## Introduction

“In addition to winning the Electoral College in a landslide, I won the popular vote if you deduct the millions of people who voted illegally”
- Donald Trump, via Twitter, November 27, 2016

With recounts underway and Republican President-elect Donald Trump publicly alleging voter fraud involving “millions of people who voted illegally,” the credibility of the 2016 General Election is under intense scrutiny. Although the rate of voter fraud in American elections is extremely low, this did not prevent Trump, his vice presidential running mate, and others allied with him from claiming during the campaign, during early voting, and now post-election that Trump might be the victim of fraudulently cast votes, either at the polls or via mail. Trump’s and others’ many campaign-period claims about fraud and assertions that the 2016 election is rigged motivated us to develop a project on voter fraud, the end product of which is this report.

The rationale behind our project is as follows. Academic investigations of voter fraud are typically time- and data-intensive, and it is hard to imagine that a truly comprehensive and definitive study of voter fraud in any presidential election could be carefully executed prior to deadlines imposed by institutions such as the Electoral College. Given severe time constraints in conjunction with constraints on the sort of data that is typically available immediately post-election, during the summer of 2016 we wondered, what could we as academics say about election fraud in the aftermath of the then-upcoming 2016 presidential election? Our particular concern was that Trump might suffer a close loss in the presidential election and react by leveling widespread accusations of voter fraud that, in such a parable, could explain his defeat at the polls. As of last summer, we believed that accusations in this vein would only be politically germane in the event that the 2016 presidential election was close, and we thought it unlikely that a close victory by Trump would spur allegations that the final election outcome was tainted by fraud.

There are a variety of ways that one could initiate a post-election voter fraud investigation. We wanted our approach to be national in scope and, most importantly, not conditioned on a particular post-election fraud claim. In addition, our approach needed to be feasible given data typically available in the immediate aftermath of a presidential election. We did not want to rely on comparisons between election returns and pre-election polls or exit polls because comparisons of polls and results are inevitably confounded with questions about sampling frames and representativeness—it was important to us that our analysis not depend on polling quality. With these exigencies in mind, we assembled an extensive county-level data set on historical election returns and other county features. Our plan was, starting on the evening of November 8 and as returns began to trickle in, to seek a rationalization of the 2016 presidential election using tools typically applied by academics studying federal elections. We would then ask, do we observe across the country large deviations from our rationalization in a way that is redolent of the types of voter fraud that Trump regularly cited during this presidential campaign? Did we observe, for example, evidence at the county level that the presence of certain classes of non-citizens was associated with decreased support for Trump, as Trump himself intimated might happen? If so, this in principle could indicate the presence of fraud. If we observed nothing like this, we would be more confident—albeit not completely confident—that voter fraud in the fashion envisioned by Trump did not play a pivotal role in the 2016 presidential election.

Our research project is not an audit of the 2016 election in which one might randomly sample cast ballots for inspection and then compare results from the sample with actual election returns. Rather, our goal is to be able to comment on the sort of campaign-period fraud accusations, leveled by Trump, that, if true, would have taken place on a truly massive scale (i.e., millions of votes), a scale that could have tipped an otherwise not-close election. Our county-level approach does not have any power over minor fraud, something on the order of a single voter casting two votes for one candidate. All instances of voter fraud, regardless of scope, should be considered troubling, but not all frauds are pivotal. In the course of this project, we decided to focus solely on the possibility of massive and widespread fraud because this sort of fraud, in principle, has the potential to be pivotal and because, as the Republican nominee for president, Trump’s assertions carried weight.

Given extant academic literature and journalistic accounts of voter fraud, some of which is discussed by Goel and colleagues in their study of double-voting (they uncover almost none), we anticipated in the summer of 2016 that our research project would find essentially nothing. That is, we expected that there would be no evidence of the type of extensive voter fraud that was posited repeatedly by Trump and his allies. Non-findings are important when claims of election fraud are rampant. Moreover, and as is becoming clear in social science more broadly, conditioning a research project on the assumption that one would develop a positive finding is not appropriate.

In light of the high stakes inherent in any presidential election, not to mention the vitriol of the 2016 contest, we moved forward with our study during the summer and early fall of 2016. Trump won the 2016 presidential election in a way that was not particularly close, and we believed that this obviated the need for a report like ours to combat a recalcitrant loser. Instead, we face a recalcitrant winner. There are, however, some concerns regarding abnormalities in three Midwestern states, namely, Michigan, Pennsylvania, and Wisconsin. Moreover, Trump himself as recently called attention to three states, California, New Hampshire, and Virginia, where he believes massive voter fraud has occurred. We address these two groups of states later in this report even though, strictly speaking, they fall beyond the purview that initially motivated our research.

