Justifying the assumptions

Before proceeding to the theoretical analysis, we justify and elaborate on the two key theoretical assumptions that drive the results. Our first assumption is that incumbents exhibit some degree of ideological policy persistence. Relative to challengers, it is more difficult for incumbents to change the ideological positions that they have adopted. There are a number of sources of ideological persistence among incumbents. First, legislators may have personal ideological convictions that they are not willing to change under normal circumstances. Second, even if there are changes in the electoral environment, voters may penalize politicians that change ideological positions. A central reason that incumbents may choose to persist in their ideological positions despite changing electoral circumstances is that voters negatively evaluate candidates who adjust their policy positions. In the context of US politics, Tomz and Van Houweling (Reference Tomz and Van Houweling2012) use experimental vignettes and DeBacker (Reference DeBacker2015) employs data from US Senate elections to show that voters penalize candidates who engage in ideological adjustment. As a result, it can be electorally costly for an incumbent to locate at a new ideological position. An incumbent will not change positions if the costs of adapting are larger than the benefits of adopting a new policy stance.Footnote ⁵

The ideological position of general election challengers, on the other hand, is more flexible than that of incumbents for at least two reasons. First, the average challenger has a less well-defined public record of policy positions than the average incumbent (Peskowitz Reference Peskowitz2019). Therefore, they can more easily change ideological positions to adapt to a new electoral environment after a reform is implemented or removed. Second, general election challenger candidates are typically selected among a large pool of potential challengers through a primary election process. An electoral reform may impact which challenger from the pool of potential challengers is ultimately successful in the primary, which will affect the ideological position of a general election challenger.

The empirical evidence on whether incumbent legislators in the United States change their ideology as measured by their roll-call voting behavior is ambiguous. While some scholars argue that members of Congress do not change their roll-call voting over time (Poole Reference Poole2007), others have found evidence of ideological adaptation to changing electoral circumstances (Stratmann Reference Stratmann2000; Kousser et al. Reference Kousser, Lewis and Masket2007; Kogan Reference Kogan2016). While we assume that incumbent policy positions are completely persistent in our discussion of the results below, the extreme version of this assumption is not necessary to derive our key results. As we show in the Online Appendix, in the more realistic case that incumbents have some ability to change their ideological positions but cannot perfectly adjust to the location of their party's challengers, then our main results still hold.

The other essential assumption is that increasing the ideological distance between a candidate and the pivotal voter decreases the probability that the candidate wins the election. We do not assume that ideological proximity is the only factor that affects voter utility. Our results still hold if citizens have preferences over candidates’ non-ideological valence characteristics or if voters choose between candidates probabilistically. There is compelling evidence to support this assumption.Footnote ⁶ Burden (Reference Burden2004) finds a positive association between the spatial proximity of House candidate to the voters and the candidate's electoral performance. At the state level where we perform our empirical application, Birkhead (Reference Birkhead2015) finds that in the average state legislature moderates electorally outperform extremist candidates. Hall (Reference Hall2015) uses a regression discontinuity design to show that when ideologically extreme US House candidates narrowly win a partisan primary over a moderate candidate, their party performs dramatically worse at the ballot box in the subsequent general election.

Electoral reforms, incumbent reelection rates, and polarization dynamics

We develop a simple decision theoretic framework to show how electoral reforms affect the dynamics of partisan polarization. The only optimizing agent in our framework is the voter who chooses between candidates on the basis of ideological proximity and a stochastic error term. We do not microfound the spatial locations of the incumbents and challengers and instead take these locations as exogenous parameters. Our framework analyzes electoral reforms in the context of a general election with a left party candidate facing off against a right party candidate.Footnote ⁷

We present the intuition behind our main theoretical results in a series of figures for a simple parametric example.Footnote ⁸ In the Online Appendix, we generalize these results to the setting where incumbents have asymmetric ideological positions.Footnote ⁹ We compare two states that both have the same level of initial partisan polarization, but differ in their electoral institutions. In one state, the moderation-inducing reform will be implemented, and in the other state, the reform will be removed. We assume that electoral reforms operate by shifting the location of the challenger. When a moderation-inducing reform is implemented, the challengers shift toward the pivotal voter by a factor of τ/2, while when a moderation-inducing reform is removed, the challengers shift away from the voter by the same τ/2 distance.

