- Email: [email protected]
The location of american presidential candidates: An empirical test of a new spatial model of elections
Math1 Comput. Modelling, Vol. 12, No. 415, pp. 461-470, Prmted m Great Britain. All rights reserved
19X9 Copyright
9
0895.7 177,‘89 $3.00 + 0.00 1989 Pergamon Press plc
THE LOCATION OF AMERICAN PRESIDENTIAL CANDIDATES: AN EMPIRICAL TEST OF A NEW SPATIAL MODEL OF ELECTIONS-f JAMES Department
of Political
Science,
MELVIN University
M.
ENELOW
State University
of New York,
Stony
Brook,
NY 11794, U.S.A
J. HINICH
of Texas at Austin,
Austin,
TX 78712, U.S.A.
Abstract-Using a scaling procedure that directly estimates the key parameters of a new spatial model of elections described previously, we recover the locations of voters and candidates in the three waves of the 1980 Major Panel File. The candidates lie relatively close to the average voter and exhibit little movement over the campaign. Voter locations are more volatile. A unique equilibrium is computed for each wave of the panel, based on a probabilistic model of individual voting decisions. This equilibrium stays relatively close to the Republican candidates, finally settling in a position between the Democrats and Republicans, but closer to Reagan than Carter.
INTRODUCTION
While long admired for its theoretical sophistication, the spatial theory of elections has often been criticized for the scarcity of empirical testing. The empirical side of spatial theory is certainly not barren. Beginning with Weisberg and Rusk (1970) followed by Rabinowitz (1973, 1978) Aldrich and McKelvey (1977) Hinich (1978) Enelow and Hinich (1984a) and Poole and Rosenthal (1984) a body of literature has emerged that establishes several propositions. First, a number of metric and non-metric scaling procedures are capable of generating multidimensional maps of voters and candidates based on survey data that voters provide about themselves and the candidates. These maps almost invariably are two-dimensional and provide sensible pictures of where the candidates and voters are located in American Presidential contests. Second, spatial choice models perform rather well at postdicting the reported preference data given by the voters. The distance between each voter and each candidate in the recovered space is a fairly accurate measure of the voter’s preferences among the candidates. Third, it is possible to interpret the dimensions of the recovered space. Not surprisingly, the dominant dimension in American Presidential politics is seen as the economic left-right dimension that is the New Deal basis for the last major party realignment. The second dimension of Presidential politics is generally viewed as the “unorthodox issues” dimension, composed since 1968 largely of lifestyle issues such as abortion and women’s rights that fail to fit into traditional New Deal categories. There is one theoretical question that has been addressed in the above studies and that is the question of candidate location. Rabinowitz (1978) and Poole and Rosenthal (1984) recover maps that show American Presidential candidates located on the periphery of the electorate. The candidates ring the voters, who sit in the center by themselves. Enelow and Hinich (1984a), on the other hand, recover maps of the 1976 and 1980 American Presidential elections in which the major candidates are much closer to the center of the space. On the economic left-right dimension, all candidates lie within 1 SD (standard deviation) of the voter most-preferred points and on the social left-right dimension nearly all do so. There is some confusion about the relationship between convergence and spatial theory. While spatial theory predicts candidate convergence in one-dimensional two-candidate elections, twodimensional multicandidate contests are quite different in nature. It is well-known that the median voter result applies to multidimensional voting models only under very stringent conditions, raising the possibility that non-convergence among the candidates in American elections is consistent with TDelivered
at the 1984 Annual Meeting
of the American
Political 461
Science Association,
Washington,
D.C.
JAMESM. ENELOW and MELVIN J. HINICH
462
spatial theory. Still, a giant hole in the middle of the electorate is a strange occurrence, to say the least, since presumably a new candidate would emerge to plug it up. We are left then with two questions: first, does empirical evidence support the convergence hypothesis in American Presidential elections; and second, does the answer to the first question verify or disconfirm the predictions of spatial theory? These are the two questions this paper is designed to answer. In the sections that follow we will describe a methodology appropriate for testing the newest form of the spatial theory of elections described by Enelow and Hinich (1984a). We will show how this methodology-classical factor analysis-allows us to estimate candidate and voter locations in a multidimensional Euclidean space as well as most of the other parameters of the spatial model. The maps we recover (based on issue data from the 1980 CPS Major Panel File), show significant differences among candidate locations but find the candidates to be relatively close to the center of the space, confirming the results of Enelow and Hinich (1984a) based on thermometer data and a different methodology. The 1980 panel data are also useful for another purpose. These data allow us to track the dynamics of the 1980 campaign from January to November of that election year. Probabilistic voting models that permit the voter to cast his vote with some probability for other than a spatially most-preferred candidate yield a unique equilibrium location for two candidates in a multidimensional contest. We show how this equilibrium can be computed from the survey data of the 1980 panel file, and we watch its behavior across the three waves of the study (January-July-November). We find that the probabilistic equilibrium moves across the three waves, finally settling in a position between the Democrats and the Republicans, but 60% closer to Ronald Reagan than to Jimmy Carter. Not only is this result strongly intuitive (since Reagan won the election), it provides support for the predictive accuracy of probabilistic voting models in particular and spatial theory in general. Our results demonstrate that the major candidates in the 1980 Presidential race are centrally located, relative to the voters we analyze, and that spatial theory can both identify this center and explain why the candidates want to be there.
AN
APPROPRIATE
METHODOLOGY
FOR
TESTING
SPATIAL
THEORY
We now proceed to outline an empirically-inspired spatial model of elections in terms of the mathematical structure of factor analysis. In this model, the candidates are variables that load on a small set of underlying dimensions. The voters possess preferences over a larger set of campaign issues but can be represented on the set of underlying dimensions by projecting their positions in the larger space onto the smaller space. The voters and issues are sampling units of the observed variables and so can be assigned factor scores. These scores are useful not only for estimating the location of the voters in the factor space, but also for computing a probabilistic equilibrium for two-candidate elections, derived in Enelow and Hinich (1984b). We are thus able to represent in the same space: the voters, the candidates and an optimal location for the candidates. By fixing the location of the candidates across the three waves of the 1980 panel study, we can observe how changes in voter opinion are reflected in the changing location of this equilibrium. We should expect that as election day nears, this equilibrium should move toward the location of the winning candidate. A full exposition of the spatial model we employ can be found in Enelow and Hinich (1984a). For this reason, we will be somewhat brief. In the language of factor analysis, we observe M values (i=l,..., N;j=l,..., n). The random variables, m = 1, . . . , M, that assume Nxn random variables are candidates, N is the number of voters and n is the number of issues. If C,,, is the m th such variable, then we assume that cijm, the i,jth observation of C,, fits the model ciim - b, = v; pm+ eijm)
(1)
where b, is the origin of the i,jth observation on each variable, v: = (vi,, , . . . , vijr) are the values of the r common factor variates for the i, jth observation, pm= (pm,, . . . ,P,,,,)~ are the loadings of the mth variable (candidate) on the r derived factors and eiim is a random error term. In the language of the spatial model, the ith voter’s perception of the mth candidate’s position
A new spatial
model
of elections
463
on the jth issue (c,,) is assumed to be a linear function of the candidate’s position on a small set of r predictive dimensions (p,, , . . . ,p,,) with a residual term eiim to capture non-systematic influences that are uncorrelated with p, or vii. If an economic and a social left-right dimension underlie the perceived issue positions of American national politicians, then each voter uses a simple, idiosyncratic rule for translating the of the candidates into positions on the issues about which he is economic and social “ideologies” concerned. Given the disincentive to acquire direct issue information about the candidates, this procedure is cost efficient and explains why voters may have “honest” differences about the same candidate’s issue positions. The spatial model of elections we have described is equivalent to assuming that a factor structure underlies the perceived positions of candidates across a set of issues. Given the following additional assumptions, a factor analysis of the sample covariance of the 1.h.s. of model (1) yields consistent estimates of pm: (1) uijk is a random
for i = 1, . . . , N; j = 1, . . . , n; k = 1, . . . r;
variable
and (2)
U,,k
is distributed
for i and k.
