Elemental Tests of the Traditional Rational Voting Model Part 2

In summary, these are the key differences between the two types of elections: C C C C C C UNION CERTIFICATION ELECTIONS POLITICAL ELECTIONS small numbers of voters high benefit of voting, low cost issues are primarily economic, not political homogenous electorates ties and near ties not infrequent elections generated by petition large numbers of voters high cost of voting, uncertain benefit issues are primarily political heterogenous electorates ties and near ties quite rare elections held at regular intervals Model Implications. The asymptotics in equation (2) assume large electorates, while we have advocated the study of small electorates. This is not much of a problem. Equation (2) applies when N >> 1/F², which should be satisfied even with only one thousand voters. When there are fewer, as in our data, P is still approximately normal in M*, but “sampling error” inflates the variance of M above F². (See the Appendix, Claim 2.) Thus, the functional form in equation (6) is still valid, only F cannot be strictly interpreted as defined. Implications 1-4 depend only on this functional form and will continue to hold. To illustrate, the predicted TI,M,S relationship in our data is generated numerically, maintaining the same degree of uncertainty about candidate preferences (F = 0.10) as before, but without using the large-electorate formula for P.8 This is graphed in Figure 1b, using the same format as Figure 1a. The two figures are not identical, and the vertical scale differs, as expected, but the shapes of the curves and the relations between them, which generate Implications 1-4, remain. To confirm the similarity in functional forms, we treat equation (6) as a regression in the 8 As in equation (10) in the Appendix, conditional P is calculated using the binomial distribution, with the number of voters used in this calculation calculated from M and S (in effect instrumenting the number of voters with M and S), using the estimates in row 2 of Table 3. parameters ß and F, and “fit” it to the points plotted in Figure 1b (scaling instrumental turnout so its maximum is half of non-instrumental turnout, as in the actual data). The results are presented in the fifth row of Table 3. The true value of F is not obtained, reflecting the broader horizontal scale in Figure 1b, but the fit is excellent. With just one free parameter (other than the constant, ß), R² is 0.98, and the correlation between actual and predicted TI is virtually one. Equation (6) can be used for specification tests in very small electorates, though the structural parameters cannot be recovered. III. Estimation. First Order Implications. Estimates of equation (1), which tests Implication 1, are presented in the top panel of Table 3, for the full sample and for the subsample for which S $ 100, with and without controls.9 The coefficients on M and S are always negative, as predicted, and highly significant; there is no material difference between the full sample and the subsample, as is the case with the rest of the estimates we present. A ten percentage point increase in the winning margin, or two hundred additional eligible voters, reduces turnout by about one percentage point. The predicted difference in turnout between a hotly contested election involving one hundred voters and a lopsided election involving one thousand voters is nearly twenty percentage points, one-fifth of the mean. Here, many individuals vote only in close elections. As we have argued, this is reasonable given the importance of the issue being decided and the significant possibility of a tie in a close election. 9 The model is estimated using OLS, which is traditional. Though some classical model assumptions are not strictly met, results are not sensitive to the estimation method used. Weighting the observations by electorate size did not materially affect the results; neither did a tobit estimator. Repeat elections in the same bargaining unit generate correlated errors, but there are few of these. Further perspective on the turnout/closeness relationship can be obtained by comparing it to the relation between turnout and other key factors: state, union, and industry. This is done by calculating the standard deviation of the appropriate set of fixed effects.10 A one standard deviation change in M and S yields a 4.7 percentage point change in turnout. In contrast, the state fixed effects have a standard deviation of 2.6 percentage points; the union fixed effects, 4.3 percentage points; the industry fixed effects, 4.4 percentage points. The turnout/closeness relationship estimated here is not just empirically significant; it is the strongest empirical regularity observed in these data. Second Order Implications. To test the second order implications, we must uncover the TI,M,S relation in the data without imposing unnecessary functional form, as the functional form implied by theory generates these implications to begin with. No natural reduced form specification will do this.11 But semiparametric regression accurately reveals this relation without restriction. Here the controls enter parametrically, as before, but not the winning margin. Rather, turnout is an arbitrary function of M, fg(M), for each of the five electorate size groups defined in Table 1: (7) where i indexes elections and g electorate size groups, and the dummy Di,g equals one if election i 10 Sampling error in estimating these fixed effects inflates this standard deviation. To reduce this problem, the calculations omit unions and industries appearing in at most five observations. 