Measuring Closeness in Proportional Representation Systems

Simon Luechinger†‡, Mark Schelker§¶, Lukas Schmid†¶

May 13, 2021

Abstract

We provide closed-form solutions for measuring electoral closeness of candidates in proportional representation systems. In contrast to plurality systems, closeness in proportional representation systems cannot be directly inferred from votes. Our measure consistently captures electoral closeness for both open- and closed-list systems and for both main families of seat allocation mechanisms. This uniﬁed measure quantiﬁes the vote surplus (shortfall) for elected (non-elected) candidates. It can serve as an assignment variable in regression discontinuity designs or as a measure of electoral competitiveness. For illustration, we estimate the incumbency advantage for the parliaments in Switzerland, Honduras, and Norway.

JEL Classiﬁcation: C18, D72 Keywords: Regression discontinuity design, proportional representation systems, assignment variable

†University of Lucerne, Faculty of Economics and Management, Frohburgstrasse 3, 6002 Lucerne, Switzerland. ‡ETH Zurich, KOF Economic Institute, Leonhardstrasse 21, 8092 Zurich, Switzerland. §University of Fribourg, Department of Economics, Boulevard de P´erolles90, 1700 Fribourg, Switzerland. ¶University of St. Gallen, Department of Economics, SIAW-HSG, Bodanstrasse 8, 9000 St. Gallen, Switzerland.

For helpful comments and discussions, we thank Pietro Biroli, David Dorn, Jon Fiva, Sebastian Garmann, Benny Geys, Simon Hug, Gary King, Jaakko Merilainen, Manuel Oechslin, Panu Poutvaara, Marko Terviö,Simon Weschle, participants of the Swiss Workshop on Local Public Finance and Regional Economics in Lugano, Switzerland, the Workshop on Parliamentary Careers in Comparison in the Hague, the Netherlands, the Workshop of the Swiss Network on Public Economics in Zurich, Switzerland, the Swiss Workshop on Political Economy and Development in Weggis, Switzerland, the International Institute of Public Finance virtual congress 2020, and seminar participants at the University of Zurich, Switzerland. For excellent research assistance, we thank Anita Bossard Hou, Roxane Bründler,BénédicteDroz, Luca Fässler,Rino Heim, Jana Jarck, Nicolas Mauri, Jonas Röllin,Gabriela Rychener, AurélienSalli, Vasco Schelbert, Florence Stempfel, and Marlen Walthert. Finally, we thank Madeleine Schneider and Corinne Straub from the Swiss Federal Statistical Office for providing data and Sugarcube Information Technology Sàrlfor help with digitizing the election data. We thank Jon Fiva for sharing the Norwegian data. The research was supported by the Swiss National Science Foundation (grant 100017 163135). 1 Introduction

This paper develops a measure of closeness for elections in proportional representation systems. Measuring electoral closeness is important for two strands of the literature. One focuses on close elections to identify the causal eﬀects of winning elections using regression discontinuity designs both at the party and the candidate level (Lee, Moretti, and Butler

2004; Lee 2008; Lee and Lemieux 2010; Cattaneo, Frandsen, and Titiunik 2015; Fiva and

Smith 2018). The other considers closeness as an expression of a competitive political system (Besley, Persson, and Sturm 2010; Cox, Fiva, and Smith 2020). Both strands of the literature have primarily focused on plurality systems due to the challenges of measuring closeness in proportional representation systems. However, proportional representation is the most common electoral formula for legislative elections around the world. Figure 1 documents that 119 countries use a proportional representation or mixed system, while 86 countries use a plurality or majority system to elect their national parliament.

Figure 1: Electoral formulas in the world

PR and mixed Plurality/Majority Not applicable, other, and in transition

Note: The ﬁgure depicts a classiﬁcation of electoral formulas based on data of IDEA, the International Institute

for Democracy and Electoral Assistance. Countries shaded dark grey use a proportional representation system

or a mixed system. Countries shaded light gray use a plurality or majority rule. Countries shaded white are

either in transition or for them the classiﬁcation is not applicable. Datasource: www.idea.int/data-tools/;

accessed November 25, 2019.

1 How can we measure closeness in elections? Some elections are obviously close. In rare cases, several candidates tie and a lottery decides the election outcome (Hyytinen et al. 2018b). But to characterize the competitiveness of elections more generally and to com- pare barely elected and barely non-elected candidates in regression discontinuity designs more speciﬁcally, we need to measure closeness systematically and consistently for all elections and all candidates.

In a plurality voting system, an intuitive and widely used measure of closeness is simply the difference in votes (or vote shares) to the marginally elected or non-elected candidate. In proportional representation system, however, measuring closeness is not straightforward as seats are assigned in two steps. In a first step, seats are distributed to parties according to a transformation of party votes depending on the specific seat allocation mechanism. The two main families of seat allocation mechanisms are the highest average method and the largest remainder method. In a second step, these seats are assigned to individual candidates within a party based on the candidates’ vote ranking (open-list systems) or on their list position (closed-list systems). These characteristics of proportional representation systems pose three challenges in defining a measure of closeness analogous to plurality systems. First, party votes are not directly informative about marginal winners and losers for different seats considered (Folke 2014; Freier and Odendahl 2015). Second, the measure of closeness is not simply the vote difference to these marginal winners or losers. Third, from the perspective of a candidate, there are potentially multiple margins to win (or lose) a seat as she can win (or lose) a seat for her party (in open- and closed-list systems) or pull ahead of (or fall behind) copartisan candidates (open-list systems). In the following, we refer to these margins as the party and the candidate margin.

Let us illustrate the ﬁrst two points using a simple example in a district with three seats and three parties P1, P2, and P3 with 45, 35, and 20 votes. We assume that the D’Hondt

2 method is used to distribute the seats. With this method and three seats, we calculate

D’Hondt ratios by dividing the votes by 1, 2, and 3 (see Table 1) and then distribute the three seats by choosing the three highest D’Hondt ratios. In this example, P1 receives two seats with D’Hondt ratios of 45 and 22.5 and P2 obtains one seat with a D’Hondt ratio of

35. To illustrate the first point from above, let us calculate P2’s vote surplus for its seat. For this, we need to find out which party is next in line to win this seat. This cannot be inferred from raw votes but only from the other parties’ D’Hondt ratios. The transformation from raw votes to D’Hondt ratios reveals that P3 is closest of winning a seat with a D’Hondt ratio of 20. The vote surplus for P2’s seat is the difference in the relevant D’Hondt ratios which, in this case, corresponds to the vote difference between P2 and P3. However, we generally have to re-transform the D’Hondt ratio differences to calculate the vote shortfalls or vote surpluses. This is our second point mentioned above. We illustrate this point for the vote shortfall of P2’s second seat. P2 would have to win this seat at the expense of the second seat of P1. The difference in D’Hondt ratios to this marginal seat is the difference of the second-highest D’Hondt ratios of P1 and P2, which is equal to 5. To express this difference again in terms of actual votes instead of D’Hondt ratios, we have to multiply the D’Hondt difference by two as we consider the second seat of P2.

