The distance test statistic is motivated by the key feature of the dynamics in
region II of Figure 3, which is the existence of at least two fixed
points, one being a spiral source and one a saddle point. As noted in the
text, flows of system (1) that start near the source in general
approach the saddle point before wandering unboundedly. An empirical
implication is that the observed data should be concentrated near a point
distinct from the estimated origin of the dynamics. The sample mean
of the observed data should be distinct from the estimated
origin, 
. Of course, the origin and the sample mean
should be distinct even if the dynamics are not in Figure 3's region II,
in part because flows in the four-dimensional data cannot be expected to be
confined to a plane,
and
in part because the orbits in regions I and III of Figure 3 cannot be
expected to be circular. But if the distance between the origin and the mean
is greater for one election period than another, it is reasonable to conclude
that the dynamics are more unstable during the former period than during the
latter. If the distance is dramatically different between periods, then the
most likely explanation would be that during the less stable period the
dynamics are occurring in region II while during the more stable period they
are occurring in region I or region III.
Conditioning on the MLE and treating the sample
mean as random, a measure of the distance between the
origin and the mean for election period j is
where 
is the number of observations and
,
with k=28 being the number of parameters in model (3). Under
the hypothesis that 
, 
has the
distribution. As noted above, however, such a hypothesis of
equality is not reasonable for model (3). The distribution for
should therefore be taken as noncentral 
with
noncentrality parameter 
. The degree of instability between
election periods can be compared by comparing the magnitudes of for the
respective periods, via the ratio 
. 
is the value of
for the period 
that is predicted to be more unstable and 
is
the value for the period 
that is predicted to be more stable. In
general, 
has the doubly noncentral F distribution,
(Johnson, Kotz and Balakrishnan 1995, 480). The
hypothesis 
, which asserts that the election period is neither
more nor less unstable than the period, implies 
.
Under the hypothesis, therefore has the distribution
. Values of significantly greater than
1.0--i.e., 
for test level
--indicate departures from equality in the theoretically predicted
direction.
The divergence test statistic is motivated by the contrasting effects flows in
regions I and III of Figure 3 have on the volumes of bounded sets near
the fixed point. In region I, flows decrease the volume of such a set, while
in region III flows cause the volume of such sets to increase. By Liouville's theorem (Arnold 1978, 69-70), the rate of
change that system (1) induces in the volume of a bounded set is
equal to the integral of the divergence of system (1)'s
vector field over that set. The divergence of a vector field at each point is the trace of its
Jacobian matrix evaluated at that point (Weibull 1995, 251). Writing the
vector field for system (1) as 
, the
divergence is
To estimate the divergence for each observed data point 
, I
reverse the approach used to derive the statistical model (4) from
system (2) and treat each value 
as an
estimate of the value of the vector field at . I estimate
the divergence at by using finite differences to compute
. The test statistic 
is the
t-statistic for the difference of means between the set of values 
for the election period that is predicted to be more
unstable and the set of values for the period that is predicted to be more
stable. The theory predicts that the differences will be significantly
positive.