District magnitude, electoral coordination, and legislative fragmentation

ABSTRACT Theories of party systems suggest that more restrictive rules should lead to fewer parties. According to this line of reasoning, parties and voters should strategically respond to reforms in electoral rules, such as changes in district magnitude. That is, a decrease in district magnitude should decrease the size of the party system, whereas an increase in magnitude should enlarge the party system. Using a series of difference-in-differences models based on data from Brazilian municipalities before and after an exogenous reform in magnitude, I study the effects of this electoral reform on both electoral coordination and legislative fragmentation. Contrary to the expectation, the number of lists did not change after a decrease in magnitude. However, parties formed more pre-electoral coalitions in municipalities that lost seats. Voters also coalesced around fewer lists. Lastly, as expected, the reform produced a decrease in legislative fragmentation. Both mechanical and psychological effects were responsible for this modification in fragmentation.


Introduction
Electoral rules influence the structure of political competition (Duverger 1954;Cox 1997;Leys 1959;Sartori 1968). Proportional systems with districts that award many seats allow many parties to flourish. Conversely, where one or a small number of seats are handed out by a majoritarian principle, political forces are encouraged to coalesce into a small number of electoral options (Duverger 1954). A fundamental challenge in evaluating this line of thinking empirically is that electoral rules are endogenous (Benoit 2007;Shugart 1992;Benoit 2002;Colomer 2005;Remmer 2008). The very politicians who will eventually compete under new rules are usually responsible for selecting them. Thus, party system size might explain the electoral rules rather than vice versa.
As a solution to this issue, political scientists have increasingly employed research designs that exploit exogenous variation in rules to produce better estimates. For example, variations in electoral calendars (e.g. Crisp and Demirkaya 2020;Lucardi 2019), changes in rules triggered by population thresholds (e.g. Fujiwara 2011;Singer 2015), and reforms promoted by external actors (e.g. Fiva and Folke 2016;Górecki and Kukołowicz 2018) have been used to evaluate hypotheses about electoral rules and political behaviour. In this paper, I join this literature and analyze an exogenous electoral reform in Brazil, a country where parties are often characterized as weak and unorganized (Mainwaring 1999;Ames 2002). The reform I analyze modified district magnitude in 2,425 municipalities (43% of the total) and stripped local legislators of the right to define the size of their assemblies. I take advantage of this exogenously imposed change to examine its consequences on electoral coordination and legislative fragmentation.
I find that the number of electoral lists entering did not change in municipalities that lost seats. Nevertheless, I find evidence that the reform changed elites' behaviour. Crucially, the number of pre-electoral coalitions increased in municipalities that lost seats, indicating that party elites opted for combining their efforts to stay in the competition after a reform that shrunk legislative chambers. I also find that voters coalesced around fewer lists, suggesting that even in a very personalitic system such as the Brazilian, voters are able to coordinate around fewer entrants after a restriction in electoral rules. The reform also produced the expected results on legislative fragmentation. Fewer lists gained representation in municipalities that lost seats. Using a counterfactual analysis, I demonstrate that both the mechanical and psychological effects collaborated to the changes in the distribution of seats.
This paper contributes to the literature in four distinct manners. First, although scholars have analyzed exogenous variations of electoral rules, just a few of these focus on developing countries (e.g. Crisp and Demirkaya 2020;Lucardi 2019;Fujiwara 2011). This lack of studies on developing countries is problematic due to differences in behaviour between political actors in developed and developing democracies. As shown by Moser and Scheiner (2012), while voters and elites in established democracies behave strategically when facing constraining rules, their counterparts in new democracies are less likely to do it, leading to a larger party system in these countries. To evaluate the effect of reforms on electoral outcomes in developing democracies, therefore, is key to extending our knowledge of the consequences of electoral rules. Second, the 2004 reform produced different changes in magnitude, which makes this an opportunity to evaluate heterogenous effects of a modification in magnitude. Third, while this paper draws from data based on Brazil, the characteristics of the Brazilian electoral system may inform the literature on electoral reforms more broadly. As detailed below, Brazil employs a proportional representation systemthe most common electoral system used to select members of national legislatures (IDEA 2018). Further, voters can vote either for a party label or a candidate, a feature that countries are increasingly including in their systems (Renwick and Pilet 2016). As a result, the widespread use of electoral system like the one employed in Brazil may help the findings of this paper to travel to other political systems. Lastly, this paper also contributes to the literature on Brazilian politics. Brazil's party system is characterized by the presence of weak parties, high volatility, and extreme fragmentation (Mainwaring 1999). These attributes can harm the country's governability and weaken the linkage between parties and voters (Ames 2002). By evaluating whether elites and voters react to changes in electoral rules as predicted by Duverger (1954), I shed some light on the consequences of changes in rules for the Brazilian party system.

