Proportionality and Turnout

Abstract

Many studies find that voter turnout is higher in proportional representation (PR) elections than in plurality elections, but because the two systems differ in multiple ways and are used in different contexts it is difficult to know precisely why. I focus on municipal elections in France, where cities above a certain population threshold are required to use a PR system while those below use a type of plurality rule; this setting allows me to compare political outcomes across electoral systems while holding fixed a large set of social and political features. I find that the PR system noticeably increases turnout compared with plurality. I provide evidence suggesting that it does so in part by encouraging turnout in lopsided races and in part by inducing entry of new candidates. The findings highlight the importance of electoral proportionality in explaining cross-national differences in voter turnout.

Keywords

elections public opinion and voting behavior representation and electoral systems

Introduction: Voter Turnout and Electoral Systems

What explains voter turnout? Few questions have attracted more attention, both theoretical and empirical, in political science. Early efforts to explain turnout from a rational choice perspective (Downs, 1957; Riker & Ordeshook, 1968) treated voters as strategic actors who vote to influence election outcomes. More recently, attention has shifted to political parties as the locus of strategic activity: Citizens’ voting decisions may be driven largely by social norms such as a sense of duty, but strategic elites decide whether to activate those norms based in part on the probability that their mobilization efforts will tip the outcome (Aldrich, 1993; Cox, 1999). Both approaches imply that turnout should be higher when the marginal voter has a higher probability of casting the pivotal vote, a prediction that has largely been borne out in studies showing higher turnout in smaller electorates and in closer elections.¹

The idea that turnout depends on pivotality has also been used to explain the consistent finding that turnout is higher in systems using proportional representation (PR) than in systems using plurality or majoritarian formulas.² Intuitively, the idea is that voters in a plurality election only stand a chance of affecting the outcome when the race is very close between the top two candidates, whereas voters in a multi-member PR district can affect which party wins a seat in a variety of possible competitive scenarios.³ Depending on competitiveness, then, the ex ante probability of being pivotal may (all else equal) be higher on average under PR, which would tend to boost the incentives for citizens to vote and for parties to mobilize voters.

Although the higher probability of being pivotal under a more proportional electoral formula may well account for higher turnout in PR, prior empirical studies leave considerable doubt that proportionality is in fact an important part of the explanation. The fundamental problem is that proportionality is usually just one of many factors that differ between PR and plurality systems. PR elections typically involve voters choosing among party lists that compete in geographically large districts (sometimes as large as the entire country) within systems with multiple parties and frequent coalition governments; plurality elections, by contrast, typically involve voters choosing among individual candidates who compete in relatively small districts within systems with few parties and less frequent coalition governments. With the two types of elections occurring in settings that differ in these and other ways (including, for example, citizens’ attitudes toward the fairness of the electoral system; Banducci, Donovan, & Karp, 1999), it is difficult to know which specific factors explain differences in turnout. This concern is echoed in Blais (2006), who summarizes the literature on electoral systems and turnout by saying that “Most of the literature supports the view that PR fosters turnout, but there is no compelling explanation of how and why.”

More troubling still, there is reason to question the usual causal interpretation of the positive correlation between the use of PR and voter turnout. It is widely appreciated that electoral systems are chosen strategically (Boix, 1999; Colomer, 2005; Rokkan, 1970) and it is reasonable to think that voter turnout may be related to the factors (strategic and otherwise) that affect a given political system’s electoral rules. This suggests that PR and plurality systems may differ not just in ways that are either integral to the electoral system (e.g., party lists vs. individual candidates, large vs. small districts) or effects of the electoral system (e.g., many vs. few parties, frequent vs. infrequent coalition governments, perceptions of fairness) but also in features that are fundamental causes of the electoral system (e.g., the nature of social cleavages, the type of party system, the prevalence of norms of inclusion). The fact that it is difficult if not impossible to adequately control for these factors, and that they all could affect turnout, casts doubt on our ability to infer from observational data what would happen to turnout if a given system were to change from plurality to PR, let alone which aspect of the electoral system accounts for any effects we find.⁴

In this article, I examine a setting in which I can address many of these problems and thus provide unusually clear answers about how and why PR increases turnout. French electoral law prescribes a PR electoral system for municipal council elections in municipalities of at least 3,500 inhabitants and a plurality system for smaller municipalities. I use a regression discontinuity design to compare turnout in municipalities near this threshold as a way of measuring the effect of the electoral formula while holding other important factors constant. This research design has two clear advantages compared with the usual observational study of turnout and electoral systems. First, because the electoral system used in each municipality is determined by an arbitrary population cutoff, I can credibly estimate the effect of the electoral system on turnout (for cities near the population threshold at least) without worrying about unobserved factors that may have led one city to adopt PR while another did not. Second, the plurality and PR systems used in these cities are similar to each other in ways in which plurality and PR systems often differ (notably the arrangement of districts and the prevalence of coalition government), which allows me to focus on a smaller set of possible channels through which the electoral formula could affect turnout.

What I find is that the PR system used in French municipal elections increases turnout (relative to the plurality system) by about 1 percentage point, from about 69% to 70%. This effect is small in comparison with cross-national turnout differences but substantial given that the particular electoral systems I am comparing differ relatively modestly in proportionality (as I detail below) and are identical in many other respects that usually differ between PR and plurality systems. Consistent with theoretical predictions, the effect is concentrated in less competitive settings (where the more proportional formula does most to increase the chances that a voter will affect the outcome). I also find that the PR system reduces the variance of turnout across municipalities, again consistent with predictions. I show that higher proportionality seems to increase the number of lists in competition, as suggested by Duverger’s psychological effect. Although the unusual setting I examine allows me to sidestep many of the typical problems of cross-sectional turnout studies, there is the important shortcoming that other municipal policies change at the same population threshold. I provide evidence indicating that these other policy changes are not responsible for the effects I see: The heterogeneity in treatment effects across municipalities is less consistent with the idea that these other policies play a role, and similar effects are not seen at other population thresholds where some of these same policies change. Although no test can completely rule out the possibility that one or more of these other policy changes accounts for the jump in turnout I see, these tests cast doubt on such alternative explanations.

The findings of this study are clearly of most direct relevance to French local elections. Over the past two decades, there have been periodic proposals to apply the PR system to smaller municipalities,⁵ and in May of 2013 a reform was finally passed changing the relevant threshold from 3,500 to 1,000.⁶ My analysis suggests that, in addition to the other benefits usually attributed to PR in debates over these proposals, applying the PR system to smaller cities will promote both participation and competition in local elections. More broadly, the findings provide strong evidence that greater proportionality increases turnout even in the absence of changes to the arrangement of districts and the format of the ballot. I return in the conclusion to assessing how relevant these findings are for researchers who are primarily interested in explaining cross-country variation in political participation.