Table 1: An example of party votes and D’Hondt ratios

Party Votes D’Hondt ratio

First Second Third

P1 45 45.0 22.5 15.0

P2 35 35.0 17.5 11.7

P3 20 20.0 10.0 6.7

3 We propose the construction of a closeness measure for proportional representation, which is applicable in open- as well as closed-list systems and we propose solutions for both main families of seat allocation mechanisms. The intuition of our approach is to compute the vote surplus of elected candidates to losing their seat as well as the vote shortfall of not elected candidates to winning a seat. We take into account that candidates of a party can potentially gain or lose a seat along two margins. First, they can win or lose by winning or losing seats for their party (open- and closed-list systems). Second, they can surpass or fall behind other candidates within the same party (open-list systems). We explicitly calculate these margins and then deﬁne closeness as the minimal necessary movement along both margins to gain or lose a seat. Our approach can be extended to capture other features of the electoral system such as party alliances.

For illustration purposes, we implement our approach and estimate the individual incumbency advantage in three countries with electoral systems diﬀering along two main dimensions of proportional representation systems: the seat allocation mechanism (highest average versus largest remainder) and the list structure (open versus closed lists). We use data from 22 elections to the Swiss National Council in the years 1931 to 2015 (open lists; highest average), three elections to the Honduran National Congress in the years 2009 to

2017 (open lists; largest remainder), and eight elections to the Norwegian Storting in the years 1953 to 1981 (closed lists; highest average). For Switzerland, we identify a substantial incumbency advantage of 30 to 45 percentage points. For Norway, we observe an incumbency advantage of a similar size. For Honduras, we ﬁnd no overall incumbency advantage, but an increased probability of participating in the subsequent election.

Our ﬁndings are relevant for the literature measuring electoral closeness, especially the literature using regression discontinuity designs. Previous research has primarily focused on majoritarian systems (Lee, Moretti, and Butler 2004; Lee 2008; Gagliarducci and Paserman

4 2012; Eggers et al. 2015; Anagol and Fujiwara 2016; Pons and Tricaud 2018). For proportional representation systems, researchers have proposed three diﬀerent ways to measure closeness. The ﬁrst approach focuses only on the party margin for one extra seat, the second approach only considers the candidate margin, the third approach uses simulation methods to measure overall closeness.

The ﬁrst approach is laid out by Blais and Lago (2009), Selb (2009), and Folke (2014).1

As in their contributions, our approach is based on identifying the marginal seat of the other parties and calculating the vote distance to this seat. But our approach goes well beyond this: First, we calculate the vote shortfall or vote surplus for all parties and for all seats a party can possibly achieve and not only for marginal seats. This allows researchers to address a wide range of issues, such as quantifying a party’s vote distance to winning a seat majority or the vote shortfall of any non-marginal candidate in closed-list proportional representation systems to winning a mandate. Second, we present solutions for both highest average and largest remainder methods (not only for the former).2 Third, we provide an overall measure of closeness at the candidate level for open-list proportional representation systems, which requires considering and aggregating the party margin and the candidate

1. Blais and Lago (2009) and Selb (2009) propose the vote margin for the marginal seat and the party closest to winning this marginal seat as a measure for district-level competitiveness. Folke (2014) uses the vote margin for parties’ marginal seats to assess the importance of parties in a regression discontinuity design. Subsequent research has implemented the latter research design to explore diﬀerent aspects of coalition governments (Meril¨ainen2019), party fragmentation (Carozzi, Cipullo, and Repetto 2019; Palguta

2019), policies (Carlsson, Dahl, and Rooth 2019), geographical representation (Fiva, Halse, and Smith 2019), the effect of parties on voter attitudes (Solé-Olléand Viladecans-Marsal 2013), party favoritism (Curto-Grau,

Sol´e-Oll´e,and Sorribas-Navarro 2018; Carozzi and Repetto 2019), and political dynasties (Fiva and Smith

2018).

2. Blais and Lago (2009) do propose a closeness measure for largest remainder methods. However, they assume a constant quota and thus constant ratios for competing parties. As our discussion in Section 2.1.2 shows, this assumption does not hold.

5 margin.

The second approach only considers the candidate margin and ignores the party margin.

Scholars who apply this approach basically calculate the distances of votes, vote shares or ranks to marginally elected or marginally not elected candidates.3 This approach suﬀers from the fact that the marginal candidate is not ﬁxed. If the candidate of interest receives more votes, she not only catches up to other copartisan candidates, but she may also win an additional seat for her party and thereby change who is the marginal candidate. By ignoring the party margin, the approach makes many candidates appear to be less close than they are. In addition, this approach cannot measure closeness for candidates of parties winning no seat and for candidates of parties with all candidates elected. Figure A.1 in the

Appendix illustrates these problems with data from our application for Switzerland. In this application, we ﬁnd that for many candidates the party margin, not the candidate margin, is binding.

The third approach uses simulation methods to measure closeness in terms of election probabilities rather than in terms of vote distances. A major conceptual drawback of simulated election probabilities is that they inherently depend on arbitrary choices of the simulation setup. For example, Kotakorpi, Poutvaara, and Tervi¨o(2017) simulate elections by resampling votes from the actual vote distribution. They apply the relevant seat allocation mechanism, count the number of times a candidate is elected, and divide this number by the number of simulations to calculate the candidate’s election probability. This probability

3. Papers using only the candidate margin analyze questions related to the individual incumbency advantage (Fiva and Røhr 2018; Hyytinen et al. 2018b; Redmond and Regan 2015; Golden and Picci 2015), political careers (Lundin, Nordstr¨om-Skans, and Zetterberg 2016), the separation of powers (Poulsen and

Varjao 2019), public sector employees in parliaments (Hyytinen et al. 2018a), and politicians’ income (Berg

2018).

6 essentially serves as a measure of closeness.4 However, this probability crucially depends on the size of the simulation samples. The probability approaches a candidate’s election status with increasing sample size, it approaches the actual vote share of a candidate’s party with decreasing sample size. We illustrate this problem by applying the approach of Kotakorpi,

Poutvaara, and Tervi¨o(2017) to our previous numerical example with the three parties P1,

P2, and P3. Figure A.2 demonstrates the arbitrariness of simulated election probabilities.

With one resampled vote, the simulated election probabilities of all candidates coincide with their party’s vote share of 0.45 for P1, 0.35 for P2, and 0.2 for P3. Already for around 5,000 resampled votes, the simulated election probabilities of all candidates converge to zero for non-elected candidates or to one for elected candidates. Thus, for example, for the ﬁrst

5 ranked candidate of P2, the simulated election probability ranges from 0.35 to 1.

In sum, we provide the ﬁrst analytical measure of overall closeness at the candidate level in proportional representation systems, applicable for both open- and closed-list systems and for both main families of seat allocation mechanisms.

Our paper also relates to an earlier literature that deﬁnes lower and upper bounds of

4. An application of this approach is Dahlgaard’s (2016) examination of the incumbency advantage in

Denmark. Freier and Odendahl (2015) suggest another simulation-based approach that adds noise to the actual vote shares to estimate the party margin. Baskaran and Hessami (2017) and Fiva, Folke, and Sørensen

(2018) use a simulation-based approach to measure the distance of a party block to a seat majority. They successively add or subtract votes to or from a party block and randomly redistribute these votes among the parties within the block in simulated elections until the majority changes in at least half of the simulated elections.

5. With three parties, more extreme examples would be possible: A candidate’s election probability could range from 0.2 to 1. In addition, this conceptual problem, there are also technical problems with the third approach. With our data for Switzerland, for many election districts and years, the simulation method fails to meet the necessary convergence criterion (see footnote 3, Kotakorpi, Poutvaara, and Tervi¨o2017), even after over 2,000,000 resamplings.