Electoral rules and party system size
Electoral systems produce both mechanical and psychological effects that shape the party system and electoral competition (Duverger 1954). The mechanical effect consists of how electoral rules convert votes into seats. Because no electoral system perfectly translates the votes received by entrants into seats, parties end up with a proportion of seats that is nonequivalent to their percentage of votes. The mechanical effect is responsible for this difference between the proportion of votes received and of seats won.
Conscious of this mechanical effect, elites and citizens should modify their behaviour to avoid electoral punishment. This acting in anticipation is Duverger's psychological effect. Specifically, entering elites and voting citizens should coordinate to prevent wasting resources on non-viable candidacies. Party should decide not to field candidates when the likelihood of winning is low. Consequently, the number of political alternatives available on the ballot should decline. Voters should cast ballots only for the set of viable candidates, thereby diminishing the dispersion of votes. It is worth noting that this logic assumes that parties and voters are only interested in the current election (Cox 1997:158) and not playing a long-term or a multi-level game, such as entering an election to increase support in future contests or to make a statement about their platforms. If this assumption holds, then electoral rules should shape the party system. Permissive rules should create a large number of parties and a substantial dispersion of votes, whereas constraining rules should lead to fewer parties (Cox 1997). 1 Scholars have found mixed results for Duverger's (1954) hypotheses. For example, whereas the effective number of parties monotonically rises with the increase in district magnitude in Switzerland, such an association does not seem to occur in Spain (Grofman and Selb 2011); both countries use proportional representation systems. Studies on plurality systems have reached conflicting findings, as well. Using district-level data from democracies that employ first-past-the-post voting, Singer (2013) shows that third-place parties do not disappear.
One challenge in this area is that confounding variables may drive the relationship between the electoral rules and party system. Social cleavages are commonly cited as one of these possible confounding variables (Ordeshook and Shvetsova 1994;Lipset and Rokkan 1967;Clark and Golder 2006;Neto and Cox 1997). The party system and electoral system might reflect underlying characteristics of the society, such as racial, ethnic, or other social cleavages. More precisely, political elites would choose electoral rules that allow for a greater number of parties to flourish when there is a greater number of social cleavages. Social cleavages, electoral system, and political parties would form a "triangle of interdependence [that] overshadows purely unilateral cause-effect linkages" (Taagepera 1999, 544).
Recognizing such challenges to their inferences, scholars have examined the effects of electoral systems using quasi-experimental research designs (Crisp, Potter, and Lee 2012;Lago 2012;Górecki and Kukołowicz 2018;Pellicer and Wegner 2014;Singer 2015;Blais et al. 2012). With noticeable exceptions (e.g. Fujiwara 2011; Lucardi 2019; Crisp and Demirkaya 2020), scholars have mostly used such designs to study developed countries, where parties are strong, and voters are identified with parties. Given that elites and voters in developing countries are less likely to behave strategically, compared to their counterparts in developed democracies (Moser and Scheiner 2012), it is important to examine whether elites and voters in developing countries respond to changes in electoral rules as predicted by theoretical reasoning (Duverger 1954;Cox 1997).

Brazilian local elections
Brazil holds municipal elections every four years. Voters choose mayors and local assembly members in these elections. All municipalities use an openlist proportional representation system to select local legislators in which the municipality is the single at-large district. The only difference from one municipality to another is the number of seats to be filled, which varies from 9 to 55.