The assumption that challengers shift toward the pivotal voter in response to the implementation of a moderation-inducing reform is more straightforward to justify on the basis of candidate electoral considerations than the assumption that challengers shift away from the pivotal voter when a moderation-inducing reform is removed. However, there are settings where this assumption is more plausible. For example, if implementing a limit on PAC contributions increases the probability that extreme candidates will win primaries then we will observe more extreme general election candidates on average when the reform is removed. More importantly, our main results on the relative magnitudes of the effects of electoral reforms on aggregate polarization still hold if the challengers do not change their location when the moderation-inducing reform is removed. If we were to assume that challengers are located at the incumbents’ locations when the moderation-inducing reform is removed, there would be no effect of removing the moderation-inducing reform on incumbent reelection rates or aggregate polarization.

In Figure 1, we plot the deterministic spatial component of the voter's utility function for incumbents and challengers. Based on our assumption of incumbent policy persistence, the incumbent's ideological location remains the same regardless of the reform and is indicated by I _L and I _R for the left and right party incumbents, respectively. However, the challenger's ideological location is shifted by the institutional reform. When the reform is removed, the challengers are more ideologically extreme than the incumbents and the ideological position is denoted by $\widetilde {C}_L$ and $\widetilde {C}_R$. In contrast when the reform is implemented, the challengers are more moderate with positions $\hat {C}_L$ and $\hat {C}_R$.

Figure 1.

Voter's spatial utility.

As an immediate consequence of our assumptions, the probability that the incumbent is reelected differs across the two states. In the state where the moderation-inducing reform is implemented, the reelection probability of the incumbent is reduced relative to the baseline while this probability increases when the reform is removed. In Figure 2, we plot the probability that an incumbent wins reelection as a function of the treatment effect magnitude under the alternative electoral reform scenarios. Figure 2 also reveals that the magnitude of the change in the reelection rates depends upon the strength of the ideological location treatment effect. When the electoral reform leads to a dramatic change in challenger ideological positions, incumbents will enjoy higher reelection rates when the reform is removed and suffer lower reelection rates when it is implemented relative to a reform that has a smaller effect on challenger ideology.

Figure 2.

Incumbent reelection rates.

This example motivates our first result.

While this result is substantively important and we empirically examine whether it holds in state legislative elections, our primary interest in this result is its implications for how electoral reforms affect aggregate partisan polarization in the legislature. Each party's mean ideology in the legislature is a weighted average of the ideology of the party's incumbents who successfully secure reelection and the challengers who secure a seat. The weights are determined by the incumbent and challenger election rates.Footnote ¹⁰ In Figure 3, we plot the locations of each party's mean legislative ideology as a function of the treatment effect magnitude for the different reform scenarios. The figure shows that effects of the reform on each party's ideology are dramatically different depending upon whether the reform is implemented or removed. The magnitude of the change is relatively small when the reform is removed. Additionally, when the reform is removed, there is a non-monotonic relationship between the magnitude of the challenger ideology treatment effect and the party's mean ideology. For sufficiently high treatment effects, the reduction in challenger win probabilities is so dramatic that the party's legislators are almost entirely incumbents.

Figure 3.

Mean party ideological positions.

We can now characterize the aggregate level of partisan polarization in the legislature under the different scenarios. Partisan polarization is simply the difference in the mean ideology of the right party's legislators and the left party's legislators. Figure 4 displays these quantities for our two scenarios. While polarization increases when the reform is removed, the effects are quite small in magnitude. To see this directly, we next plot the magnitude of the polarization changes relative to the baseline period.

Figure 4.

Partisan polarization.

As seen in Figure 5, the magnitudes of the change in polarization are not equivalent when the reform is implemented and removed. The magnitude of the change is always larger when the reform is implemented than removed. We can now express our second result.

Result 2: Expected partisan polarization increases in reform-removing states and decreases in reform-implementing states relative to the baseline period. For a fixed treatment effect size, the magnitude of the change in expected partisan polarization is greater in reform-implementing states than in reform-removing states.

Figure 5.

Change in partisan polarization.

In the Online Appendix, we generalize the results from this parametric example. We allow for any spatial component of the utility function that is single-peaked, symmetric, and strictly decreasing in the distance between the voter's ideal point and the candidate's ideology and any preference shocks that are independent and identically distributed with support over the entire real line. We also allow the incumbent to partially adjust her ideological location. Complete incumbent policy persistence is often an unrealistically strong assumption and we show that our results hold whenever there is a friction that prevents incumbents from adjusting completely to their party's challenger position.