Being written within three weeks of the 2016 presidential election, we recognize that the analysis that follows is rough in many ways, not the least of which is its focus on aggregate units—counties—across the country. Ideally, we would like to be able to conduct an analysis on every individual voter, identifying those voters who voted illegally in some fashion. However, data for such an individual-level analysis is simply unavailable to us. As a result, we draw inferences about voter fraud using aggregate election returns.

We expect that other scholars may investigate potential avenues of presidential election voter fraud in the upcoming weeks and months. To the extent that this happens, and we hope it does, our results will complement fraud studies that focus on differences between exit polls and observed election returns as well as studies that analyze unusual patterns in digits based on the work of Mebane, Mebane, et al., Beber and Scacco, and others.

## Research design

Our research design consists of two main steps. In the first, we provide a rationalization of the 2016 presidential election. The reason this step is important is because our analysis of fraud in the sense of Trump’s allegations requires that we establish a baseline for expected Trump vote across the country. As will be clear shortly, for most of our analysis we use Republican presidential candidate Mitt Romney’s vote share from 2012 as our baseline. In the course of inductively developing our rationalization, we present a series of plots and arguments that together offer a picture of the county-level features associated with votes cast in the 2016 presidential election. In the second step, we look for deviations from the rationalization we have developed and in particular for deviations consistent with Trump’s fraud allegations offered during the 2016 presidential campaign and thereafter.

It is important to recognize that we are not offering a general approach aimed at detecting all types of voter fraud in presidential elections. Notwithstanding recent concerns that focus on voting in Michigan, Pennsylvania and Wisconsin, and then later on California, New Hampshire, and Virginia, we are concerned primarily with Trump’s campaign-period allegations in particular and whether there is evidence in favor of them.

This said, our second step considers four types of voter fraud:

• Non-citizens casting votes. We have already noted allegations made by Trump that non-citizens might play a role in the 2016 presidential election.
• Individuals casting votes in the names of deceased individuals. This claim was made by Trump as well.
• Fraud carried out by election officials. During the campaign, Trump alleged on numerous occasions that the 2016 presidential would be rigged. One interpretation of this claim is that the officials who collectively run the 2016 General Election are biased against Trump, and we explore this possibility by looking for patterns in the timing of election returns that were promulgated on November 8 and thereafter.
• Fraud associated with vote tabulating and electronic voting in particular. To the best of our knowledge, Trump and his allies did not argue specifically during the 2016 presidential campaign that issues with tabulating machines would lead to voter fraud directed against him. However, given the extent to which electronic voting is often blamed for election problems in conjunction with Trump’s regular invocation of the word “rigged,” and in light of recent concerns regarding voting in Michigan, Pennsylvania, and Wisconsin, we believe that it would be remiss of us not to consider, albeit briefly, voting machines as possible sources of fraud in the 2016 presidential election.

## Should we take Trump’s voter fraud allegations seriously?

Before we consider the two steps that jointly constitute our research design, we consider openly the question as to whether we, or anyone else for that matter, should take Trump’s allegations about voter fraud seriously. The existence of this document is evidence of our affirmative answer to this question, but at the same time we recognize that there may be countervailing opinions on this subject.

We can envision two arguments against taking Trump’s claims seriously. One, the evidence on voter fraud in American federal elections is clear: there is essentially none. Despite this fact, many Americans perceive voter fraud as somewhat common. The historical non-existence of fraud certainly militates against an extensive post-election fraud analysis in 2016. Second, and related to the first point, this report bears some risk of lending credibility to serious and potentially inflammatory fraud claims that are otherwise without merit.

Regarding arguments in favor of a post-election fraud analysis in 2016, there are several. First, simply because there is no history of massive voter fraud in American presidential elections does not imply that massive fraud was not present in 2016. Second, while studying what many consider to be outlandish claims does risk lending credibility to said claims, we hope that this risk can be mitigated through careful writing. And, third, if the 2016 election had been close, we suspect that there is some subset of the population that could be swayed by scientific reasoning on fraud as opposed to arguments made without reference to election returns. Another way to put this is as follows. We can divide Americans who were truly concerned about a potential Trump victory being stolen via voter fraud into two groups, those whose opinions on the prevalence of voter fraud are not a function of evidence and those otherwise. Insofar as the former group is not likely to be persuaded by anything we or others write on the subject of voter fraud, it behooves us to focus on the latter. We do not know how large this group is, but members of such a group, along with any other individuals who in principle can be persuaded by evidence about voter fraud, constitute a key audience for our project.