7 the vote share that a party needs to win to receive a particular number of seats (Rokkan

1968; Rae 1971; Rae, Handby, and Loosemore 1971; Lijphart and Gibberd 1977; Gallagher

1992). The aim of this literature is to characterize diﬀerent seat allocation mechanisms, not to measure closeness in real elections. Therefore, the bounds are only a function of the seat allocation mechanism, district magnitude, and number of parties and not a function of actual vote distributions. Finally, our empirical application contributes to the vast literature on the individual incumbency advantage (Gelman and King 1990; Ansolabehere, Snyder, and

Stewart 2000; Lee 2008).

2 Calculation and aggregation of vote margins

In this section, we present our measure of closeness. It captures how many votes away a particular candidate is from winning or losing her seat. In a proportional representation system, seats are allocated in two steps. In a first step, seats are distributed to parties. In a second step, seats won by a party are allocated to individual candidates. Therefore, winning or losing votes affects a candidate’s election prospects in two ways: It affects her party’s total number of seats and, in addition in open-list proportional systems, her individual ranking within the party.6 The construction of our vote margin variable mirrors this two- step procedure. First, we construct a party margin variable that captures the vote change necessary so that the party obtains a particular number of seats (open- and closed-list

6. In the following, we assume that every vote for a candidate counts towards her party’s vote total. This is the standard case for open-list proportional representation systems, even though they can diﬀer in many other dimensions. For example, voters might cast from one vote (e.g., Brazil and Finland) to as many votes as there are seats (e.g., Honduras and Switzerland). Another distinction is that voters either must choose from a list of candidates (e.g., Honduras) or can opt to forgo the possibility to choose candidates and can cast a party vote only (e.g., Switzerland).

8 proportional representation). Second, we calculate the candidate margin that captures vote diﬀerences to other candidates from the party (open-list proportional representation). Third, we aggregate the two margins to construct an overall measure of a candidate’s closeness.

Throughout, negative numbers indicate vote shortfalls and positive numbers vote surpluses.

2.1 Party margin

In the following, we construct the party margin for both families of seat allocation mechanisms, the highest average methods and largest remainder methods. Let us consider a district with J parties Pj indexed by j ∈ P ≡ {1, 2, ..., J}. The party index of all parties other than j is −j ∈ P\{j}. A total number of n seats are allocated. The index i ∈ {1, 2, ..., n} denotes the potential seats of party Pj. Similarly, ˜i captures the potential seats of each of the other parties P−j. We denote the total number of votes of a party Pj by votesj.

2.1.1 Highest Average Methods

We use the D’Hondt method to illustrate the calculation of the party margin for highest

7 average methods. For every party, a total of n D’Hondt ratios, votesj/i, are calculated.

The ﬁrst seat goes to the party with the highest D’Hondt ratio, the second seat to the party with the second highest D’Hondt ratio and so on.

We deﬁne the variable party margini,j. This variable captures the necessary change in the number of votes such that party Pj is at the margin of winning exactly i seats. This happens if the ith-highest D’Hondt ratio of party Pj equals the (n−i+1)th-highest D’Hondt

7. Other highest average methods, such as the Sainte-Laguë,modified Sainte-Laguëor Imperiali method, only differ by the denominator used to calculate the ratios.

9 ratio of all other J − 1 parties and, thus, if the ith seat for party Pj is decided by a coin ﬂip.

The (n−i+1)th-highest D’Hondt ratio is the one with the order statistic (n(J −1)−(n−i)).

Formally, the party margini,j is deﬁned as:

votes party margin = votes − i −j i,j j ˜ i (n(J−1)−(n−i))

In our example, the seats are allocated to three parties. Party P1 has D’Hondt ratios of

45.0, 22.5, and 15.0, party P2’s ratios are 35, 17.5, and 11.7, and party P3’s ratios are 20,

10, and 6.7 (see Table 1). The three highest D’Hondt ratios achieve a seat and thus in this case party P1 receives two seats and party P2 obtains one seat.

Let us illustrate the calculation of the party margin for all three potential seats of party

P2. The party’s ﬁrst seat is close if its highest D’Hondt ratio is equal to the third highest

D’Hondt ratio of the other parties P1 and P3 which is 20 for the ﬁrst seat of P3. The party could lose 15 votes to secure their ﬁrst seat (i = 1) against party P3. Thus, its party margin is 15 for i = 1. If party P2 wanted to win two seats, the second D’Hondt ratio of party

P2 would have to be equal to the second-highest D’Hondt ratio of all other parties which is

22.5. To achieve this, party P2 would need 10 additional votes and thus the party margin for i = 2 is −10. Similarly, if party P2 wanted to achieve all three seats, the third-highest

D’Hondt ratio of party P2 would have to be equal to the highest D’Hondt ratio of all other parties which is 45. Thus, the party margin is −100 for i = 3.

2.1.2 Largest Remainder Methods

Proportional representation systems that apply the largest remainder method distribute seats by dividing the total votes of a party by a quota. For example, the Hare-Niemeyer method, which we employ for illustration, uses the division of the total number of party votes by the number of seats in a district as a quota. The division of party votes by the

10 quota results in a party-speciﬁc Hare-Niemeyer ratio that consists of an integer part plus a fractional remainder. In the ﬁrst round of the seat allocation, parties receive a total number of seats equal to their integer. In the second round of the seat allocation, parties obtain at most one of the remaining seats in decreasing order of their remainder. We denote the

votesj votesj integer part of the Hare-Niemeyer ratio as ﬂoor quota and the remainder as mod quota .

We follow our notation for the highest average method and denote the total votes of our party of interest as votesj, the votes of all other parties as votes−j, and the (Hare-Niemeyer) P votesj + −j votes−j quota as n .

In what follows, we quantify the vote shortfall for seat i of party Pj. In a first step, our procedure establishes the necessary votes such that party Pj receives i − 1 seats in the first round of the seat allocation and has the same remainder as another party P−j. In a second step, we use these votes to determine the number of free seats to be distributed in the second round of the seat allocation and the ranking of the other parties’ remainders. We repeat the first two steps for all parties P−j and their possible seats. In the third step, we select the relevant solution from the first step. The relevant solution is the one for which the rank of party P−j’s remainder equals the number of seats to be distributed in the second round of the seat allocation. Intuitively, the remainder that has the same rank as the number of seats to be distributed in the second round obtains the last seat. In a final step, we calculate the party margin as the difference between the actual number of votes and the necessary number of votes from the solution selected in the third step.