Our framework can be extended to allow for subsequent elections. In each period, the expected fraction of incumbents who served in the initial period will decrease as they are defeated by challengers. After many elections, the expected effect of aggregate polarization will converge to the challenger-level treatment effect τ because the fraction of the legislature consisting of candidates who did not serve in the initial period will converge toward 1. This dynamic analysis also makes clear that the replacement rate of initial period incumbents by challengers is an essential variable for understanding how electoral reforms affect polarization.

When estimated on the sample of new entrants to the legislature, the expectation of the estimator of the effect of the electoral reform on polarization is equal to τ so this procedure generates an unbiased estimator for the long-run change in polarization. Of course, this quantity does not allow us to understand the dynamic path of legislative polarization because it ignores both how incumbents adjust their behavior to changing electoral circumstances and the probability that new entrants will replace the baseline period incumbents in the legislature. Below, we discuss how to combine estimates of τ with information on incumbent policy persistence and the replacement rate of incumbents to estimate the dynamic path of aggregate polarization.

Our theoretical framework makes clear that the reelection rate of incumbent legislators is an important factor driving the aggregate polarization consequences of an electoral reform. This insight raises the important issue of how other institutional reforms, such as term limits, that affect the probability that incumbents continue in office, but may not have a direct effect on the policy positions of incumbents and challengers, interact with reforms that change the ideological location of challengers to jointly determine the aggregate level of polarization.

Data description

We combine data from several sources to examine the implications of the theory in the context of campaign contribution limits and polarization in state legislatures. Following Barber (Reference Barber2016), we examine the relationship between changes in PAC contribution limits in state legislative elections and incumbent electoral performance and polarization. Barber (Reference Barber2016) argues that individual donors are ideologically motivated and more likely to support ideologically extreme candidates while PACs are more interested in obtaining access to legislators. As a result, he expects that when candidates are able to receive an unlimited amount of money from PACs, legislators will be less polarized. Using data on campaign finance laws in the US states and measures of state legislator ideology, Barber (Reference Barber2016) finds that in states that allow unlimited PAC contributions, the ideological extremity of legislators in their first term decreases. Meanwhile, in states that allow unlimited contributions from individual donors, first-term legislators are more polarized.

Based on his analysis, we identify the implementation of unlimited PAC contribution limits as a moderation-inducing reform and use his data to identify the presence of PAC campaign contribution limits in each state from 1990 to 2012. We use this information to create the three main independent variables used in our analyses. We first generate an indicator that is equal to one in each year that a state allows unlimited PAC contributions. Then, we create two variables indicating periods in which a state implements unlimited PAC contributions or removes unlimited PAC contributions in order to assess how the effects of electoral reforms vary across these scenarios. The first of these two variables is coded such that it is equal to one for each year after a state switches from having a PAC contribution limit to allowing unlimited PAC contributions.Footnote ¹¹ The second variable is equal to one in each year after a state switches from allowing unlimited PAC contributions to a PAC contribution limit being in place.Footnote ¹² Table A.1 in the Online Appendix indicates the states and years that are classified as removing and implementing unlimited PAC contributions.

Our first set of dependent variables are two measures of incumbents’ electoral success. In particular, we generate indicators of whether or not an incumbent runs for reelection and whether or not an incumbent wins the election using the data from Klarner et al. (Reference Klarner, Berry, Carsey, Jewell, Niemi, Powell and Snyder2015). Next, we measure the ideological position of incumbent legislators and challengers using two sources of data. First, we utilize Shor–McCarty scores, which are ideal points that are scaled using legislators’ roll-call votes and bridged across states using candidate responses to the National Political Awareness Test (Shor and McCarty Reference Shor and McCarty2011). Because Shor–McCarty scores are not available for challengers who do not win their election, we also use CF-scores from Bonica (Reference Bonica2016) to measure ideal points of challengers. CF-scores are generated for both incumbents and challengers using campaign contributions (Bonica Reference Bonica2013).Footnote ¹³

