Any effort to test the empirical reality of the dynamics predicted by the solution of the game faces two difficulties. First, system (1) is certainly not an exactly correct description of the dynamics of a real campaign. At best, system (1) captures important features of the true dynamics. Second, the variables in terms of which system (1) is defined cannot be measured in a truly dynamic fashion during a campaign. Time series can be constructed to record the times at which candidates received particular monetary contributions. But it is not clear how well such data would measure the evolving intentions of individual contributors. And in any case the succession of service commitments that the candidates may be making--offers and counter-offers--would be extremely difficult if not impossible to observe. It will be necessary to use cross-sectional data to assess the game's predictions regarding the dynamics.
A solution to both problems is to use a statistical model that represents key
features of system (1) in a way that is robust even to gross errors
of functional form and measurement. To develop such a model I focus on the
fact that system (1) exhibits Hopf bifurcations for (g, h)
values near 
. For empirical analysis, 
is the most important of the equilibrium outcomes, because
it is the only outcome that leads to a race in which the incumbent runs and
faces opposition. By the theory of the game, all actual campaigns in which
the incumbent runs for reelection and faces at least minimally funded
opposition ought to have dynamics like those of system (1) for
with 
. All such campaigns ought to have contribution levels,
vote totals and post-election service amounts that are generated by processes
that are qualitatively similar to system (1) near
and 
.
Using local bifurcation theory's method of normal forms, it can be shown that any bivariate system of differential equations that exhibits Hopf bifurcation has a Taylor series expansion of degree 3 that can be brought into the following form by smooth changes of coordinates,
where 
and 
(Guckenheimer and Holmes 1983,
150-151). When 
, system (2) has a circular
limit cycle. The theorems of local bifurcation theory that justify system
(2) as a generic representation of Hopf bifurcation imply the
crucial point about its robustness: the qualitative property of exhibiting
Hopf bifurcation is not affected by terms of degree higher than 3 in the
Taylor series expansion (Arnold 1988, 270-275; Guckenheimer and Holmes 1983,
151-152). As a representation of Hopf bifurcation, system (2) is
perfectly robust. In practical terms, the robustness of system (2)
means that no matter how complicated the system may be that actually generates
the data we can observe from a competitive campaign, if that generating system
is near a fixed point and exhibits Hopf bifurcation at that point, then almost
any set of measurements of the system can be smoothly transformed into a set
of coordinates such that the equations of (2) are adequate to
characterize the qualitative properties of interest for testing the
predictions of the game. If the theory of the game is qualitatively correct
in predicting that all competitive campaigns occur near a continuum of Hopf
bifurcations, then empirical models built on the formulation of system
(2) should accurately and reliably approximate the qualitative
features of the true dynamics.
In the Appendix I show that the theoretically crucial qualitative properties
of system (2) can be recovered from cross-sectional data by using a
simultaneous statistical model for four observed variables, denoted 
,
, 
and 
. Using 
, 
,
, 
, the functional form of the model is
The random vector 
is assumed to be
normally distributed with mean 
and covariance matrix
. Unknown parameters, to be
estimated, are 
, 
, 
, 
and
, 
, 
and 
for 
, with 
. I refer to this model
as the four-dimensional Hopf (4DH) model.
The bifurcation set shown in Figure 3 exhibits Hopf bifurcations in
system (1) only crossing the open segment O-C. The other
bifurcations shown in the figure are saddle connection bifurcations rather
than Hopf bifurcations. There is no guarantee that the equations of
(2) will provide good approximations to the system's qualitative
properties as (g,h) varies in the direction of the saddle connection
bifurcations. Unfortunately for the goal of empirical testing, a saddle
connection bifurcation is a global rather than a local phenomenon
(Guckenheimer and Holmes 1983, 295). No normal form such as (2)
exists that can generically represent the qualitative properties of such a
bifurcation. But in light of the quantitative similarities across all three
regions of Figure 3 in the flows that begin with , at least not too far from
, it is plausible that for actual campaigns
the different kinds of dynamics will be quantitatively sufficiently similar
that model (3) will nonetheless provide a good approximation.