If the following condition holds, the parties Pj and P−j achieve exactly i − 1 and ˜i − 1

11 seats in the ﬁrst round of the seat allocation and their remainders are identical8:

votes × n i,˜i,j,−j votes−j × n ˜ P − (i − 1) = P − (i − 1) −j votes−j + votesi,˜i,j,−j −j votes−j + votesi,˜i,j,−j

To get the necessary votes for party Pj for the ﬁrst step outlined above, we can rewrite this condition as:

˜ P votes−j × n + (i − i) −j votes−j votes ˜ = where 1 ≤ ˜i ≤ n − i + 1 (1) i,i,j,−j n − (i − ˜i)

For the second step, we substitute votesi,˜i,j,−j for the actual votes of Pj and re-calculate the ratios of the J − 1 other parties. Because we repeat the ﬁrst two steps for all parties indexed by −j, we need to introduce a new index −˜j with the same values as −j to express these ratios. The ratios are:

votes−˜j × n P . −˜j votes−˜j + votesi,˜i,j,−j

Based on these ratios, we determine the number of seats that remain to be distributed in the second round of the seat allocation

! X votes−˜j × n Fi,˜i,j,−j = n − (i − 1) − ﬂoor P −˜j votes−˜j + votesi,˜i,j,−j −˜j and the ranking of all the remainders

!! votes−˜j × n Ri,˜i,j,−j,−˜j = rank mod P . −˜j votes−˜j + votesi,˜i,j,−j

As mentioned above, we repeat the ﬁrst two steps for all J − 1 parties and all possible seat diﬀerences i −˜i. Therefore, we have (J − 1) × (2n + 1) possible solutions. In the third step,

8. The condition only changes with (i − ˜i).

12 we select the relevant solution, votes∗ , which satisﬁes the following conditions: i,˜i,j,−j

Fi,˜i,j,−j = Ri,˜i,j,−j,−˜j,

−˜j = −j, and ! votesi,˜i,j,−j × n (i − 1) = ﬂoor P . −˜j votes−˜j + votesi,˜i,j,−j

In the fourth step, we deﬁne the party margin as the diﬀerence between the actual votes and the necessary votes selected in the third step:

∗ party margini,j = votesi,j − votesi,˜i,j,−j.

In our example with three seats and 45, 35, and 20 votes for the parties P1, P2, and

P3, the Hare-Niemeyer quota is 33.33. The party P1 has a ratio of 1.35, the party P2 of

1.05, and the party P3 of 0.6. Thus, the parties P1 and P2 receive one seat each in the first round of the seat allocation and the remaining seat is distributed to party P3 in the second round based on its largest remainder. Let us demonstrate our procedure to find the party margin for the second seat (i = 2) of party P2. In the first step, we calculate the necessary number of votes such that party P2 gets one seat (i − 1) in the first round, the comparison party, P1 or P3, gets zero or one seat(s) (˜i − 1 where 1 ≤ ˜i ≤ 3 − 2 + 1) in the first round, and the party P2 and the comparison party have the same remainder. For example, for the comparison party P3 and ˜i = 1, the solution is 62.5. If party P2 had 62.5 votes, the new

Hare-Niemeyer quota would be 42.5, and the ratios of the three parties would be 1.06, 1.47, and 0.47. In the second step, we determine that one seat would remain to be allocated in the second round of the seat distribution and that the remainder of the relevant comparison party P3 would rank ﬁrst. In the third step, we select 62.5 as the relevant solution because the number of seats to be distributed in the second round and the rank of the comparison

9 party P3 are both equal to one. Finally, we calculate the party margin for the second seat

9. Consider another example with the comparison party P1 and ˜i = 2. The solution in the ﬁrst step is

13 of P2 as 35 − 62.5 = −27.5.

2.2 Candidate margin

We now construct the candidate margin that measures how far an individual candidate is from reaching the same position as another copartisan candidate. This variable is only relevant for open-list proportional representation systems. We extend the setting and introduce candidates Ch of party Pj. They are indexed by h ∈ H ≡ {1, ..., nj} where nj is the number of candidates in party Pj. Similarly, all other candidates of party Pj are indexed by

−h ∈ H\{h}.

The vote change required to be on par with the ith highest copartisan candidate, i.e., the one with the order statistic (nj − i + 1), is simply the diﬀerence between the number of votes of the two candidates. The candidate margin for candidate h in party j to position i is deﬁned10:   votesh,j − (votes−h,j) if i < nj  (nj −i+1) candidate marginh,i,j =  votesh,j if i = nj

We continue our example and calculate the candidate margin for candidate C2 of party

P2. We assume the following individual vote distribution: 22 votes for C1, 8 votes for C2, and

5 votes for C3. Candidate C2 needs 14 votes to be on par with the highest-ranked copartisan candidate, she can lose three votes to be on par with the second-ranked candidate among all other copartisans, and she can lose all her eight votes and still be the lowest-ranked

45.0, resulting in ratios of 1.23, 1.23, and 0.55. In the second step, we determine that one seat would remain to be distributed and that the remainder of the relevant comparison party would rank second. Because these two numbers are diﬀerent, we do not select 45.0 as a solution for party P2 and i = 2 in the third step.

Instead of party P2, party P3 would win the remaining seat.

10. To be the lowest-ranked candidate (i = nj ), candidate Ch could lose all her votes.

14 candidate. The candidate margin for candidate C2 of party P2 is −14 for i = 1, 3 for i = 2, and 8 for i = 3.

2.3 Aggregation

In the final step, we aggregate the party and candidate margin to our overall measure of closeness, i.e., the vote margin. For each candidate Ch there are n possibilities such that her election outcome is decided by a coin flip: the party gains one seat and candidate Ch is on par with the best-ranked copartisan candidate, the party receives two seats and Ch is on par with the second-best-ranked copartisan candidate, and so on. A final possibility is that

Ch is the lowest-ranked candidate and the party is at the margin of winning n seats.

We can think of our aggregation as a two-step process. In the first step, we identify the binding margin such that party Pj obtains exactly i seats and candidate Ch is on par with the candidate on position i. The binding margin is the minimum of the party margin and the candidate margin. In the second step, we identify the combination with the narrowest binding margin for each h. Among all binding margins, we choose the smallest absolute vote change that would alter the election status of candidate Ch. This is the maximum among all binding margins identified in the first step. Our overall vote margin of candidate Ch in party Pj is defined as:

vote marginh,i,j = max {min{party margini,j, candidate marginh,i,j}}. i∈{1,...,n}

In our example, the candidate C2 of party P2 has three possibilities to win a seat by a coin

ﬂip. First, party P2 receives one seat (i = 1) and candidate C2 makes up the vote diﬀerence to the highest-ranked copartisan candidate. The respective party margin and candidate margins are 15 and −14. In other words, the candidate C2 could lose 15 votes in order for party P2 to still get one seat but she would require 14 additional votes to have a chance to

15 win this seat. The binding margin for this ﬁrst possibility is −14. We underline this binding margin in the ﬁrst row of Table 2. Second, party P2 obtains two seats and candidate C2 ties with the second-ranked copartisan candidate. The respective party margin and candidate margins are −10 and 3. The binding margin for this possibility is the party margin of −10.

Third, party P2 receives three seats and candidate C2 is the lowest-ranked candidate. For this possibility, the party margin is −100, the candidate margin is 8 and thus the binding margin is −100. Candidate C2 is not elected and her easiest possibility to get elected is the second possibility. It is the one with highest of the binding margins which is −10, overlined in Table 2. Thus, the overall vote margin is −10.

Table 2: An example of party and candidate margins

Party Candidate Seat i Margin

Party Candidate

P2 C2 1 15 −14

P2 C2 2 −10 3

P2 C2 3 −100 8

2.4 Extension

The electoral systems of many jurisdictions are characterized by additional institutional features. We can introduce such features into our approach. As an illustration, we introduce party alliances in the D’Hondt system because this setting is relevant for our empirical application.11 In the context of our application for Switzerland, parties can form alliances

11. The possibility to form party alliances is rather common around the world though details diﬀer. Our extension captures the situation in Israel, the Netherlands, and Switzerland, among others.

