Elemental Tests of the Traditional Rational Voting Model Part 3

Table 1. Descriptive Statistics by Unit Size. Unit Size 51-100 101-200 201- 400 2,976 1,827 815 0.87 0.87 0.88 (0.129) (0.123) (0.116) 0.39 0.35 0.33 (0.263) (0.242) (0.227) 0.46 0.39 0.34 52 14 4 Full 25-50 401+ Observations (elections) 9,854 3,764 472 Participation Rate (Turnout) 0.88 0.88 0.86 (standard deviation) (0.125) (0.122) (0.140) Margin (percent pts/100) 0.39 0.43 0.32 (standard deviation) (0.265) (0.282) (0.237) Result (pct. union wins/100) 0.44 0.47 0.36 Near Tie Votes (number) 199 126 3 Fiscal Year (fraction in category) 1995 0.15 0.15 0.15 0.14 0.15 0.14 1996 0.14 0.14 0.13 0.14 0.13 0.14 1997 0.15 0.16 0.16 0.16 0.13 0.15 1998 0.16 0.15 0.17 0.16 0.16 0.14 1999 0.15 0.15 0.15 0.16 0.17 0.14 2000 0.13 0.13 0.13 0.14 0.15 0.17 2001 0.12 0.13 0.11 0.11 0.11 0.11 Unit Type (fraction in category) 0.54 0.55 0.54 0.46 0.43 0.47 Industrial 0.26 0.20 0.17 0.21 0.19 0.19 Professional 0.07 0.09 0.11 0.14 0.18 0.14 Trucking 0.13 0.25 0.18 0.19 0.21 0.19 Other Region (fraction in category) 0.33 0.34 0.32 0.31 0.30 0.31 Midwest 0.30 0.29 0.24 0.21 0.19 0.22 South 0.21 0.23 0.25 0.27 0.28 0.26 Northeast 0.16 0.14 0.19 0.21 0.23 0.21 West Industry (fraction in category) 0.51 0.48 0.43 0.36 0.33 0.38 Goods Producing 0.02 0.02 0.03 0.04 0.06 0.05 Construction 0.47 0.44 0.38 0.31 0.25 0.32 Manufacturing 0.57 0.52 0.49 0.64 Service Producing 0.62 0.67 0.13 0.06 0.16 0.17 0.21 0.17 Trans. and Public Utilities 0.09 0.08 0.10 0.09 0.11 0.13 Trade Union (fraction in size category) Teamsters 0.46 0.30 0.15 0.06 0.03 Service Employees 0.29 0.35 0.25 0.07 0.04 United Food and Commerce 0.34 0.34 0.17 0.09 0.06 Workers Steel Workers 0.29 0.31 0.22 0.10 0.08 Auto Workers 0.19 0.24 0.28 0.22 0.06 Note: Data for units with 25 or more members eligible to vote. The professional unit type includes professional, clerical, professional and clerical, and departmental units. Other unit types include guards, craft, and other. Near-tie votes include ties and one-vote victories for certification. Source: National Labor Relations Board. Table 2. Descriptive Statistics by Year, Unit Type, and Region. Full Sample Fiscal Year 1995 1996 1997 1998 1999 2000 2001 Unit Type Industrial Professional Trucking Other Region Midwest South Northeast West Industry Goods Producing Construction Manufacturing Service Producing Trans./Public Util. Trade Union Teamsters Service Employees United Food and Commerce Workers Steel Workers Auto Workers Eligible Voters 117.3 (173.1) Participation Rate 0.88 (0.125) Margin 0.39 (0.265) 1,458 1,346 1,516 1,542 1,500 1,327 1,165 113.6 113.3 114.0 117.0 115.7 132.0 116.2 (157.4) (163.1) (165.4) (167.9) (154.0) (211.4) (193.0) 0.88 0.88 0.88 0.87 0.88 0.87 0.87 (0.118) (0.125) (0.127) (0.123) (0.117) (0.137) (0.125) 0.38 0.41 0.40 0.38 0.39 0.37 0.39 (0.261) (0.261) (0.272) (0.270) (0.272) (0.260) (0.258) 0.44 0.41 0.46 0.45 0.47 0.39 0.41 4,669 1,917 1,411 1,856 125.4 131.0 89.4 104.1 (170.1) (211.1) (135.9) (157.0) 0.89 0.87 0.89 0.84 (0.120) (0.119) (0.104) (0.148) 0.37 0.38 0.39 0.44 (0.256) (0.261) (0.253) (0.293) 0.42 0.46 0.35 0.52 3,070 2,169 2,593 2,021 123.2 132.9 108.3 103.0 (190.7) (179.1) (155.5) (157.5) 0.89 0.89 0.86 0.86 (0.110) (0.129) (0.129) (0.130) 0.35 0.39 0.43 0.39 (0.249) (0.261) (0.284) (0.262) 0.39 0.42 0.48 0.48 3,716 444 3,148 6,137 1,711 1,054 135.9 74.1 143.9 106.0 90.4 104.4 (199.6) (90.2) (207.0) (153.8) (118.3) (154.6) 0.90 0.78 0.90 0.86 0.87 0.89 (0.118) (0.214) (0.083) (0.126) (0.118) (0.098) 0.36 0.51 0.34 0.40 0.37 0.40 (0.249) (0.306) (0.233) (0.273) (0.257) (0.252) 0.34 0.42 0.33 0.49 0.44 0.37 1,996 575 494 93.9 (140.5) 115.2 (136.6) 129.6 (210.3) 0.89 (0.100) 0.83 (0.142) 0.88 (0.097) 0.38 (0.244) 0.43 (0.296) 0.37 (0.237) 0.34 0.67 