French Municipal Electoral Systems and Implications for Turnout

The commune, or municipality, is the lowest level of French government. Municipalities in France maintain roads and schools, manage local development, and administer cultural programs and some social welfare functions (Loughlin, 2007, pp. 90-91). They are responsible for almost a quarter of all public expenditure, amounting to about 6% of GD P (Loughlin, 2007, pp. 184-185). Each municipality is governed by a municipal council, ranging in size from 9 in the smallest municipalities to 163 in Paris; the council is in turn led by a mayor, who is a member of the council and ordinarily the leader of the party or list that won the previous election. Municipal elections are held approximately once every 6 years simultaneously in each of the more than 36,000 municipalities of France.⁷

Since 1946, French electoral law has specified a different system of elections for large and small cities, with the population threshold that separates the two groups of cities varying over time. The current electoral law, enacted in 1982, prescribes a single-district, multi-member plurality system for municipalities with fewer than 3,500 inhabitants and a form of “fortified” proportional representation for those with 3,500 or more inhabitants. In particular, in the larger cities the council is elected using a list-PR system with a 50% winner’s bonus: The list with the most votes is awarded half of the seats on the council, and the remaining seats are distributed proportionally among all of the lists including the winning list.⁸ In the smaller cities, candidates appear on lists with as many members as there are seats on the council and voters can vote for a whole list, a subset of the candidates from one list, or candidates from more than one list;⁹ seats are awarded to the top individual vote-getters. In both systems the election can take up to two rounds.¹⁰ From the standpoint of the electoral rules, then, what differs between cities above and below the 3,500 population threshold is whether voters are permitted to express candidate-level preferences (yes in smaller cities, no in larger cities) and whether seats are allocated to candidates using a plurality formula (as is the case in smaller cities) or a proportional formula (as is the case in larger cities).

Figure 1 depicts the typical relationship between vote shares (horizontal axis) and seat shares (vertical axis) for cities just above and below the 3,500 population threshold that divides plurality and PR cities. (I assume two lists are competing, which is the modal case.) In both sets of cities around 25 seats are at stake. In the PR system (gray line), a party can win a seat with just over 10% of the vote; as a party’s vote share increases from there, its seat share rises at regular intervals before jumping at 50%, where it receives the winner’s bonus. In the plurality system (black line), it is very unusual for a party to win any seats with less than 40% of the overall vote; if one list wins at least 60% of the vote, it typically wins all of the seats.¹¹

Figure 1.

Proportionality in French municipal electoral systems.

As noted in the introduction, the proportionality of an electoral system can be linked to turnout through both a direct and indirect channel. By “direct channel” I refer to the fact that the proportionality of the electoral system shapes beliefs about how likely a voter is to cast a decisive vote. As highlighted by Figure 1, different electoral formulas imply different hypothetical vote shares at which a voter could cast a decisive vote. Under the plurality system used in the smaller cities, for example, in a two-party contest a voter could plausibly only cast a decisive vote if party x wins between 40% and 60% of the votes; under the PR system, by contrast, the hypothetical vote shares at which a voter could cast a decisive vote are spread more evenly from less than 20% to more than 80%. Whether the ex ante probability of casting a decisive vote is higher under the plurality system or the PR system depends on expectations about likely electoral outcomes: The probability may be higher under plurality if the election is likely to be close, whereas it might be higher under PR if one party is likely to win easily. This implies two predictions about the effect of proportionality on turnout. First, if PR boosts turnout relative to plurality, the effect should be concentrated in less competitive municipalities, because these are the ones in which the PR electoral rule most increases the probability of casting a pivotal vote. Second, whether or not PR boosts turnout on average, it should reduce the variance of turnout across municipalities, because the ex ante probability of casting a decisive vote depends less on the competitive context in PR than in plurality elections.¹²

Some of the effect of proportionality on turnout may operate through indirect channels: A more proportional system may lead more lists to enter the election, which could in turn affect turnout by increasing aggregate mobilization efforts and by providing voters with a more compelling set of options. The idea that proportional representation favors the entry of more parties (compared with plurality systems) has a long history; it is notably a component of the “psychological effect” described by Duverger (1954).¹³ Given that lists can win seats with far less support under the PR system (as indicated by Figure 1), we should expect more lists to enter in cities just above the 3,500 population threshold. We might also expect the leaders and members of non-dominant lists to be of higher “quality” in the PR system, given that the top members of a modestly successful list can only expect to win seats under the PR system. Thus, we have a third prediction: The PR system should lead to more lists in competition.

What is unusual about the setting I examine is that while electoral proportionality differs between cities just above and below the 3,500 population threshold, other important features of the electoral environment that typically go along with the electoral formula remain fixed. As noted above, the district magnitude and the arrangement of districts is basically the same in the two sets of cities. The size of the electorate in each district is obviously essentially the same when we focus on cities very close to the population threshold. Given the winner’s bonus in the PR system and the nearly winner-take-all nature of the multi-member plurality elections, coalition government in both sets of municipalities is basically unknown: The list that wins the plurality of votes always chooses the mayor and deputy mayors in the PR system and essentially always does so under the plurality system as well. The organization of the municipal council and the benefits of holding a council seat are comparable in the two systems.¹⁴ Campaign regulations that would affect the effectiveness of mobilization efforts are also essentially the same in the two sets of municipalities.¹⁵ These similarities between the PR and plurality cities suggest that in comparing them I can rather narrowly measure the effect of electoral proportionality on turnout while holding other important factors fixed; by looking for the effect of the electoral rule on competitive outcomes I can also provide suggestive evidence about some of the channels through which this effect might operate.

In interpreting the results of my analysis, it is important to emphasize that, as indicated by Figure 1, the electoral systems I compare in this study are actually more similar in terms of proportionality than is true of the usual comparison of PR and plurality districts. Figure 2 illustrates the point. Not only is the “fortified” PR system used in larger cities considerably less proportional than the typical PR system, the plurality system is also somewhat more proportional than a typical winner-take-all plurality system because of the large number of seats and the possibility for more than one list to win seats (Eggers & Fouirnaies, 2014). In going from the plurality system to the PR system, then, French cities experience a relatively weak dose of proportionality compared with the effect of moving from a single-member district (SMD) to a large-magnitude PR district. The weakness of this “dose” suggests that it may be difficult to detect an effect of the electoral system on turnout in this setting. As discussed above, however, the problem with studying the effect of a larger dose of proportionality is that it typically comes along with other differences in the political setting that are likely to have their own impact on turnout, making it difficult to measure the effect of either proportionality or the electoral system more generally. To the extent that we do find an effect of the electoral system on turnout in the French municipal setting, it is reasonable to expect that a larger change in proportionality would have a correspondingly larger effect.