independently
If the factors are correlated, the recovered candidate map will be distorted. The distortion will be a continuous function of the factor correlation. Thus, a small amount of correlation will not cause much distortion in the estimated map. (3) var(ai,,) = . . . = var(u,,,). Since the units of pmk are undefined, we can set this variance equal to one. If the variances are unequal, the candidate map will be stretched along the axis with the largest variance. The standard deviation of the factor multiplies the factor loading coefficient and thus is not estimable since the parameters are not identified. The assumption of equal variances is an identification restriction. Such restrictions are needed to estimate the parameters of models such as factor models which have unobserved variates. The next assumption is also an identification restriction: (4) covar(e,,, Given assumptions to show that
, eljn>
(l)-(4),
=
0
for m #n.
if dt,m= cij,,,- btj and dijn= ciin- b,, , then it is a straightforward covar(&,,
= P: pn
for m #n
= P%P, + +??I
for m = n,
,&,I
exercise
where Ic/, = var(eij,) is the specific variance of the m th variable and pm are the loadings of the m th candidate on the r underlying dimensions. In matrix form, covar(d,,,, di,,) is the m,nth element of the A4 x M covariance matrix C, where c=ppr++. P is the A4 x r matrix of loadings of the M candidates on the r underlying dimensions an M x M diagonal matrix whose m th diagonal term is $,(m = 1, . . . , M). The diagonal of PPT are called the communalities of the variables. The sample covariance of dilmand d,ju is s mn= (Nn - 1))’ i
(2) and Y is elements
5 (dijm- d,)(dijn - d,),
j=li=1
where d, = (Nn)-’
i
2 diJq
of the m,n th The m,n th element of the sample covariance matrix s,, is an unbiased estimator element of C. It is also a consistent estimator as N+co. A factor analysis of S yields a consistent estimator of PT, where T is an r x r orthogonal rotation matrix.
464
JAMES M. ENELOW
SCALING
THE
CANDIDATES
IN
and MELVIN
THE
1980
J. HINICH
PRESIDENTIAL
ELECTION
The factor loadings of the candidates are their positions in the factor space corresponding to the predictive dimensions that are postulated in the spatial model described by Enelow and Hinich (1984a). In order to estimate these loadings, we require data that tell us where the voters think the candidates are located on some set of campaign issues. For the 1980 CPS Election Study, these data are given by the issue scales on which voters are asked to place themselves and the candidates. These scales are labelled at the end points and typically subdivide a line segment into seven equal subintervals that constitute the seven permissible locations for the respondent to place himself or a given candidate. For the 1980 Major Panel File, there are four issues scales that appear in every wave from January to November. These are: Defense Spending; Government Spending; Government Antiinflation Plan; and Russia. The end-point labels for these four scales, are, respectively: (l-greatly decrease defense spending, 7-greatly increase defense spending); (l-Government should provide many fewer services; reduce spending a lot, 7-Government should continue to provide services; no reduction in spending); (l-reduce inflation even if unemployment goes up a lot, 7-reduce unemployment even if inflation goes up a lot); (l-important to try very hard to get along with Russia, 7-big mistake to try too hard to get along with Russia), In each of the three waves, the respondent is asked to locate himself and a set of candidates on each of these issues. The second set of data are, of course, the cj,, to be used in the factor analyses. Interestingly, the respondent was also asked to place “What the Federal Government is doing at the present time” on each scale, providing a natural measure of b,,, the origin on issue j as perceived by voter i. Before limiting ourselves to these four issues, however, a factor analysis was performed for each wave averaging across all issues in the wave (four in wave 1, five in wave 2, seven in wave 3). In addition, an issue-at-a-time factor analysis was also performed for each issue and wave. The point of doing these analyses was to see how robust the candidate loadings were to variations in the number and type of issue that were used. The results were striking. Very little change could be detected across these factor analyses in the unrotated candidate loadings. These results support an important component of the issues and dimensions model of elections that we employ. If perceived issue positions are a function of a common set of underlying candidate positions, then these same underlying positions should be recovered from a factor analysis of the issue data, no matter which subset of issues we analyze. The stability of the candidate loadings across waves does not necessarily follow from our model, but is consistent with the results of Poole and Rosenthal (1984) who apply a metric unfolding technique to thermometer data given by the voters about the candidates. Based on this cross-wave stability, we follow the decision of Poole and Rosenthal to fix the candidate loadings across the three waves of the panel study. This step allows us to disentangle the effects of voter change from candidate change in the maps we will examine. In order to maximize the number of candidates we can scale, the unrotated loadings of the candidates were averaged across issues and waves. The resulting candidate map is presented in Fig. 1, where the origin is the position of the Federal Government in the factor space. A 0.91
'BROWN
KENNEDY*
0.78
t DEMOCRATS
0.65 0.53
t
ANDERSON
*CARTER
;,r;,, ,,
o.92 ,:,
-0.20
0.01
0.17
0.36
0.55
0.73
Fig. 1
CONNfijI;
465
A new spatial model of elections
two-dimensional solution was chosen on the basis of the size of the first two eigenvalues of the covariance matrix and the proportion of the total variance explained by the first two factors. In increasing order, the distances of the candidates from the origin are Carter (0.60), Democrats (0.87) Kennedy (1.17), Anderson (1.24), Brown (1.37), Baker (1.41), Connally (1.47) Bush (1.49) Republicans (1.66) and Reagan (1.71). Since the origin of the space is, in a real sense, the “status quo”, the candidate distances correspond to what we would expect. Jimmy Carter, the incumbent, is closest to the status quo. That he is not identical to the status quo indicates that some separation exists in the voters’ minds between current policies and what Carter-the-candidate represents. With the exception of Anderson, whose policy positions are quite different from any of the other Republicans, all Democrats are closer to the status quo than any of the Republicans, with Reagan occupying the most extreme position. ADDING
THE
VOTERS
TO
THE
SPACE
In order to judge how well the candidates represent voter opinion we must scale the voters and candidates together. In Enelow and Hinich (1984a), it is shown that if xi = (x,, , . . . , x,,)‘~ are voter i’s ideal positions on issues 1, . . , n and A, is i’s n x n symmetric positive definite salience matrix that measures how i makes trade-offs between any two issues; then i’s most-preferred point in the underlying predictive space is zi = (VBA, Vi)-’ VTA,y,,
(3)
where L’, is an n x r matrix whosej,kth element vlik is, in the factor analytic model, the score of the i,jth sampling unit on the kth common factor; or, in the spatial model, the slope of the line that translates changes on the kth dimension into changes on the jth issue. The n-dimensional vector yi = (xi, - h,, , . . , xi” - l~,)~ is i’s ideal point in the issue space as measured from bi. We can derive the factor scores of the Nn sampling units from the identity F = DS-‘P (see Morrison 1967). The matrix F is Nn x r, where the first N rows are the factor scores for issue 1 and so on. The matrix D is Nn x M, where d,,,, = c,,, - b,. The matrix S ’ is the inverse of the M x M sample covariance matrix, and P is the M x r matrix of estimated (unrotated) candidate V, by taking the ith, The matrix F is then used to construct loadings. (N + i)th, . . , [N(n - 1) + i]th rows of F. The y, vector is obtained from the raw issue data. The salience matrix cannot be measured, and so we set it equal to the n x n identity matrix. Otherwise, we are able to estimate all the parameters necessary to locate the voters in the same space with the candidates. By fixing P across waves, any changes in voter location over time must be due to changes in voter opinion, either regarding their own issue positions or those of the candidates. In our maps, the candidates are fixed and the voters move. Convergence, then, must be redefined as movement of the voters toward the candidates instead of movement of the candidates toward the voters. If some location in the electoral space best represents the voters, then for convergence to occur, this location