The Swing Voter Paradox: Electoral Politics in a Nationalized Era

tury -appear to have given way to strong party discipline among candidates and nationalized partisanship among voters. Observers of modern American politics are therefore tempted to write off the importance of the swing voter, defined here as voters who are indifferent between the two parties and thus likely to split their ticket or switch their party support.

By assembling data from historical elections (1950 -2020), surveys (2008 -2018), and cast vote record data (2010 -2018), and through developing statistical methods to analyze such data, I argue that although they comprise a smaller portion of the electorate, each swing voter is disproportionately decisive in modern American politics, a phenomenon I call the swing voter paradox. Historical comparisons across Congressional, state executive, and state legislative elections confirm the decline in aggregate measures of ticket splitting suggested in past work. But the same indicator has not declined nearly as much in county legislative or county sheriff elections (Chapter 1). Ticket splitters and party switchers tend to be voters with low news interest and ideological moderate. Consistent with a spatial voting model with valence, voters also iii become ticket splitters when incumbents run (Chapter 2). I then provide one of the first direct measures of ticket splitting in state and local office using cast vote records. I find that ticket splitting is more prevalent in state and local elections (Chapter 3). This is surprising given the conventional wisdom that party labels serve as heuristics and down-ballot elections are low information environments.

A major barrier for existing studies of the swing voter lies in the measurement from incomplete electoral data. Traditional methods struggle to extract information about subgroups from large surveys or cast vote records, because of small subgroup samples, multi-dimensional data, and systematic missingness. I therefore develop a procedure for reweighting surveys to small areas through expanding poststratification targets (Chapter 4), and a clustering algorithm for survey or ballot data with multiple offices to extract interpretable voting blocs (Chapter 5). I provide open-source software to implement both methods. These findings challenge a common characterization of modern American politics as one dominated by rigidly polarized parties and partisans. The picture that emerges

| Acknowledgments

Ten years ago in the Spring of 2011, I was taking my first statistics class, my first lecture for Introduction to American Politics, and my first political science seminar. I owe all that I have learned since that semester, including this dissertation, to the people who have helped me and enriched my life along the way.

My main dissertation advisors were formidable. From Steve Ansolabehere, with whom I took my first American Politics class at Harvard in my first semester, I learned how to formulate long term research while being excited to come to work every day.

From Kosuke Imai, I learned to push myself just the right amount to meet higher standards and the importance of helping fellow students while learning immensely from them in turn. From Jim Snyder, I learned how to spot an interesting research question and go about turning that into a paper, and an appreciation for hours of data collection to find a single statistic. From all three of them, I learned how to think clearly and write elegantly.

Over the course of my time at Harvard, I also benefited from the advice from many other faculty and teachers, including Matt Blackwell, Amy Catalinac, Ryan Enos, Andrew Hall, Gary King, Dan Levy, Xiao-Li Meng, Susan Pharr, Jon Rogowski, Maya Sen, Dan Smith, Teddy Svoronos, Dustin Tingley, and Ista Zahn. Even in their busiest moments, their comments led to one discovery after another.

One of the best experiences as a graduate student at the Government department was the time spent with amazing peers. I learned immensely from Pam Ban, Peter x Bucchianeri, Dan De Kadt, Connor Huff, Chris Lucas, Michael Morse, Daniel Moskowitz, and Melissa Sands, who all guided my path as I entered the program. Those who entered at the same time or after me also were sources of inspiration and fun, including Jake Brown, Chris Chaky, Angelo Dagonel, Naoki Egami, Shusei Eshima, Max Goplerud, Jaclyn Kaslovsky, Chris Kenny, Hanno Hilbig, Michael Olson, Yon Soo Park, Hunter Rendleman, Albert Rivero, Tyler Simko, Diana Stanescu, and Michael Zoorob. Outside of the department I had the good fortune of knowing Alexander Agadjanian and Shun Yamaya. I am especially thankful to Andy Stone and Soichiro Yamauchi, whose friendship kept me going on more days than they probably realized. Conversations with both lightened up any day. In addition, two of the chapters in this dissertation would not have been possible without Soichiro's substantial help.

In my personal and pre-graduate school life, I would first like to thank Susan Fiske, my undergraduate advisor and one of my academic heroes. Christina Davis, who taught that first political science seminar and wrote my first letter of recommendation, has been a generous advisor. I am grateful for Youngsuk Chi for believing in my future in academia and life in the United States. After graduation I was lucky to join the Analyst Institute, through which I started to learn from Jonathan Robinson, Josh Rosmarin, Aaron Strauss, and Annie Wang, and others. Special appreciation goes to Yi Yang for her warm support as well as to Jacky Cheng and Chris Cochran, with whom I have effectively lived together for nearly a decade. I credit Chris for many of the life habits that helped me stay health and happy through graduate school.

I have failed to list many others who have helped me through this process. My final and deepest thanks goes to my family, especially my mother. Her commitment to my education and personal happiness came with sacrifices to her time and career, I cannot overemphasize my appreciation of how much that has shaped my trajectory into graduate school and beyond.

| Introduction

Changes in the vote choice of a small subset of the electorate can dramatically swing election results, making voters who deviate in any way from a straight party ticket a perennial interest for social scientists and political campaigns alike. This volatile electorate are often called swing voters. Understanding the prevalence and characteristics of these swing voters is a cornerstone to understanding why strategic parties and re-election seeking politicians take the positions they do. This dissertation collects data from multiple sources -historical election results, survey data, and cast vote records -to provide a sytematic and in-depth accounting of the prevalence and voting pattenrs of swing voters. 1 Of particular interest is whether (or in which eras and elections) the groups of voters I identify as swing voters is consequential for who wins an election.

In this introductory chapter, I provide a definition for the swing voter and a theoretical rationale for how that latent concepts relates to observable behavior such as ticket splitting and party switching. Without a clear definition, a "swing voter" is a rather elusive term that social science disciplines and journalistic coverage define in differing ways. For example, some include decisiveness as part of the definition of the swing voter, while in other definitions a swing voter is a easily persuadable voter regardless of whether they are pivotal or not. I define the swing voter in a spatial voting framework where voters choose candidates by comparing the utility they derive from 1 Data and code to replicate some of this analysis is provided in Kuriwaki (2021c).

1 candidates, whose positions are defined on a left-right scale.

The notion of swing voters has been a focus in the study of electoral behavior and in theoretical models of political economy (Burden and Kimball 2002;Persson and Tabellini 2000, ch.8). Indeed I argue that this simple framework is appropriate because the definitions are clear, widely applicable, and illuminating in its own right.

The framework justifies the use of ticket splitting and party switching as the key behavior I study empirically in the rest of this dissertation. Being a swing voter is a latent quality, but these two voting patterns are logical implications of a swing voter in a general spatial voting framework.

This formalization clarifies the difference with other ways scholars have defined the swing voter, as well as where the logic overlaps. One of the latest formalizations of the swing voter is by Mayer (2008), who focused on identifying the swing voter in the context of Presidential elections. His definition relies on a survey instrument often called the feeling thermometer, where respondents indicate how favorable or unfavorable they feel towards each presidential candidate. Mayer operationalizes the swing voter as voters who give the exact or approximately same value for both candidates, and argues that other related measures, such as party switching, being undecided, or independents, are less desirable mainly for measurement and interpretability. In a similar spirit, Hillygus and Shields (2009) investigated the characteristics of the persuadable voter, operationalized as a political independent or having issue positions that conflict with the party platform.

While the general strategy of this body of work is reasonable, taken literally its generalizability is limited. For example, both focus on a single office and the theoretical framework provides only suggestive guidance about modeling voting across multiple offices. The measurement strategy of a feeling thermometer is of limited applicability; most political surveys measure vote choice and partisanship but few consistently provide a feeling thermometer. And while, as Mayer argues, other measures such as being a moderate or undecided have their own measurement challenges, a black-orwhite statement about whether these are correct ways to measure the swing voter makes it difficult to aggregate evidence across multiple classic studies, including V.O.

Key's final work on floating voters (Key 1966).

In what follows, I provide a definition and measurement strategy for swing voters in a simple spatial voting framework. The general idea of the definition is consistent with what scholars such as Mayer (2008) have outlined, i.e. that swing voters are generally indifferent to the two alternatives. The spatial voting framework posits a leftright ideological spectrum and voters who make choices based on a cardinal measure of utility. This utility framework is in fact quite similar to the feeling thermometer measure. There is less conflict between my definition and Mayer as it might seem at first. The spatial voting framework can flexibly incorporate factors other than ideology. Here I consider two: a valence term and uncertainty.

In sum, the spatial voting framework turns out to be a simple and fruitful model to operationalize the concepts and justify why ticket splitting is a reasonable measure to operationalize the swing voter. As with all models, the goal of the model is not to fully predict behavior or describe the psychological process by which voters make the choices they do. Yet the model is illuminating because it logically shows how being a swing voter relates to ticket splitting or party switching. Perhaps more useful is that it also shows how the propensity for a voter to split a ticket is increasing in the ratio of two well-known factors in electoral politics: a candidate's valence advantage and the spatial distance, or polarization, between the two candidates.

Model of Vote Choice

Each of a voter's outcome y ij ∈ {l, r} refers to a party choice by voter i in office j. Voters prefer candidates with closer policy positions, but they may also prefer more valence over less. Therefore in contest j a voter considers a quadratic utility for each party a with random measurement error:

The key parameter of interest is θ ∈ R, which indicates the weight voters value valence relative to party. If θ = 0, then voters only ignore valence and only vote party.

If θ > 0, some voters may defect from their party allegiance to vote for a high quality candidate. For simplicity θ is left constant for all voters, but we now let it vary by i in order to identify the model (see end of this section).

Decision Rule For tractability let both errors have a Normal distribution with the same mean, and let the variance of the difference of the two distributions be 1, i.e.,

( ijr ij ) ∼ Normal(0, 1).

Then voter i votes for the Democrat (candidate l) in office j if

where Φ(•) is the cumulative density function of a standard Normal distribution. In other words, a voter's choice for a particular election depends on the spatial differential ∆x j ≡ x l jx r j combined with the cutpoint κ j ≡ (x l j +x r j ) 2

, a valence differential ∆v j ≡ v l jv r j :

Valence differentials can therefore lead voters to split their ticket, which in this model corresponds to choosing a candidate that would not have been chosen based on the spatial cutpoint κ j . The impact of valence is determined by its size relative to the spatial differential. To see this, consider the one-dimensional case and solve for Pr(y ij = l) > 0.5. Rearranging terms and assuming x l j < x r j without loss of generality, Vote for l if:

The intuition for this result is that considering valence moves the cutpoint in l's favor from the original (κ j ) by an additive factor that is increasing in θ i and (v l jv r j ), and decreasing in the positive difference (x r jx l j ). The figure below sketches out an example where the voter chooses l only because of l's valence advantage, with the contribution of valence highlighted in blue.

x r j Voter type x * i that votes l (with valence) (without valence)

Valence as Clarity of in Ideal Points

We can also consider how uncertainty factors into this decision by treating the candidate position x a j as a random variable. For simplicity consider the one-dimensional case and let E(x a j ) = µ a j , Var(x a j ) = η a j > 0.

Then the voter's expected utility from party a, E(U i (x a j , y a j )), becomes

Therefore, a large variance η a j can be interpreted as having the opposing effect as high valence θv a j . In other words, high uncertainty in policy position effectively lowers valence.

Connection with Ticket Splitting and Party Switching Thus far, I have limited the exposition to a case where a voter makes one office, rather than voting on the long ballot. The theoretical results make the extension straightforward if we fix the national, top of the ticket office to be polarized and contested by candidates equally matched in valence. In essence, vote choice is a function of the relative difference between candidates, not offices. With a simialr logic, we can capture party switching overtime in this model by considering offices at time 0 and time 1 as two choices the same voter makes.

In the figure below, we show three cases: the first case repeats the single-office above, which can be thought of as a local office. In the second case, we introduce a "national" office in j = 1 where candidates are equally separated and assumed to have equal valence.

In the ticket splitting case, the voter's ideology x * i remains the same across offices, because the voter is voting for the two offices in the same ballot. As before, the tickmark is the cutpoint between candidates (κ) and the blue region indicates the type 0, θ(v l j -v r j ) 2(x r j -x l j ) which is all the possible values of x * i under which he will vote for the Left candidate. Local Office (j = 2)

National Office (j = 1) Local Office (j = 2)

The figure reinforces the point that ticket splitting for a candidate a is increasing in the valence advantage of candidate a and decreasing in the degree of moderation (or convergence) between the two candidates.

To model party switching, we can replace the diagrams above with offices at two different time periods. An additional moving piece in the party switching model is that the voter's ideal point itself may shift over time. This adds an additional degree of complication, but still can be thought of in the same framework.

Candidates who run in local offices may be different from national elections because there is less information about them in the media. Recall that uncertainty in the information about candidate b's position also effectively enters the decision rule as negative valence for candidate b. This implies that candidate a being more well-known (perhaps through news coverage, advertising, incumbency) than candidate b in a local office also draws ticket splitting towards candidate a. All this can happen even if voters cared equally about party across offices (θ is constant across offices) and candidate's positions were nationalized (x l j = x l j for all j, and the same for r).

Identification Although I do not attempt to estimate the parameters of the model from the data in this dissertation, it is worth noting how that would be possible. On its own, this model is not identified; an additional constraint is needed to distinguish between the contribution of the valence differential and the spatial differential in the data. To see why, following Bailey and Maltzman (2008), consider adding and subtracting some arbitrary θ∆y j to the right-hand side of equation 2. Then we would be able to rewrite the equation as Φ 2(∆x j ) (z iκj ) + θ(∆v j ) where κj = κ j + θ∆v j β j and θ = (θ i -θ), making it indistinguishable from the original equation despite having a different coefficient on ∆v j . Bailey and Maltzman get around this problem by adding Members of Congress to the dataset and assuming that their equivalent of θ is 0 for those individuals. In my case, I can take the voters who choose the party lever (and therefore do not consider candidate-specific valence) and set θ i = 0 for those individuals.