Figure 2.

French municipal electoral systems in the spectrum of proportionality.

It is of course possible to think of channels other than proportionality through which the electoral system may affect turnout in these cities. For example, voters may find the PR system to be normatively superior from a fairness perspective, or they may prefer to vote in the plurality system where they have the opportunity to vote for individuals; the incentives of candidates to devote effort to mobilization may also be higher or lower in the PR system depending on where they stand on the list and how much they care about their individual vote total.¹⁶ These channels cannot be dismissed in analyzing the effect of the electoral system in this setting. To the extent that electoral proportionality affects turnout through strategic mobilization and entry decisions, however (i.e., the direct and indirect channels introduced above), I may be able to uncover evidence of these effects empirically. As noted above, for a fixed set of lists in competition, we expect PR to increase mobilization efforts especially in less competitive contests, which can be tested; to the extent that PR increases turnout through encouraging entry, we should also see an effect of PR on the number of lists, which again can be tested. I now turn to empirical analysis in which I test these predictions.

Turnout and the 3,500 Population Threshold

To measure the effect of the electoral system on voter turnout in small French cities, I apply a regression discontinuity design (RDD; Hahn, Todd, & Van der Klaauw, 2001; Lee, 2008; Thistlethwaite & Campbell, 1960), taking advantage of the fact that the electoral system is determined at a population cutoff.¹⁷ Before presenting results, I describe my implementation of RDD and provide evidence that it will produce credible estimates in this setting.

Preliminaries

My approach to the regression discontinuity design is to use local linear regression and to show that the results are robust to various modeling choices. To estimate the discontinuity at the 3,500 threshold, I define a sample of cities close to the threshold and regress the dependent variable on the log of population within that sample¹⁸ (recentered at the population threshold) interacted with an indicator for whether the city is above the threshold (Imbens & Lemieux, 2008). In some specifications, I include as covariates the municipality’s turnout in a previous presidential election (1995 or 2002) as well as a set of socioeconomic factors (proportion retired, proportion unemployed, proportion possessing a baccalaureate degree, proportion employed in agriculture, population in the previous census), the geographical area of the municipality, and the region; given that cities just above or below the 3,500 threshold should be similar on average in these and other dimensions, the purpose of including covariates is primarily to increase precision rather than to control for confounding factors.

One key choice parameter is the window within which one conducts the local linear regression. In principle, the RDD should depend exclusively on cities very close to the threshold, such that the effect estimate is minimally dependent on modeling choices; in practice, carrying out a regression using a wider window may improve estimates of the conditional expectation at the threshold. One widely used method for choosing a window within which to carry out analysis is a cross-validation procedure described by Imbens and Lemieux (2008) that searches for a window that minimizes mean squared error in the vicinity of the threshold; another algorithm suggested by -Imbens and Kalyanaraman (2012) approximates the optimal bandwidth by estimating the curvature of the conditional expectation in the neighborhood of the threshold.¹⁹ Using either procedure, I obtain surprisingly wide optimal windows, the majority of which are above 50% of population (i.e., a window of 1,750 to 5,250 for a threshold of 3,500).²⁰ The reason why the optimal bandwidths are so large appears to be that most of the outcomes I examine are quite linear in log population, such that the bias resulting from using a larger bandwidth is minimal. As explained below, I present the results in a variety of ways to show how my findings depend on the bandwidth and procedure chosen.

The validity of the RDD could be called into question if mayors of cities near the population threshold can manipulate their official population numbers (e.g., by allowing or disallowing housing permits) and thus effectively choose their electoral system; if that were the case, then cities on either side of the threshold may differ not just in their electoral system but also in other (possibly unobservable) features such as their preferences for one system or another. One standard way of checking the validity of the RDD, due to McCrary (2008), involves testing for a jump in the density of the forcing variable at the threshold; in this case, McCrary (2008)’s test fails to reject the null (p = .127). Another standard validity check is to carry out RDD analysis in which pre-treatment covariates serve as outcome variables. Table A1 in the appendix reports RDD effect estimates at varying population windows (25%, 50%, and 75%), showing that there is (as one would expect) no “effect” of crossing the 3,500 population threshold on the vast majority of placebo outcomes. These tests suggest that cities just above and below the population threshold are indeed comparable in not just observed but also unobservable features (e.g., local political culture).

Results: Turnout Overall

The top three rows of Table 1 report RDD estimates of the effect of crossing the 3,500 population threshold on turnout in the 2001 and 2008 municipal elections.²¹ In each case, I report the estimated coefficient $\hat{τ}$ from a regression like

\begin{array}{l} {Turnout}_{i} = β_{0} + τ 1 {{Pop}_{i} \geq 3500} + β 1 {ln(Pop)}_{i} + β_{2} 1 {{Pop}_{i} \geq 3500} \times \\ {ln(Pop)}_{i} + ε_{i} . \end{array}

Table 1.

Effect of Crossing 3,500 Population Threshold on Turnout in Municipal and Higher-Level Elections.

		Effect estimates
Outcome	Mean turnout	(1)	(2)	(3)	(4)	(5)	(6)
Municipal, 2001	70.73	0.989	1.537**	1.525***	1.075^†	1.685***	1.525***
Municipal, 2001	70.73	(0.778)	(0.538)	(0.433)	(0.598)	(0.413)	(0.33)
Municipal, 2008	69.14	0.763	0.929^†	1.476***	0.71	1.128**	1.504***
Municipal, 2008	69.14	(0.765)	(0.523)	(0.423)	(0.582)	(0.414)	(0.338)
Municipal, 2001 & 2008	69.96	0.878	1.242**	1.502***	0.946^†	1.346***	1.518***
Municipal, 2001 & 2008	69.96	(0.71)	(0.481)	(0.385)	(0.528)	(0.359)	(0.285)
Presidential, 2002	74.95	−0.04	−0.189	−0.038	0.11	−0.121	0.012
Presidential, 2002	74.95	(0.413)	(0.29)	(0.241)	(0.265)	(0.184)	(0.153)
Regional, 2004	63.38	−0.448	−0.7^†	−0.241	−0.384	−0.401	−0.177
Regional, 2004	63.38	(0.583)	(0.414)	(0.341)	(0.398)	(0.288)	(0.239)
Presidential, 2007	86.33	−0.248	−0.439^†	−0.253	−0.099	−0.13	−0.095
Presidential, 2007	86.33	(0.326)	(0.224)	(0.185)	(0.18)	(0.124)	(0.103)
Window:	25%	25%	50%	75%	25%	50%	75%
Covariates?					✓	✓	✓

Regression discontinuity design estimates are shown for the effect of crossing the 3,500 population threshold on turnout in two municipal election years (2001 and 2008, separately and pooled) and three higher-level elections. Robust standard errors are shown in parentheses; for the pooled regression, standard errors are clustered at the municipality level. Sample sizes depend on bandwidth: about 1,400 at 25%; 3,200 at 50%; and 7,500 at 75% (double those for the pooled regressions).