Henceforth, we treat Trump’s allegations about massive voter fraud seriously and without prejudice.

## Step 1. Rationalizing the 2016 presidential election

Our ability to ascertain whether there was widespread voter fraud in the 2016 presidential election in the sense of Trump’s allegations depends on a baseline expectation for either Trump or Clinton vote share. We focus primarily on the former because it was Trump who was responsible for the voter fraud allegations that have motivated our work. And in much of what follows, we compare at the county level Trump vote share from 2016 to Mitt Romney’s vote share in 2012. Henceforth, when we say Trump vote share, we mean the fraction of the so-called two-party vote that was cast for Trump, i.e., total Trump votes divided by the sum of Trump votes plus Clinton votes. A similar comment applies to Romney vote share. Broadly speaking, this means that our study of Trump’s vote share is for the most part a study of the difference between Trump’s vote share from Romney’s vote share. The objective of our rationalization, then, is not generating explanations for Trump per se but rather for the extent to which Trump’s share exceeded or fell beneath Romney’s share. Implicit in this idea is that there is a base of Republican candidate support in each county in the United States and that deviations from this base are conceivably important in the matter of voter fraud and also in the sense of understanding trends in American voting behavior across presidential elections.

Our historical election data are from Dave Leip’s Atlas of United States Presidential Elections, and returns from the 2016 presidential election are drawn from the Associated Press Election Services. We drop all counties in Alaska, where election results at the county level are not available. We also exclude Kalawao County, Hawaii, which is not reported by the Associated Press, as well as Oglala Lakota County, South Dakota, which experienced changes to its Census coding after 2010. Therefore, out of the 3,142 counties and county equivalents in the United States, we conduct our analysis on the remaining 3,111. Our calculations were carried out using the R statistical computing environment, and this document was produced using R Markdown.

### Map of Trump-Romney differences

We start forming a rationalization of the 2016 presidential election by considering a county-level map that displays differences between Trump’s vote share and Romney’s vote share. In the map, red counties are those where Trump gained support over Romney with darker red connoting greater relative support. The presence of blue indicates where Trump’s vote share was less than Romney’s, and darker blue is consistent with greater relative Romney support.

The map shows that Trump was very successful in the upper Midwest: Minnesota is quite red, for example, as are most of Wisconsin and most of Iowa. In addition, there is a band of Trump support in New England that starts in Maine and heads south west, maintaining some distance from the coast. Moreover, Trump did well compared to Romney in parts of the West, notably in Nevada but for the most part not in Utah, where it appears Romney had particular support from the large Mormon community that resides there.

Perhaps the most notable pattern displayed by this map is the difference between urban and rural locations. Metropolitan and urban centers across the country seem to have shifted away from Republican support (in the Midwest, for example, Chicago is shaded blue in the map, as are Detroit, Milwaukee, and Minneapolis-St. Paul) while less-urban counties in the periphery shifted in a pro-Republican direction. In other words, our map seems to suggest that Trump managed to gain an advantage in 2016 by securing the votes of the electorate living outside of major city centers. However, we must be cautious in drawing conclusions about the 2016 presidential vote simply by eyeing a map. A county-level choropleth like ours tends to over represent conditions in non-urban counties, which are geographically expansive and hence relatively visible but at the same time lightly populated.

Our county-level map does not have much to say about voter fraud per se, but the geographical patterns in it suggest that Trump’s vote share in the 2016 presidential election was fairly systematic. Had his support been scattered willy-nilly throughout the United States, our exercise in creating a rationalization for his victory would be ever more challenging.

### Trump-Romney differences by race

As is well known, the areas highlighted in the above map have notable racial profiles. Parts of New England are heavily White as are sections of the Midwest, and it is fair to say that the matter of race received significant attention during the 2016 presidential campaign. For these reasons, we now break down our county results by race.

We say that a county is majority Black or Hispanic if the percentage of its citizen age voting population that is Black or Hispanic is at least 50%, and for the purposes of this report citizen voting age population (CVAP) is defined using 2010-2014 American Community Survey five year estimates. With this in mind, we plot Trump’s vote share from 2016 against Romney’s vote share from 2012, and the two figures above focus on majority Hispanic (left) and majority Black counties (right). Both plots include 45-degree lines, and the size of county dots correspond to the total number of presidential votes cast for Trump and Clinton.