Operationalization

We cannot observe a voter's utility for each alternative, let alone the specific component of the utility that is about the party bundle. At best, we only observe the revealed preference of the voter. Scholars have used two measures in particular: vote switching and ticket splitting. Both measures involve the same voter voting for both parties, either across time, across office, or both.

Party Switching: Surveys and panels of respondents can ask respondents to recall their party choice for two different elections, often spaced across time. This corresponds to the common definition of swing districts. V.O. Key's seminal study of electoral change focuses on this operationalization as well. Through analyzing a series of national surveys Key (1966) showed how voters transitioned between the Republican and Democratic presidential candidates during 1940 -1960, finding that as much as 15

to 20 percent of voters in this era switched parties to align their vote with their policy views on key issues of the day. This is both an observable implication and a operationalization of being a swing voter. If the same person switches parties, that suggests voters derive similar utilities from both parties generically.

Ticket Splitting: The pair of elections to be compared can also be across offices within the same election. The American federalist system abounds with direct elections of various elections. The majority of these are partisan elections (i.e. candidates have party labels on the ballot) in the same general election cycle. Beck et al. (1992) examine ticket splitting in state executive offices and Burden and Kimball (2002) examine ticket splitting in the US House election. But as noted by Burden and Helmke (2009), a general definition of ticket splitting applies to what others will call vote switch-ing.

Persuadability is another operationalization of the swing vote that conforms with one way the term is used colloquially. In this definition, a voter is a swing voter if they are one of the first voters to switch their vote under the counterfactual that some factor, like candidate valence, candidate ideology, or national information shocks, changed.

Unfortunately, this counterfactual quantity is difficult to measure reliably. Work in estimate heterogeneous treatment effects have been applied successfully in turnout (Imai and Strauss 2011), but less in candidate vote choice, perhaps because effects are noisy and experimental samples for vote choice are smaller than turnout experiments.

Coppock , Hill, and Vavreck (2020) report a large collection of survey experiments to examine heterogeneity and subgroup effects, and report little heterogeneity in persuadability. However, their settings focus on the 2016 Presidential election and therefore do not test variation in candidate attributes. Other machine learning methods may be able to reliably identify subgroups with high treatment effects, but this is an evolving methodological field (Künzel et al. 2019) and beyond the scope of most of the evidence presented in this dissertation. differed from the Presidential voteshare by 2 to 3 percentage points, the proportion of ticket splitting this implies was large enough to have reversed party control in Congress. I also find that statewide executive and state legislative elections trend the same as Congress. However, gubernatorial and countywide elections do not show the same trend, or have larger discrepancies from the Presidential vote. This suggests that the swing voter is not a single bloc, but varies by the office and candidate. * I thank Jim Snyder, Gary Jacobson, Carl Klarner, Steven Rogers, Michael Zoorob, Chris Warshaw, and Justin de Benedictis-Kessner for sharing their electoral data, most of it unpublished or before public release. I also thank those at Daily Kos for their values of Presidential voteshare at legislative districts that enable many of these comparisons.

All states in the 2016 U.S. Senate Election produced "straight-ticket" delegations with Presidential results. In all states where a Republican candidate won the U.S.

Senate race, Donald Trump, the Republican, won. And in all states where a Democratic candidate won the U.S. Senate race, Hilary Clinton, the Democrat won. The 2020 elections were similar. Except for Senator Susan Collins (R-ME) winning reelection, no state with a U.S. Senate election that year delivered a split delegation between President and Senate. Earlier, Jacobson (2015) noted that "[w]ith little fanfare, the electoral advantage enjoyed by U.S. representatives has fallen over the past several elections to levels not seen since the 1950s," a pattern of a declining personal vote that would be consistent with congressional election results increasingly mirroring Presidential support.

Observers took this as evidence that ticket splitting was less consequential, that allegiance to national parties now dominated electoral behavior, and differences across candidates and offices were increasingly minor (Stein 2016;Enten 2016). This became a pattern for the next few elections in 2018 and 2020 (Skelley 2018;Rakich and Best 2020). Moreover, the decline in ticket splitting has co-occurred with the increase in polarization between parties' voting behavior in Congress. Putting these trends together, one might expect highly disciplined political parties with extensive degrees of partisanship found among voters, cutting across different levels of government (Drutman 2018).

In this chapter I document and interrogate these overtime trends from the lens of the swing voter. Disentangling the contribution of parties, candidates, and voters from a set of election results is a source of numerous debates in political science (Fiorina and Abrams 2008;Ansolabehere, Snyder, and Stewart 2001). Large electoral trends that decide which parties and politicians win elections and set policies should be traceable to specific blocs of voters. But simple statistics often mask this inter-pretation. For example, the statistic that "no states split their Senate-presidential vote for the first time ever" (Stein 2016) is only loosely indicative of the prevalence of ticket splittings in the electorate because it only accounts for who won an electionmasking the margins.

Another potential pitfall when inferring voting behavior from these election returns in Congress is that it generalizes across offices too easily. The United Stats features a "long ballot" (Key 1963), where offices ranging from President to Sheriff to County Council are up for re-election at once, often affiliated with a major party. Have ticket splitting rates in these state and local elections also declined? Because of the limited availability of state and local election data, only recently have scholars tracked rates across the long ballot.

Through my analysis, I argue for two modifications to this common interpretation of modern elections. First, while ticket splitting indeed has likely declined, this does not mean that ticket splitting has become electorally irrelevant. In fact, the modern era of nationalized Congressional elections is also an electoral politics where ticket splitting has even more disproportionate electoral influence to determine the party control over elections. I label this the swing voter paradox because one might think that as swing voters decline, electoral results become more predictable. The basic idea of this pattern is that ticket splitting has declined but parties have been equally balanced, and, as we see in Chapter 2, ticket splitting is not concentrated in any particular district. Unstable majorities have been highlighted in past work as well, but my focus on the low rates of ticket splitting does not hinge on the argument (as in Fiorina 2017) that a large portion of voters are moderate.

Second, I collect electoral data in offices other than Congress, and show varying degrees of decline. If swing voters are a single group that is indifferent to either party, one might expect to see ticket splitting for all offices declining across for state and lo-cal offices as well. I do find decline in ticket splitting rates in offices such as Governor elections and statewide legislative elections, but the drop is not as precipitous. Moreover, the evidence is inconclusive in county offices such as Sheriff and county council. There is value in comparing electoral results from different offices because it can start to disentangle a secular trend that applies to all voters, or particular aspects of the candidates and the districts they run in (for a similar design, see Ansolabehere and Snyder 2002). Under the spatial voting framework with valence considerations, I would expect that whether or not a voter casts a split ticket is not something inherent to the voter, but something that varies across offices and candidates.

The goal of this chapter is to be broad and represent electoral behavior with a common metric that can be compared across different offices and different decades.

On the other hand, the simple measure has its flaws. I use the difference in a pair of vote shares to measure the prevalence of swing voters. This is a classic ecological inference problem. In the two-party case I focus on here, the difference in two vote shares almost certainly underestimates the proportion of voters who split their ticket in the population. Nevertheless, by using a simple measure allows for a more transparent comparison across different datasets. Moreover, because the direction of mismeasurement is largely known, the evidence is sufficient to make an important observation in support of the paradox I propose: Even by a conservative estimate, the proportion of ticket splitters in Congressional elections in the modern era has been sufficient to reverse the party control of both chambers of Congress.

Data and Measures

To provide a broad picture of ticket splitting, I collected and pooled together existing datasets to cover as wide a set of offices and years as possible. Ansolabehere and Snyder (2002) and Eggers et al. (2015). State legislative elections are provided by Rogers (2021) and Klarner (2021), with Presidential vote at the state legislative district computed by Rogers or the organization Daily Kos. For county Sheriff elections, I rely on a dataset of historical Sheriff elections collected by Zoorob (2020). For county legislative elections, I rely on an ongoing data collection effort by de Benedictis Kessner and Warshaw, which collect results from municipal and county election results from 2000 to 2018, including those published in de Benedictis-Kessner and Warshaw (2020). I do not use municipal elections in this analysis because most of these offices are nominally non-partisan. That is, these candidates do not participate in a party primary to appear on the general election ballot, and party affiliation is not listed on the ballot. For any election for office j in constituency i, I compute the simple vote share difference from the Presidential vote,

where D ij indicates the number of votes for the Democratic candidate in constituency i for office j, and R ij indicates the number of votes for the Republican candidate. The overall goal of this section is to provide good enough measures for as wide a historical range as possible.

Throughout this section, I use the Presidential vote as the reference office because the President -Vice President ticket is the only office elected by the entire county.

It is also conveniently comparable: always contested by the two parties, giving each voter in every state the same two choices. Because a goal of the analysis is to compare different congressional, state, and local offices with each other, I fix the reference category in all these comparison of pairwise differences. There are exceptions to this general pattern: strong third party candidates in 1960, 1968, and 1992 make the twoparty presidential vote share in those states harder to interpret.

Therefore, high values of the Vote Share Difference indicate that one candidate has outperformed expectations relative to how a national candidate in the Presidential race did in that same district. The lowest value of 0 indicates that the Democratic and Republicans got exactly the same two-party voteshare as their respective Presidential candidates. For an example of this measure used in the literature, see Darr, Hitt, and Dunaway (2018). Moskowitz (2020) shows that this measure correlates highly from actual ticket splitting rates estimated from ballots.foot_0

Of course, even if it is correlated with the quantity of interest, the vote share difference measure suffers from an ecological inference problem in measuring the degree of ticket splitting or party switching. The three simplified cases in Figure 1 The subsequent chapters improve upon this vote share difference by using individual- level data such as surveys and cast vote records. The simple vote share is nevertheless useful for historical comparisons where survey data do not exist. Moreover, in the twoparty case, the direction of the measurement error is known. Even this lower bound measure is sufficient to demonstrate the point that there are enough ticket splitting voters to reverse the party control in modern Senate and House elections. Because in a midterm year there is no Presidential election and therefore no single pair of candidates that the nation as a whole votes on, I use the Presidential election House, 1952House, -2020) ) result two years prior in the same constituency. This introduces some complications in interpretation. Voters loyal to the President's party may stay at home in a midterm year, hurting that party's Democratic Senate candidate up for election rather than reflecting an increase in ticket splitters in the population. That is at least partially why the average difference is higher in midterm elections than presidential elections. Note: Each line represents different ways to measure Independents. The gray line is a wider measure that includes Independents who lean towards a particular party.

The Rise and Decline of Ticket Splitting in Congress

The black lines estimate the proportion of "pure" Independents who do not lean towards a particular party. The dotted line restricts the sample to the turnout electorate. The trends do not correspond to the rise and fall of ticket splitting in Congressional Elections.

Source: Pew Research Center (2015), ANES Cumulative File (1948-2016), using the variable VCF0305. For 2020, I use the preliminary release of the time series dataset.

of traditional parties (Klar and Krupnikov 2016). A better measure that isolates the indifference to parties on the voter-side may be the Independents that do not lean to- The next important time period in the history of the two parties was the 1980 election. Lee (2016) traces the polarization of parties to this time period and specifically this election, where Republicans unexpectedly won a majority of the U.S. Senate.

At this time, the Democrats still appeared to have a solid grasp of the U.S. House.

But once the control of a chamber was in play, Lee argues, party leaders had additional incentive to differentiate their platforms from the other party. Congressional

Republicans therefore increased their use of messaging bills and Congressional hardball instead of emphasizing consensus building. The strategy also required the entire caucus to stick to the party line. The legislative dynamics that Lee traces fits the trends in electoral results rather well. As Congressional candidates repositioned themselves to align more closely with the President's party, the spatial voting framework would predict that voters will accordingly be less likely to split their ticket.

Similar trends show up in the rise and fall of the incumbency advantage in the U.S. House. Various estimates suggest that by the 1980s, incumbents gained an additional 8-10 percentage points above and beyond the normal party allegiance (Levitt and Wolfram 1997), that this increase was similar in state executive and state legislative office as well (Ansolabehere and Snyder 2002), and the advantage slowly declined thereafter. The incumbency advantage is closely tied to ticket splitting because it often involves some voters defecting from their general party identification to vote for the incumbent, which happens to be of a different party (Jacobson 2015). It is therefore hard to say whether the declining incumbency is a cause of ticket splitting. A declining incumbency advantage is often a manifestation of an increase in ticket splitting, which can be seen both in overall trends, the theoretical model outlined in the introduction of this appendix, and the results in the rest of this dissertation.

Other work points to the realignment of the former Confederate states, increased centralization of power to the federal government, and the nationalization of the media as contributing factors to the decline in ticket splitting and the nationalization of politics (Hopkins 2018). Testing each of these factors is beyond the scope of this chapter on electoral politics. From a broad view of Congressional elections reviewed here so far, a change in party brands appears to be an important factor. This trend

The Closeness of Elections

For the rest of the analysis, it would be useful to set some benchmarks about how much of a swing vote is large enough to be politically important. I propose one benchmark: the magnitude of uniform swing it takes to reverse the party that wins the most seats in the legislature.

That is, the "minimum swing", δ for a given U.S. House election is given by

where s ∈ {1, ..., S} indexes seats, V s indicates the two-party vote share in seat s of the party that ultimately won the majority of seats in the election. Therefore, a δ = -0.01 indicates that a 1 percentage point uniform swing against the majority party would have cost the winning party enough seats to cost them the majority. Another way to think of this value is the majority party's lead in the tipping point district.