†

.05 < p < .1. *.01 < p < .05. **.001 < p < .01. ***p < .001.

In columns 4, 5, and 6, I add municipal covariates²² and in row 3 I pool the results for 2001 and 2008. Standard errors are robust in all cases and clustered in the pooled regressions. Within columns 1 to 3 and 4 to 6, each column reports estimation for a different bandwidth of population. The results indicate an effect of crossing the threshold on turnout that ranges between about 1 percentage point and 1.5 percentage points. The effect is generally statistically significant when the threshold is 50% or higher; in the pooled sample including covariates the effect is significant at the .1 level.

To give a sense of how the estimated effect depends on modeling assumptions, Panel A of Figure 3 depicts the non-parametric relationship between population and turnout around the 3,500 population threshold (left plot) and the estimated jump as a function of bandwidth (right plot) for the 2008 municipal elections. The left plot shows the local linear regression line using a 50% bandwidth; in the right plots, a vertical gray line identifies the bandwidth chosen by cross-validation. The left plot indicates that the conditional expectation seems to be shifted slightly upward at 3,500; the right-hand plots show that the point estimate does not depend much on the bandwidth chosen. (Table A2 in the appendix reports the RDD estimates at the optimal bandwidths for this and all other analysis in the article; Table A3 reports estimates using the algorithm proposed by Imbens and Kalyanaraman, 2012.²³) For comparison, in Panel B I show the same analysis for the 2001 elections, showing results that are similar and if anything stronger than the results for 2008.

Figure 3.

The effect of crossing the 3,500 population threshold on 2001 and 2008 municipal turnout.

Results: Turnout in Higher-Level Elections

The bottom three rows of Table 1 report the average effect of crossing the 3,500 threshold on turnout in higher-level elections between 2002 and 2007 based on the same procedures used to estimate the effect on municipal election turnout. Because the only electoral rule that changes at the 3,500 population threshold is the municipal electoral rule, one would not expect turnout to increase in higher-level elections at this threshold. This is in fact what I find. In the presidential elections of 2002 and 2007 and the regional elections of 2004, I do not find higher turnout in cities just above the 3,500 population threshold; when covariates are included none of the effects are significant.

The absence of a positive effect on turnout in higher-level elections is particularly noteworthy in light of Ladner and Milner (1999), whose main finding is that turnout in municipal elections is higher in Swiss municipalities that adopt PR than in those that use plurality methods. Remarkably, Ladner and Milner (1999) also find that PR cities have higher turnout in higher-level elections (even though all cities use the same system for these higher-level elections), which they interpret as evidence that PR is adopted in municipalities that are “culturally predisposed toward higher political participation” (p. 248).²⁴ The absence of a higher-level “effect” of the electoral system in the French context highlights the uniqueness of this setting: because French cities did not choose their electoral system, we can be confident that the turnout differences we see in municipal elections are the result of the municipal electoral system rather than other underlying differences between cities.

Results: Turnout and Competitiveness

As discussed in the previous section, the votes–seats relationship under the two electoral systems used in French municipalities suggests that PR should especially increase turnout in less closely contested elections. To test that prediction, I divide municipalities into terciles based on predicted competitiveness²⁵ and separately estimate the effect of crossing the 3,500 threshold on turnout for these different groups. Figure 4 depicts the change in turnout in 2008 separately for the most competitive (Panel A, top) and least competitive (Panel B, bottom) terciles of municipalities, using the presentation method of Figure 3. (The analysis is restricted to 2008 because candidate-level vote totals in the 2001 election were not collected by the Ministry of the Interior.) These plots indeed suggest that the turnout jump is concentrated in less competitive municipalities: There is a clear level shift in turnout at 3,500 in less competitive municipalities (bottom left) but not in the more competitive municipalities (top left). The bandwidth sensitivity plot at the top right of Figure 4 indicates that the effect of crossing the 3,500 threshold in competitive cities is about zero for most bandwidths and does not approach significance at any bandwidth; the bottom right plot indicates that in the least competitive cities, the estimated effect is at or above 1 percentage point for all bandwidths, with an effect that is at least borderline statistically significant (even without covariates) for all bandwidths above 40%.

Figure 4.

The effect of crossing the 3,500 population threshold on 2008 municipal turnout for most and least competitive municipalities.

In a separate analysis, I carry out a regression in which I interact tercile indicators with the treatment. When including covariates, I find a significant difference (at the .1 level) between the effect estimated for the least competitive tercile and the most competitive tercile; the estimated effect for competitive municipalities is basically zero. These analyses are consistent with the theoretical prediction that proportionality should increase turnout in this setting by increasing the probability of casting a pivotal vote in contests in which one list is very likely to win.

Results: Variance in Turnout Across Municipalities

As noted by Cox (1999), studies of the variance of electoral outcomes are far less common than studies of levels.²⁶ In part, this must be due to the fact that political scientists are less accustomed to explaining the variance of outcomes than they are to explaining the mean of outcomes (Braumoeller, 2006). My approach is to extend the RDD framework to study how moving above the 3,500 population threshold affects the magnitude of residual turnout after accounting for mean shifts. To be more specific, I first use local linear regression (with a population bandwidth of 50%) to estimate mean turnout as a function of population on either side of the 3,500 population threshold. I then extract the absolute value of the residuals—that is, the size of the difference between average turnout at a given population level and the turnout in a given city of that size. Finally, I apply the standard RDD analysis to this vector of residuals.²⁷

Figure 5 presents the results for analysis pooling 2001 and 2008. The analysis indicates that, consistent with Cox’s (1999) prediction, higher proportionality leads to lower variation in turnout. The mean residual is lower in PR cities by between .5 and 1, depending on the bandwidth used in the regression; this constitutes a drop of between 10% and 20% in the average magnitude of the residuals. This drop in variance (like the jump in average turnout I documented above) may occur because, for a fixed set of lists in competition, the greater proportionality of the PR system makes incentives to mobilize or vote more consistent across municipalities; the greater proportionality of the PR system may also encourage more lists to enter, which might create a more consistent level of competition across municipalities and thus make turnout more consistent across municipalities. I now turn to examining whether the more proportional system does indeed encourage entry of new lists.