16 that inﬂuence the seat allocation but have no further implications beyond this. If parties form alliances, seats are allocated to parties in two rounds, whereby each round uses the

D’Hondt method. In the ﬁrst round, seats are allocated to alliances. In the second round, the seats won by the alliances are distributed to parties. We show how to adapt the construction of the party margin to reﬂect this two-round procedure.

We continue with a district with J parties and n seats. The parties form L disjoint alliances Al indexed by l ∈ A ≡ {1, 2, ..., L}. An alliance is formed by one or more parties.

Therefore, an alliance Al is a subset of the set of parties. The index k ∈ {1, 2, ..., n} denotes

˜ the potential seats of an alliance Al. Similarly, k captures the potential seats of each of the other alliances indexed by −l ∈ A\{l}. We denote the total number of votes of a party Pj of an alliance Al by votesj,l.

In the ﬁrst round, seats are allocated to alliances based on their total number of votes, votes = P votes . The construction of the ﬁrst round margin is equivalent to the l Pj ∈Al j,l construction of the party margin in Subsection 2.1.1 with alliances replacing parties:

votes ﬁrst round margin = votes − k −l . k,l l ˜ k (n(L−1)−(n−k))

In the second round, all seats won by alliance Al are allocated to the parties Pj ∈ Al.

This means that we have to calculate the second round margin for all potential seats i and all potential ﬁrst round results k as long as i ≤ k.12 Again, we construct the second round margin analogously to the party margin in Subsection 2.1.1, but replace the number of seats in a district, n, by the numbers of seats won by the alliance, k, and the overall number of parties, J, by the number of parties in alliance Al, card(Al). For seat i of party Pj of

12. If n is even, the number of second round margins for a particular party is n2/2, otherwise (n + 1)n/2.

17 alliance Al that received k seats in the ﬁrst round, the second round margin is

votes second round margin = votes − i −j,l . i,j,k,l j,l ˜ i (n(card(Al)−1)−(k−i))

We aggregate the ﬁrst round margin and the second round margin to get the party margin in a setting with alliances. The party marginij for a seat i of party Pj in alliance Al is deﬁned as:

party margini,j = max {min{ﬁrst round margink,l, second round margini,j,k,l}}. i∈{1,...,n}

Let us introduce alliances in our example and assume that party P1 forms the singleton alliance A1 and parties P2 and P3 form another alliance A2. In the ﬁrst round, the three

D’Hondt ratios for A1 are 45, 22.5, and 15.0 and for A2 55, 27.5, and 18.3. Thus, alliance

A1, and therefore party P1, gains one seat and alliance A2 receives two seats. In the second round, the two seats of alliance A2 are distributed to parties according to their party-speciﬁc

D’Hondt ratios of 35, 17.5, and 11.7 for party P2 and 20, 10, and 6.7 for party P3. Thus, parties P2 and P3 each obtain one seat.

As before, we construct the party margin for party P2. We begin with the ﬁrst round margin and the ﬁrst seat, k = 1, of alliance A2. If alliance A2 lost 40 votes, its highest

D’Hondt ratio would be equal to third highest of alliance A1 and it would still win exactly one seat. Thus, the ﬁrst round margin for k = 1 of alliance A2 is 40. Equivalently, the ﬁrst round margin of alliance A2 is 10 for k = 2 and −80 for k = 3.

For the second round margin, we have to consider six (4 × 3/2) diﬀerent cases for party

P2 with i ≤ k. If the alliance won only one seat, the party could lose 15 votes to secure this seat against party P3. Thus, the second round margin is 15 for i = 1 and k = 1. If the alliance won two seats, the party would have a chance to get one seat if it lost 25 votes and its highest D’Hondt ratio were equal to the second highest D’Hondt ratio of its ally. This

18 implies a second round vote margin of 25 for i = 1 and k = 2. The remaining four cases are

−5 for i = 2 and k = 2, 28.3 for i = 1 and k = 3, 15 for i = 2 and k = 3, and −25 for i = 3 and k = 3. The following table summarizes the vote margins from the two rounds.

Table 3: Party vote margins for party P2

Margins

i k First round Second round

1 1 40.0 15.0

1 2 10.0 25.0

1 3 −80.0 28.3

2 2 10.0 −5.0

2 3 −80.0 15.0

3 3 −80.0 −25.0

We aggregate these margins to get the party margin in a setting with alliances. We start with the aggregation for the first seat, i = 1, of party P2. The first three rows in Table 3 depict cases for which party P2 receives one seat. In the first step, we determine whether the first or the second round margin is binding by taking the minimum for each row. For example, the alliance A2 could lose 40 votes and would still get one seat (k = 1), but party

P2 could only lose 15 votes to win this seat (i = 1). Thus, the binding margin is 15 votes.

The binding margins for i = 1 and k = 2 is 10 and for i = 1 and k = 3 is −80. The maximum of these binding margins is the one for i = 1 and k = 1 of 15. This implies that party P2 could lose up to 15 votes and therefore, the party margin for the ﬁrst seat, i = 1, of party P2 is 15. Similarly, the party margin is −5 for i = 2 and −80 for i = 3.

19 3 Empirical illustration

We assess the individual incumbency advantage for three countries with proportional representation to illustrate how we can use our measure of closeness in regression discontinuity designs as an assignment variable. Our main focus is on open-list systems, which require calculating and aggregating party and candidate margins. As examples we assess the prospects of election winners in subsequent elections for the Swiss National Council with an electoral formula from the highest average family and for the Honduran National Congress with one from the largest remainder family. For completeness, we also estimate the incumbency advantage for a closed-list system, which only involves measuring the party margin. Here we use the case of Norway previously analyzed by Fiva and Smith (2018).

3.1 Highest average methods: Switzerland, 1931-2015

3.1.1 Institutional background

The National Council is the larger chamber of the Swiss parliament. Since 1963, it consists of 200 members, for the earlier years in our sample, its size varied between 187 and 196.

The electoral districts are the 26 cantons (25 before 1979). Their magnitude ranges from one to 35 seats.

The D’Hondt method and open ballots are key elements of the electoral system. Every voter has as many votes as her canton has seats. Voters can cast their votes for candidates

(at most two votes per candidate) from diﬀerent parties. Votes for candidates always count for the respective parties (party votes). Voters can also give their votes to a party without specifying candidates. Therefore, the number of votes for parties is always (weakly) larger than the number of votes for candidates. For purely electoral purposes, parties can form

20 alliances and suballiances (see Section 2.4).13

We use data from the years 1931 to 2015. In these years, elections take place every four years and National Council members face no term limit. Women have the right to vote only since the 1971 election. In our sample period, the number of participating parties per canton and election is 7.670 on average and ranges from one to 35.

3.1.2 Data

We have data on all candidates’ votes, election outcomes, names, parties, age, and gender and on all parties’ votes and alliances from the Federal Gazette for the early years and from the Swiss Federal Statistical Oﬃce for later years.14 We link observations from the same candidate and canton across election years to construct a panel at the candidate × canton level.15 Starting with 41, 555 observations (excluding vote totals of unnamed individual candidates in single-member districts), we delete 92 observations from tacit elections. We remain with 41, 463 observations from 26,629 candidates.

To account for the variation in district magnitude, we divide the vote margin by the number of eligible voters (following the discussion in Cox, Fiva, and Smith 2020). The data on voters is from the Swiss Federal Statistical Oﬃce. The variable relative vote margin is the assignment variable in our regression discontinuity analysis:

vote margin relative vote margin = . eligibles

13. In some cases, an alliance comprises different parties. In others, alliances link different ballots of the same party if the party files candidates on different ballots.