must move closer to the candidates over time. Before addressing the convergence question, however, let us examine the three maps that are obtained by fixing the candidates and scaling the voters in each of the three waves of the 1980 panel study. To increase our confidence in the estimates of the voter ideal point locations, we will scale only those voters with factor scores on at least three issues. This subsample represents the public most attentive to the issues of the campaign, and, consequently, most responsive to the issue appeals of the candidates. Our first observation is that in all three maps, the candidates lie not on the periphery of our subsample of voters, but squarely in the middle. For the first wave, the voter mean is (1.05, -0.27); for the second wave (1.70, 0.20); and for the third wave (1.20, 0.14). The three respective vectors of voter SDS on the two dimensions are (2.06, 3.97) (2.38, 3.62), and (2.83, 5.30). These statistics compare with the candidate mean and SD vectors of (1.15, 0.25) and (0.43, 0.46) respectively. The voter mean stays relatively close to the candidate mean while the voter SDS are anywhere from 4.8 to 6.6 times the candidate SD on the first dimension and anywhere from 7.9 to 11.5 times the candidate SD on the second dimension. Although we are dealing with a subsample of the electorate, we do not find the “empty center”
JAMES M. ENELOW and MELVIN J. HINICH
466 4.00
WAVE 1
.
. . 2.40
.
.
.
.
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.
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Fig. 2. 0 Denotes
- 0.08
I
I
I
I
I
-2.40
-4.00
2.40
0.08
4.00
overlap; M denotes mean; 0, Carter; 1, Reagan; 2, Kennedy; 3, Anderson; 5, Connally; 6, Brown; 7, Baker; 8, Republicans; 9, Democrats.
4, Bush;
depicted by Rabinowitz (1978) and Poole and Rosenthal (1984). In fact, the increase in the voter SD vectors across waves indicates that, relative to the voters, the candidates are moving closer together. in an aggregate sense, we To confirm that the scaled positions of the voters are “reasonable” 4.00
I
w AVE
2.4C
.
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2
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.
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Fig. 3. Key as in Fig. 2.
I 2.40
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A new spatial
model
of elections
461
4.00 WAVE
.
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Fig. 4. Key as in Fig. 2.
computed the location of the (axis-by-axis) median strong Democrat, weak and independent Democrat, pure independent, weak and independent Republican and strong Republican. The median strong Democrat is located near the Democrats, the median weak-independent Democrat is located near Kennedy, the median pure independent is near the northeast corner of the map, about equidistant from Kennedy and Baker, while both the weak-independent and strong Republicans are located near Reagan. Breaking down the voters by other issues, such as “liberals, moderates and conservatives” yields similar results. These results provide an internal check on the validity of the estimated voter ideal points. We do not wish to overinterpret these results. Monte Carlo analysis of a related scaling procedure for mapping voters and candidates (Palfrey and Poole 1987) suggests that estimates of voter locations are quite noisy relative to estimates of candidate locations. Three types of noise affect the estimated voter locations in our analysis. With two predictive dimensions, two factor scores are estimated for each voter on each issue. Given 10 candidates, each score is estimated with a sample size of 10, leaving only 8 surplus degrees of freedom. This first type of noise will be reduced as the number of candidates increases. The second type of noise derives from a lack of model fit for individual voters. Our assumption that the candidate locations in the predictive space are identically perceived by all voters is clearly violated for some voters. In order to fit these voters in the predictive space, their locations will be distorted. A third type of noise results from the unreliability of the answers voters give to the issue questions. Given that many voters are unfamiliar with the issues about which they are questioned and the I-l-point scales which define the possible set of answers, any given answer a voter provides is a sample point from a distribution of responses that he may give. In their analysis of Aldrich and McKelvey’s (1977) scaling procedure, Palfrey and Poole (1987) reach several conclusions which are germane to our analysis. First, they find that “individuals with a high level of information tend to be more extreme than those with low levels and are much more likely to vote” (p. 511). They also find that uninformed voters tend to be mapped toward the center of the space “regardless of their ‘true’ distribution” (p. 516). In fact, some bias toward the center also affects the estimated location of extremist voters.
468
JAMES M.
ENELOWand MELVIN
J. HINICH
These results are consistent with the voter locations we recover. The voters in our maps are highly informed. It is quite likely that were we to include less informed voters, they would be mapped toward the center of the space, “pushing” the candidates away from the center. But, from Palfrey and Poole’s conclusions, the estimated locations of uninformed voters are hghly unreliable. Furthermore, we expect these individuals are less likely to vote than those we have mapped. We conclude that from the candidates’ standpoint, our maps describe both the more relevant and more accurately measured public. A PROBABILISTIC
EQUILIBRIUM
FOR
THE
CANDIDATES
We have yet to address the question of where spatial theory predicts that the candidates will be located. One answer is given by probabilistic voting models. These models typically assume that the voter attaches probability weights to the alternative actions: vote for candidate A; vote for candidate B; abstain. The weights placed on the first two actions are typically a function of the voter’s utilities for the candidates, while the weight placed on the last action depends on both the utility difference between the candidates (the indifference factor) and the utility associated with the candidate the voter most prefers (the alienation factor). Fnelow and Hinich (1984b) examine two-candidate elections under the assumption that the voter’s probability of voting for candidate A, prob(A), and his probability of voting for B, prob(B), are given by prob(A)
= eu[u(A)J + (1 - e)b[u(A)
- u(B)]
= ea[u(A)]
if u(A) > u(B) otherwise,
and prob(B)
= ea[u(B)] + (1 - e)b[u(B)
- u(A)]
= ea[u(B)]
if u(B) > u(A) otherwise,
where e is the relative importance of alienation vs indifference as a cause of abstention (0 < e < l), a is a concave function of its argument, u is the voter’s utility function and b is a positive constant. All parameters are subscripted by voter, but the subscript has been dropped for simplicity. Given this probabilistic voting model, Enelow and Hinich (1984b) show that regardless of the shape of the voter utility functions or the dimensionality of the electoral space, a unique location exists that maximizes the expected plurality of each candidate in a two-candidate race. For the case of quadratic voter utility and equal relative importance of alienation vs indifference among voters, this unique equilibrium is: p* = [E(V/Tyl)]-‘E(Vfy;),
(4)
where E is the arithmetic average. Since we have estimated all the terms on the r.h.s. of this equation, we can compute this equilibrium for each wave of the 1980 panel study and see how closely it approaches the positions of the candidates. Figure 5 represents the probabilistic equilibrium for all three waves, along with the mean voter, in the same space with the eight candidates and two major parties. The number of voters on which p* and the mean voter are based is 168 for wave 1,263 for wave 2 and 357 for wave 3. To preserve comparability across waves, only the four issues that were common across the three waves were used in the analysis. For a voter to enter into the analysis, he would need to have factor scores on at least three of these issues. Firstly, p* shows significant movement across the three waves. Beginning near Bush, the probabilistic equilibrium moves in a northeast direction away from the Republican candidates and then moves back to a position near Baker. The mean voter describes a similar path but is generally further from Reagan and the Republicans. Most importantly, however, the final location of the mean voter is nearly equidistant between Carter and Reagan, while the final location of p* is significantly closer to Reagan. The distances between each candidate and the mean voter, as well as the distances between each candidate and p* are listed for each wave in Table 1, along with the means and SDS of the distances for each wave.