Figure 5.

Effect of crossing 3,500 threshold on mean absolute deviation of turnout in 2001 and 2008.

Results: Number of Lists and Support for Lesser Lists

The top row of Table 2 and the top panel of Figure 6 report RDD analysis of the effect of crossing the 3,500 threshold on the number of lists in competition in the 2001 and 2008 elections. (Analysis here is restricted to northwest France, because list-level results for sub-3,500 cities are not available elsewhere.) The results suggest that the number of lists jumps by about 0.2 on average, from just below 2 to just above 2. As indicated by Figure 6, the point estimate depends on bandwidth but is positive and significant for any bandwidth above about 35%. (As indicated by Table A3, the Imbens–Kalyanaraman estimate is also positive and significant; note however that the cross validation-based algorithm (Table A2) yields a narrow bandwidth and a negative and borderline significant estimate.) The binned averages plot (top left of Figure 6) gives the clear impression of a jump in the number of lists per contest.

Table 2.

Effect of Crossing 3,500 Population Threshold on Number of Lists and Support for Lists.

		Effect estimates
Dependent variable	M	(1)	(2)	(3)	(4)	(5)	(6)
Number of lists	2.02	−0.016	0.177***	0.315***	−0.007	0.162***	0.299***
Number of lists	2.02	(0.074)	(0.053)	(0.04)	(0.063)	(0.048)	(0.037)
Vote share (%) for top two lists	95.42	−0.659	−2.456***	−3.996***	−0.994	−2.25***	−3.872***
Vote share (%) for top two lists	95.42	(0.911)	(0.706)	(0.56)	(0.786)	(0.654)	(0.532)
Window:	25%	25%	50%	75%	25%	50%	75%
Covariates?					✓	✓	✓

Analysis is based on cities in northwest France in the 2001 and 2008 elections. See Table 1 for details of the table presentation. Sample sizes depend on bandwidth: 573 at 25%; 1,406 at 50%; 3,080 at 75%.

†

.05 < p < .1. *.01 < p < .05. **.001 < p < .01. ***p < .001.

Figure 6.

Effect of crossing 3,500 population threshold on number of lists and support for top two lists.

The second row of Table 2 and the bottom panel of Figure 6 reports the effect of crossing 3,500 on the share of the vote (in %) won by the top two lists. The binned averages plot (bottom left of Figure 6) suggests a drop at the threshold, and for larger bandwidths we have a statistically significant drop of about 3% on average, from around 98% to around 95%. Both of the bandwidth selection algorithms yield a significant effect of around −3%.²⁸

Both findings suggest that the PR system may have increased turnout in part by encouraging the entry of new lists that increase aggregate mobilization effort and appeal to previously unengaged parts of the electorate.²⁹ It may have also increased turnout by increasing the quality or differentiation of the lists in competition: Conditional on two lists competing, for example, we might expect more experienced candidates to run on the less dominant list when the leaders of that list are more likely to win a seat.

Addressing Multiple Treatments

So far I have provided evidence that average municipal election turnout in cities just above the 3,500 population threshold is slightly higher and less variable than that in cities just below the threshold; I have shown that the effect is only found in less competitive municipalities (as would be predicted if voters and candidates are acting based on the probability of being pivotal) and that the number of lists also appears to jump at the same threshold. Because the electoral formula changes from plurality to PR at that threshold, it seems reasonable to attribute these effects to the more proportional electoral formula used in the PR municipalities. There is, however, a complication. As in many cases in which policies depend on population thresholds (Ade & Freier, 2011), other features of municipal government change at the 3,500 population threshold: A gender parity rule is imposed on electoral lists, the number of councilors increases from 23 to 27, and the mayor’s salary increases by about €400 a month. In this section, I consider whether these other treatments may also have contributed to the jump in turnout I observe at the 3,500 threshold.

First, consider the gender parity rule that, since 2001, has applied to cities of 3,500 or more inhabitants. The rule aims to achieve gender parity on municipal councils by requiring that each list include an equal number of men and women. The fact that the electoral system and the gender parity rule go into effect at the same population threshold is not an accident; the assumption among policy makers seems to be that parity would be much less effective under plurality rule because voters in that system could disproportionately cast votes for male candidates even if lists are gender-balanced.³⁰

In principle, the increase in average turnout at the 3,500 threshold could be explained by the gender parity rule: Perhaps the newly included women are able to mobilize a new set of voters, or perhaps their presence in politics spurs turnout by increasing the perceived legitimacy of municipal politics.³¹ One can even think of a logic by which the effect of the parity rule on turnout might be larger in less competitive elections: Perhaps voters who would be mobilized by the parity rule in less competitive contests are already mobilized in competitive contests. To test whether the parity rule might be responsible for the increase in turnout I find at 3,500, I carry out subgroup analysis based on the idea that, if the parity rule were in fact responsible, the jump in turnout would be largest in places where the parity rule had the largest effect—cities where women would be most under-represented in the absence of the parity rule. I use the observed level of female representation in 2008 in cities below the 3,500 threshold to predict the level of female representation for cities on both sides of the threshold (based on socioeconomic covariates and voting outcomes in the 2007 presidential election) in the absence of parity requirements. I then use the predicted level of representation to divide cities into terciles based on the size of the predicted effect of the parity rule (where the largest effect is predicted for those cities with the lowest predicted level of female representation). After confirming that the actual effect of the gender parity rule (estimated via RDD) is in fact strongly related to the predicted effect, I estimate the effect of crossing the 3,500 population threshold separately for the three groups of cities. The results are reported in Table 3. (Appendix Figure A1 reports graphical results.) It does not appear to be the case that the jump in turnout at 3,500 is largest where the gender parity law would have the largest effect. If anything, turnout jumps more in places where the effect of the gender parity rule would be smaller, which suggests that the parity law is not responsible for the turnout jump at the 3,500 threshold.

Table 3.

Effect of Crossing 3,500 Population Threshold on 2008 Turnout, as a Function of Effect of Gender Parity Law.