14. We digitize the candidate-level election data for the years 1931 to 1971, and the candidate characteristics and the party-level data for the years 1931 to 1967 from the Federal Gazette.

15. We complement Stata’s record linkage procedure reclink with several rounds of manual checks by research assistants.

21 Figure 2 depicts the smooth distribution of the assignment variable at the threshold. In contrast to other settings, the density of the assignment variable is not informative about its manipulation. Election manipulation moving one candidate above the threshold necessarily moves another candidate below the threshold because the number of seats is ﬁxed. This symmetry prevents bunching.16

Figure 2: Distribution of the assignment variable for Switzerland

0.03

0.02

0.01

0.00 −10.0% −5.0% 0.0% 5.0% 10.0%

Note: This ﬁgures depicts the density (y-axis) in equally-sized bins of width 0.2 percentage points of the

assignment variable (x-axis) for around three times the optimal bandwidth based on the standard settings with

ﬁrst-order polynomial and triangular kernel.

3.1.3 Results

Figure 3 presents our main results for Switzerland. Panel (A) depicts the relationship between a candidate’s relative vote margin in the election in t and her probability of being

16. The density test developed by Cattaneo, Jansson, and Ma (2018; 2019) rejects the smoothness of the density around the threshold with optimal bandwidth (p-value=0.004), but not with half the optimal bandwidth (p-value=0.870).

22 elected in the subsequent election in t + 1. For both elected and non-elected candidates, this probability increases with the relative vote margin. However, there is a clear discontinuity between the two groups of candidates at the relative vote margin of zero. According to the estimates in column (1) of Table A.1 in the Appendix, this discontinuity is 42 percentage points. Given the average election probability of 11% for the years 1931-2011 this amounts to a substantial incumbency advantage. In column (3) we show that this eﬀect is relatively robust to using second order-polynomial regressions. If we consider only a smaller range of the assignment variable, the estimates decrease slightly with the linear model in column

(2) and the quadratic model in column (4). Panel (B) of Figure 3 depicts the estimates for varying bandwidths scaled relative to the optimal bandwidth. It demonstrates that the eﬀect robustly lies between 30 and 45 percentage points.

One important aspect of the incumbency advantage is the increased probability of running in the subsequent election. Panel (C) and (D) of Figure 3 show that the eﬀect of being elected on this probability is positive. According to Table A.1 in the Appendix it amounts to 19 to 26 percentage points.

23 Figure 3: Main results for Switzerland

(A) Elected in t + 1 (B) Eﬀect for diﬀerent bandwidths

.8 .5

.3 .2

0 .2 -.04 -.02 0 .02 .04 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5

.7 .3

.25 .6

.15

.4 .1

.3 .05 -.04 -.02 0 .02 .04 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5

Note: Panel (A) depicts the relationship between the assignment variable in t (x-axis) and the probability of

being elected in t + 1 (y-axis), whereby t indexes elections, with the average values in quantile-spaced bins

and linear regressions on both sides of the threshold. On both sides, we use the same optimal bandwidth for

the election probability in t + 1 suggested by Calonico et al. (2017) and a triangular kernel. Panel (B) depicts

point estimates and 95% conﬁdence intervals (y-axis) for bandwidths ranging from 10% to 150% of the optimal

bandwidth (x-axis). Panels (C) and (D) presents the corresponding information for participating in t + 1. In

panel (C) and (D), we use the same bandwidths as in panels (A) and (B).

For the validity of our regression discontinuity design it is necessary that only the probability of being elected changes discontinuously at the relative vote margin of zero. We examine this assumption by looking at discontinuities of candidates’ characteristics. Ta- ble A.1 in the Appendix shows that most characteristics are balanced, especially for the

24 smaller bandwidths for which our results are robust (see Panel (B) of Figure 3).

3.2 Largest remainder methods: Honduras, 2009-2017

3.2.1 Institutional background

The National Congress is the unicameral parliament of Honduras with 128 members from

18 departments that constitute the electoral districts. There are between one to 23 repre- sentatives per department.

Honduras employs the Hare-Niemeyer method and open ballots for the parliamentary elections. As Swiss voters, Honduran voters have as many votes as their department has seats. They can vote for candidates from diﬀerent parties by marking the respective candidates on a ballot that contains all the candidates from the department.

This system is only in place since 2005 (Mu˜noz-Portillo 2013). However, for 2005 no candidate-level data is available. Therefore, we have only the three elections 2009, 2013, and 2017. The number of parties per department and election varies between four and ten with a mean of 7.852.

3.2.2 Data

We have all the information on candidates’ votes, election outcomes, names, and parties from the Honduran Tribunal Supremo Electoral. We code candidates’ gender from the pictures on the ballots and their ﬁrst names.17 For the elected candidates in 2009 and all candidates in the elections 2013 and 2017, the data from the Honduran Tribunal Supremo Electoral contain unique candidate identiﬁers. For the remaining candidates, we use a similar procedure as

17. Following the oﬃcial decision of the electoral authority, we code one transgender woman as male.

25 in Switzerland to construct the candidate-level panel.18 We have 3, 025 observations from

2, 644 candidates.

Again, we divide the vote margin by the number of eligible voters to account for the diﬀerence in district magnitude. Figure 4 depicts the density of the assignment variable.19

Figure 4: Distribution of the assignment variable for Honduras

0.04

0.03

0.02

0.01

0.00 −6.0% −4.0% −2.0% 0.0% 2.0% 4.0% 6.0%

Note: This ﬁgures depicts the density (y-axis) in equally-sized bins of width 0.2 percentage points of the

assignment variable (x-axis) for around three times the optimal bandwidth based on the standard settings with

ﬁrst-order polynomial and triangular kernel.

3.2.3 Results

Figure 5 illustrates the main ﬁndings for Honduras. According to Panel (A), a close election victory in the election in t does not increase the probability of being elected in the subsequent

18. We have ﬁve observations of the 2009 elections with missing candidate information. We cannot know if these candidates run in later years.

19. We do not reject the smoothness of the assignment variable at the threshold with the optimal bandwidth

(p-value=0.265).

26 election in t+1. However, according to Panel (C), it does increase the probability of running in t + 1. Both these results are confirmed with different bandwidths and models (Figure 5 and Table A.2 in the Appendix). The estimates in Table A.2 suggest that the positive effect on participation is between 22 and 26 percentage points.

Figure 5: Main results for Honduras

(A) Elected in t + 1 (B) Eﬀect for diﬀerent bandwidths

.8 .4

.6 .2

.4 0

.2 -.2

0 -.4 -.02 -.01 0 .01 .02 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5

1 .8

.8 .6

.4 .6

.2 .4

0 .2

-.2 0 -.02 -.01 0 .01 .02 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5

Note: Panel (A) depicts the relationship between the assignment variable in t (x-axis) and the probability of

being elected in t + 1 (y-axis), whereby t indexes elections, with the average values in quantile-spaced bins

and linear regressions on both sides of the threshold. On both sides, we use the same optimal bandwidth for

the election probability in t + 1 suggested by Calonico et al. (2017) and a triangular kernel. Panel (B) depicts

point estimates and 95% conﬁdence intervals (y-axis) for bandwidths ranging from 10% to 150% of the optimal

bandwidth (x-axis). Panels (C) and (D) presents the corresponding information for participating in t + 1. In

panel (C) and (D), we use the same bandwidths as in panels (A) and (B).