469
A new spatial model of elections 0.91
l
-
DEhfOCRATS 0.65
-
0.40
-
l
KENNEDY
BROWN
l
l
l
ANDERSON
CARTER 9 E
0.14
F
-
C + -0.11
BAKER
CONNALLY D -0.36
l
t BUSH A
-
-0.2 Fig.
5.
,
I
0.22
0.64
A=p:,
B=p:,
1.1
C=p;,
D=M,,
l
REP”;UCA REAGAN* I
I
1.5
E=M,,
A’S 1.9
F=M,
Inspection of Table 1 reveals that p* is closest to Bush in waves 1 and 2, and is closest to Baker in wave 3. This finding suggests that while Reagan was certainly a stronger candidate than Carter, a “moderate” Republican like Bush or Baker would have made an even stronger head for the Republican ticket. Since the closer a candidate’s position is to p* the better, we might think of a candidate’s distance from p* as a measure of the success of his campaign. From this standpoint, both Carter and Reagan’s campaigns faltered between January (wave 1) and July (wave 2) but picked up support by election day. Locating at p* maximizes the candidate’s expected plurality and so should not be taken as a direct measure of estimated support. Still, among survey respondents, Carter’s relative support across the three waves is 50, 29 and 46%, respectively, while Reagan’s support is 11, 29 and 43%, respectively. The change in Reagan’s support from wave 1 to wave 2 is the only change in support which does not follow the path of p*. As measured by the mean and SD of the candidate distances either from the voter mean or from p*, it is fair to conclude that particularly over the latter part of the 1980 Presidential campaign, candidate convergence occurs among the voters in our subsample. By this we mean, the center of this electoral subsample moves toward the candidates. In fact, between wave 2 and wave 3, p* moves closer to every candidate. Thus, we conclude that strong centripetal tendencies existed in the 1980 American Presidential election. For those voters most informed about the issues of the campaign, we find movement of public opinion toward the candidates in general and the winner in particular. CONCLUSION The spatial theory of elections is a that political candidates are motivated is whether this theory can contribute The purpose of this paper has been
Table
purposeful actor model of elections, based on the premise by the desire to win votes. A legitimate concern, however, to our understanding of real elections. to argue that spatial theory can answer empirically-based
candidate
to
P:
I. Distance
P:
from
each
P:
MI
M,
MI
Carter
1.1209
1.3954
0.9054
0.8145
I .2078
0.7218
0.6230
0.5680
0.6908
1.2015
1.1714
0.8750 0.3489
I
p* and M
RGlgdIl
0.2799
0.6803
0.551
Kennedy
1.2772
1.2947
I .0066
Anderson Bush
0.7210 0.0654
0.7852 0.5391
0.4497 0.2062
0.7559 0.4734
0.6303 0.3504
0.3580
Connally BrOWI
0.1223 1.1440
0.6675 .0707
0.4032 1.1742
0.4932
0.4459
I
0.3362 0.8755
0.9737
0.7847
Baker
0.1740
0.5621
0.1051
0.4429
0.3582
0.2499
Republicans
0.1428
0.5417
0.4062
0.5943
0.4096
0.5637
DtWIOCWIS
1.3878
1.5246
1.1330
1.1828
1.3698
0.9664
Meall
0.6435
0.9062
0.5976
0.7666
0.7533
0.6005
SD
0.5147
0.3617
0.3392
0.3006
0.3693
0.2316
470
JAMES M. ENELOW and MELVIN J. HINICH
questions about actual elections. We have argued that convergence, which is a dynamic phenomenon, did occur in the 1980 American Presidential election with respect to those voters most informed about the issues. We have also argued that the point of this convergence is best described by a probabilistic equilibrium. We find no support for the “empty center” results of Rabinowitz (1978) and Poole and Rosenthal (1984) but rather find a set of candidates that is centrally located relative to the voters whose opinions we have scaled. As we mentioned earlier, if less informed voters were scaled along with more informed voters, our conclusions may require modification. Given the paucity of data on these voters, with the attendant increase in error and the expected bias toward the center, our view is that the gains from analyzing an increased sample of voters would be negated by the unreliability of the results. Moreover, spatial theory assumes that voters are responsive to issue appeals. Thus, it seems best to limit our tests to those voters for whom this assumption is reasonable. Voters who are uninformed about the candidates are outside the scope of the theory. Acknowledgements-This
work
was supported
by the NSF under
Grant
SES-8310591
REFERENCES Aldrich, John and Richard McKelvey. 1977. A Method of Scaling with Applications to the 1968 and 1972 Presidential Elections. American Political Science Review 71: 11 l-30. Enelow, James and Melvin Hinich. 1984a. The Spatial Theory of Voting: An Introduction. New York: Cambridge University Press. Enelow, James and Melvin Hinich. 1984b. Probabilistic Voting and the Importance of Centrist Ideologies in Democratic Elections. Journal of Politics 46: 459478. Hinich, Melvin. 1978. Some Evidence on Non-voting Models in the Spatial Theory of Electoral Competition. Public Choice 33: 83-102. Morrison, Donald. 1967. Multivariate Statistical Methods. New York: McGraw-Hill. Palfrey, Thomas and Keith Poole. 1987. The Relationship Between Information, Ideology, and Voting Behavior. American Journal of Political Science 31: 51 I-530. Poole, Keith and Howard Rosenthal. 1984. U.S. Presidential Elections 196&80: A Spatial Analysis. American Journal of Political Science 28: 282-3 12. Rabinowitz, George. 1973. Spatial Models of Electoral Choice: An Empirical Analysis. Chapel Hill: Institute for Research in Social Science. Rabinowitz, George. 1978. On the Nature of Political Issues: Insights from a Spatial Analysis. American Journal of Political Science 22: 793-817. Weisberg, Herbert and Jerrold Rusk. 1970. Dimensions of Candidate Evaluation. American Political Science Review 64: 1167-l 185.