Only a third of U.S. Senate seats up for election in a general election, so the formula for swing in the Senate takes into account the number of seats that the winning party already has locked in and not defending. The seats required for a majority also depend on the party of the Vice President, who casts a tie-breaking vote. Because every marginal seat matters for the majority, independents are coded as essentially belonging to the party they caucus for.foot_2 Uncontested elections and seats in California and Washington where two candidates of the same party win the primary are coded as safe seats for that party.

This measure of swing has several advantages over existing measures of competitiveness that are easier to find. Typically, analysts use a party's seat share or the pop-ular vote to track how close a party is from capturing a majority. But using seat share may mask the heterogeneity in the vote share of pivotal seats. And the popular vote is ambiguous because an electoral system's swing ratio varies by context. The ratio has ranged from 2 to 3 in the modern era (Linzer 2012), meaning that a 1 percent increase in the popular vote can translate to roughly a 2 to 3 point increase in the proportion of seats. In this application, we care about the seat swings directly. Moreover, the presence of some uncontested seats and seats contested by two candidates of the same party makes the computation of the popular vote ambiguous. The distinctiveness of elections in the modern era is clear from the historical comparison. In the early 2000s to 2020, control of both chambers could have been reversed

by less than a percentage point swing against the Democratic Party. One would need to go back to the 1950s to find a time when that small a swing would have been decisive for legislative control. The historical trends highlight two prior turning points where deepening Democratic control reversed -the Gingrich Revolution of 1994 which ushered in a two-decade stretch of near-Republican control of the U.S., and the Republican takeover of the Senate in 1980. Lee (2016) Note: Each point represents the smallest uniform swing against the party that eventually won control of the chamber necessary to reverse the majority party. In 2020, for example, an 1 percentage point uniform swing against the Democratic party would have reversed their majorities in both the House and the Senate.

In 1990, in contrast, it would have required a 9 point swing for the House and a 7 point swing in the Senate to reverse the Democratic majority. * In 1966, the Democratic Senate majority held 67 votes and only 20 were up for election. Therefore, no amount of swing would have been sufficient to vote Democrats out of their majority.

their party strategy towards distinguishing the party from the Democratic party.

Combining the findings on the prevalence of the swing voter and the closeness of party control produces Figure 1.5. This is the main relationship in the swing voter paradox: elections with fewer swing voters are the closest ones. The relationship is stronger for U.S. House elections, where all seats are up for election each year. Interpretation is more complex for the Senate, where roughly two thirds of the majority party seat share in any given year is actually not reflected in the ticket splitting estimates (which only use the year's 33 or so elections) represented on the horizontal axis.

The slope coefficient in the U.S. House is 0.68 with a robust standard error of 0.14.

This implies that every one percentage point decrease in the prevalence of swing vot- Note: The horizontal axis shows the values used in Figure 1.2, and tracks the degree of net ticket splitting from the President that year. The vertical axis shows the share of seats the majority party won in the election. Slope coefficients show coefficients and robust standard errors of the bivariate relationship. The two measures roughly correlate, especially for the House, where all seats are up for re-election. Somewhat paradoxically, the decline of ticket splitting and the increase in party polarization has occurred as any swing voter has become more likely to be pivotal.

ers is associated with a 0.68 percentage point decrease in the winning party's margin, thus making their party's victory more precarious.

These statistics of minimum swing provide useful context for subsequent analysis of the swing voters because it establishes in a concrete way how much their vote is electorally important. Swing voters that comprise three percent of the electorate would not be electorally important in the U.S. Senate election of 1966, where the Democratic party majority was so large that no amount of swing could have put them out of their majority. But the same three percent constituency is more decisive in a year like 2020, which could have made the difference between a Democratic trifecta and a Republican trifecta (The tipping point state in the 2020 Presidential election is either Wisconsin or Pennsylvania, where Biden won by less than a percentage point of the two-party vote).

Patterns in State and Local Offices

Almost all of past work on ticket splitting has focused on comparisons of Congressional races with the President. This is not sufficient evidence to attribute that the decline in ticket splitting is due to changes in the voter's preferences towards generic parties. This is why comparison across multiple levels of government is important.

If the generic swing voter has declined, we would expect ticket splitting and party switching to decline across all offices similarly, because a single voter often votes for multiple partisan offices on the same ballot. On the other hand, if patterns differ by office even in the same state and same election, this suggests declines in ticket splitting may be particular to the office, and perhaps shaped by the ideological and valence candidates that run in those elections.

Data limitations have prevented even simple comparisons, however, below the statewide level of Governor. In this section, I walk through each set of offices outlined in Table 1.1. I retain the Presidential race as the reference category for the vote share difference.

Moving to state and local offices changes the interpretation of ticket splitting in one important respect, because these offices do not deliberate on the same legislation with the President. Because Congress and the President negotiate over the same legislation, it may be the voters prefer to balance their vote, ensuring that no party gains a trifecta (Fiorina 1992). Concerns about weak, unaccountable party government (American Political Science Association 1950) are also largely predicated on the common legislative setting. These concerns are not at play, for example, when a voter is voting for a Governor and a President. However, it is precisely because of this lack of a legislative connection that makes nationalization and party polarization of particular concern (Hopkins 2018). If voters cast their ballot for a single party regard-less of the particularities of each level of government, this is one (but certainly not a sufficient) indication that partisan voting occurs without considerations of important qualifications.

Governors and Statewide Executives

Figure 1.6 starts by comparing the offices of statewide offices. Past research shows that compared to Senators, there are a few more Governors who come from opposite parties as their Presidential candidates (Sievert and McKee 2018). The data in this project allows a broad comparison of all executive offices. Some care is necessary to provide a informative comparison of the rates found here with that of Congress, because election cycles vary. Statewide executive elections occur every four years in almost all states. About two thirds of these states hold elections in midterm years, about a third in Presidential years, and two states, New Jersey and Virginia, in odd years. Therefore, the difference between the year-by-year averages of Governor and Congress in Figure 1.2 may be due to the comparison of different states. To avoid this confounding, for each year I subset the Congressional statistics to the states which hold a Governor (or other statewide executive) election that year. Sometimes there is no Senate election in that year, so I use the U.S. House values as well. I generate a Congressional average baseline by upweighting the U.S.

Senate values with the number of Congressional Districts the state has that year as a rough measure of population.

Figure 1.6 shows that statewide executive offices follow the same trend as Congress, but the office of Governor is more resilient in the modern era to the trend. In 2016, the average Democratic vote share in a Governor race (weighted roughly by a state's population size) differed from the state's Presidential election by 6 percentage points, while a Congressional race in those same states was about 3.5 percentage points. In 2018, with a different set of states with Governor elections in midterm years, the rates were 4.5 percentage points and 4 percentage points, respectively. This may in part be due to the fact that Governors in most states are elected in midterm years, therefore facing different electorates. Governors may also be able to free themselves from national party discipline as executives of their own states, dealing with a different set of issues.

The remaining statewide executive offices shown in Figure 1.6 include races for Attorneys General, Secretaries of State, Auditors, and State Treasurer. In the "Lottery of the Long Ballot," Key (1963) expressed concern that the separate elections of these offices would lead to state executive cabinets of different party members, and dampen accountability. At least in the modern era, these offices tend to align even closely with the national party lean of the state.

Regional Differences

Regional and state-specific differences in these trends may also be an important factor, in addition to office, in explaining these trends. Mayhew (1986) showed vary-ing levels of strength in party organization and machine politics across states. Strong party states may have been able to run and elect a party slate that differed in ideology from the party platforms in the Presidential elections. Regions such as the former Confederate South may also have been unique in resisting the convergence of state party platforms to national ones.

We can investigate the extent to which state and regional differences explain variation in ticket splitting through the analysis of statewide elections, where we have a long time series of over 70 years. Figure 1.9 shows the trends of each state separately.

There are typically one to five statewide elections in each state and each general election. To reduce the variance from idiosyncratic races, I therefore take the simple average across statewide offices and across years within a decade. Each state -full decade observation, then, contains an average of 14 elections.

There are no clear regional differences in the statewide trends of the difference from the Presidential vote. Figure 1.9 shows the trends of all 50 states in light gray over the course of these decades, show trends that are roughly in line with the aggregate trend.

There are only two states where trends in divergence have bucked the rest of the country: West Virginia and Maine. West Virginia started out as having a close correspondence between statewide and Presidential vote, but gradually diverged, with Democrats winning statewide in 2010 but with the same voters voting for Republican presidential candidates. The same pattern can be said for Maine, although independent candidates (which are excluded in the voteshare calculation) complicate this picture. Overall, almost all states appear to have experienced a trend for convergence to the Presidential vote, a story consistent with theories of nationalization.

The South in particular is a region that stands out in Figure 1.9 as well as in the scholarship on realignment (Hopkins 2018;Aldrich 2000). In Figure 1.10 I show more clearly the difference between the South and non-South with the same statewide data.

Here I group the differences by every pair of elections that use the same Presidential election (for example, 2016 and 2018), and compute differences separately by office.

I compute the difference by regressing an election's average difference on a binary variable separating Southern vs. non-Southern states and show the regression coefficient on the binary variable. I use two operationalizations of the South: one with core Southern states including those that so diverged from the national Democratic party that they ran third party Presidential candidates, and a more expansive definition including more former Confederate states listed in the figure.

The differences reinforce how the South drove the aggregate trends before the 1980s, but this is no longer the case in the modern era. As we saw with Mississippi and South Carolina, by the 2000s Southern states tended to have lower divergence than non-Southern states across all statewide offices. In other words, there appears to be more party loyalty in statewide Congressional and executive elections in the South than in the non-South.

Conclusion

In The main implication of this chapter is to establish this new equilibrium, which hews closely to the idealized notion of two-party government but is a break from most of the last past century. In this politics, fewer voters appear persuadable, but electoral results vary widely. This electoral landscape has several implications for the strategy of campaigns that seek to gain power, although many are not as clear-cut as they may appear at first. The dearth of swing voters might naturally imply that campaigns should focus instead of mobilizing their base to turn out. In a 50-50 election, every voter matters. But if the two parties have starkly polarized, partisans are less persuadable than the sliver of swing voters. It is also the case that persuading a swing voter to switch parties is twice as efficient for the electoral margin as persuading a core supporter to turn out to vote.

One reason it may not be obvious to characterize this regime as a two-party system is because, contrary to theories of spatial convergence in two-party elections, parties do not appear to have converged their platforms to the median voter. Recent research suggests that this is not a coincidence: minority parties have an incentive to distinguish themselves from the ruling party by exaggerating their differences, especially when their power is equally balanced (Lee 2016). These dynamics may in fact paint a normatively troubling picture of unproductive governance (Kustov et al. 2021),

or what Drutman (2020) calls a "Two-Party Doom Loop" in which two polarized parties engage in a protracted back-and-forth of control without convergence. Using a large survey dataset with samples in every Congressional District allows me to test two hypotheses from the previous chapters: that the swing vote can be explained by a spatial voting model with valence, that the degree of swing voters in each district is often large enough to flip the Congressional election results. I find that Independents, White Voters, and voters who do not frequently follow the news are more likely to be swing voters. In a panel regression of congressional districts, I also find that candidates who become incumbents net more votes through increases in ticket splitting from out-partisans. The district-specific levels of ticket splitters were enough in 2018 to reverse party control.

Introduction

The first chapter of this dissertation showed a decline in the aggregate rate of ticket splitting from around the 1980. It also made the case that, even as the proportion of swing voters were declining, a switch in their vote choice was still likely to be pivotal, if note more so. By using a simple difference in aggregates for the measure of ticket splitters and using the assumption of uniform swing to measure pivotality, however, the chapter still did not definitively answer the seemingly straightforward question of how many swing voters there are in each legislative district.

How common are swing voters, why do they swing, and when are swing voters decisive? This chapter provides systematic answer to these questions in the U.S House.

Conventional wisdom suggests swing voters are thought to have all but disappeared.

My measurement approach does confirm that swing voters are a small portion of the electorate, about 3 to 5 percent. This level is within the margin of error in a typical Hall 2015). Candidate visibility is in fact one of the key predictors of ticket splitting found in early work (Burden and Kimball 1998;Beck et al. 1992). Hall and Thompson ( 2018) is one exception, which tests the effect of US House candidate extremism on differential turnout at the congressional district level. However, the authors do not make small area estimation adjustments to their analysis of survey outcomes.

After reporting descriptive statistics from the measurement strategy, I conduct three sets of analyses to understand the drivers of the swing vote in the modern era.

To estimate the causal effect of candidate quality -a key parameter in theoretic models -I employ a panel regression. Together, the evidence identifies a relatively small subset of the population of swing voters who respond to candidate quality, and may, especially in districts that are largely balanced in partisan strength, be decisive.

The Prevalence and Geographic Distribution of Swing Voters

To provide estimates for each Congressional district in each election of the prevalence of ticket splitters or party switchers requires additional modeling adjustments.

Because there can often be only about 60 respondents in a given Congressional district and the national rate of ticket splitting can be in the low single digits, there is a decent probability that no ticket splitters are sampled into a Congressional district subset by chance alone. In such cases, no amount of weighting will move the point estimate of ticket splitting in that district from 0, even though there is likely a small per- centage of ticket splitters in the population. To avoid this problem, survey researchers have used partial pooling regressions such as random effects to smooth out small sample idiosyncracies (Gelman and Little 1997;Ghitza and Gelman 2013). I estimate an outcome model similar to the one in the previous section, but I remove variables which cannot be post-stratified (such as news interest) and add district level variables such as past Presidential voteshare and the incumbency status of the House candidate that help partial pooling. I investigate the variability in these Congressional district level MRP models in Chapter 2.