Voters have the choice of casting a vote for a party label or for a specific candidate. The allocation of seats is made proportionally to the number of votes that each list receives (the sum of the party label votes and personal votes). Parties have the option of running solo or in a pre-electoral coalition (PEC). Seats are allotted using an electoral quotient combined with the D'Hondt method to assign the remaining seats. If parties run as part of a PEC, their votes are pooled together to determine how many seats the coalition wins.
In its original version, Brazil's Constitution provided ranges for the size of local assemblies based loosely on population. Municipalities were free to choose an assembly size (district magnitude) anywhere between a minimum and a maximum number of seats. For example, municipalities with population smaller than 1 million could opt to have between 9 and 21 local legislators. 2 This rule was followed until 2004, when, six months before election day, the Tribunal Superior Eleitoral (TSE), the Brazilian Electoral Authority, published a resolution (RES 21.702/2004) that modified the criteria to set the district magnitude in municipal elections.
The TSE's decision came after the Supremo Tribunal Federal (STF), the Brazilian Supreme Court, judged that a municipality in the State of São Paulo, Mira Estrela, should have an assembly of nine seats. 3 According to the STF's decision, the principle of proportionality to population size should be strictly used to define the number of seats in a municipal council. After the STF's decision, the TSE determined that the principle of proportionality should be respected in all municipalities. 4 To ensure it, the TSE defined that one seat would be added to each assembly for every 47,619 inhabitants, with a minimum assembly size of 9 and a maximum of 55 seats. This decision resulted in an abrupt change in the number of seats in 2,425 out of 5,558 local assemblies. Of those municipalities that had their magnitude changed, 2,406 had their assembly size decreased and only 19 increased. The decision led to alterations that vary from a loss of 12 to a gain of 12 seats, with the number of municipalities in each of these categories ranging from 1 to 1,122 observations. The most common changes were decreases of 2 (1,122 municipalities), 4 (510 municipalities), and 6 seats (234). The entire distribution of changes in magnitude is available in Appendix B.
One concern when analyzing the 2004 electoral reform to evaluate the effects of electoral systems is that the TSE resolution could be a response to the political elite's desires to decrease magnitude in municipal elections, thereby reducing the political competition. There are a few reasons to believe that this was not the case. First, in an attempt to revert the TSE's decision, parties and associations formed by local politicians contested the constitutionality of the new rule before the Supreme Court (da Rosa 2006; Guerzoni Filho 2010). Additionally, in an attempt to revert the court's decision before the 2004 election, Congress tried, without success, to approve a constitutional amendment (Jornal da Globo 2004). Further, in 2009, Congress approved Constitutional Amendment n.58 that again modified the rules on the definition of the number of seats in local assemblies. The amendment restored to the local assemblies the right to define district magnitude. Specifically, Constitutional Amendment n.58 defined 24 upper limits to district magnitude based on municipal population size. When given the opportunity, local legislators made widespread use of the possibility of changing the number of seats in their assemblies: 1,696 legislatures changed their size between the 2008 and 2012 electionsonly two municipalities decreased their magnitude during this period. Concerning the municipalities that lost seats in 2004, 1,428 of them increased their magnitude after the constitutional amendment. Further, 46% of these municipalities increased magnitude by the same number of seats that they had lost in 2004. Clearly, local legislators were not in favour of changing district magnitude and the 2004 reform was exogenously imposed in opposition to elites' preferences.