19X9 Copyright
9
0895.7 177,‘89 $3.00 + 0.00 1989 Pergamon Press plc
THE LOCATION OF AMERICAN PRESIDENTIAL CANDIDATES: AN EMPIRICAL TEST OF A NEW SPATIAL MODEL OF ELECTIONS-f JAMES Department
of Political
Science,
MELVIN University
M.
ENELOW
State University
of New York,
Stony
Brook,
NY 11794, U.S.A
J. HINICH
of Texas at Austin,
Austin,
TX 78712, U.S.A.
Abstract-Using a scaling procedure that directly estimates the key parameters of a new spatial model of elections described previously, we recover the locations of voters and candidates in the three waves of the 1980 Major Panel File. The candidates lie relatively close to the average voter and exhibit little movement over the campaign. Voter locations are more volatile. A unique equilibrium is computed for each wave of the panel, based on a probabilistic model of individual voting decisions. This equilibrium stays relatively close to the Republican candidates, finally settling in a position between the Democrats and Republicans, but closer to Reagan than Carter.
INTRODUCTION
While long admired for its theoretical sophistication, the spatial theory of elections has often been criticized for the scarcity of empirical testing. The empirical side of spatial theory is certainly not barren. Beginning with Weisberg and Rusk (1970) followed by Rabinowitz (1973, 1978) Aldrich and McKelvey (1977) Hinich (1978) Enelow and Hinich (1984a) and Poole and Rosenthal (1984) a body of literature has emerged that establishes several propositions. First, a number of metric and non-metric scaling procedures are capable of generating multidimensional maps of voters and candidates based on survey data that voters provide about themselves and the candidates. These maps almost invariably are two-dimensional and provide sensible pictures of where the candidates and voters are located in American Presidential contests. Second, spatial choice models perform rather well at postdicting the reported preference data given by the voters. The distance between each voter and each candidate in the recovered space is a fairly accurate measure of the voter’s preferences among the candidates. Third, it is possible to interpret the dimensions of the recovered space. Not surprisingly, the dominant dimension in American Presidential politics is seen as the economic left-right dimension that is the New Deal basis for the last major party realignment. The second dimension of Presidential politics is generally viewed as the “unorthodox issues” dimension, composed since 1968 largely of lifestyle issues such as abortion and women’s rights that fail to fit into traditional New Deal categories. There is one theoretical question that has been addressed in the above studies and that is the question of candidate location. Rabinowitz (1978) and Poole and Rosenthal (1984) recover maps that show American Presidential candidates located on the periphery of the electorate. The candidates ring the voters, who sit in the center by themselves. Enelow and Hinich (1984a), on the other hand, recover maps of the 1976 and 1980 American Presidential elections in which the major candidates are much closer to the center of the space. On the economic left-right dimension, all candidates lie within 1 SD (standard deviation) of the voter most-preferred points and on the social left-right dimension nearly all do so. There is some confusion about the relationship between convergence and spatial theory. While spatial theory predicts candidate convergence in one-dimensional two-candidate elections, twodimensional multicandidate contests are quite different in nature. It is well-known that the median voter result applies to multidimensional voting models only under very stringent conditions, raising the possibility that non-convergence among the candidates in American elections is consistent with TDelivered
at the 1984 Annual Meeting
of the American
Political 461
Science Association,
Washington,
D.C.
JAMESM. ENELOW and MELVIN J. HINICH
462
spatial theory. Still, a giant hole in the middle of the electorate is a strange occurrence, to say the least, since presumably a new candidate would emerge to plug it up. We are left then with two questions: first, does empirical evidence support the convergence hypothesis in American Presidential elections; and second, does the answer to the first question verify or disconfirm the predictions of spatial theory? These are the two questions this paper is designed to answer. In the sections that follow we will describe a methodology appropriate for testing the newest form of the spatial theory of elections described by Enelow and Hinich (1984a). We will show how this methodology-classical factor analysis-allows us to estimate candidate and voter locations in a multidimensional Euclidean space as well as most of the other parameters of the spatial model. The maps we recover (based on issue data from the 1980 CPS Major Panel File), show significant differences among candidate locations but find the candidates to be relatively close to the center of the space, confirming the results of Enelow and Hinich (1984a) based on thermometer data and a different methodology. The 1980 panel data are also useful for another purpose. These data allow us to track the dynamics of the 1980 campaign from January to November of that election year. Probabilistic voting models that permit the voter to cast his vote with some probability for other than a spatially most-preferred candidate yield a unique equilibrium location for two candidates in a multidimensional contest. We show how this equilibrium can be computed from the survey data of the 1980 panel file, and we watch its behavior across the three waves of the study (January-July-November). We find that the probabilistic equilibrium moves across the three waves, finally settling in a position between the Democrats and the Republicans, but 60% closer to Ronald Reagan than to Jimmy Carter. Not only is this result strongly intuitive (since Reagan won the election), it provides support for the predictive accuracy of probabilistic voting models in particular and spatial theory in general. Our results demonstrate that the major candidates in the 1980 Presidential race are centrally located, relative to the voters we analyze, and that spatial theory can both identify this center and explain why the candidates want to be there.
AN
APPROPRIATE
METHODOLOGY
FOR
TESTING
SPATIAL
THEORY
We now proceed to outline an empirically-inspired spatial model of elections in terms of the mathematical structure of factor analysis. In this model, the candidates are variables that load on a small set of underlying dimensions. The voters possess preferences over a larger set of campaign issues but can be represented on the set of underlying dimensions by projecting their positions in the larger space onto the smaller space. The voters and issues are sampling units of the observed variables and so can be assigned factor scores. These scores are useful not only for estimating the location of the voters in the factor space, but also for computing a probabilistic equilibrium for two-candidate elections, derived in Enelow and Hinich (1984b). We are thus able to represent in the same space: the voters, the candidates and an optimal location for the candidates. By fixing the location of the candidates across the three waves of the 1980 panel study, we can observe how changes in voter opinion are reflected in the changing location of this equilibrium. We should expect that as election day nears, this equilibrium should move toward the location of the winning candidate. A full exposition of the spatial model we employ can be found in Enelow and Hinich (1984a). For this reason, we will be somewhat brief. In the language of factor analysis, we observe M values (i=l,..., N;j=l,..., n). The random variables, m = 1, . . . , M, that assume Nxn random variables are candidates, N is the number of voters and n is the number of issues. If C,,, is the m th such variable, then we assume that cijm, the i,jth observation of C,, fits the model ciim - b, = v; pm+ eijm)
(1)
where b, is the origin of the i,jth observation on each variable, v: = (vi,, , . . . , vijr) are the values of the r common factor variates for the i, jth observation, pm= (pm,, . . . ,P,,,,)~ are the loadings of the mth variable (candidate) on the r derived factors and eiim is a random error term. In the language of the spatial model, the ith voter’s perception of the mth candidate’s position
A new spatial
model
of elections
463
on the jth issue (c,,) is assumed to be a linear function of the candidate’s position on a small set of r predictive dimensions (p,, , . . . ,p,,) with a residual term eiim to capture non-systematic influences that are uncorrelated with p, or vii. If an economic and a social left-right dimension underlie the perceived issue positions of American national politicians, then each voter uses a simple, idiosyncratic rule for translating the of the candidates into positions on the issues about which he is economic and social “ideologies” concerned. Given the disincentive to acquire direct issue information about the candidates, this procedure is cost efficient and explains why voters may have “honest” differences about the same candidate’s issue positions. The spatial model of elections we have described is equivalent to assuming that a factor structure underlies the perceived positions of candidates across a set of issues. Given the following additional assumptions, a factor analysis of the sample covariance of the 1.h.s. of model (1) yields consistent estimates of pm: (1) uijk is a random
for i = 1, . . . , N; j = 1, . . . , n; k = 1, . . . r;
variable
and (2)
U,,k
is distributed
for i and k.