Descriptive statistics in Table 2.1 show that about 6 percent of a general election electorate are swing voters under my definition: 3 percent who both vote for a Republican President and a Democratic House candidate, and 3 percent who vote for a Democratic President and Republican House candidate. For the average district, MRP estimates that about 10 percent of the consecutive electorate are swing voters. These numbers average across years, but each year typically has a national swing. In 2018, the national tide advantaged Democrats: 1.4 percent were Clinton voters who then switched to a Republican House candidate, and 2.6 percent were Trump voters who voted for a Democrat. Analysis of the turnout and vote by Ghitza (2019) suggest that in 2018, this vote switching was the critical piece that best explained the change in seat control. In 2014, more Obama voters defected to Republican House candidates than did Romney voters to Democratic House candidates. In addition, Ghitza (2019) identifies 2014 as an election year where turnout differentials also made a difference.

The turnout differentials also advantaged Republicans, with more 2012 Obama voters not turning out to vote compared to 2012 Romney voters.

Each year's geographic distribution of swing voters is shown in Figure 2.2. These choropleth maps show both geographic and year-to-year variation in our main outcome measure of interest. In a given year, the range of swing voters is rather limited.

For example, all districts had less than 5 percent of vote switching to Republicans in 2018. Table 2.2 lists the Members of Congress who represent the districts with the highest amount of swing in 2018. Some members of this list are expected -they include well-known moderates such as Dan Lipinski (IL-09, more conservative than 85%

of his Democratic colleagues in the House by NOMINATE), Collin Peterson (MN-07, the third most conservative Democratic member), and Paul Cook (CA-08, more liberal than 80% of his Republican colleagues). However, some of the other members who draw large proportions of ticket splitters are at the extreme ends of their party.

The map, combined with the standard deviations presented in Table 2.1, shows that the cross-district variation in swing voters is not as high as one might think. In Table 2.1 the standard deviation in the district-level proportion of swing voters is about 0.03 for all three proxies, whereas the standard deviation for two-party vote share is five times larger. Although political observers classify districts and states as swing vs. safe districts, the difference between those types of districts is only on the order of a percentage point or two.

The range of values estimated in Figure 2.2 may also strike most observers as exceedingly small. They are indeed an order of magnitude smaller than Key's estimates from the 1940s and 1950s, although Key compared switching between Presidential elections 4 years apart. My estimates are in line with the estimates by multiple other sources -for example, Jacobson (2019) finds similar numbers using a collection of polls. That the typical rate of individual-level swing is on the order of 5 to 10 percent is consistent with the narrative of increasing party loyalty and nationalization.

How do we square this small number with sizable seat swings -that for example in 2018, when we see the lowest level of individual-swing in the MRP estimates, the most number of congressional districts changed party control? A common explanation is differential turnout. 2018 saw a historic surge in turnout, so it could be that these new voters gave Democrats their critical votes, while 2016-2018 voters actually did not change their preference. This explanation, however, is not supported by the data in 2018. Ghitza (2019) estimates that about 90 percent of the total vote margin in 2018 was due to persuasion, not turnout.foot_3 They estimate that the gain in vote margin of drop-off and surge voters effectively offset each other, at least nationally. Historically, midterms have a pro-Republican turnout bias because Democratic voters stay home.

In 2018, the midterm surge had a pro-Democratic lean, and so the turnout effect of each group cancelled out in the final vote margin. In contrast, their analysis finds that differential turnout was likely a decisive factor in 2014.

I argue instead that the association between individual-level swing and seat-level swing depends on whether swing voters in each district are pivotal. When a district is lopsided with partisan straight-ticket voters of a particular party, even a large bloc of swing voters that comprise 20 percent of the electorate may not be enough. But in a battleground where each party holds 47 percent of the electorate, even a small group that comprises 2 percent of the electorate is decisive. For example in 2016, Donald

Trump won Michigan with a vote margin of two-tenths of a percent, won Wisconsin with five-tenths of a percent, and Pennsylvania with seven-tenths of a percent, and consequently won the electoral college. In Chapter 1, I showed that the control over Congress was similarly close in the modern era. The Presidency, the U.S. House, and U.S. Senate have slightly different dynamics -the U.S. House has more districts so its control hinges less on a single district, and the U.S. Senate only puts a third of its members for election -but each has had close control change rapidly.

In Section 2.6 I quantify pivotality by taking the observed vote share and computing a simple hypothetical vote share had swing voters switched backed their vote as a bloc. Under this definition, swing voters were probably decisive in 125 out of the 435 congressional districts, especially in suburban ones.

The Role of Candidate Incumbency

In a given year, the estimated prevalence of swing voters in US House districts ap- I estimate the contribution of incumbency in ticket splitting by an fixed effects approach. While the predictive model in the previous section suggest that the fact that a candidate is an incumbent leads to more ticket splitting, it may be the case that unmeasured confounding between the type of voters in a district and the tendency for incumbents to stay in office in that district biases the estimates. One way to account for this possible confounding is to group data at the constituency level (or district, indexed by c) and year level (indexed by t), and estimate the following two-way fixed effects model:

where S l ct is the percentage of voters in the district that split their ticket for candidate l as measured by MRP, I l ct is a binary treatment variable for whether the Democratic (l) candidate is an incumbent, and γ c and γ t are constituency and time fixed effects.

The least squares estimator for β identifies the one-shot effect of the Democratic candidate becoming an incumbent on splitting, controlling for time-invariant characteristics of the district.

The two-way fixed effects estimator is equivalent to a weighted average of differencein-differences estimators that each estimate the Average Treatment Effect on the Treated (ATT) from a matched set of control units, but where each match is not optimal (Imai and Kim 2020). To improve the quality of the matches, I also estimate a matched difference-in-difference estimator that matches pre-trends based on covariates (Imai and Kim 2019).

The current dataset covers all contested congressional districts each observed in four cycles (2012)(2013)(2014)(2015)(2016)(2017)(2018). When a district is uncontested, the observation is set as missing. When estimating each treated unit's match, I also include the missingness patterns as a matching criterion, as well as lagged values for presidential vote share and lagged values for the treatment indicator.

Table 2.3 shows the main coefficient of the panel regressions. Across specifications, being a House incumbent is associated with about a 2 percentage point increase in the percent of ticket splitting. This estimate is on the same order of magnitude as the predictive models estimated in the previous section. The columns labelled with 2FE

show the two-way fixed effects estimate, and columns with PM show the PanelMatch method. Using sets matched on pre-treatment covariates generate similar sized estimates, except when three pre-treatment periods are used to generate matches. This is not surprising because there are only four time periods in the data and so the effective sample size for such a match is small.

Hypothetical Vote Margins

The district estimates of the proportion of a President's supporters who vote for the opposing part in the U.S. House is a useful representation because it can also be compared on the same units as actual election outcomes. This allows us to consider some hypothetical scenarios for election outcomes if vote switchers had not switched their vote. By providing district specific estimates, moreover, I avoid the uniform swing assumption used in Chapter 1.

As a concrete example, consider a congressional district where the Democrat nar-rowly won by 4 percentage points, i.e. the two party voteshare was 52 to 48. Suppose that 3 percent of those who voted were crossover voters in favor of the eventual winner (the Democrat), and 1 percent of those who voted were crossover voters in favor of the eventual loser (the Republican). Now consider the hypothetical where friendly crossover voters had not crossed over and stayed with their previous vote for the Republican party. Shifting over the mass to the other side, the two party voteshare is now 49 to 51 and the Democrat loses by 2 points. Next suppose that the unfriendly crossover voters did not switch as well. Then the margin is 50 to 50 and the Democrat ties. It is common knowledge in political campaigns that persuasion is worth "twice" more than mobilization because changes in vote choice will have double the effect in margin.

In Figure 2.3 I align pairwise comparisons of the actual win margin and hypothetical margin of each of the 390 contested 2018 House districts. Each comparison is depicted with an arrow. The starting point of the arrow is the win margin observed from the election result, and the end point of the arrow is the hypothetical win margin under two conditions. In other words, let M 0i ∈ [0 + , 1] be the observed win margin for district i. Then, using the survey data, we attempt to estimate the proportion ψ i ∈ [0, 1] of the friendly crossover voters in district i. Separately, we estimate the mass of "unfriendly" crossover voters are of mass ψi . Then I define the hypothetical margin, which I denote M 1i as the margin, still for the same candidate, as:

when the counterfactual is that only friendly cross-over voters swing back, and

for the counterfactual that both types of crossover voters switch back to their 2016 vote. Figure 2.3 shows the same value of M 0i but draws the line to either M 1i or M 1i .

Both hypotheticals are important in their own ways. The first indicates the worst case scenario if the winner's coalition asymmetrically defected. The second indicates a partisan polarized case where no one defects. However, because elections tend to have roughly uniform swing that advantages one party over the other, it may be less plausible to imagine a symmetrical decline in both types of switches either.

The cross-over voter is pivotal when the arrow ends up reaches at our beyond 0. In The first key implication from this first hypothetical is that although 1-3 percent of vote swing may at first seem small, it can still more than explain the consequential seat swings in the Congress as a whole (of 30 flips) here. The second implication is that whether or not swing voters are consequential depends not only their size in the population but also their spatial position in the district. For example, in suburban districts, win margins were sufficiently low to begin with -i.e. the districts were probably more competitive -that small amounts of cross-over voters were more likely to be decisive in those districts.

The second panel in Figure 2.3 shows a similar, if more moderate, picture. Because friendly and unfriendly crossover voters cancel out, the lengths of the arrows are always smaller than the top figure, and can go in different directions. Interestingly, Democrat candidates across the board tended to win thanks to vote switchers, whereas winners of Republican candidates tended to win despite vote switchers.

These figures use ψ i as fixed but there is of course uncertainty around these estimates. To account for this, I approximate confidence intervals around my estimate in the following way. I take the standard error around ψ i implied by the MRP credible intervals, and then multiply it by √ 2 and then by the Normal cumulative density Φ(0.9) to estimate the 80 percent (as is standard for Bayesian models) confidence interval around 2 ψ i . I take the observed win margin as constant. In Figure 2.4, I drop the observed margin and simply plot M 1i with those confidence intervals. If the confidence intervals cross 0, that means I fail to reject the null hypothesis that M 1i < 0 with α = 0.20, implying (loosely) that swing voters were likely in that district. I naturally get larger estimates of pivotality based on these estimates.

Potential Explanations for Why Voters Split their Ticket

Analysts typically interpret the lack of ticket splitting as a measure of the nationalization and polarization of voter preferences, but the literature on ticket splitting has long shown that voters split their ticket for many other reasons. I will ultimately focus on a valence advantage explanation, which provides some of the clearest theoretical predictions.

A spatial voting framework bears out the logic of existing explanations. In a canonical spatial model, citizens choose the candidate whose policy position is closest to them on an ideological spectrum. Voting on the long ballot is akin to a citizen in these models making a series of choices between candidates of two parties. If all Republican candidates for these choices held identical policy positions and all Democratic candidates held another set of identical positions, a voter would cast a straight ticket vote.

This setting mimics a completely nationalized politics: co-partisan candidates for local and national office run on identical platforms and voters vote accordingly for a party slate.

There are at least three classes of explanations for why a voter might split their ticket. First, voters could cross party lines in races where the candidates are less polarized. This is the simplest explanation because we still presume a single issue dimension. A second explanation for ticket splitting posits that state and local politics is contested over different issue dimensions. For example, Oliver, Ha, and Callen (2012) document how local politics revolves around contestations over land use, economic development, and other issues specific to that locality. And recent survey evidence

shows that voter's preferences over those policy debates often do not align with their partisanship (Jensen et al. 2019). If local elections feature candidates with differing views on the environment, for example, while candidates for national office all take the same position on the environment, environmentally conscious voters would vote for the same party within national offices but then defect from that party allegiance in local races to vote for the pro-environment candidate (Besley and Coate 2008).

The third reason voters may split their ticket, the one I focus on the most in this paper, is that voters care about valence and one candidate has a valence advantage.

Valence is an attribute that all voters prefer more of to less, such as candidate compentence, effort, or experience for the job. The main insight from spatial models with valence is that ticket splitting for a candidate would be increasing in that candidates' valence advantage relative to the candidate's spatial distance. I show these results for-mally in the Introduction of this dissertation (equation 3). Furthermore in this setup, more certainty around a candidate's policy position effectively constitutes as a valence advantage as well. In this sense, candiate visibility, or salience, is captured within the concept of valence as well. profile offices also lends support for the model that the higher quality candidate nets more by voters splitting their ticket. Beck et al. (1992) showed that highly visible candidates draw more split ticket voting, and Burden and Kimball (2002) showed that congressional candidates with larger campaign expenditures appear to compel more voters to split their ticket.

Information is crucial in these valence-based accounts. A candidate's valence advantage cannot factor into voter's decisions unless that information reaches voters before they vote, for example through campaigns and press coverage. In all but three U.S. states, ballots do not contain anything other than the candidate's name and party.foot_7

That is why theories of information processing may lead one to predict more straight ticket voting in low-salience elections (Darr, Hitt, and Dunaway 2018;Moskowitz 2020) . Peterson (2017) showed systematically that the lack of candidate-specific information increases the likelihood that voters vote straight ticket

In this paper, I primarily test the valence hypothesis for theoretical clarity, as well as to show how ticket splitting could occur even when candidates have, as conventional wisdom goes, nationalized. Of course, multiple issue dimensions may be at work as well. But spatial models with multiple issue dimensions are notoriously intractable, so their theoretical predictions are less clear. And while moderation may be a factor, valence is an explanation that is theoretically plausible even if all candidates of the same party are polarized, as an account of nationalization would stipulate.foot_8

State and Local Elections and Politics in South Carolina

South Carolina is comparable to other states in the number of elected offices on each voter's general election ballot. The state has 7 congressional districts, 124 state house districts, 46 state senate districts, and 46 counties each with a county council often elected through single member districts. Statewide, Attorneys General, Secretaries of State, agricultural commissioners, and superintendents are elected in conjunction with the Governor's race in midterm years. Countywide elections include the partisan offices of sheriff, county clerk, treasurer, and probate court judges.