Electoral coordination and the distribution of seats I use data from the 2000, 2004, and 2008 municipal elections to evaluate whether a change in district magnitude affected both parties and voters' behaviour. From the raw data, I calculated four dependent variables to evaluate electoral coordination and another two to analyze the distribution of seats. 5 Regarding the former, I compute the number of lists running for office, the number of pre-electoral coalitions (PECs), effective number of lists running for office (ENLv), and the vote share received by the top two lists (Vote Share). For the latter, I calculate the number of legislative lists and effective number of legislative lists (ENLs). 6 I examine the effect of a change in district magnitude using a series of difference-in-differences models, in which the treated group is composed of municipalities that lost at least one seat after the 2004 reform and the control group is composed of municipalities that were not affected by the 2004 reform (i.e. where the difference in magnitude between elections was equal to zero). These two groups represent 86% of all municipalities in the country. 7,8 The models are written as: where y mt is the dependent variable in municipality m and time t; M mt is the district magnitude in municipality m and time t. The parameters μ and δ are municipality and election fixed effects. The quantity of interest is captured by g 1 which represents the average effect of magnitude in the outcome y mt considering all elections and treatment sizes (changes in M). 9 The key assumption of the difference-in-differences design is the parallel trends assumption. In the absence of the treatment, the difference between groups needs to be constant over time after the intervention (Angrist and Pischke 2009). Two ways to violate this assumption are self-selection and targeting. The first issue occurs when units are able to select themselves into the treatment or control group, whereas the second problem happens when the decision behind the intervention targets units that were performing in a specific manner. It is unlikely that these issues affect the analysis in this paper. First, though it is unlikely that politicians would be able to change the population enough to gain or lose an additional seat, the fact that the TSE used the population in 2003 to define the number of seats in each assembly guarantees that politicians could not manipulate the assignment of the treatment. Second, the STF's decision aimed to apply the principle of proportionality described in the Constitution. This decision provoked both increases and decreases in district magnitude, suggesting that the court was not concerned about the party system size when it made its decision.
Nevertheless, the parallel trends assumption may not hold because of omitted time-variant variables. Features of municipalities may be correlated with the outcomes and the probability of belonging to either treatment or control groups. That is, some of these features may explain why some politicians decided to increase magnitude in the periods before the 2004 reform. Unfortunately, data on PECs are not available for the 1996 election, which precludes me to evaluate the plausibility of the parallel trends assumption using pre-intervention periods in the models (Autor 2003). 10 To mitigate concerns related to the parallel trends assumption, I employ two different strategies. First, I only analyze municipalities where there was no change in the magnitude since the 1996 election. 11 Second, I use Covariate Balancing Generalized Propensity Score (CBGPS) to match treatment and control groups (Fong, Hazlett, and Imai 2018), a matching algorithm that maximizes the prediction of treatment assignment and covariate balance. The use of this matching algorithm improves the comparison between groups (1) by guaranteeing that comparable observations have more weight in the estimation and (2) by reducing the influence of observations located out of the region of common support.
I match observations using six variables: GDP per capita (Log), Average Schooling Years, % of Urban Population, Family Income, HDI, and state. These variables aim to account for the fact that politicians from more urban and wealthier municipalities may be more interested in increasing their magnitude because of political competition. Figure E.1 shows that CBGPS largely reduced the imbalance between treated and control groups. Figure 1 presents the results from the models for the effect of the 2004 reform on electoral coordination and seat distribution. Tables are available in Appendix E.1. Note that municipalities lost between −12 to −1 seats, as a result, the coefficients need to be multiplied by minus one to indicate the average effect of losing one seat after the reform.

Results
The findings in Figure 1 reveal similar results for the models using the unmatched and matched datasets. In both panels, the coefficient for the number of lists entering is small and statistically insignificant, indicating that the number of entrants was not affected by the change in magnitude.
Nevertheless, electoral coordination was affected by the reform. Specifically, more pre-electoral coalitions entered elections after the decrease in magnitude. Substantively, the number of PECs increased, on average, between 0.155 (panel a) and 0.148 (panel b). Given that the average change in seats was equal to −3.53, this result suggests that the number of PECs in legislative elections increased by 0.54 in the average treated municipality (unmatched results).
In terms of voters' behaviour, Figure 1 shows that voters started to concentrate votes in fewer lists after the reform. More precisely, the loss of one seat led to an average decrease in ENLv between 0.029 (unmatched data) and 0.075 (matched data). Even though these effect sizes are tiny (they represent between 1% and 3% of the average pre-reform ENLv), an improvement in coordination is somewhat surprisingly given that most of Brazilian voters cast ballots at the candidate-level (Zucco and Nicolau 2016). Regarding the vote concentration in the top-two lists, the findings are not robust when using the matched dataset. More precisely, whereas there is evidence in panel a that a decrease in magnitude led to an average decrease in Vote Share equal to 0.234, the coefficient when using the matched dataset suggest an increase in Vote Share, though the coefficient does not reach the conventional levels of statistical significance. Consequently, there is no robust evidence that a decrease in magnitude produced a change in the vote share received by the top-two lists in treated municipality.