independently
If the factors are correlated, the recovered candidate map will be distorted. The distortion will be a continuous function of the factor correlation. Thus, a small amount of correlation will not cause much distortion in the estimated map. (3) var(ai,,) = . . . = var(u,,,). Since the units of pmk are undefined, we can set this variance equal to one. If the variances are unequal, the candidate map will be stretched along the axis with the largest variance. The standard deviation of the factor multiplies the factor loading coefficient and thus is not estimable since the parameters are not identified. The assumption of equal variances is an identification restriction. Such restrictions are needed to estimate the parameters of models such as factor models which have unobserved variates. The next assumption is also an identification restriction: (4) covar(e,,, Given assumptions to show that
, eljn>
(l)-(4),
=
0
for m #n.
if dt,m= cij,,,- btj and dijn= ciin- b,, , then it is a straightforward covar(&,,
= P: pn
for m #n
= P%P, + +??I
for m = n,
,&,I
exercise
where Ic/, = var(eij,) is the specific variance of the m th variable and pm are the loadings of the m th candidate on the r underlying dimensions. In matrix form, covar(d,,,, di,,) is the m,nth element of the A4 x M covariance matrix C, where c=ppr++. P is the A4 x r matrix of loadings of the M candidates on the r underlying dimensions an M x M diagonal matrix whose m th diagonal term is $,(m = 1, . . . , M). The diagonal of PPT are called the communalities of the variables. The sample covariance of dilmand d,ju is s mn= (Nn - 1))’ i
(2) and Y is elements
5 (dijm- d,)(dijn - d,),
j=li=1
where d, = (Nn)-’
i
2 diJq
of the m,n th The m,n th element of the sample covariance matrix s,, is an unbiased estimator element of C. It is also a consistent estimator as N+co. A factor analysis of S yields a consistent estimator of PT, where T is an r x r orthogonal rotation matrix.
464
JAMES M. ENELOW
SCALING
THE
CANDIDATES
IN
and MELVIN
THE
1980
J. HINICH
PRESIDENTIAL
ELECTION
The factor loadings of the candidates are their positions in the factor space corresponding to the predictive dimensions that are postulated in the spatial model described by Enelow and Hinich (1984a). In order to estimate these loadings, we require data that tell us where the voters think the candidates are located on some set of campaign issues. For the 1980 CPS Election Study, these data are given by the issue scales on which voters are asked to place themselves and the candidates. These scales are labelled at the end points and typically subdivide a line segment into seven equal subintervals that constitute the seven permissible locations for the respondent to place himself or a given candidate. For the 1980 Major Panel File, there are four issues scales that appear in every wave from January to November. These are: Defense Spending; Government Spending; Government Antiinflation Plan; and Russia. The end-point labels for these four scales, are, respectively: (l-greatly decrease defense spending, 7-greatly increase defense spending); (l-Government should provide many fewer services; reduce spending a lot, 7-Government should continue to provide services; no reduction in spending); (l-reduce inflation even if unemployment goes up a lot, 7-reduce unemployment even if inflation goes up a lot); (l-important to try very hard to get along with Russia, 7-big mistake to try too hard to get along with Russia), In each of the three waves, the respondent is asked to locate himself and a set of candidates on each of these issues. The second set of data are, of course, the cj,, to be used in the factor analyses. Interestingly, the respondent was also asked to place “What the Federal Government is doing at the present time” on each scale, providing a natural measure of b,,, the origin on issue j as perceived by voter i. Before limiting ourselves to these four issues, however, a factor analysis was performed for each wave averaging across all issues in the wave (four in wave 1, five in wave 2, seven in wave 3). In addition, an issue-at-a-time factor analysis was also performed for each issue and wave. The point of doing these analyses was to see how robust the candidate loadings were to variations in the number and type of issue that were used. The results were striking. Very little change could be detected across these factor analyses in the unrotated candidate loadings. These results support an important component of the issues and dimensions model of elections that we employ. If perceived issue positions are a function of a common set of underlying candidate positions, then these same underlying positions should be recovered from a factor analysis of the issue data, no matter which subset of issues we analyze. The stability of the candidate loadings across waves does not necessarily follow from our model, but is consistent with the results of Poole and Rosenthal (1984) who apply a metric unfolding technique to thermometer data given by the voters about the candidates. Based on this cross-wave stability, we follow the decision of Poole and Rosenthal to fix the candidate loadings across the three waves of the panel study. This step allows us to disentangle the effects of voter change from candidate change in the maps we will examine. In order to maximize the number of candidates we can scale, the unrotated loadings of the candidates were averaged across issues and waves. The resulting candidate map is presented in Fig. 1, where the origin is the position of the Federal Government in the factor space. A 0.91
'BROWN
KENNEDY*
0.78
t DEMOCRATS
0.65 0.53
t
ANDERSON
*CARTER
;,r;,, ,,
o.92 ,:,
-0.20
0.01
0.17
0.36
0.55
0.73
Fig. 1
CONNfijI;
465
A new spatial model of elections
two-dimensional solution was chosen on the basis of the size of the first two eigenvalues of the covariance matrix and the proportion of the total variance explained by the first two factors. In increasing order, the distances of the candidates from the origin are Carter (0.60), Democrats (0.87) Kennedy (1.17), Anderson (1.24), Brown (1.37), Baker (1.41), Connally (1.47) Bush (1.49) Republicans (1.66) and Reagan (1.71). Since the origin of the space is, in a real sense, the “status quo”, the candidate distances correspond to what we would expect. Jimmy Carter, the incumbent, is closest to the status quo. That he is not identical to the status quo indicates that some separation exists in the voters’ minds between current policies and what Carter-the-candidate represents. With the exception of Anderson, whose policy positions are quite different from any of the other Republicans, all Democrats are closer to the status quo than any of the Republicans, with Reagan occupying the most extreme position. ADDING
THE
VOTERS
TO
THE
SPACE
In order to judge how well the candidates represent voter opinion we must scale the voters and candidates together. In Enelow and Hinich (1984a), it is shown that if xi = (x,, , . . . , x,,)‘~ are voter i’s ideal positions on issues 1, . . , n and A, is i’s n x n symmetric positive definite salience matrix that measures how i makes trade-offs between any two issues; then i’s most-preferred point in the underlying predictive space is zi = (VBA, Vi)-’ VTA,y,,
(3)
where L’, is an n x r matrix whosej,kth element vlik is, in the factor analytic model, the score of the i,jth sampling unit on the kth common factor; or, in the spatial model, the slope of the line that translates changes on the kth dimension into changes on the jth issue. The n-dimensional vector yi = (xi, - h,, , . . , xi” - l~,)~ is i’s ideal point in the issue space as measured from bi. We can derive the factor scores of the Nn sampling units from the identity F = DS-‘P (see Morrison 1967). The matrix F is Nn x r, where the first N rows are the factor scores for issue 1 and so on. The matrix D is Nn x M, where d,,,, = c,,, - b,. The matrix S ’ is the inverse of the M x M sample covariance matrix, and P is the M x r matrix of estimated (unrotated) candidate V, by taking the ith, The matrix F is then used to construct loadings. (N + i)th, . . , [N(n - 1) + i]th rows of F. The y, vector is obtained from the raw issue data. The salience matrix cannot be measured, and so we set it equal to the n x n identity matrix. Otherwise, we are able to estimate all the parameters necessary to locate the voters in the same space with the candidates. By fixing P across waves, any changes in voter location over time must be due to changes in voter opinion, either regarding their own issue positions or those of the candidates. In our maps, the candidates are fixed and the voters move. Convergence, then, must be redefined as movement of the voters toward