Both of these ideal point measures are polarized such that lower scores indicate that a legislator is more liberal and higher scores indicate that a legislator is more conservative. Zero is an arbitrary neutral point on the ideological scale and does not have a substantive interpretation. We create a dependent variable that measures ideological extremism for our analysis by taking the absolute value of the Shor–McCarty scores and CF-scores so that smaller values of the variable indicate that legislators are more ideologically moderate and larger values indicate that legislators are more ideologically extreme. However, due to a lack of substantive meaning for zero in the Shor–McCarty scores and CF-scores, as the dependent variable decreases, it does not necessarily mean that a legislator is approaching the ideological position of a perfect moderate. As an alternative, we could generate a measure of within-party ideological extremism using a transformed version of the ideology scores. For example, Barber (Reference Barber2016) measures ideological polarization of state legislators by multiplying Shor–McCarty scores by negative one for Democrats and using the original Shor–McCarty scores for Republicans. We use the absolute value of the ideology measures in our main analysis because we are interested in the ideological extremism of legislators regardless of their party affiliation.Footnote ¹⁴ We present results in Table A.11 of the Online Appendix with the same measure of the Shor-McCarty scores as Barber (Reference Barber2016) and the results are substantively similar to those presented in the body of the paper.Footnote ¹⁵

Using these two sources of data, we create a dataset that includes all incumbent state legislators along with their Shor–McCarty score and CF-score as well as a dataset with all challengers and their CF-score. The coverage of the Bonica (Reference Bonica2016) data begins in 1990 while the Shor and McCarty (Reference Shor and McCarty2011) data starts in 1993. We include the both upper and lower chambers of state legislatures in the sample and exclude multi-member districts. Finally, because term limits affect the rate at which incumbent legislators can be replaced and, consequently, moderate the effects of electoral reforms, we also compile data on the presence of term limits in state legislatures using information from Kousser (Reference Kousser2005) and the National Council of State Legislatures. These data are used to create an indicator equal to one in years in which state legislators are subject to a term limit.

Empirical approach

We examine the central implications of the theoretical framework in the context of an important electoral reform in state legislatures: the implementation and removal of unlimited PAC contributions.Footnote ¹⁶ As discussed, Barber (Reference Barber2016) argues that implementing unlimited PAC contributions causes legislators to behave more moderately, because it results in candidates focusing less on obtaining funds from ideologically extreme individual donors. On the other hand, removing unlimited PAC contributions should result in more extreme legislators as they have to appeal to these more extreme individual donors.

Our first specification examines how the level of the dependent variable changes when states have no limits on PAC contributions. Denoting legislatorsFootnote ¹⁷ with the subscript l, states with the subscript s, and time period with the subscript t, we first estimate the standard specification with an indicator for whether PAC contributions are unlimited in state s in year t:

$$Y_{lst} = \alpha_{s} + \gamma_{t} + \beta {\rm PACUnlimited}_{st} + \epsilon_{lst}$$

The state fixed effects allow for unobserved time-invariant heterogeneity within states and we include time period fixed effects to accommodate differences in the level of the dependent variable across time periods. The above specification cannot identify how treatment effect magnitude varies with whether a PAC contribution limit was implemented or removed. To examine this possibility, we separately include indicators for each of these scenarios in our next specification.

$$Y_{lst} = \alpha_{s} + \gamma_{t} + \beta_1 {\rm ImpPACUnlimit}_{st} + \beta_2 {\rm RemPACUnlimit}_{st} + \epsilon_{lst}$$

In all of our analyses, we employ heteroskedasticity robust standard errors and cluster the standard errors at the state level because the PAC regulations vary at the state level, not the individual legislator level.

As in Barber (Reference Barber2016), our key identifying assumption is that, after conditioning on state and year fixed effects, the error term is conditionally independent of PAC contribution limit regulations. While the state fixed effects account for time-invariant unobservables, there may be time-varying unobservables that are correlated with the presence of PAC contribution limits. For example, the assumption could be violated if states that are becoming more polarized are more likely to change their PAC contribution regulations. One approach to make this conditional independence assumption more plausible is to allow for trends in the state-specific level of the dependent variable. In each table, we also include the results of a regression specification with a state-specific linear time trend. We also interact the year fixed effects with an indicator for whether the legislator is a Democrat to allow for partisan waves that influence the level of the dependent variable. While these approaches cannot rule out richer dynamics that might confound the relationship between treatment assignment and the outcome of interest, they do give us more confidence in the credibility of our estimates.

Our main analysis of polarization uses a measure of ideology that does not vary within legislators over time. As a result, we cannot directly estimate the extent of incumbent ideological policy persistence and how it responds to changes in PAC contribution regulations.Footnote ¹⁸ If legislators do change their ideological behavior in response to electoral reforms, these static measures of legislator ideology will result in underestimates of the magnitude of the treatment effects because some of the change will be averaged across all of the years that the legislator serves in office. Our estimates of polarization effects can be interpreted as lower bounds on the true magnitude.