Turning to the effect of a change in district magnitude on the distribution of seats, the estimates in Figure 1 indicate that the number of legislative lists and effective number of legislative lists (ENLs) decreased in municipalities that lost seats. A loss of one seat led to a decrease between 0.184 and 0.150 (unmatched and matched datasets, respectively) in the number of legislative lists. Again, considering that the average change in seats was equal to −3.53, the decrease in the number of legislative lists was between 0.65 and 0.53 in the average treated municipality. The results for the ENPs follow similar lines. A loss of two seats led to a decrease in ENLs between 0.122 and 0.130. These decreases represent roughly 4% of the average value of ENLs in the pre-treatment period ( m = 3.11). Note that the small size of these effects is likely related to the fact even after the reform, these electoral systems were still very permissive (the lowest magnitude defined by the TSE was 9 seats). As a result, a change in magnitude may have had a limited effect on fragmentation.
As detailed above, the reform did not produce the same change in magnitude across all treated municipalities. One may wonder whether the estimates in Figure 1 are driven by specific groups of municipalities. I address this concern in Appendix E.2, where I report models for each treated group defined by the sized of the change in seats produced by the 2004 reform. I examine groups of municipalities with at least 10 unitseleven groups in total with treatment varying from −1 to −11 seats. The results from these models show that the number of lists entering decreased only in municipalities that lost five, seven, and nine seats. The number of PECs increased in all treated groups, except for those municipalities that lost one seat. Regarding voters' behaviour, I find that ENLv decreased in municipalities that lost five, seven, nine, and ten seats. Moreover, the top-two lists' vote share only increased in municipalities that lost seven seats. Lastly, in terms of distribution of seats, the estimates indicate that in all treated groups of municipalities, except for those that lost one seat, the number of legislative lists decreased. Similarly, the ENLs also decreased in all treated groups, except for those municipalities that lost one or two seats.

Disentangling the mechanical and psychological effects on the distribution of seats
To disentangle the mechanical and psychological effects of the reform on the number of legislative lists and the effective number of legislative lists (ENLs), I follow Fiva and Folke's (2016) approach. Specifically, I created counterfactual scenarios for the elections using the distribution of votes and magnitude before and after the electoral reform. To illustrate this approach, I use the municipality of Mira Estrela as an example. Cells A and D in Figure 2  To calculate the psychological effect, I utilize B→D. I keep magnitude fixed to its value in 2004 and compute the statistics using the vote distribution in both 2000 and 2004. Therefore, the estimated differences between pre-and post-reform periods are solely produced by a change in the distribution of electoral support. Again, using the example of Mira Estrela, the psychological effect on the number of legislative lists and ENLs was equal to −1 and −1.02, respectively. The psychological effect can be split into two components (Fiva and Folke 2016). The first one is the effect of a change in the support for specific lists. For example, after a decrease in magnitude, voters' support may have largely favored large lists. If this change is large enough; the party system would shrink even if the district magnitude remained constant. To calculate this quantity, I estimate the difference between cells A, the actual electoral returns in 2000, and C, the counterfactual electoral returns for the 2004 electionvote distribution from 2004 and magnitude from 2000. The comparisons between cells A and C in Figure 3 indicate that the first component had an effect equal to −1 and −1.07 on the number of legislative lists and ENLs, respectively.