		Effect estimates
Subgroup: Effect of parity law	Mean turnout	(1)	(2)	(3)	(4)	(5)	(6)
High	69.82	−0.503	0.166	0.14	0.334	0.837	0.975^†
High	69.82	(1.242)	(0.867)	(0.683)	(0.996)	(0.697)	(0.556)
Medium	69.33	1.029	1.984*	2.385**	0.575	1.934**	2.035***
Medium	69.33	(1.313)	(0.924)	(0.753)	(0.928)	(0.695)	(0.566)
Low	68.25	1.561	0.573	1.859*	0.992	1.037	1.555*
Low	68.25	(1.431)	(0.918)	(0.761)	(1.118)	(0.751)	(0.623)

Window:	25%	25%	50%	75%	25%	50%	75%
Covariates?					✓	✓	✓

As described in the text, cities are divided into terciles (“High,” “Medium,” and “Low”) according to the size of the predicted effect of the gender parity law; the effect of crossing the 3,500 population threshold is separately estimated for each group. See Table 1 for details of the table presentation.

†

.05 < p < .1. *.01 < p < .05. **.001 < p < .01. ***p < .001.

Two other important policies also change at the 3,500 threshold: The salary of the mayor increases (from about €1,600 per month to about €2,060 per month) as does the size of the council (from 23 to 27). In both cases, one can imagine a mechanism by which the policy change would increase turnout: A higher salary makes it more valuable to win office and should thus induce greater mobilization efforts; a larger council means longer lists and thus more people with direct involvement in the elections. I use two approaches to determine whether either or both of these treatments could be responsible. First, I take advantage of the fact that both the salary and the council size increase at other population thresholds (500 and 1,000, for salary; 500 and 1,500, for council size) where we can conduct RDD analysis.³² Under the assumption that the effect would be roughly similar at other thresholds where the same policy changes, we can evaluate whether the change at 3,500 could be attributable to one or both of these policies.³³ In Table 4, I report RDD results for the effect of crossing the 500, 1,000, and 1,500 population thresholds on turnout in 2001 and 2008 (pooled); I also show the effect at three placebo thresholds (750, 2,150, and 4,520), where no policy change takes place.

Table 4.

Effect of Crossing Other Population Thresholds on Turnout in 2001 and 2008 Municipal Elections.

			Effect estimates
Population	Policy changes	Mean turnout	(1)	(2)	(3)	(4)	(5)	(6)
500	S,C	78.95	0.735*	0.359	0.299	0.705**	0.419*	0.392**
500	S,C	78.95	(0.323)	(0.228)	(0.184)	(0.263)	(0.186)	(0.144)
750		77.21	−0.179	−0.383	−0.226	0.071	−0.332	−0.303^†
750		77.21	(0.356)	(0.252)	(0.206)	(0.289)	(0.204)	(0.163)
1,000	S	75.99	0.916*	0.82**	0.446*	0.7*	0.586**	0.075
1,000	S	75.99	(0.398)	(0.275)	(0.224)	(0.311)	(0.219)	(0.179)
1,500	C	74.08	0.426	−0.275	−0.07	0.274	−0.185	−0.101
1,500	C	74.08	(0.463)	(0.335)	(0.273)	(0.369)	(0.266)	(0.218)
2,150		72.06	0.19	−0.191	−0.825*	0.399	−0.095	−0.578*
2,150		72.06	(0.569)	(0.404)	(0.328)	(0.44)	(0.31)	(0.253)
4,520		69.33	−0.451	−0.192	0.448	−0.358	−0.064	0.744*
4,520		69.33	(0.796)	(0.537)	(0.426)	(0.57)	(0.391)	(0.318)
Window:		25%	25%	50%	75%	25%	50%	75%
Covariates?						✓	✓	✓

Regression discontinuity design estimates are shown for the effect on turnout of crossing the population thresholds indicated in the left-most column on turnout in 2001 and 2008. The second column from the left indicates the policies that change at each threshold; “S” indicates an increase in the salary of the mayor and “C” indicates an increase in the size of the municipal council. See Table 1 for other notes.

†

.05 < p < .1. *.01 < p < .05. **.001 < p < .01. ***p < .001.

At none of the thresholds in Table 4 do I find an effect on turnout that is either as large or as consistent as the effect of crossing the 3,500 threshold. There is some evidence, however, of a jump in turnout at 500 and 1,000, the two population thresholds where the mayor’s salary increases. This suggests that mayor salary could play a role in the jump at 3,500. A careful consideration of the way in which a higher mayor salary would affect mobilization incentives suggests this role is small, however. If a higher salary increases turnout by inducing greater mobilization effort by mayoral hopefuls, it would be most likely to do so when it is most unclear which list will prevail—that is, in elections expected to be close. In Section III, however, it was seen that the largest effect of crossing the 3,500 threshold on turnout occurred in the least competitive races. Table 5 reports the same subgroup analysis at the 500 and 1,000 population thresholds.³⁴ The results depend on the specification but indicate a different pattern from the one I find at 3,500: Across the two thresholds, the evidence of a turnout jump seems to be strongest in the highest competitiveness tercile. Together with the analysis reported in Section III, this analysis casts doubt on the idea that the effect of crossing the 3,500 threshold on turnout can be attributed to the change in salary: The jump in turnout at 3,500 occurs where the mayor’s salary should have the smallest effect and the electoral rule should have the largest effect.

Table 5.

Effect of Crossing Population Thresholds on 2008 Turnout, as a Function of Competitiveness.

			Effect estimates
Population	Subgroup: Competitiveness	Mean turnout	(1)	(2)	(3)	(4)	(5)	(6)
500	High	79.81	1.777**	1.246**	0.517	1.402**	0.947*	0.494^†
500	High	79.81	(0.625)	(0.432)	(0.341)	(0.529)	(0.372)	(0.296)
500	Medium	78.31	0.27	−0.281	0.052	0.526	−0.207	−0.008
500	Medium	78.31	(0.637)	(0.446)	(0.36)	(0.563)	(0.399)	(0.328)
500	Low	76.26	0.515	0.572	0.834	0.609	0.78	0.885
500	Low	76.26	(0.662)	(0.476)	(0.387)	(0.601)	(0.434)	(0.354)
1,000	High	77.81	1.944**	1.458**	1.396**	1.878**	1.061*	0.827*
1,000	High	77.81	(0.707)	(0.501)	(0.406)	(0.598)	(0.426)	(0.354)
1,000	Medium	75.13	0.221	0.021	−0.234	−0.258	−0.239	−0.539
1,000	Medium	75.13	(0.761)	(0.525)	(0.435)	(0.666)	(0.469)	(0.387)
1,000	Low	72.9	1.238	1.364*	0.906*	1.46*	1.373**	0.75^†
1,000	Low	72.9	(0.822)	(0.554)	(0.452)	(0.703)	(0.488)	(0.4)
Window:		25%	25%	50%	75%	25%	50%	75%
Covariates?						✓	✓	✓

Regression discontinuity design estimates are shown for the effect on turnout of crossing the population threshold indicated in the left-most column, separately by tercile of predicted competitiveness. See Table 1 for presentation notes. See text for construction of competitiveness terciles.