0.47 440 340 142.2 (180.0) 171.2 (197.9) 0.93 (0.068) 0.92 (0.089) 0.30 (0.221) 0.28 (0.234) 0.38 0.49 Elections 9,854 Result 0.44 Note: Standard deviations in parenthesis. Data are for units with 25 or more eligible voters. The professional unit type includes professional, clerical, and departmental units. Other unit types include guards, craft, and other. One extremely large election at Boeing in 2001 (17,000 eligible voters) is omitted because it substantially skews means and standard deviations in the categories in which it is included. Result and margin are defined as in Table 1. Source: National Labor Relations Board. Table 3. Regression Results (coefficient estimates, with standard errors in parentheses). Implied F (percentage points) Eligible Voters (S) in hundreds Winning Margin² (M²) Eligible Voters, in logs (ln S) RESET Term (1st stage prediction squared) Implied ß Regression Descriptor Winning Margin (M) in percentage points R² TRADITIONAL MODEL Eligible Voters $ 25 No Controls -0.147* (0.004) -0.438* (0.049) ---- ---- ---- ---- ---- 0.102 Full Controls -0.100* (0.004) -0.552* (0.044) ---- ---- ---- ---- ---- 0.340 Eligible Voters $ 100 No Controls -0.185* (0.009) -0.359* (0.052) ---- ---- ---- ---- ---- 0.136 Full Controls -0.129* (0.008) -0.432* (0.047) ---- ---- ---- ---- ---- 0.460 STRUCTURAL MODEL Theory (Figure 1b) ---- ---- -6.56* (0.11) -1.00 by restriction ---- 0.64 19.5 0.979 Data (Figure 3) ---- ---- -1.23* (0.09) -1.00 by restriction ---- 2.38 45.1 0.295 Data (Figure 3) ---- ---- -0.83* (0.03) -0.15* (0.01) ---- 2.82 52.3 0.542 Data (Figure 3) ---- ---- 4.11* (0.16) 0.70* (0.03) -1.93* (0.06) ---- ---- 0.753 Note: N = 9,854. A constant also estimated in all regressions. The dependent variable is actual turnout in the traditional model, and a complex function of instrumental and non-instrumental turnout, described in the paper, for the structural model. * indicates significance at the 5% level. Figure 1. Instrumental Turnout, Winning Margin, and Electorate Size: Theory. Figure 1a. Relation between T, M, and S as described by equation (6). Electorate sizes are arbitrary, but follow the rule that the number of Figure 1b. Relation between T, M, and S generated as described in footnote 8. Units on the vertical axis are not equivalent eligible voters in electorate g is twice that in electorate g-1. between the two graphs. Figure 2. Semiparametric Estimates of the Closeness/Turnout Relation Using Grant’s (1998) Congressional Election Data. Note: N = 868 congressional elections held during 1982, 1986, or 1990 in all states electing governors in those years. Elections with winning margins exceeding eighty percentage points omitted, as in the original study. Controls as in the original: state and year fixed effects, real per capita spending by both candidates, a dummy for a contested senate race, and the following districtwide variables: the unemployment rate, the percentage of the adult population over age 65, real median income, the percentage of workers who are blue collar, the percentage of the adult population with a college degree, the percentage of housing that is owner occupied, the percentage of the population residing in a rural area, and the percentage of the adult population that is black. Figure 3. Instrumental Turnout, Winning Margin, and Electorate Size: Semiparametric Estimates. Note: Confidence intervals are not presented in the graph, for clarity, but are described here. They are generally narrow, and are smaller for smaller electorates and smaller margins, where there are more observations. At M = 10 percentage points, the 95% confidence intervals in Lines A-E have ranges of ± 0.5, 0.6, 1.0, 1.0, and 1.3 percentage points, respectively. At M = 50 percentage points, the 95% confidence intervals in