Springer - Read

6.3 Unit Roots in Time Series Models 193
Example 6.2.2 The lake data
6.3 Unit Roots in Time Series Models
then reoptimize, we obtain the model,
1 − .270B − .224B 5 − .149B 8 + .099B 11 + .353B 12 Xt Zt,
with {Zt} ∼WN(0, 0.0138) and AICC value −172.49.
In order to check more general ARMA(p, q) models, select the option Model>
Estimation>Autofit and specify the minimum and maximum values of p and
q to be zero and 15, respectively. (The sample ACF and PACF suggest that these
limits should be more than adequate to include the minimum AICC model.) In a
few minutes (depending on the speed of your computer) the program selects an
ARMA(1,12) model with AICC value −172.74, which is slightly better than the
subset AR(12) model just found. Inspection of the estimated standard deviations of
the MA coefficients at lags 1, 3, 4, 6, 7, 9, and 11 suggests setting them equal to zero
and reestimating the values of the remaining coefficients. If we do this by clicking on
the Constrain optimization button in the Maximum Likelihood Estimation
dialog box, setting the required coefficients to zero and then reoptimizing, we obtain
the model,
(1 − .286B)Xt 1 + .127B 2 + .183B 5 + .177B 8 + .181B 10 − .554B 12 Zt,
with {Zt} ∼WN(0, 0.0120) and AICC value −184.09.
The subset ARMA(1,12) model easily passes all the goodness of fit tests in the
Statistics>Residual Analysis option. In view of this and its small AICC value,
we accept it as a plausible model for the transformed red wine series.
Let {Yt,t 1,...,99} denote the lake data of Example 1.3.5. We have seen already
in Example 5.2.5 that the ITSM option Model>Estimation>Autofit gives
the minimum-AICC model
Xt − 0.7446Xt−1 Zt + 0.3213Zt−1, {Zt} ∼WN(0, 0.4750),
for the mean-corrected series Xt Yt − 9.0041. The corresponding AICC value is
212.77. Since the model passes all the goodness of fit tests, we accept it as a reasonable
model for the data.
The unit root problem in time series arises when either the autoregressive or movingaverage
polynomial of an ARMA model has a root on or near the unit circle. A
unit root in either of these polynomials has important implications for modeling.
For example, a root near 1 of the autoregressive polynomial suggests that the data
should be differenced before fitting an ARMA model, whereas a root near 1 of