11 For example, a linear regression of total turnout on M², to test Implication 2, an M*S interaction, to test Implication 3, and 1/S, to test Implication 4, will not work. The sign of M²T/MM² varies with M, so the hypothesized coefficient on the quadratic term is indeterminate, and the M*S term muddies the coefficient on 1/S. A double-log specification relating ln(T) to ln(N(M)), ln(S), and controls implies total turnout approaches zero as P vanishes, and is also unacceptable. falls within group g and zero otherwise. This specification takes the form of a generalized additive model (Hastie and Tibshirani, 1990), which allows quick estimation via spline smoothing techniques (Yatchew, 2003).12 The theory is supported only if Implications 2-4 are all confirmed. The results–estimates of f1 - f5–are graphed in Figure 3, using the same format as Figure 1b, with which they can be compared. Turnout is scaled to zero at the largest margin for the largest electorate size group, so the vertical axis represents instrumental turnout, at least putatively. Confidence intervals are discussed in the note but omitted from the graph, for clarity; the standard errors, usually less than one percentage point, are small in absolute and relative terms. All second-order implications of the model are contradicted. The negative relation between winning margin and turnout obtains throughout the full range of winning margins, in contrast to Implication 2; indeed, it is strongest for large margins, not small ones. It is also strongest in large electorates, in contrast to Implication 3. Finally, the inverse relation between size and instrumental turnout asserted in Implication 4 is not observed. Larger electorates do exhibit lower turnout, as in Implication 1, but instrumental turnout is not halved when electorate size doubles. Turnout responds to election closeness, but not in the way rational voting theory predicts.13 12 The more general option is to estimate a surface representing instrumental turnout in T×M×S space. But estimation suffers from the curse of dimensionality and is infeasible, and the approach adopted here is a more convenient way of depicting the data’s defiance of the theory in any event. However, electorate size was divided into more than five groups for the structural estimation conducted below. Groups were based on ln(S), which was partitioned into intervals of 0.1. 13 Figure 3 could be “distorted” by selection bias stemming from the requirement that 30% of employees in the bargaining unit support a petition in order to hold the election. Units disinclined to unionize and large units will find it more difficult to meet this requirement. Of these units, those holding elections will have workers that are especially willing to participate in the certification decision. One way to determine whether selection bias drives the results is to distinguish union wins from union losses: selection problems should be smaller in units that support certification, because it should be easier to satisfy the petition requirement. When Figure 3 was replicated (separately) for There is formal statistical support for these assertions. The “average slope” of each line in Figure 3 is easily obtained; these can be compared using an F-test to formally assess Implication 3. Here, the F statistic of 8.83 rejects the null of equal slopes at p < 0.01, but except for line D, these slopes monotonically increase in magnitude, in contrast to theory. Statistical tests of the significance of the nonlinearities in lines A-E are available in our statistical package (SAS). Each is statistically significant at p << 0.01, but is also the opposite of that implied by theory. Formal tests of Hypothesis 4 await the final step, structural estimation. These tests easily reject Implication 4. Structural Estimation. One need not return to the original data to conduct structural estimation of equation (6). With large samples, one can recast this equation as a regression employing the semiparametric estimates as dependent variables, essentially “estimating” Figure 3. Least squares can be used to do this, so computation is far simpler than in other general equilibrium voting models. Using a general semiparametric form T = f(M,S) + 7X + , for notational simplicity, recombine the terms estimated in equation (7) as follows: (8) as long as P vanishes in unanimous elections in the largest electorates, of size SMAX. The residual, included in calculating T$ N, is assumed to represent factors known to the agent but not the econometrician. As discussed in Section I, however, parameter estimates will be very similar if T$ N excludes the residual, and this is easily confirmed with our data. the subsamples of union wins and of union losses, however, the patterns described in the text were always clearly visible. Therefore the results are not attributable to selection bias. When sampling error in the semiparametric estimates, 0, is small, ln(x+0) . ln(x) + 0/x. Substituting into equation (6), and defining T0 to be the empirical analog of E0, yields: (9) This can be estimated using least squares. Heteroskedasticity exists, as the error term < contains both sampling and specification error. But the latter may dominate given small sampling error, so weighting using the standard errors on the semiparametric estimates may not be desirable. Estimating (2 instead of restricting it to be -1 allows Implication 4 to be formally tested. As previously discussed, the structural parameters ß and F can be directly recovered from this regression in all but the smallest electorates, and specification tests can still be conducted even in these. OLS estimates of equation (9) are presented at the bottom of Table 3. The coefficient estimate on ln(S) is tiny, reflecting the weak relation already observed. The R² value, 0.57, reflects the tremendous misspecification in equation (9); otherwise, R² would deviate from one only due to sampling error in the semiparametric estimates, which is minute. Formal specification tests, against parametric or nonparametric alternatives, can be conducted using standard methods; the covariance matrix of the semiparametric estimates should be employed to give the tests proper power. These are superfluous here, as Implications 2-4 were each decisively rejected, but for completeness Table 3 presents a simple RESET test, which strongly rejects the null of no specification error. IV. Comparison with Other Voting Models. The semiparametric regression reveals that the turnout/margin relationship in our data is especially strong in large electorates and when the winning margin is large, and that instrumental turnout is negatively related but not inversely proportional to electorate size. The failure of the rational voter model to explain these heuristics motivates an inquiry into alternative explanations. One final advantage of the methodology introduced here is that it facilitates such comparisons. The brute force approach would be to formally test each alternative theory with our data. This is prevented by their variety (which precludes an exhaustive comparison), their computational complexity (many new theories require nonlinear, structural estimation), and their inexactness (some theories do not specify the relation between turnout and election closeness). Our approach, instead, is to compare the relationships predicted by alternative theories to the patterns revealed by the