The second component measures how the strategic behaviour of voters and elites may indirectly affect the party system through the mechanical effect. For instance, it is possible that, before the reform, there were fewer lists in municipalities with low magnitude than those with high magnitude. In such a scenario, an equal size decrease in magnitude should lead to a larger negative effect in municipalities that had a larger number of lists (and magnitude) before the reform. That is, the strategic behaviour of voters and elites before the reform limits the effect of a change in magnitude. Taagepera and Shugart (1991)  Then, I calculate the difference between these quantities. Note that vote distribution may have changed between elections for two reasons: (1) the set of lists may have changed; or (2) the set of lists remained stable, but voters changed their preferences. Therefore, both components of the psychological effect are a function of both entry and voters' behaviour. In the example in Figure 2, the calculations reveal that the second component was equal to zero for the number of legislative lists and to 0.05 for the ENLs.

Results: mechanical and psychological effects
The coefficients in Table 1 indicate the mechanical and psychological effects of one-unit change in magnitude on the number of legislative lists and effective number of legislative lists (ENLs). Again, because the municipalities lost between −12 to −1, the coefficients need to be multiplied by minus one in order to provide the direction of effect. Tables with complete results are in Appendix E.1.2. The estimates in panels A and B are similar to each other. For both outcomes, I find that the mechanical and psychological effects were responsible for the changes in legislative fragmentation detected in the previous section. Further, the psychological effect was the driving force behind these decreases in fragmentation. 13 Regarding the mechanical effect, the estimates in panel (a) suggest that each seat lost in the 2004 reform led to a mechanical decrease between 0.072 and 0.067 in the number of legislative lists. These values represent 38.5% and 47.3% of the total effect of the reform. The results in panel (b) display that the contribution of the mechanical effect to the decrease in ENLs after the reform. Crucially, a loss of one seat led to a mechanical decrease in ENLs between 0.039 and 0.038, meaning that the mechanical effect was responsible for roughly 30% of the total effect.
Turning to the psychological effect, Table 1 indicates a psychological effect of a decrease in magnitude on legislative fragmentation. One seat decreased in magnitude produced a psychological effect between −0.113 and −0.083 (unmatched and matched datasets) the number of legislative lists. These estimates suggest that the psychological effect accounted for more than half of the total effect produced by the 2004 reform on the number of legislative lists. It is worth noting that the psychological effect was roughly twice the size of the mechanical effect on the ENLs. Using the unmatched sample, the psychological effect on the ENLs was equal to 0.084, while using the matched sample, it was equal to 0.093. This finding shows that actors' anticipation of the mechanical effect was the driving force behind the reduction in fragmentation found in the previous section. In terms of the components of the psychological effect, the first componentthe direct effect of a change in voters' support for specific listsis statistically significant in three of the four models. That is, the support for small lists seems to have decreased after the reform. Note that this result is consistent with fact that the ENLv decreased in these municipalities (Figure 1). 14 The estimates for the second componentthe indirect effect of the strategic behaviour of elites and votesare statistically significant only when using the unmatched dataset. Due to the differences in the estimated effect size of each component between the unmatched and matched models, it is not possible to point out the driving force behind the psychological effect.

Conclusion
Questions regarding the relationship between the effects of electoral rules on party system size are among the most common but also the most challenging in political science. The nearly impossible task of using experiments to disentangle the effect of institutions from other covariates led researchers to search for opportunities for quasi-experimental research designs. In this paper, I analyzed an exogenously imposed change in district magnitude in Brazil to evaluate whether changes in magnitude affected the electoral coordination and distribution of seats. The results indicate that a decrease in magnitude led to a subsequent decrease in legislative fragmentation. Both mechanical and psychological effects were responsible for the changes. I found that the reform did also affect coordination. Even though the number of lists entering the election did not change, my analysis suggests that more pre-electoral coalitions were created in municipalities that lost seats. Moreover, I also voters coalesced around fewer lists in municipalities that lost seats. This finding indicates that even in country with weak party system as the Brazilian one, voters are able to coordinate after a restriction in electoral rules.
Despite the advantage of analyzing an exogenous reform, there are limitations to this research. First, there are reasons to believe that characteristics of the municipalities affected their probability of having an excess (or deficit) of seats before the reform. Although I only utilized data from municipalities that did not change their magnitude since 1996 and employed a matching algorithm, as in all observational studies, I cannot completely exclude the chance of unmeasured confounding factors. Second, the low number of municipalities that gained seats prevented an analysis of the effect of an increase in magnitude. The inferences drawn from this reform, therefore, are limited to how decreases in magnitude impact the party system size. Third, the fact that the local legislators regained the right to define magnitude in 2009 precludes the possibility of evaluating the long-term effect of the 2004 reform.