Existing research suggests that state legislatures, which in South Carolina deliberate on issues including education spending, environmental regulation, and abortion, to be as polarized as Congress.foot_11 However, other offices tend to focus on administrative matters (Oliver, Ha, and Callen 2012). County councils are legislative bodies that often discuss transportation infrastructure, public facilities, and sales taxes. Sheriffs are the chief law enforcement officer and manage county jails, auditors calculate millage rates, treasurers collects taxes and oversees the disbursement to other jurisdictions, the clerk of court manages court dockets and manages the collection of fines and fees, and coroners perform independent investigations of deaths. In the judicial branch, circuit solicitors (known as district attorneys in other states) serve as the chief prosecutor of state government, and probate court judges have jurisdiction over civil cases such as estate inheritance. Despite their administrative functions, all of these offices are directly elected through partisan elections in general elections. Almost all candidates register for the Republican or Democratic party and win a party primary to be elected.

While South Carolina is a solidly Republican state in national elections, Democrats win seats at considerable rates on the long ballot. For example, just over a half of all countywide executive offices elected on a partisan ballot in 2016-2018 were Democrats The available survey evidence suggests that straight ticket voting in South Carolina is comparable to the national average. Among the 4,512 respondents in the CCES between 2010 and 2018 voting for major party candidates in the state, 93 percent voted for the same party between the Presidency and the U.S. House, 91 percent between the U.S. Senate and the Governor, and 92 percent between the U.S. House and Governor. All three numbers are within one percentage of their respective national average (n = 318,346).

Voter-Level Straight Ticket Voting

Throughout this analysis, I refer to straight-ticket voting as the action of choosing candidates of the same party for all partisan offices under consideration. There are two subtleties to this operationalization. First, uncontested races do not offer voters a real choice to either vote straight or split ticket, and therefore I will limit my analysis to contested races. Throughout I will use contested to mean that the contest features both a Democratic and Republican candidate. For example, if a contest features a Republican candidate, a Green party candidate, and a Libertarian candidate, I still count that as an uncontested race. In the discussion, I consider the implication for this restriction when generalizing to voter behavior in other districts.

Second, South Carolina is one of the few states in which voters have an option to explicitly cast a straight ticket. A voter can either click through the entire touchscreen ballot, or he can select the "Straight Ticket Party Option" that appears as the first question on every ballot (See Appendix A.1 for an example). To avoid confusion of terms, I refer to this latter option as using the party lever, a slightly dated phrase originating from the era when voters pulled a physical lever on a voting machine to the same effect.foot_12 Pulling the lever for a particular party auto-fills the voter's ballot to select that party's candidates for every applicable contest. These selections are reversible case-by-case before the ballot is cast, and in my dataset I find slightly below 3 percent of voters who use the Republican or Democrat party lever later switch their vote in a contested race.

The Appendix Table A.4 shows the prevalence of straight ticket voting in my entire dataset. The number of contested races on a voter's ballot can range between 1 to 12. I compute the proportion of straight ticket voters among those contested choices only, and show the proportion as well as the general distribution of party loyalty. In the modal case of a ballot with 5 contested races, 77 percent of voters are straight ticket voters. The proportion drops to the 60s among voters who happen to face a ballot with more contested races. These numbers arguably inflate the proportion of full straight ticket voting because it includes those who pulled the party lever and likely gave less consideration to each office. Among the half of the electorate that opted out of the party lever, the prevalence of straight ticket voters is about 10 to 30 percentage points less than the full sample.

Party Loyalty by Office

Are defections from a straight party ticket more prevalent in some offices than others? As a first-order description, Figure 3.2 sets the office at the top of each election's ballot as the reference category and shows the overall rate of split ticketting by office.

In red are the rates of voting among "Republican" voters: those who voted for Romney, Trump, or the gubernatorial candidates Haley or McMaster, depending on the year. In blue are "Democratic" voters who voted for Obama, Clinton, or the gubernatorial candidates Sheheen or Smith. For example, the figure shows that among voters who voted for a Republican President or Governor candidate in a congressional district where the House race was contested, only 4 percent of them split their ticket, or voted for the Democrat.

Because Figure 3.2 does not make within-person comparisons, I use a tailored clustering algorithm to summarize the data into interpretable prototypes of voting patterns while still leveraging the full distribution of voting patterns that the cast vote records reveal. Clustering is an attractive approach several reasons. Like ideal point estimation, clustering efficiently incorporates data in which the same individual makes multiple choices, and it can do so even when there is no information about the individual other than their choices (as in cast vote records).foot_13 In contrast, simple comparison of ticket splitting rates by office will treat votes for each office separately without In this clustering method, the user picks the number of clusters to divide the voters into, and a fast Expectation Maximization (EM) algorithm identifies the set of cluster assignments that best fits the data. Formally, I posit that each voter i belongs to one of K clusters, but that cluster membership Z i is unobserved. Instead we only observe a vector of J choices Y i = [Y i1 , ..., Y iJ ] for each voter: a straight ticket, split ticket, or abstain for each of the J offices on the long ballot. In its simplest form, a clustering algorithm uses only the matrix Y and a simple model of vote choice to esti-mate two quantities: The overall prevalence of cluster k:

and the probability that a member of cluster k votes for choice in office j:

The model of vote choice underlying this representation is that a vote in one office is independent of each other, within each cluster, i.e.,

This also serves as the identification assumption to estimated the parameters in the clustering algorithm.

The algorithm is designed to be tailored to three features of the ballot data, with formal derivations in Chapters 5 and C. First, to account for abstentions and third party votes, outcomes are allowed to be unordered categorical variables. Second, unlike a canonical clustering model, I account for the fact that uncontested races provide voters with a limited pool of candidates, the method incorporates these varying choice sets by an independence of irrelevant alternatives assumption. Third, to handle over a million votes, the algorithm uses a C++ backend to perform internal calculations quickly. Both the choice of the number of clusters (K) and the substantive interpretation of each cluster is determined by the user. Although this leaves room for some ambiguity when implementing the clustering algorithm, one does not need to commit to the view that there exists a single correct number of clusters in the data. One cluster can often be divided into two slightly more homogeneous clusters. To provide some guidance, I cluster the same data with values of K between 2 and 10, and compute the BIC fit statistic. I ultimately choose to present results with K = 4 given that is where the fit statistics start to level off in 2016 (Appendix A.3). The BIC statistic uses the observed log likelihood that the EM algorithm tries to maximizes, penalized by the number of parameters it is asked to estimate. I then provide a label for each of the clusters according to the values of the estimated values of the vote choice parameters µ.

In 2016, a bare majority of both Republican and Democratic voters (as inferred from their presidential vote choice) are solid partisans in their votes, because they vote solidly for the same party up and down the ticket. In 2012, only about 40 percent of the electorate is classified as solid partisans. Even in 2016, this group is not large enough a group to deliver a election-winning majority for a particular candidate, as Romney and Trump comprised about 55 percent of the state's electorate.

The second largest cluster of voters vote for the same party in congressional races but are more likely to split their ticket in state elections. This pattern is particularly noticeable among Republican voters, where for example 5 percent of Trump voters in cluster 2 split their ticket in the U.S. House but 15 to 50 percent of them split their ticket for the Democrat in their vote for the state legislature, sheriff, and county council. This cluster comprises about 35 percent of both Trump and Clinton voters, which makes them large enough a group to be pivotal even in a statewide race. The third and fourth largest cluster of voters vary in their voting patterns by year and party.

Many of these groups primarily abstain after voting for President, while a the fourth cluster among Clinton voters appears to be solid Republicans who only broke away from their party preference in the race for President.

Finally, the clustering differentiates between ticket splitting for particular offices.

Among 2012 Obama voters, for example, two out of its four clusters had substantial probabilities of splitting their ticket, but while the second largest cluster was most likely to split in the vote for State House, the third largest cluster was more likely to split in the office of Sheriff.

Therefore, despite one line of reasoning that would predict straight ticket voting to be more prevalent in down-ballot races where candidate specific information is scarce, election day voters tend to defect from their national party loyalty as much as, if not more than, national congressional races. These cast vote records reveal new patterns of voting behavior that have been not possible to measure in existing surveys and election returns.

To summarize, both Figures 3.2 and 3.3 exhibit the same general pattern. First, it is not the case that voters vote more straight ticket in state and county offices than they do for Congress. Most straight ticket rates in state legislative and county executive offices are lower than those for Congress, and especially so for sheriff and county council races. For example, while 94 percent of top of the ticket Republicans voted for a Republican congressional candidate, only 77 percent of them in contested sheriff elections voted for a sheriff. Accordingly, ticket splitting is more prevalent for the majority of these state and county offices. Roll-off is also higher further down the ballot: typically around 1 percent in congressional elections and 2 to 4 percent in down bal-

Difference in Means

Access to the full ballots allow some straightforward calculations to estimate how much ticket splitting is explained by incumbency. I first count the fraction of split ticket votes that were cast towards the incumbent. I subset the data to six offices with a sufficient number of contested contests in enough districts, and further subset to contests which featured an incumbent running for re-election against a major party challenger. After these restrictions, we are left with 566,232 split ticket votes (as defined in Figure 3.2) cast by 495,138 voters. In each of these split ticket choices where the voter could choose between the incumbent or the challenger, a clear majority of 69 percent voted for the incumbent.

Open-seat contests serve as an additional useful comparison because incumbency is not at play. In Table 3.2, I show the proportion of the straight ticket voting rates separated by the presence of an incumbent and the party affiliation of the incumbent.

If voters value the qualities associated with incumbency, we would expect to see the most same-party votes when (i) the incumbent is of the same party as the voter's top of the ticket choice, and fewer same-party votes when that (ii) entails voting against the incumbent. Finally, the rate of straight ticket voting when (iii) there is no incumbent should be lower than case (i) but higher than case (ii).

Consistent with those expectations, in all of the six offices covered in Table 3.2, straight ticket voting is highest when doing so coincides with voting for the incumbent. Among contested U.S. House races, 94 percent of voters whose party choice at the top of the ticket happened to align with the party affiliation of their U.S. House incumbent voted for that incumbent (column (i)). But when they did not align (column (ii)), only 87 percent of these voters voted straight ticket, indicating a split ticket to vote for the incumbent. The rate in open-seats where no incumbent exits (column (iii)) tends to fall in the middle of the two values for all offices. If voters did not value the qualities associated with incumbency, all three proportions for each office would have been the same. Instead, we see sizable differences.

Generalizability

One limitation of these findings is that they examine contested races in a single state. In Appendix A.4, I analyze a smaller set of cast vote records in two states -Maryland and Florida, and find similar patterns of higher ticket splitting in state and local offices. Other than that, several considerations suggest that the main implica-tions here are generalizable to most contexts in contemporary American politics.

South Carolina state legislative seats are one of the least contested in the country. For example, according to Ballotpedia, only 30 percent of state legislative districts were contested in the 2012 general election, putting the state's competitiveness index only ahead of Georgia and Massachusetts. But it is reasonable to expect that split ticket rates would be higher if parties contested more districts. Districts with no challenger tend to be those where the disadvantaged party's chance of victory is slim to begin with (Rogers 2015). Therefore, voters who value the correlates of incumbency should be even more likely to cross party lines if a disadvantaged party were to enter a lopsided race.

When extending to other states, the findings here suggest that ticket splitting would be less prevalent in down ballot races where candidate-specific information is sparse, two-party competition is high, and the incumbency advantage is weak. One might worry that South Carolina is an outlier in this regard: An uncompetitive state consisting of lopsided districts. But as Fraga and Hersh (2018) showed using congressional, statewide, and state legislative elections, it is rare for any single voter to reside in that sort of enclave. South Carolina is no exception. The long ballot and the high degree of district overlap in the U.S. electoral system all but ensure that most voters' long ballots feature competitive contests as well as uncompetitive ones.

Another concern when predicting patterns in other states is South Carolina's history as a Southern State, where Democrats such as Strom Thurmond switched to the Republican party in a massive realignment in the 1960s and 1970s (Mickey 2015;Key 1948, also documented in Table A.2). It is likely that some of the pattern here is driven by older voters who, like in the election of Dwight Eisenhower, voted for Republican national candidates but Democratic candidates in state and local candidates.

On the other hand, realignment is a common feature of a two party system and many other U.S. states outside the South experienced realignments, if not to the same degree. Moreover, the logic of the incumbency advantage and valence does not rely on such massive realignments. I now finally turn to the more speculative question of whether the dynamics of split ticket voting in state and local races documented here will eventually disappear in an era of increasing nationalization. Most studies describe nationalization as largely a top-down process (Abramowitz and Saunders 1998;Aldrich 2000). The electoral trends over the past 40 years also document a consistent trend towards Republican dominance in the state. But this realignment did not impact all levels of elections at once, nor did it progress at the same rate (Appendix A.2). Within the set of elections collected in my dataset, I do find an uptick in the rates of straight ticket voting overtime (Appendix A.3), but how these rates would change beyond the sample depends on the future party alignment in national politics, something beyond the scope of this paper. My results do suggest that the incumbency advantage is still a significant force in state and local elections and may delay the tides of nationalization that has swept congressional and gubernatorial elections.