semiparametric estimates, looking for a perfect match. Even when these relationships are not precisely specified, one can often determine whether the patterns observed in our data are plausible. We do this for three sets of self-interested voting theories that feasibly predict a positive turnout/closeness relation: information theories such as Matsusaka (1995), group voting theories such as Morton (1991), and theories of expressive voting (Brennan and Lomasky, 1993). (Ethical or altruistic voting theories are omitted because they probably do not apply to union certification elections.) None explains all the patterns in the data, for reasons that appear to be fundamental. In Matsusaka’s (1995) information theory, the likelihood of voting is the product of P and the potential voter’s confidence that his preferred choice is truly the best choice, in a world of imperfect information about the “true” attractiveness of the options on the ballot. Left unspecified is the relationship between winning margin and voter confidence. If these two quantities are independent, the relationships between T, M, and S match those in the rational voter model. If voter confidence is greater at high margins, the patterns in Figure 3 still would not be replicated.14 Furthermore, this model predicts Implications 3 and 4, as S affects turnout only through P, and these are also not reflected in the data. Group voting models posit two opposing blocks of potential voters. Each decides how many members to send to the polls in order to maximize the net expected benefits of the election to that block (the expected benefits of electoral victory minus the costs of voting). Coate and Conlin (2004) have successfully tested one such model, and Nalebuff and Shachar (1999) have successfully tested a variant in which group turnout is generated by the mobilization efforts of elites. Turnout need not be inversely proportional to electorate size in these models (though some predict turnout and electorate size are unrelated, counter to our findings). But voting is still motivated instrumentally, just at the level of the group instead of the individual, so Implication 2 should still hold. Turnout should increase when the expected margin narrows, but the greatest increases should not occur at large margins, as observed in our data. There is little benefit to increasing turnout from a block of voters when an election is expected to be lopsided. Finally, in expressive voting models, such as Brennan and Hamlin (1998), individuals vote because they derive utility from expressing themselves, not because they expect to alter the outcome of the election. The expected margin need not be of direct interest to expressive voters; any turnout/margin relationship must result from a correlation between the winning margin and the expressive benefit of voting. This is not specified in many variants of expressive voting theory, so 14 The idea being that a lopsided election means one candidate is pervasively viewed as being superior. The instrumental disincentive to vote at higher margins would be counterbalanced by increased confidence in the “correctness” of one’s vote. Thus the turnout/margin relationship would be even flatter at high margins, and possibly positive. These patterns do not match Figure 3. the relationship between turnout and election closeness is similarly unspecified. (This is not needed to test the theory; see Kan and Yang, 2001.) An exception is Schuessler (2000), which predicts higher turnout in closer elections, as we observe. Furthermore, Schuessler’s model is sufficiently general that the non-linearities between T and M observed in our data cannot be ruled out. But the relation between turnout and electorate size does not appear to match the data. The expressive motive in most models, including Schuessler’s, is independent of electorate size, so turnout should not be negatively related to electorate size, as observed here.15 No theory yields a perfect match, though some outperform the rational voting model, and might achieve success with further development (perhaps by letting electorate size influence turnout). Thus, while we reject the rational voter model here, we do not have a ready replacement. V. Conclusion. In union certification elections, like political elections, closeness matters. Turnout is strongly related to the two determinants of one’s chances of deciding the electoral outcome: the size of the electorate and the winning margin. Turnout in the closest elections in our sample is much higher than in their weakly contested counterparts, an economically reasonable finding given the high stakes in these elections and the substantial probability an individual vote will be decisive in a close contest. Rigorous tests require asking not just ask whether turnout responds to election closeness, but 15 In the elections studied here, a larger margin may have utilitarian value by enhancing the union’s negotiating power. The authors experimented extensively with “augmented” expressive models of this type, in which the motivation to vote is to provide information about the preferences of the electorate. These predict the observed relation between T and M, but Implication 3, which is not observed, still holds. how. In our data the turnout/margin relationship is especially strong for large electorates and large winning margins, counter to rational voting theory, which predicts stronger relationships for small electorates and small winning margins. And, while instrumental turnout is negatively related to electorate size, the inverse relationship postulated is not observed. Something else must generate these turnout patterns. A survey of alternative theories reveals none that can fully do so. Still, the simple heuristics we have articulated can help guide future theoretical developments. With straightforward, distinct theoretical predictions and abundant data, rational voting is an auspicious area for empirical research. The existing literature has not fully capitalized on these advantages: the interpretation of empirical results is often subject to question, because of the small P problem or because the first-order implication being tested is not unique to rational voting theory, while the complexity of other models can hinder their application elsewhere and obscure the role of key modeling assumptions. The methodology presented here moves the literature closer to the ideal: empirical tests that are transparent, rigorous, easily executed, and unambiguously interpreted.