The findings in this paper contribute not only to the literature on electoral reforms but also to the literature on Brazilian politics and its discussions about electoral rules and party system fragmentation. Considering that parties formed more pre-electoral coalitions to stay in the competition after the decrease in magnitude, it will be interesting to observe the long-term effect of the recent prohibition of PECs in legislative elections. One may wonder whether local legislators will increase magnitude to compensate for this limitation. . The effects of the STF and TSE's decisions and attempts to revert them were publicized in newspapers and television news shows. See, for example, the coverage of these events on Rede Globo, the major media group in Brazil, and Folha de São Paulo, one of the major newspapers in the country (Folha de São Paulo 2004; Jornal da Globo 2004) 5. As noted in the previous section, there are two types of entrants in these elections: parties and pre-electoral coalitions. All dependent variables, in the main analysis, are calculated at the level of the entrant (parties or PECs). I present an analysis at the level of the party in Appendix F. The results in this appendix indicate that legislative fragmentation decreased, but more parties entered the elections. As shown below, this result is explained by the fact that parties formed more pre-electoral coalitions in the post-reform period. 6. Effective number of (legislative) parties is given by 1\ n i=1 p 2 i , where p is the proportion of votes (seats) received by the party i in the municipality (Laakso and Taagepera 1979). 7. Because the difference-in-differences design assumes that the composition of treated and control groups is stable over time, I restrict the analysis to municipalities that were created before the 1996 election and existed at least up to 2008. Moreover, I also remove municipalities that had their magnitude changed between the 2004 and 2008 elections because of population changes. The analyzed sample include 1,992 out of the 2,425 municipalities affected and 2,833 out of the 3,133 municipalities not affected by the 2004 reform. See Appendix B for the distribution of changes in magnitude in the final sample. 8. Although the creation of population thresholds by the 2004 reform makes the regression discontinuity design (RD) a possible choice, the sudden change in magnitude provoked by the TSE's decision precludes the use of such a design. The RD assumes that observations close to the cutoff point should be similar in terms of observed and unobserved characteristics (Angrist and Pischke 2009). The abrupt change in magnitude is an obvious confounder. In Appendix C, I show that municipalities above the threshold lost, on average, 1.1 fewer seats than those municipalities below the threshold. This indicates that the increase in magnitude at the cut-point is confounded by the change in magnitude produced by the 2004 reform, making the RD unsuitable for the analysis. One alternative would be to use the population required to lose a given number of seats as the running variable. However, there are few observations on the right-hand side of the cutoff (municipalities that could have lost seats). It occurs because most of the municipalities that did not lose seats already had the minimum number of seats allowed (see Table B.1). As a result, these municipalities could not lose any additional seats. Excluding these observations, there are only 23 municipalities that were not affected by the 2004 reform. Moreover, most of the municipalities (1,873 out of 2,415) lost the number of seats necessary to make their magnitude equal to nine seats, the minimum number of seats allowed by the constitution. Therefore, the majority of the municipalities could not lose more seats. 9. I present analysis for different treatment groups and elections in Appendix E.2. 10. Remember that the dependent variables are measured at the list level (PECs or parties running solo) 11. Ideally, I would consider only municipalities that did not change their magnitude since the first election after the promulgation of the 1988 Constitution. Unfortunately, data for elections before 1996 are not available. 12. Appendix D contains an example for the construction of the counterfactual scenarios. 13. I report estimates for subgroups in Appendix E.2.7. These reveal that the psychological effect was the main force behind the changes in most of the treated municipalities. 14. Note that the lack of statistical significance for some of components of the psychological effect does not mean that voters and elites behave in exactly the same manner before and after the reform. However, it means that changes in behavior were not enough to produce meaningful impacts on legislative fragmentation.