27 As in the case of Switzerland, the validity tests raise no concerns. All candidate characteristics are balanced between elected and non-elected candidates (Table A.2).

3.3 Closed-list highest average methods: Norway, 1953-1985

In this last application, we brieﬂy illustrate how our measure of closeness can be applied in the context of closed-list electoral systems, where only the calculation of the party margin is required. We use the data of Fiva and Smith (2018) for Norwegian parliamentary elections.

The Norwegian Storting is a unicameral parliament with between 150 and 157 members in the relevant period. Elections are based on closed lists and a highest average method

(modiﬁed Sainte-Lagu¨e).

Our approach allows us to calculate the vote margin for all candidates and, thereby, to go beyond the focus on candidates on marginal seats. The focus on marginal candidates is an a priori restriction of the estimation sample. Non-marginal candidates can locate as close to the threshold as marginal ones. Figure 6 illustrates this point by presenting the relative vote margin in the vicinity of the threshold for marginal (dark-shaded) and non- marginal (light-shaded) candidates. In this application, closer to the threshold marginal candidates dominate. However, in other application this pattern might be less pronounced.

More importantly, using all, marginal as well as non-marginal, candidates permits a data driven sample selection.

28 Figure 6: Distribution of the assignment variable for Norway

Marginal candidate Non−marginal candidate 0.015

0.010

0.005

0.000 −36% −24% −12% 0% 12% 24% 36%

Note: This ﬁgures depicts the density (y-axis) in equally-sized bins of width 0.5 percentage

point of the assignment variable (x-axis) for around three times the optimal bandwidth based

on the standard settings with ﬁrst-order polynomial and triangular kernel. The dark-shaded

part represents the marginal candidates, the light-shaded part non-marginal candidates.

Figure 7 depicts the incumbency advantage for marginal candidates in Panel (A) and for all candidates in Panel (B). In this application, the a priori restriction of the sample to marginal candidates makes a substantial diﬀerence. With the inclusion of all candidates the eﬀect is larger.

29 Figure 7: Results for Norway

(A) Eﬀect for marginal candidates (B) Eﬀect for all candidates

.8 .8

.6 .6

.4 .4

.2 .2

0 0 -.04 -.02 0 .02 .04 -.1 -.05 0 .05 .1

Note: This ﬁgure depicts the relationship between the assignment variable in t (x-axis) and the probability of

being elected in t + 1 (y-axis), whereby t indexes elections, with the average values in quantile-spaced bins and

linear regressions on both sides of the threshold. On both sides, we use the same optimal bandwidth for the

election probability in t + 1 suggested by Calonico et al. (2017) and a triangular kernel. Panel (A) restricts

the sample to the marginal candidates, Panel (B) includes all candidates.

4 Conclusion

Proportional representation systems are the dominant electoral formula around the world.

Hence, our measure of electoral closeness in proportional representation systems opens up rich research opportunities. It allows researchers to investigate the determinants and consequences of electoral competitiveness at the candidate level. We propose a unified measure for different proportional representation systems and provide a consistent assignment variable in candidate-level regression discontinuity designs and thus makes this credible research design applicable in proportional representation systems. In addition, our closed-form solution is flexible. As we demonstrate in the illustration for Switzerland, it can accommodate particularities of an electoral system such as party alliances. This flexibility is important because hardly any two proportional representation systems look the same.

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35 Appendix

Figure A.1: Comparing our overall margin with the candidate margin

Note: This figure documents the relationship between our overall vote margin (x-axis) and the candidate margin (y-axis). Both variables are rescaled by the number of eligible voters in a canton. For 44% of the candidates, the candidate margin is not defined as their party won no seat or as all their copartisan candidates were elected. These candidates are depicted at the bottom of the figure “NA”). For 3% of the candidates the party margin, not the candidate, is binding. These are the candidates off the 45-degree line.

36 Figure A.2: Arbitrariness of simulated election probabilities

1.00

0.75

0.50 0.45

0.35

0.25 0.20

0.00

1 10 100 1000 10000

Note: The figure depicts simulated election probabilities (y-axis) for different sizes of simulation samples on a logarithmic scale (x-axis). The election probabilities are simulated for our numerical example with vote shares of 0.45 for party P1 (blue lines), 0.35 for P2 (red lines), and 0.2 for P3 (green lines). For each party, the election probabilities are simulated for the first-ranked (solid line), second-ranked (dashed line), and third- ranked (dotted line) candidate. The vote shares of the individual candidates are 0.25, 0.15, and 0.05 for P1,

0.22, 0.08, and 0.05 for P2, and 0.08, 0.07, and 0.05 for P3. The simulation is based on the approach suggested by Kotakorpi, Poutvaara, and Tervi¨o(2017).

Table A.1: Results and balance tests for Switzerland

(1) (2) (3) (4)

Elected in t+1 0.418∗∗∗ 0.345∗∗∗ 0.407∗∗∗ 0.351∗∗∗

(0.020) (0.029) (0.022) (0.030)

Participation in t+1 0.255∗∗∗ 0.203∗∗∗ 0.251∗∗∗ 0.194∗∗∗

(0.021) (0.030) (0.023) (0.031)

37 Vote margin in t-1 0.005 0.001 0.006 -0.001

(0.003) (0.005) (0.004) (0.005)

Age 2.031∗∗∗ 1.521∗∗ 1.883∗∗∗ 1.183∗

(0.463) (0.634) (0.494) (0.658)

Sex -0.008 -0.021 -0.010 -0.038∗

(0.015) (0.021) (0.016) (0.022)

Year 2.527∗ 2.247 2.603∗ 4.452∗∗

(1.303) (1.719) (1.379) (1.776)

Number of seats -2.141∗∗∗ -0.174 -1.409∗∗ -0.217

(0.592) (0.761) (0.622) (0.791)

Aargau 0.026∗∗ 0.017 0.021∗ 0.013

(0.011) (0.015) (0.012) (0.016)

Appenzell Innerrhoden 0.000 0.000 -0.000 0.000

(0.000) - (0.000) -

Appenzell Ausserrhoden 0.001 0.000 0.001 0.000

(0.002) (0.003) (0.002) (0.003)

Bern -0.075∗∗∗ -0.028 -0.056∗∗ -0.038

(0.023) (0.029) (0.024) (0.030)

Basel Landschaft 0.003 -0.001 -0.002 -0.000

(0.005) (0.008) (0.005) (0.008)

Basel Stadt 0.009 0.007 0.006 0.006

(0.008) (0.011) (0.009) (0.011)

Fribourg 0.006 -0.001 0.006 -0.000

(0.007) (0.008) (0.007) (0.009)

Geneva 0.009 0.004 0.008 -0.004

38 (0.015) (0.019) (0.015) (0.019)

Glarus -0.000 0.000 0.000 0.000

(0.000) (0.000) (0.001) (0.001)

Graub¨unden 0.005 0.000 0.006 0.002

(0.004) (0.004) (0.004) (0.005)

Jura 0.002 0.001 0.001 0.002

(0.003) (0.004) (0.003) (0.004)

Lucerne -0.003 0.000 -0.005 -0.001

(0.009) (0.011) (0.009) (0.011)

Neuchˆatel 0.003 -0.000 0.001 -0.001

(0.005) (0.006) (0.006) (0.006)

Nidwalden 0.000 0.000 0.000 -0.000

(0.000) - (0.000) (0.000)