Conclusion

The picture of the electorate that emerges from these analyses is one that votes largely along party lines, but still with important variations across offices and candidate's incumbency. A new dataset that provides an unprecedented view into voters choices showed that about seven out of ten voters in South Carolina vote a complete ticket, and in any given office, about eight to nine out of ten voters vote the same party as the President or Governor. Ticket splitting is especially prevalent in sheriff contests, and state and local contests are more varied in their level of split-ticket voting.

Should we consider the statistic that 80 percent of Trump (Clinton) voters voted for a Republican (Democratic) county sheriff candidate to be a large or small number? On the one hand, as I have suggested, this is smaller than what we would expect from a fully nationalized politics. A district in which 20 percent of voters is up for grabs is by most measures a volatile one. On the other hand, there are also good grounds to interpret party loyalty of such degree as too high. The traditional view of local politics has been that it is void of partisanship altogether, with recent work updating that view (Tausanovitch and Warshaw 2014;Bucchianeri 2020). Another study of close elections between Democratic and Republican sheriffs finds that party control does not cause changes in how sheriffs implement policy (Thompson 2019), suggesting that partisan splits in voter's supports for that office is disconnected from the policy preferences of the candidate.

Regardless of one's interpretations, the findings presented here would not have been obvious without empirical investigation. Extending the nationalization literature into state and local politics, one might have predicted that the rate of straight ticket voting to be equally high for all pairs of offices. And extending theories of partisanship and political communication, one might have predicted that, if anything, straight ticket voting rates would be higher in down ballot races than in national ones because voters would have less candidate specific information to inform their choices for county council than they would for the U.S. Senate.

The first part of the empirical findings do not lend support for these predictions, at least for the average voter and the average election. The second part starts to reveal why. Incumbents systematically netted more votes from party defection than challengers or contestants in an open race (Table 3.2), even after controlling for a voter's own party allegiance (Figure 3.5). This pattern is consistent with a model of elections with nationalized partisan candidates in state and local offices with differ-ent levels of experience or quality. In other words, even if we assume that nationalization has so thoroughly polarized candidates such that even candidates for state and local offices follow the national ideological platforms of their respective parties (Hopkins 2018;Shor and McCarty 2011), some voters know and care enough about nonideological aspects of the candidates such that they split their ticket.

By constructing the first database of cast vote records spanning an entire state across multiple elections, this study has overcome some common challenges researchers face in studying vote choice for state and local office. Cast vote records will prove valuable for understanding electoral behavior more widely. In addition to ticket splitting, they allow researchers to study vote choice in party primaries, ballot measures, and elections for non-partisan offices such as school board elections. Existing studies of these three types of elections are limited by the same sort of measurement problems that surveys and election returns have for studying ticket splitting, and therefore can benefit from wider use of cast vote records.

The findings of this paper are not without their limitations. Cast vote records reveal how people vote in state and local offices, but they reveal much less about the demographic characteristics of those voters. And partly because of this limitation of survey or demographic evidence, it becomes difficult to disentangle potential mechanisms underlying the findings, i.e., a valence advantage, candidate moderation, or multidimensional voting. Future research that combines cast vote records with precinct-level data could help distinguish more carefully the process through which voters form their preferences for state and local offices. This paper does, on the other hand, establish some baseline expectations for state and local elections in a nationalized politics. On election day, U.S. voters must make a series of choices with limited information beyond party labels. But after an accumulation of campaign outreach, media coverage, and information acquired through everyday observation, a considerable number of vot- them further by about 1 to 2 percentage points (to 5.9 points). * I acknowledge the support of NSF Grant 1926424 and thank Steve Ansolabehere, Andrew Gelman, Yair Ghitza, Lauren Kennedy, Jonathan Robinson, and especially Soichiro Yamauchi for numerous discussions on the findings related to this chapter. I thank Douglas Rivers, Eddie Mertz, and Brandon Bertelsen for sharing the summary statistics from YouGov's database to enable extensions.

Surveys continue to be the main way through which scholars study electoral behavior. As survey samples have become increasingly larger, social scientists have turned to estimating quantities at particular subgroups of the entire data (Broockman and Skovron 2018;Kalla and Porter 2020;Hertel-Fernandez, Mildenberger, and Stokes 2019). At the same time, the political survey community has also become more conscious about selection bias and unrepresentativeness in these data. Accurate subgroup estimates of voter behavior at the state and legislative district level are crucial for studies of electoral politics like the one I explore in this dissertation.

As ubiquitous as the use of pollster's survey weighting is, however, the construction of weights is still an open discussion in social science research for which little guidance exists. The observation that "survey weighting is a mess... the construction of weighting itself is an uncondified process" (Gelman 2007) still rings true. The situation has undoubtedly improved, with pollsters documenting their own complex weighting process (Ansolabehere and Rivers 2013) and review texts that connect weighting methods in a single framework (Caughey et al. 2020). However, many of the methods are still out of reach for applied researchers who want to adjust existing weights to their own subgroup of interests.

Moreover, the suitability of a set of weights is not a black-and-white issue. Because much of the validity of an estimated weight depends on the quality of imperfect data, an empirical accounting of how well a set of survey weights adjust a sample to various geographies is therefore necessary for applied researchers.

Methods for reweighting, and the estimation of synthetic population targets that are required to enable to such a weighting, has wide applications to modern survey research. For example, it is crucial for improving Multilevel Regression and Poststratification (MRP) models as well as non-MRP estimates. MRP combines the traditional study of shrinkage and partial pooling that is mostly concerned with variance reduc-tion with standard infrastructure for poststratification weighting that is mostly concerned with bias reduction (Gelman and Little 1997). Much of the recent research on MRP has exclusively focused on the former: improving the model that induces partial pooling in the outcome. It has held constant the poststratification table constant, often with off-the-shelf Census datasets that do not include the variables pollsters typically use. Improving the estimation procedure for population targets for subnational geographic units can benefit both traditional weighting and any MRP model.

Here I conduct a validation and outline an open-source method that encompasses all these aspects through a concrete example, the Cooperative Congressional Election Study (CCES). I discuss the sampling and small area problem in the CCES for congressional districts, where the survey sample for each district is only around 50 respondents. Specifically, I propose a workflow to construct a poststratification target that approximates the joint distribution of six standard variables: age group, sex, education, race, turnout, and congressional district (which are nested in states). A multinomial logit model with simultaneous calibration properties implemented by Yamauchi and Kuriwaki (2021) allows this joint estimation, which is more scalable and has better theoretical guarantees than existing attempts for using synthetic distributions in MRP (Leemann and Wasserfallen 2017;Ghitza and Steitz 2020). I then show how such a reweighting improves the estimates of vote choice at the congressional district level in the 2016 CCES, while off-the-shelf poststratification with only a few demographic variables does not noticeably improve the aggregate error of estimates relative to a simple raw average. Partial pooling alone, which precedes the postratificaiton step in MRP, also does not improve the overall accuracy of the estimates. The proposed workflow is open-source and draws from datasets that can be downloaded by a user-friendly interface (Kuriwaki 2021a). In an extension, I show how non-public data such as party registration statistics in voterfiles can be used to further

The Rise of Online Surveys and Calibration Weighting

As the typical sample size of data have grown larger through technological innovation, one might think that survey researchers are no longer befuddled by small sample problems. But this is not so for two main reasons. Even with large datasets, scholars have turned to estimating population quantities at smaller and smaller subnational geographies. Table 4.1 compares the sample sizes of two common datasets at the national level, state level, and sub-state congressional district level. Even with a survey like the CCES which is an order of magnitude larger than the typical national poll, there are fewer than a hundred observations from a given congressional district (which represents more than half a million people). Second, as sample sizes have become larger, response rates have also plummeted, raising the danger that the survey samples we do collect are less representative (Meng 2018).

This work contributes to recent literatures in political science and survey statistics that has arised to keep up with the technical realities of polling. Three bodies of work are particularly relevant. Applied examinations of calibration weighting, recent statistical innovations in estimating calibration weights, and the existing small area estimation literature in political science which has largely focused on MRP.

An overview of weighting methods appears in Caughey et al. (2020). They identify the problem of target estimation as a challenging task, for which "how best to approach the problem is still an open question and a subject of ongoing research." I provide such an extension in estimating synthetic population data for a turnout electorate. I also focus on the issue of small area estimation, a topic Caughey et al. only discuss in passing and leave for further research. A concrete description of poststratification in the CCES is given in Ansolabehere and Rivers (2013). However, their benchmarks to election results and benchmarks stop at the state level, where survey samples are large (about 1000 respondents) and the weighting specifically target demographic distributions at that level. In this chapter, I investigate smaller areas of geography that the pre-computed weights are not adjusted to.

Target estimation and calibration weighting is a broad field, featuring classic studies that have enabled now standard tools such as rake weighting (Deming and Stephan 1940). But statistical methods in this area are continuously evolving, seeking to improve the stability of estimated weights and adding more calibration constraints to an approximation of the propensity score model. Contrary to the canonical model of inverse probability weighting typically associated with survey weighting, "survey weights are not in general equal to inverse probabilities of selection" (Gelman 2007).

Instead, the population distribution that the weights target needs to be estimated itself, through a series of statistical imputation methods (Caughey et al. 2020). Because many of these constraints are not observed in practice, there is room for improved modeling (Ben-Michael, Feller, and Rothstein 2020;Zubizarreta 2015;Imai and Ratkovic 2014). This chapter draws from the insights that have recently emerged in this statistical research, summarized most recently by Chattopadhyay, Hase, and Zubizarreta (2020). The central idea is that the calibration estimation is an approximation to the true propensity model, and bias-variance trade-offs exist in choosing an optimal set of weights.

Finally, MRP is an increasingly common method for survey inference at small subgroups, especially in political science (Lax and Phillips 2009;Warshaw and Rodden 2012;Buttice and Highton 2013). While MRP is a general procedure that covers many of the practical issues in subgroup analysis, it is important to remember it is essentially "a modification of the conventional poststratification estimator" (Caughey et al. 2020, p.70) and its main innovation, the Multilevel Regression, does not directly address concerns for nonresponse bias. As I clarify in the next section, the multilevel regression stage of MRP uses a shrinkage method to deal with the high variance of small samples, but the identification assumption to validate this step is distinct from that of representativeness. Put another way, if the poststratification stage of MRP is biased or insufficient, so will MRP. MRP is also a data-intensive method. Practically all of the numerous studies that validate MRP or improve with machine learning methods innovate on the regression model and does not vary the post-stratification dataset.

Even those that do (Leemann and Wasserfallen 2017) propose fairly simple methods for extending target areas, either assuming away the ecological inference problem or applying iterated proportional fitting.

Methodological Foundations of Calibration Weighting

The fundamental problem in survey inference as well as the general idea in most survey adjustment methods is shown in of the data in this population is observed, and a sample is drawn through an unobserved selection mechanism to make inferences. The binary variable for selection, S i , is 1 if the individual ends up in the sample survey, and 0 otherwise. S denotes the set {i : S i = 1}. I use n to denote the sample size N i=1 S i . Suppose that the primary subgroup of interest is a subnational geography, such as congressional district, which I denote with the random variable A i ∈ {1, ..., J}.

The subgroups need not be geographic but can easily be demographic subgroups or the interaction of the two. Geographic subgroups are of common interest for political science research and a subgroup where election results can be validated from official election results.

We denote the area of interest by j, and the quantity of interest as the population average in each area µ j is therefore represented as

where

is the population size for area j. To introduce the calibration (or post-stratification) weights that are now the norm in online surveys, it is useful to begin with the classic inverse probability model. Without complete random sample the sample average is no longer an unbiased estimator of the population, but if the selection probability for every individual is known, then an inverse probability weighting renders the estimator unbiased. This is the core of most survey weighting approaches as well as the power of propensity score weighting in causal inference (Dehejia and Wahba 1999). The standard correction weight, w i , then is proportional the inverse of the selection probability π i :

In practice, the weight would be normalized by multiplying by a constant so that w is mean 1.

A common question in practice is if a weight for a national survey estimating a national population is valid when applied to a subset of the survey to estimate that subgroup population. In the ideal situation where the propensity score is known, the answer is yes. To see this, we can first see how the weighted proportion of the entire sample is consistent for the population mean. Because Y i is constant in the finite pop-ulation setting and π i is constant if observed,

after which Pr(S i = 1 | X i , A i )w i will cancel and generate µ. The constant rescaling of the weights combine with the denominator 1/n is designed to adjust to the correct scaling. While a ratio estimator like this are not unbiased in general, it is asymptotically unbiased. In practice the statistical bias is often small and with more data the estimator converges. Now if we target the subgroup quantity in similar fashion, we simply keep the conditioning of A so that

will also equal µ j . Here n j = N i=1 1 (A i = j, S i = j) is the survey sample size for area j.

However, modeling the selection probability is difficult. In the causal inference setting, the propensity scores estimated through, for example, a logit regression do not guarantee balance in the particular sample (Imai and Ratkovic 2014). In the survey setting, the population of S i = 0 is unobserved so running such a regression is impossible.

That is why any survey weights computed after a survey is run are estimated through calibration methods (Zubizarreta 2015;Chattopadhyay, Hase, and Zubizarreta 2020).

Calibration methods in one form or another compute a vector of weights that meet a balancing constraint the user defines comparing the target population and sample. Constraints can handle joint distributions through a distinct metric (Hainmueller 2012), but in surveys the weighted only feasible constraint are moment conditions for observable covariates. That is, we find a vector of weights such that 1 n i∈S w i X i = 1 N i X where the value of the right hand side is observed in the Census and other larger datasets. Because X covers multiple categorical covariates such as age group, education, and race, it is convenient to index all the poissle joint combinations of each level of the covariates and denote them as cells. Specifically, C i is a deterministic function of the covariate vector of respondent i and returns a number in {1, ..., C}.