†

.05 < p < .1. *.01 < p < .05. **.001 < p < .01. ***p < .001.

As a final exercise, I check whether the drop in the variance of turnout (documented for the 3,500 threshold in the previous section) is also seen at the thresholds where salary increases. This could be the case if increasing the salary of mayors intensifies competition in the least competitive places and thus makes turnout more uniform across thresholds. In fact, I find no effect of crossing these other thresholds on variance in turnout. Using the same procedure described above, I do not find a robust effect of crossing the 500 and 1,000 thresholds on the variance of turnout. (Figure A2 in the Appendix reports graphical results.) Thus, while there is intriguing evidence of turnout jumps at other thresholds where the mayor’s salary increases, these jumps seem to reflect a different process from the one responsible for the increase in turnout at 3,500: They do not happen in the less competitive cities, and they do not coincide with a drop in the variance of turnout. Especially given that the change in salary at 3,500 is substantially smaller in percentage terms than that at 500 and 1,000 (see Note 33), this provides evidence that the jump in turnout at 3,500 is due to the electoral system and not other factors that change at the same population threshold.

Conclusion

This article takes advantage of an unusual natural experiment in small French municipalities to investigate the relationship between electoral proportionality and turnout. Theory suggests that greater electoral proportionality could induce higher turnout through both a direct channel and an indirect channel: Greater proportionality could lead directly to higher turnout by inducing parties to mobilize and voters to vote even when competition is not particularly balanced between the leading parties; it could also lead indirectly to higher turnout by inducing more (and possibly better) lists to enter competition, which could in turn lead to higher turnout. Using a regression discontinuity design, I show that turnout is indeed higher on average in small French cities that use a form of PR than in similarly sized cities that use a form of plurality. I also provide evidence about the channels through which this effect operates: I show that turnout jumps especially in places where competition is relatively low (which highlights the role of proportionality in changing expectations about pivotality, that is, the direct channel), and I show that the number of lists is also higher in the PR cities (which shows that proportionality may affect turnout by increasing competition, that is, the indirect channel). Finally, I show that (consistent with findings of Cox, 1999) turnout varies less across PR cities than it does across plurality cities.

To what extent do these results help us understand the role of electoral systems in explaining turnout outside the specific context of small French cities? To be sure, it would be difficult to use this study to make a confident prediction about the effect of, say, introducing a PR system to British parliamentary elections. The evidence I have presented about proportionality and turnout suggests that such a reform would increase turnout in British elections by at least 1 percentage point—possibly substantially more, considering that such a reform would involve a larger change in district-level proportionality than the one I study here. This prediction requires several caveats, however. The effect of such a reform would depend not only on how proportionality changes (the focus of this article) but also on how districts are drawn and how voters form expectations about coalition politics—aspects that remained constant in the unusual setting studied in this article. While theory suggests that proportionality and turnout should be similarly linked in other settings, we need a better understanding of how the effect of proportionality depends on district size as well as how proportionality interacts with other features that usually differ between plurality and PR elections. These shortcomings in the generalizability of my findings highlight the need for complementary approaches to studying the relationship between electoral systems and political outcomes. This study produces relatively clear evidence about a small set of mechanisms by focusing on a particular setting, but to draw more general conclusions about the effect of electoral systems we need to continue to accumulate both narrow and well-identified analyses like this one and broader and more descriptive studies that make more heterogeneous comparisons.

Footnotes

Appendix

Table A3.

Estimated Effects at Bandwidths Chosen by Imbens–Kalyanaraman Algorithm.

Reference	Outcome	Intercept	Window (log pop.)	Effect	SE	n
Figure 4.A	Turnout, 2008; high competition	71.717	1.25	0.619	(0.438)	2,774
	Turnout, 2008; medium competition	68.676	1.25	1.655**	(0.501)	2,279
Figure 4.B	Turnout, 2008; low competition	65.119	1.187	1.951***	(0.532)	2,148
Table 1, row 1	Turnout, 2001	70.017	1.063	1.491***	(0.314)	6,272
Table 1, row 2	Turnout, 2008	68.455	1.219	1.371***	(0.301)	7,050
Table 1, row 3	Turnout, 2001 & 2008	69.321	0.982	1.358***	(0.237)	11,096
Table 1, row 4	Turnout, 2002 presidential	74.9	1.172	−0.016	(0.168)	7,114
Table 1, row 5	Turnout, 2004 regional	63.449	1.25	−0.165	(0.236)	7,748
Table 1, row 6	Turnout, 2007 presidential	86.38	0.947	−0.231	(0.142)	5,471
Table 2, row 1	Number of lists at 3,500	1.958	1.111	0.238***	(0.053)	2,556
Table 2, row 2	Vote share for top two lists	96.544	1.195	−3.325***	(0.61)	2,780
Table 3, row 1	Turnout, 2008; high parity effect	69.593	1.25	0.083	(0.485)	2,839
Table 3, row 2	Turnout, 2008; medium parity effect	68.23	1.25	2.109***	(0.527)	2,385
Table 3, row 3	Turnout, 2008; low parity effect	67.528	1.25	1.933***	(0.531)	2,084
Table 4, row 1	Turnout, 2001 & 2008 at 500	78.595	1.062	0.343**	(0.116)	41,669
Table 4, row 2	Turnout, 2001 & 2008 at 750	77.131	1.23	−0.141	(0.118)	41,039
Table 4, row 3	Turnout, 2001 & 2008 at 1000	75.641	1.25	0.543***	(0.127)	36,680
Table 4, row 4	Turnout, 2001 & 2008 at 1500	73.998	1.25	−0.295*	(0.148)	29,083
Table 4, row 5	Turnout, 2001 & 2008 at 2150	72.048	1.04	−0.571**	(0.189)	17,957
Table 4, row 6	Turnout, 2001 & 2008 at 4520	68.988	1.2	−0.107	(0.239)	11,324
Table 5, row 1	Turnout in 2001 at 500; high comp.	75.855	0.732	0.761*	(0.358)	4,844
Table 5, row 2	Turnout in 2001 at 500; medium comp.	78.187	0.702	−0.129	(0.341)	4,788
Table 5, row 3	Turnout in 2001 at 500; low comp.	79.143	0.732	0.975**	(0.319)	5,031
Table 5, row 4	Turnout in 2001 at 1000; high comp.	72.341	0.993	1.032**	(0.356)	4,943
Table 5, row 5	Turnout in 2001 at 1000; medium comp.	75.038	1.25	−0.143	(0.308)	5,976
Table 5, row 6	Turnout in 2001 at 1000; low comp.	77.026	0.776	1.469***	(0.364)	3,523

Because the algorithm yields very large bandwidths in some cases, I set a maximum of 1.25 of log population.