Obwalden 0.001 0.001 0.001 0.001

(0.002) (0.002) (0.002) (0.002)

St. Gallen 0.011 -0.003 0.009 -0.001

(0.011) (0.015) (0.011) (0.015)

Schaﬀhausen 0.003 0.002 0.003 0.002

(0.003) (0.003) (0.003) (0.003)

Solothurn 0.010 0.000 0.008 -0.000

(0.008) (0.011) (0.008) (0.011)

Schwyz 0.003 0.000 0.004 0.001

(0.003) (0.005) (0.004) (0.005)

Thurgau 0.006 0.001 0.001 0.001

(0.005) (0.006) (0.006) (0.007)

39 Ticino 0.010 0.016 0.009 0.016

(0.008) (0.012) (0.009) (0.012)

Uri -0.000 -0.000 -0.000 -0.000

(0.000) (0.000) (0.000) (0.000)

Vaud -0.038∗ -0.052∗ -0.048∗∗ -0.033

(0.022) (0.029) (0.023) (0.029)

Valais 0.009 0.007 0.008 0.007

(0.007) (0.010) (0.008) (0.011)

Zug 0.002 0.001 0.001 0.002

(0.003) (0.004) (0.003) (0.004)

Zurich -0.002 0.028 0.016 0.027

(0.023) (0.030) (0.024) (0.031)

Social democrats (SP) 0.002 -0.004 0.003 -0.015

(0.030) (0.039) (0.032) (0.040)

Christian democrats (CVP) -0.015 0.030 -0.023 0.038

(0.025) (0.034) (0.027) (0.036)

Liberals (FDP) 0.075∗∗∗ 0.016 0.070∗∗ 0.015

(0.028) (0.039) (0.030) (0.040)

National conservatives (SVP) -0.033 -0.021 -0.033 -0.011

(0.030) (0.039) (0.031) (0.041)

Bandwidth 0.036 0.018 0.073 0.037

Left obs 12920 5694 19173 12991

Right obs 2644 1650 3437 2652

40 Note: The table reports estimates for the probability of being elected in t + 1, the probability of running in t + 1, the assignment variable in t − 1, and for several candidate characteristics in t, whereby t indexes elections. The estimates for the parties relate to the four parties represented in the Federal Council; we combine the national conservatives (SVP) with its recent splinter party

(BDP). We present bias corrected and robust point estimates and standard errors introduced by Calonico, Cattaneo, and Titiunik (2014a; b). Standard errors account for clustering at the candidate level (Calonico et al. 2017). For some small cantons and the narrow bandwidths, it is not feasible to calculate standard errors; we denote these cases with “-”. We use separate linear models on each side of the threshold in columns (1) and (2) and quadratic models in columns

(3) and (4). For the regressions, we use a triangular kernel. In each column, we use the same bandwidth for all variables. In columns (1) and (3), we employ the optimal bandwidth for the election probability in t + 1, calculated on the basis of Calonico et al. (2017), in columns (2) and (4), we use half this bandwidth. The two bottom rows refer to the observations within the bandwidth; these numbers are lower for the lagged assignment variable.

Table A.2: Results and balance tests for Honduras

(1) (2) (3) (4)

Elected in t+1 -0.011 -0.012 -0.042 -0.010

(0.066) (0.084) (0.072) (0.091)

Participation in t+1 0.239∗∗ 0.257∗∗ 0.218∗∗ 0.261∗

(0.093) (0.124) (0.102) (0.135)

Vote margin in t-1 0.013 0.009 0.019 0.001

(0.009) (0.013) (0.012) (0.015)

Sex -0.010 -0.114 -0.041 -0.134

(0.092) (0.123) (0.101) (0.133)

Year -0.363 -0.055 -0.284 0.011

(0.403) (0.536) (0.445) (0.582)

Number of seats 0.365 1.266 1.164 1.745

(1.477) (2.050) (1.640) (2.233)

Atl´antida 0.055 0.011 0.071 -0.010

41 (0.063) (0.062) (0.067) (0.061)

Choluteca -0.001 -0.024 -0.019 -0.029

(0.018) (0.020) (0.019) (0.025)

Col´on 0.018 -0.000 0.014 -0.007

(0.028) (0.035) (0.032) (0.036)

Comayagua 0.010 0.014 0.009 0.010

(0.036) (0.031) (0.036) (0.031)

Cop´an 0.001 -0.007 0.001 -0.010

(0.019) (0.013) (0.021) (0.014)

Cort´es -0.024 0.038 0.018 0.056

(0.087) (0.122) (0.097) (0.133)

El Para´ıso -0.005 0.013 -0.014 0.007

(0.035) (0.054) (0.040) (0.061)

Francisco Moraz´an 0.060 0.063 0.076 0.076

(0.081) (0.115) (0.091) (0.126)

Gracias a Dios -0.000 0.000 -0.001 -0.001

(0.001) - (0.002) (0.001)

Intibuc´a 0.002 -0.001 0.000 -0.004

(0.028) (0.028) (0.030) (0.028)

Islas de la Bah´ıa 0.000 0.000 -0.000 0.000

- - (0.001) -

La Paz 0.010 -0.002 0.006 -0.003

(0.039) (0.062) (0.044) (0.068)

Lempira 0.003 -0.003 0.000 -0.008

(0.025) (0.024) (0.026) (0.025)

42 Ocotepeque 0.004 0.007 0.005 0.010

(0.032) (0.053) (0.037) (0.059)

Olancho 0.003 -0.015 -0.014 -0.021

(0.020) (0.013) (0.023) (0.016)

Santa B´arbara -0.039 -0.020 -0.049 -0.017

(0.046) (0.059) (0.051) (0.063)

Valle -0.009 -0.002 -0.011 0.001

(0.009) (0.011) (0.010) (0.013)

Yoro -0.086 -0.071 -0.093 -0.050

(0.078) (0.109) (0.086) (0.117)

National conservatives (PN) -0.023 0.013 -0.039 0.006

(0.086) (0.117) (0.095) (0.129)

Social liberals (PL) 0.045 0.018 0.029 0.026

(0.087) (0.116) (0.095) (0.125)

Social democrats (Libre) -0.018 -0.008 -0.008 0.001

(0.088) (0.118) (0.096) (0.126)

Bandwidth 0.022 0.011 0.037 0.018

Left obs 461 189 628 399

Right obs 169 102 213 155

43 Note: The table reports estimates for the probability of being elected in t + 1, the probability of running in t+1, the assignment variable in t−1, and for several candidate characteristics in t, whereby t indexes elections. In contrast to the Swiss case, we do not have information on candidates’ age. The estimates for parties focus on the three largest parties. We present bias corrected and robust point estimates and standard errors introduced by Calonico, Cattaneo, and Titiunik (2014a; b). Standard errors account for clustering at the candidate level (Calonico et al. 2017). For some small departments, it is not feasible to calculate standard errors; we denote these cases with

“-”. We use separate linear models on each side of the threshold in columns (1) and (2) and quadratic models in columns (3) and (4). For the regressions, we use a triangular kernel. In each column, we use the same bandwidth for all variables. In columns

(1) and (3), we employ the optimal bandwidth for the election probability in t + 1, calculated on the basis of Calonico et al. (2017), in columns (2) and (4), we use half this bandwidth. The two bottom rows refer to the observations within the bandwidth; these numbers are lower for the lagged assignment variable.