Post-stratification weighting, rake weighting, iterated proportional fitting weighting falls in this broad umbrella of calibration methods because they follow this pattern as well (Caughey et al. 2020). Because online surveys through river samples generate cannot create design weights, the bulk of weighting that researchers encounter and can model fall under some sort of calibration weight.

Calibration weights implicitly model the propensity score, but inputs to calibration are almost always insufficient in the survey setting. Population moments that serve as the balancing conditions may not be observable for the covariates that are important in the propensity score. Despite the theoretical elegance of calibrating a survey to population constraints, in practice most population constraints are measured with error, measured from several years ago, or measured from yet another survey.

Conditions for a given geography may be even harder to obtain. For example, to calibrate the survey to the population distribution of religion, the CCES uses the national breakdown of religion reported from Pew's religion survey. However, Pew does not report breakdowns of religion by state, so the CCES cannot generate calibration weights that apply those constraints. These issues do not even touch on the estimation error due to functional form assumptions.

Once we frame survey weighting as an causal inference problem for observational data (Kuriwaki and Yamauchi 2021), the conditions that a calibration weight must satisfy to make the resulting area estimates unbiased is clear. The same qualifications to the validity of propensity scores apply here. Applied to each area separately:

Synthetic Estimation of Poststratification Targets

I propose the following procedure to construct a poststratification target that approximates the joint distribution of age group, sex, education, race, turnout, and Congressional District (which are nested in states). We start with the following datasets:

• The CCES survey data, which includes the outcome Y , and all covariates X and A which will be drawn in from other population datasets.

• The ACS estimates of population sizes at the congressional district level. At this level of geography, the ACS does not give a full joint distribution. We must rely on two separate tables. One that records the population counts of [age x sex

x race x CD] and another that records the population counts of [age x sex x educ x CD]).

Therefore, the main challenge is that the ACS only gives a three-way distribution of demographics while poststratification requires a single, fully joint table, and the ACS includes no data on party or turnout.

(1) Fit a multinomial logit bmlogit (Yamauchi and Kuriwaki 2021) predicting four categories of education using race, age, and sex with the CCES data, with the balancing constraint that within each CD, the estimated marginal proportions of education match the education margins reported in the separate ACS table.

( (3) Fit a logit model (again with bmlogit) predicting a binary indicator for turnout using the CCES data, where we use the indicator for voterfile match supplied by the Catalist (included in the public CCES dataset). The population constraint is given by the turnout rate among the voting age population, which can be computed from the ratio of total votes cast to the ACS estimates of the Voting Age Population. The process can falter when at least one set of survey data has at least one cell with zero observations, so here I use a simple specification of: (5) (optional) If party registration data is available and in the states where party registration is available, repeat the same process where the outcome in the multinomial regression is whether the voter in the CCES is a registered as a Democrat, a Republican, or anything else.

(6) Poststratify the survey estimates of the outcome to the resulting synthetic table.

If sample sizes for the resulting cells are too small, fit a regression model for the outcome, such as a multilevel model as in MRP.

Here, the balancing multinomial logit is a powerful population constraint. While a regular multinomial logit can fit the same sort of predictive model as in Kastellec et al. (2015), it is likely to simply propagate any bias due to unrepresentativeness into the resulting estimates. Yamauchi and Kuriwaki (2021) implements software to estimate the multinomial regressions as a constrained optimization problem, where an additional constraint that the marginal distribution of the estimated outcome must match a user-supplied population constraint. Users can set a tolerance value to control the degree to which the constraint is enforced relative to the best fitting model in the microdata.

Estimation of population targets is a rich literature of its own, and the approach I propose here is simple relative to other approaches that use proprietary data or software. For example, the CCES itself uses a sampling frame constructed by YouGov that also relies largely on the ACS (Ansolabehere, Schaffner, and Luks 2017). To this table, YouGov adds turnout estimates from the CPS and religion from Pew. However, this sampling frame is proprietary to YouGov and it is only calibrated to the state level. Ghitza and Steitz (2020) use the state-level ACS microdata to estimate onto individual census-tracts, while correcting for representativeness through a type of rake weighting. In contrast, an attractive feature of the proposed model is that it imposes a balancing constraint simultaneously with parameter estimation, and uses publicly available data and summary statistics.

Resulting estimates do not come for free. To overcome the small sample problem, the outcome model partially pools observations from multiple CDs and uses those parameter estimates to predict the outcomes in a single CD. However, this requires the assumption that the demographic predictors and the CD random intercept is sufficient to model the variations in the relationship between the outcome and predictors across the multiple districts (Si 2020). This becomes a classic bias-variance tradeoff, where pooling across districts induces bias but subsetting to specific districts in fitting the model suffers from large variance or even demographic strata with 0 observations. Another limitation of poststratification, including MRP, is that it cannot balance on important variables if its population distribution is unknown. This is an important omission for political surveys because partisanship is clearly heavily predictive of vote choice but partisan self-identification, the most commonly used measure of partisanship in surveys, is only measured in surveys themselves. Two other related measures of partisanship are vote choice from the past election and party registration where it is available. Incorporating lagged vote would require specific adjustments for voters who did not participate in the previous election. Incorporating party registration is a promising approach, especially for the CCES that includes Catalist's matched voter registration for every respondent. In this chapter, I use summary statistics of party registration in the 2016 election provided by YouGov and show that its incorporation indeed improves estimates. However, it is unclear to extrapolate this calibration to states where party registration is not recorded. The limitation is shared by virtually all methods for poststratification and is a topic of future work.

Empirical Assessment: Existing Weights

I test these strategies on the problem of measuring Donald Trump's vote share as a proportion of the two-party vote in the 2016 election. Because congressional districts cut across election reporting administrative units in complex ways, some care is needed to compute the ground truth all subsequent estimates will be compared against. I use values computed by Daily Kos (Daily Kos 2021).

The CCES is a survey of voting age adults, while the population of interest is those who voted. When estimating the outcome of interest, therefore, I subset the CCES to respondents who meet all three of the following criteria:

1. Those who responded to the post-election wave (82 percent), 2. Those who self-reported voting for either Donald Trump or Hilary Clinton after the election (76 percent of those who took the post-election), and 3. Those who matched to Catalist's voterfile as having cast a ballot for the 2016 General Election (56 percent)

This leaves a total of n = 28, 462 respondents, or 44 percent of the available 2016 CCES. When fitting multinomial models to construct the population target, I use all 64,000 respondents to match the coverage of the ACS.

I first assess how standard weighting that CCES includes applies at the subgroup level. Figure 4.2 compares these standard weighted estimates with population variables. I plot the raw proportions and weighted proportions side by side, and compute standard errors by the standard formula

where Y j is the proportion estimator (either the simple average or the weighted average) for the area of interest and n eff j is the effective sample size. For the unweighted case the effective sample size is equal to the sample size (n eff j = n j ), but for the weighted proportion it is computed with the Kish design effect correction:

This effective sample size can be rewritten as a function of the inverse of the sample variance of weights. It decreases as the weights get more variable.

The state level estimates in Panel (A) show the power of weighting. While 23 states have 95 percent confidence intervals do that include the actual result without weights, all but one (California) of the weighted state estimates include the actual election result. The root mean squared error (RMSE) of the set of estimates improves nearly three-fold. This is of course not surprising given that the weights were calibrated to statewide election returns. As for the national popular vote, the weights give an estimate of 49.5 percent while the raw average gives an estimate of 45.8 percent (Trump's two-party popular vote was 48.9 percent).

Empirical Application: Proposed Poststratification

There is no obvious way to visualize the result of a six-way cross-tabulation, but Figure 4.3 is one representation of the values resulting from the target estimation procedure. The procedure produces cell counts N cj for area j, where c ∈ {1, ..., C} indexes the multi-way table of categorical demographic variables. In this instant, c indexes the combination of age group (5 levels), sex (2 levels), education (4 levels), race (4 levels), and turnout (2 levels), so C = 320. Groupings were determined to match the levels of the ACS variables, and grouped together so that at least every state had one CCES observation of that level. The annotated point on the figure shows, for example, that each poststratification cell is around 0 to 2 percent of the estimated electorate. The estimated size is of course a function of the size of the group in the population. One interesting comparison is the proportions across the turnout and non-voting groups. Some CDs have relatively high levels of Hispanic representation, while in other CDs Hispanics comprise a relatively large group of the non-voting electorate. Non-citizens are included in the non-voting (voting age) electorate, which may explain these high numbers in Texas.

We cannot validate each of these estimates of the population quantities, given that it estimates a joint distribution of variables none of the population datasets can provide (Table 4.2). The method guarantees, however, that these estimates of the joint distribution match all population marginals. And instead of assuming that the distribution of covariates are independent and taking the product of marginals, I use individual survey data to assist in learning the joint distribution.

Weighting the outcome to this target population requires survey sample estimates of the outcome for each of the C • J cells. For each cell cj, denote the average of the outcome in the cell as Y cj . When cells are too fine such that n cj = 0 for some cells, we model a outcome regression with a shrinkage property to estimate these values, such as Y cj . As Figure 4.1 shows, the post-stratification estimator and the MRP estimator is simply the sum of these estimates reweighted to the estimated size of the population:

where µ PS j denotes the post-stratification estimator for area j and µ MRP Compared to the direct, small-sample estimators of Figure 4.2, the smoothed estimators in Figure 4.4 feature tighter credible intervals and modest reductions in the discrepancy between the true vote share. The first estimates show that simply partially pooling the survey data by congressional district does not lead to a improvement in the aggregate error in this case. However, poststratifying on a three-way demographic table appears to have no clear reduction in aggregate error. The estimates improve only in the final panel, when a four-way table is modeled so that surveys can be re-weighted to the joint distribution of race, education, age, and sex by Congressional district, within an estimated electorate instead of the all voting age adults.

Figure 4.5: Benefits of Additionally Modeling Poststratification Targets

Note: A comparison of modeled estimates using the same survey data from seven party registration states: Arizona, Florida, Iowa, Maine, North Carolina, New York, Oregon. The final model (6) uses a synthetic population target that also includes party registration breakdowns jointly with other demographic variables.

Error bar show 90 percent credible intervals from 2,000 MCMC samples.

New York, and Oregon. In order to provide a valid comparison of methods, I recompute the standard MRP and direct estimates in those same states so the underlying data is held constant. Models (1) and ( 2) repeat the finding from Figure 4.2 that using a weights cali-brated with coarser constraints may not help and even hurt direct subgroup estimates.

Model (3) -( 6) are all MRP models following the fnding with all states. Model (3) applies minimal partial pooling without any demographic poststratification, as in the first model of Figure 4.4. We see that the estimates of (3) on do not differ on aggregate from the raw averages. This is one hint that much of the improvement due to MRP is the final poststratification stage rather than the first outcome modeling stage.

Models ( 4) -( 6) vary the underlying target populations in increasing complexity.

Model ( 4) is a simple baseline, which, as in Figure 4.4, only the ACS table measuring

[age x sex x education x turnout x CD] is used. There is no apparent improvement in the aggregate error with this simple MRP. We only start to see improvements in model ( 5), which uses the proposed workflow of this chapter and creates a synthetic table of four demographic variables and models turnout, both through our balancing multinomial logit model. This improves the root mean square error from the raw average but only by a tenth of a percentage point or so.

The most noticeable improvement comes from model (6) which finally incorporates the party registration breakdown in the electorate. This model, again, ensures that the weighted proportion of registered Democrats and registered Republicans in the survey sample match those reported by the voterfile, for each congressional district.

The aggregate error decreases by about 2 percentage points compared to the raw average or the partially pooled estimators. It decreases by another percentage and a half, to 4.5 percentage points, after aggregated vote share is included as an aggregate, continuous variable in the outcome model. The strength of the party registration variable is reasonable given that the outcome of interest is voting for a Republican candidate.

Introduction

Finding and labeling voting blocs are ubiquitous in election analysis. Theories of political behavior, especially those explaining electoral change, cluster voters into interpretable prototypes and assign them labels such as core and periphery, standpatters and floating voters (Campbell 1960;Hill and Kriesi 2001;Key 1966;Smidt 2017). Almost instinctively, political consultants, journalists, and election observers latch on to labels such as the "soccer mom" or the "white working class" to construct narratives about voting behavior, even if the label may not have a uniform definition or may not be the best statistical predictor (Carroll 1999;Cohn 2019a;Carnes and Lupu 2020).

In particular, a recurring voting bloc in modern accounts of the US electorate is the "swing voter" -a pivotal (and perhaps dwindling) group of voters who are indifferent between either party to a first approximation and are therefore considered persuadable.

But existing approaches to this grouping exercise in political science are either based exclusively on pre-defined groupings, or on a series of comparisons between votes in pairs of offices. The former risks not fully leveraging the information contained in the data, and the latter simply becomes intractable with high-dimensional large-N datasets with an exceeding number of possible voting patterns.

In this chapter I offer an alternative framework: a clustering algorithm that summarizes complex individual-level voting data to interpretable blocs using a probabilis- this model to voting data,foot_17 perhaps due to concerns about interpretability and lack of substantive theory. My proposed approach has three methodological features on this point. First, by using a clustering algorithm as opposed to the more standard regression approach, I can properly leverage the information that is contained by the same voter making repeated vote choice decisions between Republicans, Democrats, and abstention (for example) in multiple offices. Second, it embraces the principle of unsupervised learning more so than ideal point models. This entails targeting the parameters of a simple model that best fit the data, instead of modeling the behavior of known or presumed voting blocs. Third, my statistical approach is grounded on a probabilistic model of political behavior (Ahlquist and Breunig 2012), instead of simply grouping observations that are close on a particular distance metric (Müllner 2013). Analysts still pick the number of clusters to estimate, and must use substantive prior knowledge to guide the interpretation of each cluster. In summary, the main virtue of the clustering approach is that it is a principled framework to leverage the information in high-dimensional voting data.