Lines A-E have ranges of ± 0.7, 0.8, 1.1, 1.4, and 1.8 percentage points, respectively. At M = 90 percentage points, the 95% confidence intervals in Lines A-E have ranges of ± 0.8, 1.2, 2.0, 2.8, and 3.4 percentage points, respectively. These intervals are narrow enough that the confidence intervals of adjacent lines sometimes do, and sometimes do not, overlap. APPENDIX Let P be one-half the probability of a tie vote (there is a 50% chance a tie would be broken in the deciding voter’s favor), p be the actual probability any randomly chosen voter chooses the more popular electoral option, and N be the actual number of voters (assumed to be even, as the extension to odd N is trivial). If p is known with certainty, P is given by the binomial distribution: (10) Claim 1. Using Stirling’s formula, the combination term can be replaced with 2N(BN/2)-½, generating: (11) Substituting the above into the identity TI = ßP, taking logs, and rearranging as in the text yields: (12) with the ex post winning margin M equal to 2p - 1 asymptotically. When TN = 0, as in Coate, Conlin, and Moro (2004), this reduces to: (13) whose solution involves Lambert’s W function. Evaluating this solution for a wide range of S and M values generates the assertion in Section I that total turnout drops considerably as M grows away from zero, so that the model cannot support realistic levels of turnout for both large and small M. A similar solution can be obtained for Hansen, Palfrey, and Rosenthal (1987), who restrict M=0 instead. Recognize, however, that both of these studies adopt slightly different set-ups from that used here, so these equations only approximate the outcomes of those other models. Claim 2. If p is not known with certainty, one must determine the unconditional probability of casting the deciding vote, P, from the conditional probability in equation (10) and the distribution of p. The asymptotics have already been proved, as noted in the text, but the small electorate results have not yet been established. To derive these, begin with a first-order normal approximation to equation (10) above, adapted from Owen and Grofman (1984): (14) Letting p be distributed normally with mean : / (1 + M*)/2 and standard deviation F/2 (the asymptotic standard deviation of M is twice that of p), the unconditional probability is then: (15) Thus unconditional P is directly proportional to the density of M* in a normal distribution with zero mean and variance F² + 1/N, and inversely proportional to the number of voters. As N 6 4, this converges to equation (2). For small electorates, the bell-curve shape between P and M* remains, but the standard deviation of the bell curve exceeds F, as claimed in Section II. Large and small electorates are distinguished by the condition N >> 1/F². It is reasonable to assume that F is in the neighborhood of 0.10 (see footnote 5), in which case N=1000 comfortably satisfies this inequality. With the equation above one could generalize equation (6) for small electorates, but this advance is not worth the additional complexity, which is considerable. Claim 3. To derive equation (6), solve equation (4) for TI in terms of TN and k* = ßN(C)/FS I < T , using the quadratic formula, then take a first-order Taylor series of this solution around k* = 0. With this, one can approximate E(TI), E(TI*TI), and E(TI)² analytically. The ratio of the second term to the third, r*, is (4/3)½exp(M²/6F²). Next, rewrite the left-hand side of (5) as E(TI)@[E(TI)r* + TN]. Expanding E(TI)r* yields the following: (16) Substituting into the left-hand side of equation (5) and taking logarithms yields equation (6). The approximation is excellent. When E0(TI) / TN = 0.1, the approximation error is 0.13% or less for M between 0 and 0.5, and essentially zero after that.