the candidates instead of movement of the candidates toward the voters. If some location in the electoral space best represents the voters, then for convergence to occur, this location must move closer to the candidates over time. Before addressing the convergence question, however, let us examine the three maps that are obtained by fixing the candidates and scaling the voters in each of the three waves of the 1980 panel study. To increase our confidence in the estimates of the voter ideal point locations, we will scale only those voters with factor scores on at least three issues. This subsample represents the public most attentive to the issues of the campaign, and, consequently, most responsive to the issue appeals of the candidates. Our first observation is that in all three maps, the candidates lie not on the periphery of our subsample of voters, but squarely in the middle. For the first wave, the voter mean is (1.05, -0.27); for the second wave (1.70, 0.20); and for the third wave (1.20, 0.14). The three respective vectors of voter SDS on the two dimensions are (2.06, 3.97) (2.38, 3.62), and (2.83, 5.30). These statistics compare with the candidate mean and SD vectors of (1.15, 0.25) and (0.43, 0.46) respectively. The voter mean stays relatively close to the candidate mean while the voter SDS are anywhere from 4.8 to 6.6 times the candidate SD on the first dimension and anywhere from 7.9 to 11.5 times the candidate SD on the second dimension. Although we are dealing with a subsample of the electorate, we do not find the “empty center”
JAMES M. ENELOW and MELVIN J. HINICH
466 4.00
WAVE 1
.
. . 2.40
.
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overlap; M denotes mean; 0, Carter; 1, Reagan; 2, Kennedy; 3, Anderson; 5, Connally; 6, Brown; 7, Baker; 8, Republicans; 9, Democrats.
4, Bush;
depicted by Rabinowitz (1978) and Poole and Rosenthal (1984). In fact, the increase in the voter SD vectors across waves indicates that, relative to the voters, the candidates are moving closer together. in an aggregate sense, we To confirm that the scaled positions of the voters are “reasonable” 4.00
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A new spatial
model
of elections
461
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computed the location of the (axis-by-axis) median strong Democrat, weak and independent Democrat, pure independent, weak and independent Republican and strong Republican. The median strong Democrat is located near the Democrats, the median weak-independent Democrat is located near Kennedy, the median pure independent is near the northeast corner of the map, about equidistant from Kennedy and Baker, while both the weak-independent and strong Republicans are located near Reagan. Breaking down the voters by other issues, such as “liberals, moderates and conservatives” yields similar results. These results provide an internal check on the validity of the estimated voter ideal points. We do not wish to overinterpret these results. Monte Carlo analysis of a related scaling procedure for mapping voters and candidates (Palfrey and Poole 1987) suggests that estimates of voter locations are quite noisy relative to estimates of candidate locations. Three types of noise affect the estimated voter locations in our analysis. With two predictive dimensions, two factor scores are estimated for each voter on each issue. Given 10 candidates, each score is estimated with a sample size of 10, leaving only 8 surplus degrees of freedom. This first type of noise will be reduced as the number of candidates increases. The second type of noise derives from a lack of model fit for individual voters. Our assumption that the candidate locations in the predictive space are identically perceived by all voters is clearly violated for some voters. In order to fit these voters in the predictive space, their locations will be distorted. A third type of noise results from the unreliability of the answers voters give to the issue questions. Given that many voters are unfamiliar with the issues about which they are questioned and the I-l-point scales which define the possible set of answers, any given answer a voter provides is a sample point from a distribution of responses that he may give. In their analysis of Aldrich and McKelvey’s (1977) scaling procedure, Palfrey and Poole (1987) reach several conclusions which are germane to our analysis. First, they find that “individuals with a high level of information tend to be more extreme than those with low levels and are much more likely to vote” (p. 511). They also find that uninformed voters tend to be mapped toward the center of the space “regardless of their ‘true’ distribution” (p. 516). In fact, some bias toward the center also affects the estimated location of extremist voters.
468
JAMES M.
ENELOWand MELVIN
J. HINICH
These results are consistent with the voter locations we recover. The voters in our maps are highly informed. It is quite likely that were we to include less informed voters, they would be mapped toward the center of the space, “pushing” the candidates away from the center. But, from Palfrey and Poole’s conclusions, the estimated locations of uninformed voters are hghly unreliable. Furthermore, we expect these individuals are less likely to vote than those we have mapped. We conclude that from the candidates’ standpoint, our maps describe both the more relevant and more accurately measured public. A PROBABILISTIC
EQUILIBRIUM
FOR
THE
CANDIDATES
We have yet to address the question of where spatial theory predicts that the candidates will be located. One answer is given by probabilistic voting models. These models typically assume that the voter attaches probability weights to the alternative actions: vote for candidate A; vote for candidate B; abstain. The weights placed on the first two actions are typically a function of the voter’s utilities for the candidates, while the weight placed on the last action depends on both the utility difference between the candidates (the indifference factor) and the utility associated with the candidate the voter most prefers (the alienation factor). Fnelow and Hinich (1984b) examine two-candidate elections under the assumption that the voter’s probability of voting for candidate A, prob(A), and his probability of voting for B, prob(B), are given by prob(A)
= eu[u(A)J + (1 - e)b[u(A)
- u(B)]
= ea[u(A)]
if u(A) > u(B) otherwise,
and prob(B)
= ea[u(B)] + (1 - e)b[u(B)
- u(A)]
= ea[u(B)]
if u(B) > u(A) otherwise,
where e is the relative importance of alienation vs indifference as a cause of abstention (0 < e < l), a is a concave function of its argument, u is the voter’s utility function and b is a positive constant. All parameters are subscripted by voter, but the subscript has been dropped for simplicity. Given this probabilistic voting model, Enelow and Hinich (1984b) show that regardless of the shape of the voter utility functions or the dimensionality of the electoral space, a unique location exists that maximizes the expected plurality of each candidate in a two-candidate race. For the case of quadratic voter utility and equal relative importance of alienation vs indifference among voters, this unique equilibrium is: p* = [E(V/Tyl)]-‘E(Vfy;),
(4)
where E is the arithmetic average. Since we have estimated all the terms on the r.h.s. of this equation, we can compute this equilibrium for each wave of the 1980 panel study and see how closely it approaches the positions of the candidates. Figure 5 represents the probabilistic equilibrium for all three waves, along with the mean voter, in the same space with the eight candidates and two major parties. The number of voters on which p* and the mean voter are based is 168 for wave 1,263 for wave 2 and 357 for wave 3. To preserve comparability across waves, only the four issues that were common across the three waves were used in the analysis. For a voter to enter into the analysis, he would need to have factor scores on at least three of these issues. Firstly, p* shows significant movement across the three waves. Beginning near Bush, the probabilistic equilibrium moves in a northeast direction away from the Republican candidates and then moves back to a position near Baker. The mean voter describes a similar path but is generally further from Reagan and the Republicans. Most importantly, however, the final location of the mean voter is nearly equidistant between Carter and Reagan, while the final location of p* is significantly closer to Reagan. The distances between each candidate and the mean voter, as well as the distances between each candidate and p* are listed for each wave in Table 1, along with the means and SDS of the distances for each wave.
