likelihood dialog box. Click OK again to obtain the parameter
estimates
ˆθ1 −0.478,
ˆ1 −0.591,