†

.05 < p < .1. *.01 < p < .05. **.001 < p < .01. ***p < .001.

Acknowledgements

Thanks to Alexander Fouirnaies, Jean Chiche, Jean-Yves Nevers, Aurélia Troupel, the seminar audience at BI Norwegian Business School, and Dominik Hangartner for advice and suggestions; to Brigitte Hazart of the French Ministry of the Interior (Bureau of Elections and Political Studies) for data; to Devin Caughey for sharing his R implementation of the optimal bandwidth algorithms; and to various mayors and municipal councilors for useful discussions and information.

Declaration of Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Notes

Author Biography

Andrew C. Eggers is an Asssistant Professor in the Department of Government at the London School of Economics. His research interests include corruption and accountability, electoral systems, and methodology.

References

Ade

Freier

(2011). When can we trust population thresholds in regression discontinuity designs? DIW Discussion Papers 1136. .

Aldrich

J. H.

(1993). Rational choice and turnout. American Journal of Political Science, 37, 246-278.

Banducci

S. A.

Donovan

Karp

J. A.

(1999). Proportional representation and attitudes about politics: Results from New Zealand. Electoral Studies, 18, 533-555.

Barone

De Blasio

(2013). Electoral rules and voter turnout. International Review of Law and Economics, 36, 205-35.

Blais

(2006). What affects voter turnout? Annual Review of Political Science, 9, 111-125.

Blais

Lachat

Hino

Doray-Demers

(2011). The mechanical and psychological effects of electoral systems a quasi-experimental study. Comparative Political Studies, 44, 1599-1621.

Boix

(1999). Setting the rules of the game: The choice of electoral systems in advanced democracies. American Political Science Review, 93, 609-624.

Braumoeller

B. F.

(2006). Explaining variance; or, stuck in a moment we can’t get out of. Political Analysis, 14, 268-290.

Brollo

Perotti

Tabellini

Nannicini

(2013). The political resource curse. American Economic Review, 103(5), 1759-1796..

10.

Colomer

J. M.

(2005). It’s parties that choose electoral systems (or, Duverger’s laws upside down). Political Studies, 53, 1-21.

11.

Cox

G. W.

(1997). Making votes count: Strategic coordination in the world’s electoral systems. New York, NY: Cambridge University Press.

12.

Cox

G. W.

(1999). Electoral rules and the calculus of mobilization. Legislative Studies Quarterly, 24, 387-419.

13.

Cox

G. W.

Rosenbluth

F. M.

Thies

M. F.

(1998). Mobilization, social networks, and turnout: Evidence from Japan. World Politics, 50, 447-474.

14.

Davidian

Carroll

R. J.

(1987). Variance function estimation. Journal of the American Statistical Association, 82, 1079-1091.

15.

Downs

(1957). An economic theory of democracy. New York, NY: Harper & Row.

16.

Duverger

(1954). Political parties: Their organization and activity in the modern state. New York, NY: J. Wiley.

17.

Egger

Koethenbuerger

(2010). Government spending and legislative organization: Quasi-experimental evidence from Germany. American Economic Journal: Applied Economics, 2(4), 200-212.

18.

Eggers

A. C.

Fouirnaies

A. B.

(2014). Representation and district magnitude in plurality systems. Electoral Studies, 33, 267-277.

19.

Fiva

J. H.

Folke

(2014). Mechanical and Psychological Effects of Electoral Reform. British Journal of Political Science, forthcoming. .

20.

Fujiwara

(2011). A regression discontinuity test of strategic voting and duvergers law. Quarterly Journal of Political Science, 6, 197-233.

21.

Fujiwara

Meng

K. C.

Vogl

(2013). Estimating habit formation in voting. NBER Working Paper No. 19721.

22.

Gagliarducci

Nannicini

(2013). Do better paid politicians perform better? Disentangling incentives from selection. Journal of the European Economic Association, 11, 369-398.

23.

Gerber

A. S.

Green

D. P.

Shachar

(2003). Voting may be habit-forming: Evidence from a randomized field experiment. American Journal of Political Science, 47, 540-550.

24.

Geys

(2006). Explaining voter turnout: A review of aggregate-level research. Electoral Studies, 25, 637-663.

25.

Grofman

Selb

(2011). Turnout and the (effective) number of parties at the national and district levels: A puzzle-solving approach. Party Politics, 17, 93-117.

26.

Hahn

Todd

Van der Klaauw

(2001). Identification and estimation of treatment effects with a regression-discontinuity design. Econometrica, 69, 201-209.

27.

Imbens

Kalyanaraman

(2012). Optimal bandwidth choice for the regression discontinuity estimator. The Review of Economic Studies, 79, 933-959.

28.

Imbens

G. W.

Lemieux

(2008). Regression discontinuity designs: A guide to practice. Journal of Econometrics, 142, 615-635.

29.

Jackman

(1987). Political institutions and voter turnout in the industrial democracies. The American Political Science Review, 81. 405-423.

30.

Ladner

Milner

(1999). Do voters turn out more under proportional than majoritarian systems? The evidence from Swiss communal elections. Electoral Studies, 18, 235-250.

31.

Lee

(2008). Randomized experiments from non-random selection in US House elections. Journal of Econometrics, 142, 675-697.

32.

Loughlin

(2007). Subnational government: The French experience. New York, NY: Palgrave MacMillan.

33.

McCrary

(2008). Manipulation of the running variable in the regression discontinuity design: A density test. Journal of Econometrics, 142, 698-714.

34.

Pettersson-Lidbom

(2012). Does the size of the legislature affect the size of government? Evidence from two natural experiments. Journal of Public Economics, 96, 269-278.

35.

Powell

G. B.

Vanberg

G. S.

(2000). Election laws, disproportionality and median correspondence: Implications for two visions of democracy. British Journal of Political Science, 30(3), 383-412.

36.

Riker

W. H.

Ordeshook

P. C.

(1968). A theory of the calculus of voting. American Political Science Review, 62, 25-42.

37.

Rokkan

(1970). Citizens, elections, parties: Approaches to the comparative study of the process of development. Oslo: Universitetsforlaget.

38.

Thistlethwaite

Campbell

(1960). Regression-discontinuity analysis: An alternative to the ex post facto experiment. Journal of Educational Psychology, 51, 309-317.

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