469
A new spatial model of elections 0.91
l
-
DEhfOCRATS 0.65
-
0.40
-
l
KENNEDY
BROWN
l
l
l
ANDERSON
CARTER 9 E
0.14
F
-
C + -0.11
BAKER
CONNALLY D -0.36
l
t BUSH A
-
-0.2 Fig.
5.
,
I
0.22
0.64
A=p:,
B=p:,
1.1
C=p;,
D=M,,
l
REP”;UCA REAGAN* I
I
1.5
E=M,,
A’S 1.9
F=M,
Inspection of Table 1 reveals that p* is closest to Bush in waves 1 and 2, and is closest to Baker in wave 3. This finding suggests that while Reagan was certainly a stronger candidate than Carter, a “moderate” Republican like Bush or Baker would have made an even stronger head for the Republican ticket. Since the closer a candidate’s position is to p* the better, we might think of a candidate’s distance from p* as a measure of the success of his campaign. From this standpoint, both Carter and Reagan’s campaigns faltered between January (wave 1) and July (wave 2) but picked up support by election day. Locating at p* maximizes the candidate’s expected plurality and so should not be taken as a direct measure of estimated support. Still, among survey respondents, Carter’s relative support across the three waves is 50, 29 and 46%, respectively, while Reagan’s support is 11, 29 and 43%, respectively. The change in Reagan’s support from wave 1 to wave 2 is the only change in support which does not follow the path of p*. As measured by the mean and SD of the candidate distances either from the voter mean or from p*, it is fair to conclude that particularly over the latter part of the 1980 Presidential campaign, candidate convergence occurs among the voters in our subsample. By this we mean, the center of this electoral subsample moves toward the candidates. In fact, between wave 2 and wave 3, p* moves closer to every candidate. Thus, we conclude that strong centripetal tendencies existed in the 1980 American Presidential election. For those voters most informed about the issues of the campaign, we find movement of public opinion toward the candidates in general and the winner in particular. CONCLUSION The spatial theory of elections is a that political candidates are motivated is whether this theory can contribute The purpose of this paper has been
Table
purposeful actor model of elections, based on the premise by the desire to win votes. A legitimate concern, however, to our understanding of real elections. to argue that spatial theory can answer empirically-based
candidate
to
P:
I. Distance
P:
from
each
P:
MI
M,
MI
Carter
1.1209
1.3954
0.9054
0.8145
I .2078
0.7218
0.6230
0.5680
0.6908
1.2015
1.1714
0.8750 0.3489
I
p* and M
RGlgdIl
0.2799
0.6803
0.551
Kennedy
1.2772
1.2947
I .0066
Anderson Bush
0.7210 0.0654
0.7852 0.5391
0.4497 0.2062
0.7559 0.4734
0.6303 0.3504
0.3580
Connally BrOWI
0.1223 1.1440
0.6675 .0707
0.4032 1.1742
0.4932
0.4459
I
0.3362 0.8755
0.9737
0.7847
Baker
0.1740
0.5621
0.1051
0.4429
0.3582
0.2499
Republicans
0.1428
0.5417
0.4062
0.5943
0.4096
0.5637
DtWIOCWIS
1.3878
1.5246
1.1330
1.1828
1.3698
0.9664
Meall
0.6435
0.9062
0.5976
0.7666
0.7533
0.6005
SD
0.5147
0.3617
0.3392
0.3006
0.3693
0.2316
470
JAMES M. ENELOW and MELVIN J. HINICH
questions about actual elections. We have argued that convergence, which is a dynamic phenomenon, did occur in the 1980 American Presidential election with respect to those voters most informed about the issues. We have also argued that the point of this convergence is best described by a probabilistic equilibrium. We find no support for the “empty center” results of Rabinowitz (1978) and Poole and Rosenthal (1984) but rather find a set of candidates that is centrally located relative to the voters whose opinions we have scaled. As we mentioned earlier, if less informed voters were scaled along with more informed voters, our conclusions may require modification. Given the paucity of data on these voters, with the attendant increase in error and the expected bias toward the center, our view is that the gains from analyzing an increased sample of voters would be negated by the unreliability of the results. Moreover, spatial theory assumes that voters are responsive to issue appeals. Thus, it seems best to limit our tests to those voters for whom this assumption is reasonable. Voters who are uninformed about the candidates are outside the scope of the theory. Acknowledgements-This
work
was supported
by the NSF under
Grant
SES-8310591
REFERENCES Aldrich, John and Richard McKelvey. 1977. A Method of Scaling with Applications to the 1968 and 1972 Presidential Elections. American Political Science Review 71: 11 l-30. Enelow, James and Melvin Hinich. 1984a. The Spatial Theory of Voting: An Introduction. New York: Cambridge University Press. Enelow, James and Melvin Hinich. 1984b. Probabilistic Voting and the Importance of Centrist Ideologies in Democratic Elections. Journal of Politics 46: 459478. Hinich, Melvin. 1978. Some Evidence on Non-voting Models in the Spatial Theory of Electoral Competition. Public Choice 33: 83-102. Morrison, Donald. 1967. Multivariate Statistical Methods. New York: McGraw-Hill. Palfrey, Thomas and Keith Poole. 1987. The Relationship Between Information, Ideology, and Voting Behavior. American Journal of Political Science 31: 51 I-530. Poole, Keith and Howard Rosenthal. 1984. U.S. Presidential Elections 196&80: A Spatial Analysis. American Journal of Political Science 28: 282-3 12. Rabinowitz, George. 1973. Spatial Models of Electoral Choice: An Empirical Analysis. Chapel Hill: Institute for Research in Social Science. Rabinowitz, George. 1978. On the Nature of Political Issues: Insights from a Spatial Analysis. American Journal of Political Science 22: 793-817. Weisberg, Herbert and Jerrold Rusk. 1970. Dimensions of Candidate Evaluation. American Political Science Review 64: 1167-l 185.