While our analysis focuses on the influence of PACs on candidates’ ideological positions, unlimited PAC contributions may also have a direct effect on the magnitude of the incumbency advantage. Primarily, access-motivated PACs contribute more of their resources to incumbents than challengers (Fouirnaies and Hall Reference Fouirnaies and Hall2014). Implementing unlimited PAC contributions may moderate challengers, but also increases incumbents’ share of campaign resources. This offsetting factor may be sufficiently large that incumbent reelection rates actually increase after the implementation of unlimited PAC contributions. Our research design does not allow us to disentangle these two effects, but we are able to investigate which effect dominates by examining how incumbent reelection rates change in response to unlimited PAC contributions.

Polarization counterfactuals

In our final set of analyses, we use the framework to investigate the dynamic path of polarization under alternative counterfactuals. The conventional polarization estimator derived from restring the sample to first-term legislators recovers the long-run effect of polarization, but does not illuminate the dynamic path of polarization across election cycles. We consider the evolution of polarization from the baseline period before the reform through the following ten election cycles under the alternative scenarios of implementing unlimited PAC contributions and removing unlimited PAC contributions with and without term limits. In the Online Appendix, we describe the exact regressions that we estimate and the fitted probabilities and polarization values that form the basis of the counterfactual exercise. The essential idea is that we use the polarization point estimates to construct the baseline level of polarization among old incumbents and the level of polarization among new entrants and then we use the fitted incumbent reelection probabilities to estimate the share of old incumbents and new entrants in the legislature in each period. We use the estimates from Column 4 of Table 6 to measure the long-run effect on the polarization of legislators.

Table 6.

Polarization (first-term legislators only)

Heteroskedasticity robust standard errors clustered at the state level in parentheses.

⁺p < 0.10, *p < 0.05, **p < 0.01, ***p < 0.001.

While the static measure of legislator ideology that we use does not allow us to estimate the extent of incumbent policy persistence, we can make alternative assumptions about the level of incumbent policy persistence in order to examine how the evolution of partisan polarization varies with incumbent policy persistence. In our first set of results, we assume complete incumbent policy persistence. We then relax this assumption and specify the level of the incumbent policy persistence parameter at 0.5 in the second set of counterfactuals.

In Figure 6, we present the results of the counterfactual analysis.Footnote ²⁰ In the left panel, we report the evolution of polarization across the electoral cycles for the four different scenarios. The right panel presents the same counterfactuals, but now defines the y-axis as the proportion of the long-run effect after each election cycle. The right panel allows us to directly compare the magnitude of the polarization effects across the four scenarios. The graphs reveal that the dynamic path of polarization differs considerably across the scenarios. After one election, the effect of removing the reform is roughly 25 percent of the total effect while the effect is approximately 50 percent when the reform is implemented. In both term-limited scenarios, the effect magnitudes are dramatically larger relative to the no-term limits case. The effect magnitudes gradually increase in the next period. Starting after election period 3, the term-limited scenarios reach the long-run level of polarization because all of the pre-reform incumbent legislators are forced to exit the legislature and the legislature is entirely composed of new entrants. By election period 4, the reform implemented effect is more than 90 percent of the long-run effect, but the reform removed effect is still about 65 percent of the long-run effect.

Figure 6.

Polarization counterfactuals, full incumbent policy persistence: λ = 1.

We now examine counterfactuals under a lower level of incumbent policy persistence. We specify the incumbent policy persistence parameter to 0.5 so that old incumbents adjust their positions to the midpoint of the old incumbent pre-reform ideological position and the new entrant ideological position. While we cannot directly estimate the degree of ideological policy persistence, this level of 0.5 represents a fairly low level of ideological policy persistence and we would expect that in real legislative settings, legislators are unlikely to exhibit such a large adjustment in their ideological policy positions. In Figure 7, we report the polarization counterfactuals with this partial incumbent policy persistence. Now the percentage of the total effect that is reached after the first election is more than 60 percent when the reform is removed and more than 70 percent when it is implemented without term limits. The effect proportions increase steadily over the next several election cycles and the dynamic path of polarization more rapidly approaches the long-run level than in the scenarios with complete policy persistence.

Figure 7.

Polarization counterfactuals, partial incumbent policy persistence: λ = 0.5