321 Coherence Consideration in Binary Time Series Analysis
The ltered data range from 142.13 to 231.73 with mean 169.05, so that level 180 is above
average. Dene the clipped series,
1, z
t
≥ 180
y
t
=
0, z
t
< 180
, t = 1, ..., 508.
This gives the binary time series
111101001000101000010000000000010000001101011111111111111110000000000000000010000100010
101000111111111111100000000000010000000010000000000000100110111111111010000000100000000
000000000000000000000000100110000000010000000000000000000000000000000000000000111111111
000000000000000000000000000000000000000000000000000111110000000000000000000000000000000
000000000010000000000000000000000000000000000000000000000000000011101100000000000000000
0000000000000010000000000111110010000000000000101000000000000000000000000
Since we intend to use C
t
with values ranging from 0.924 to 3.109, it is sensible to replace
T
t
by x
t
= T
t
/10. The residual coherence obtained from (x
t
, y
t
) is shown in Figure 14.3, sug-
gesting (past of) x
t
x
t−k
, k = 0, 2, 4 as possible interaction covariates for logistic regression.
Table 14.3 gives the Akaike information criterion (AIC) results for some models selected
out of many more. The table shows that the AIC is minimized at the model
logit(π
t
(β)) = β
0
+ β
1
y
t−1
+ β
2
y
t−2
+ β
3
C
t−1
+ β
4
x
t−1
x
t−3
(14.23)
containing the interaction x
t−1
x
t−3
(past of x
t
x
t−2
) which appears more useful than its fac-
tors. The estimates are given in Table 14.4. Apparently, x
t−1
x
t−3
is quite signicant. Model
(14.23) can be judged further from the plots of the estimated autocorrelation and cumulative
periodogram of the residuals shown in Figure 14.4.
RS(u)
0.0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5 6 7 8
9 10 11 12 13 14 15
u
FIGURE 14.3
Maximum residual coherence RS(u) obtained from scaled temperature and clipped mortality (x
t
, y
t
).
322 Handbook of Discrete-Valued Time Series
TABLE 14.3
Logistic regression models for clipped mortality data showing that the
interaction x
t−1
x
t−3
is a potentially useful covariate.
Model AIC
1. y
t−1
+ y
t−2
343.97
2. y
t−1
+ C
t−1
325.84
3. y
t−1
+ y
t−2
+ C
t−1
306.95
4. y
t−1
+ C
t−1
+ x
t−1
x
t−5
302.71
5. y
t−1
+ y
t−2
+ C
t−1
+ x
t−3
298.94
6. y
t−1
+ y
t−2
+ C
t−1
+ x
t
2
−1
298.47
7. y
t−1
+ y
t−2
+ C
t−1
+ x
t−1
298.35
8. y
t−1
+ y
t−2
+ C
t−1
+ x
t−1
+ x
t−2
+ x
t−3
297.18
9. y
t−1
+ y
t−2
+ C
t−1
+ x
t−1
+ x
t−3
297.03
10. y
t−1
+ y
t−2
+ C
t−1
+ x
t−1
x
t−3
+ x
t−1
x
t−5
295.85
11. y
t−1
+ y
t−2
+ C
t−1
+ x
t−1
x
t−3
295.08
TABLE 14.4
Logistic regression results for Model (14.23) showing the signicance
of the interaction covariate x
t−1
x
t−3
.
ˆ
β SE p-value
Intercept −3.544 1.315 7 × 10
−3
y
t−1
1.662 0.351 2.12 × 10
−6
y
t−2
1.148 0.361 10
−3
C
t−1
2.031 0.401 4.05 × 10
−7
x
t−1
x
t−3
−0.059 0.017 5 × 10
−4
0 5 10
15
Lag
20 25 0.0 0.1 0.2 0.3
Frequency
0.4 0.5
FIGURE 14.4
Autocorrelation and cumulative periodogram of the residuals from model (14.23).
0.0
0.2
0.4
0.6
0.8
1.0
ACF
0.0
0.2
0.4
0.6
0.8
1.0
323 Coherence Consideration in Binary Time Series Analysis
14.4 Discussion
In this chapter, we explored the use of residual coherence as a graphical tool for identi-
fying potentially useful interactions in logistic regression for binary time series. In many
cases, in practice, the inclusion of interaction terms leads to improved models and, as was
illustrated, an interaction covariate could be more signicant than its factors. There are sit-
uations, however, when the difference between S
2
(λ; u) and S
1
(λ) is relatively large for a
frequency band due to a local minimum in the linear coherence S
1
(λ) giving rise to a rel-
atively large residual coherence RS(u) for some u without a substantial contribution from
X
u
(t). This is another reason why we need to couple the residual coherence with some
further evidence as to the importance of an identied interaction covariate. This can be
done, for example, by hypothesis testing and model selection criteria and by some sort of
residual analysis.
Clearly, instead of prescreening the variables using residual coherence, we could sim-
ply include interaction terms in the model and apply model selection. The advantage of
prescreening using residual coherence is that the search for useful interaction terms is facil-
itated using spectral information which accommodates model selection and hypothesis
testing. Moreover, exploring potential general relationships among time series is an impor-
tant time series problem where coherence measures, including residual coherence, as well
as other measures have an important role before tting parametric models.
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