In the remainder of this paper, I describe the clustering approach and what it can reveal about swing voters in American Politics. In the models and methods section, I set up the model and show how I derive an EM algorithm to measure the parame-

ters in an open-source program, clusterCVR (Kuriwaki 2021b). In the process, I highlight the assumptions and implications of this statistical approach. I then estimate the same parameters and use a visualization that highlights the estimated parameters in a more interpretable fashion. In particular, I find in my two applications that swing voters form a clear voter type, even though the office-specific patterns of how they swing varies by the type of office and experience of the candidate. (Imai and Tingley 2012). Clustering is even more widely used in fields such as psychology and marketing (Fiske et al. 2002;Wedel and Kamakura 2000). But they are still less common than regression based methods in political science, so I start with the logic of the basic model that I implement in an open-source R package, clusterCVR, and finally respond to commonly recognized limitations of clustering as a method to analyze political behavior.

Main Logic of the Clustering Model

Instead of defining clusters of voters by predictors of vote choice such as race and education, this paper starts with the case in which we only observe the outcome of votes. Let Y be the N × J vote matrix of voters i ∈ {1, ..., N} voting in offices j ∈ {1, ..., J}. Each vote Y ij takes on a discrete, unordered categorical value ∈ {0, ..., L}.

In this paper, we focus on vote data where we have coded each vote as and the reality that in some offices in some districts are not contested by a major party further increase the number of considerations. Focusing on a pair of offices (e.g.

the President and US House, as in Burden and Kimball (2002)) effectively discards valuable information, while enumerating each potential voting pattern (Beck et al. 1992) reduces interpretability.

To address these issues, the clustering approach assumes that each voter i belongs to one of K "clusters", or latent groupings. We denote this membership as a random variable, Z i , and index clusters by k ∈ {1, ..., K}. Importantly, although different individuals may belong to different clusters, there is no differentiation of clusters within an individual even across different offices.

It then posits the following model of vote choice that incorporates our two key parameters of interest. The prevalence of cluster k in the population by π k , where π is a K-length proportion that sums to 1 ( K k=1 π k = 1), so that Z i ∼ Categorical(π).

(5.1)

Next, to characterize each cluster, µ jk represents the latent propensity for any member of cluster k to vote for a particular option in office j, so for a given cluster and given office,

For example, a political campaign may be interested in the size of the swing voter bloc and how likely that bloc is to split their ticket for a particular candidate in office j. In this case, if we had estimated two clusters, and set aside the first cluster for staunch partisans and the rest for potential swing voters, we would want to know the quantity π 2 (the size of the bloc) and µ 2,j,split .

This modeling choice maps to a theoretical notion that ticket splitting is a probabilistic function of being a swing voter. This stands in contrast to existing approaches, which is more deterministic. Although splitting one's ticket may be a sufficient indicator of being a swing voter, it is not a necessary condition because a swing voter that is indifferent to either party should have roughly equal probability of choosing one candidate over the other (Larcinese, Snyder, and Testa 2013).

Respondent Level Covariates

It is natural to posit that certain demographic groups are systematically more likely to be in particular clusters. Clustering models can incorporate such auxiliary data about the respondents in a straightforward manner by modeling the cluster assignment as a function of covariates. Suppose we have an indicator for whether every voter is an ideological moderate. The spatial voting model would predict that this indicator to positively correlate with assignment into a cluster that tends to have high rates of ticket splitting.

Respondent-level covariates like these can be incorporated into the EM algorithm's M-step by regressing the expectation of cluster assignment on covariates in what is essentially a weighted multinomial logit model. Formally, we replace 5.1 with

where X is a N × P matrix and γ k are P + 1 coefficients and an intercept. As this shows, this requires we let π be a matrix with N rows. When summarizing the data in subsequent analyses, I take the average mixing proportion for each cluster as an aggregate measure. A model with and without such covariates should produce roughly similar cluster assignments because we still infer these from the votes. But incorporating covariates can help stabilize the algorithm, and the values of the coefficients γ provide useful substantive information for interpreting cluster membership. The ECM algorithm implemented by Yamauchi (2021) makes this step fast enough to be repeated at each step in the main EM loop.

Varying Choice Sets Many elections for state and local offices are uncontested, which means that a voter still makes a choice, but from a limited menu of options.

These different settings require modelling varying choice sets (Yamamoto 2014). While existing discrete clustering models rule out this possibility and therefore require analysts to drop data that includes varying choice sets, I model these separately, with an independence of irrelevant alternatives (IIA) assumption to share information and parameter values across observations.

The added complication is that the vote choice probability must now be modeled as a function of data that varies by the choice set. Let Y ij denote the set of values that are available to voter i in option j. Such information would be clear from the candidate filings in that district, and so are directly observed. We then posit that the choice probability is generated from a ratio that is relative to the available choices for a given voter, as in a standard multinomial logit. Formally, we parameterize equation 5.2 as

where ψ is a scalar that represents the intensity of preference for option ∈ {1, 2}

relative to = 0 (abstention). To identify the MLE, for which no closed-form equation exists, I use an optimization of the likelihood with varying choice sets. The functions and derivations are provided in Appendix C.

The IIA allows us to estimate the same parameter across voters in slightly different electoral contexts, but it can be a substantial assumption to add. The multinomial probit does not explicitly make this assumption, but replaces with a distributional assumption on the errors. In general, the validity of IIA is difficult to test because each respondent's rank ordered preferences are not observed in the cases relevant here. Yamamoto (2014) shows how one can model the varying choice sets through a choice-set specific intercept and relax this assumption. Although the method in this chapter does not conduct this modeling, there is a parallel in the approach. Yamamoto's method estimates separate effects for each choice set, whereas this method partitions voters into clusters and estimates separate choice probabilities within each cluster.

Limitations and Issues of Interpretation

Before moving to empirical analyses, the implications and limitations of this methodological approach are worth some comment. While clustering algorithms are widely used in fields including computer science, marketing, and psychology, this is not the case in political science or economics. It is important to consider why.

First, the type of data this method can handle are restricted to datasets where (i) the outcome measurement is categorical and (ii) come from the roughly same choice set across offices. 2 For example, it would not be possible to analyze a set of variables that includes vote choice and numerical responses. Similarly, the model also cannot handle ballot data where some variables are partisan offices (Republican, Democrat) while others are nonpartisan offices or referendums (Yes, No), unless the analyst is willing to assume that voting "Yes" on a particular referendum represents the same underlying event as voting for a Republican or Democrat. For such cases of mixed ballots that violate (ii), an ideal point model will be more appropriate because it targets a single dimensional preference estimate and maps different votes to a single space.

For cases that violate (i), one must turn to other clustering methods like k-means for datasets with only continuous outcomes, and more involved models for a mix of continuous and categorical outcomes.

On a more important theoretical point, clustering algorithms almost always require the user to pre-determine the number of clusters K to model from the data. Therefore, one might worry that substantive findings from data may change wildly by the number of clusters. There do exist methods to pick the optimal number of clusters based on measures of model fit (Fraley and Raftery 1998) -essentially functions of the observed likelihood attempting to account for overfitting.

But we do not need to believe that there is one "correct" number of clusters the analyst has to identify in order for clustering analyses to be useful. As Broockman (2016, p. 207) argues, voter's preferences are likely formed by hundreds of small issue "dimensions", even though each one may not incrementally improve model fit.

Whether one models 2 clusters or 3 clusters from the data is not a claim about the analyst believing that 2 or 3 dimensions are enough to explain voting behavior. Instead, this method can be thought of as a principled way to summarize information and characterize prototypical voting patterns given the user's chosen level of granularity. Substantive theory, rather than only a statistical information criterion, should guide the choice of the number of clusters.

A related concern about unsupervised learning methods is that interpretation of each cluster is arbitrary. Examining the correlation of covariates with estimated clus-ter assignment is a useful way to uncover some interpretation. But the analyst must also bring some of their own substantive knowledge for this clustering algorithm to be useful. Indeed, model output should not be interpreted as anything more than as a summary of the data based on a simple probabilistic model of vote choice. In the same way that there is rarely a single "correct" number of clusters, it is actually reasonable to pick the number of clusters so that it reveals clusters whose estimated parameters µ match the theoretical quantity of interest. In my applications, the main quantities of interest are the size and voting patterns of swing voters, which the model parameters directly target.

Of course, one must start somewhere. One reasonable initial choice is K = 2, which is the simplest case and also has parallels to many theories like the black-white model or core-and-periphery. Or, one can start by setting the number of clusters to the number of response options there are. This allows the data to cluster into homogeneous response-specific clusters, if that is the underlying pattern. In the context of the core vs. swing model, one might posit that voters can be partitioned into swing voters who split their ticket regardless of the office, abstention voters who undervote regardless of the office, and so on. This is a useful null hypothesis that I test on the data, and ultimately reject.

Application to Cast Vote Records

The clustering algorithm is well suited to glean patterns from large datasets of anonymous ballots. I first illustrate the insights from the clustering approach by analyzing ballots from the Florida 2000 election, which were originally analyzed in an ideal point framework.

Both political scientists and election administrators use ballot data to understand voter behavior and ballot design, but the high-dimensional and large-N nature of these datasets makes analysis challenging. The estimated parameters offer a straightforward summary of a vast amount of voters exhibit different patterns of ticket splitting. Cluster 3, which is more likely to split their ticket than stick to their Senate choice, comprises 28 percent of Nader voters but only 11 percent of Bush voters and 8 percent of Gore voters. These findings are roughly consistent with the original paper's findings based on a IRT model, namely that Nader voters were not in fact predominantly "left" of Gore, and so it is not clear if Nader handed the Presidency to Bush by running. For example, Nader voters only supported the Democratic Senate candidate 60 -40.

The picture of the electorate that emerges from these analyses is one in which 60 percent were party loyalists and about 5 percent roll off, but where a quarter of the vote can be considered as a reasonable swing bloc. Further, the clustering model's pa-rameter estimates in both cluster 2 and cluster 3 help refute the null hypothesis that voter types vote straight regardless of the office. While a majority of voters were consistently straight partisans, the rest vote differently, with a particular difference between Congressional, state, and local offices. Across all states, we find that the majority of the electorate are straight ticket voters, who are all but certain to vote for the same candidates as their party. But the proportion of these voters ranges from close to 95 percent in Minnesota to nearly 60 percent in Massachusetts. The remaining two clusters appear to contain various types of ticket splitters.

Conclusion

In this chapter, I introduced a clustering model for election observers to draw insights from large-N , high-dimensional data. I outlined a simple model of vote choice for partisan races on the long ballot, and offered some guidance on how data analysts should interpret the model estimates of latent quantities.

In an application to the crucial vote in Palm Beach County in Florida, the clustering method found that a majority of strong partisans (although not a super-majority), about 20-30 percent of potential swing voters who split disproportionately in state and local offices, and about 8 percent of rolloff voters who still voted for their members of Congress. In the second application to survey data in 2018, the method reveals that about 80-90 percent of voters who identify with one party are straight ticket voters.

But popular Governors and some popular Senators drawing voters across the party line and effectively forming blocs that deliver their re-election.

The statistical model used here can benefit from several more additions in the future. First, one can include choice-specific covariates such as candidate ideology, candidate incumbency, and candidate gender, directly into the estimation. These coefficients are widely analyzed in multinomial logit models of consumer choice but a faster algorithm to solve such models must be derived to incorporate them into a EM algorithm for large datasets. Modeling district-specific characteristics as random effects by positing that they are drawn from a common distribution is another possible feature to add to the modeling process, although estimation of such models in multinomial regression also remains an active area of statistics research (Linderman, Johnson, and Adams 2015).

I have shown here that a straightforward application of a clustering model can be applied to illuminate patterns and identify groups from complex voting data. As I have discussed, these tools should not be a substitute for substantive theorizing and interpretation, but they can facilitate discovery, provide a more principled measurement of size and voting propensity, and improve theory building by providing databased guidance.

in each touchscreen to the voter, only the chosen candidate's name appears in the log. I merge the party affiliation to each name chosen and, given the purposes of this study.

One of the more difficult tasks in data processing is to determine which races were available to which voters' ballots, and whether or not the race was contested. The combination of different legislative, school, and special purpose districts leads to a proliferation of different ballot styles (i.e., a layout for which contested for a given voter). Each entry in the logs contain a precinct identifier as well as a ballot style identifier unique to each precinct. Although there are around 2,000 to 2,200 precincts in each general election, there are at least 5,000 different ballot styles.

3. To infer the layout of each style, I aggregate the individual logs from the bottomup. For each precinct and ballot style combination, I tabulate the votes cast for each candidate. When working with contests for offices that held elections for only a subset of voters, I denote that this office did not exist for a given precinct -ballot style if no voter in that set cast a vote for the office. This way, I distinguish abstentions from the lack of existence of the contest.

One side-effect of this procedure is that absentee votes are not counted, because the voting machine codes them with a virtual precinct at the county-level, thereby effectively erasing information about the precinct of the absentee ballot. Until 2018, South Carolina voters had to be over 65 or have an "excuse" for not be able to vote on election day to apply for an absentee ballot. In the five general elections studied here, 17.7 percent of the 8.4 million ballots cast were absentee ballots.

4. I then aggregate votes at the district level, and declare a district as contested if votes for both the Republican and Democrat exist.