In both panels above, many points lie close to their respective 45-degree lines. Broadly speaking, this indicates that Trump and Romney received very similar vote shares in majority Black and Hispanic counties across the United States. This does not mean, to be clear, that Blacks and Hispanics had an affinity for Trump. Rather, all that the two panels posit is that, roughly, Black and Hispanic counties gave similar support to Trump as they had to Romney four years prior. This is as we might expect given general patterns of consistency across the American electorate, and indeed were we to make Black and Hispanic county plots that depicted Romney vote share against John McCain share (McCain was the 2012 Republican presidential candidate who ran against, and lost to, then-incumbent President Barack Obama), we would see patterns similar to what we see above (plots available from the authors).

What about majority White counties? To address this third set of counties, consider the figure above. Although this figure is laid out in a similar way to the previous Black and Hispanic figures, its implications differ dramatically. Namely, the majority of counties pictured in it lie above the figure’s 45-degree line, and this means that, in majority White counties, Trump’s vote share tended to exceed Romney’s vote share. This pattern is illustrated by the figure’s blue smoother (a generalized additive model, like all smoothers in this report except those explicitly noted to be linear), which lies above the 45-degree line once it reaches counties that voted for Romney at a rate of at least 40%, approximately. This smoother, like others in our report, is weighted by the number of votes cast per county for the two parties. Although not shown here, a similar figure that plots Romney’s vote share against McCain’s shows that Romney received a greater vote share than McCain in majority White counties (plot available from the authors). However, Trump-Romney differences in White counties are more pronounced than corresponding Romney-McCain differences.

We have thus far argued that race appears correlated with Trump’s vote share advantage over Romney’s. Three points about this conclusion are worth noting. First, the association of race with 2016 election returns is well-known. Second, our findings do not imply that race is itself a causal factor for vote choice in 2016. Third, our results use aggregate units and thus risk ecological fallacies. One has to be very careful when drawing conclusions about individual behavior from large units like counties. We do not know—and neither do other scholars who rely on aggregate voting units—if the non-White neighbors of White voters in majority White counties were heavily pro-Trump in 2016. This could in principle confound the patterns in the White county figure, above.

Notwithstanding these points, the key implication of what we have just described is that race appears to have figured in the 2016 presidential election. Continuing with this point, we now consider an expanded version of our White county figure.

The above figure plots Trump-Romney differences against the fraction of a county’s voting population that is White, and it covers all counties in the country for which we have sufficient election data. The figure shows that Whiter counties had greater support for Trump than for Romney, and this relationship holds generally, not just in majority White counties. The connection between Trump-Romney differences and the extent to which a county is White is particularly notable in the right-hand side of the figure. We can see in the figure that near-homogeneous White counties in the United States are for the most part small, and they are associated with large Trump-Romney differences, often on the order of ten percentage points.

The White county effect is particularly pronounced when one incorporates education. As evidence of this, the above figure plots Trump-Romney differences against the fraction of a county’s over-25 population that is both White and without a college degree (our data on education is taken from the 2014 American Community Survey, five year estimates). Simply put, when this latter fraction is large, Trump-Romney differences are also large. In fact, once the percentage of a county’s over-25 population that is White and without a college degree exceeds approximately 70%, it is almost guaranteed to have a Trump-Romney difference that is greater than zero.

Given patterns of spatial variation in the map of Trump-Romney differences that we displayed earlier, one might wonder whether the relationship between Trump-Romney differences and the fraction of a county that is White and without a college degree is regional. It is certainly conceivable that the racial patterns we have already commented on are idiosyncratic to a particular set of states in the country.

In fact, the connection between White counties and education holds across the country, as the above figure illustrates. This figure divides counties by standard regions, and the positive relationship between Trump-Romney differences and the percentage of a county that is White and relatively uneducated holds essentially everywhere in the United States. This implies that Trump’s success in 2016 was not simply a Rustbelt (namely, Illinois, Indiana, Michigan, Ohio, and Wisconsin) phenomenon or, for that matter, a phenomenon that was idiosyncratic to any one particular area of the country. In general, where White, relatively uneducated voters reside, Trump did well in comparison to Romney.

### Economic rationalizations for Trump’s gain over Romney

Other examinations of the 2016 presidential election have similar findings about the role of race, education, and Trump support, and it follows from our evidence and these other findings that our rationalization of the 2016 presidential election outcome should involve race and education. With this in mind, what might explain the connection between the percent of a county that is White and Trump vote share? One possibility is that Trump’s improvement over Romney is a result of poor economic conditions that disproportionately affected counties populated primarily by uneducated Whites. If this were the case, then we should expect to see a positive relationship between Trump-Romney differences and measures of local economic activity, for example, unemployment. And if so, it would further follow that our rationalization for the 2016 presidential election outcome should include a measure, or measures, of county-level economic conditions.

To this end, in the figure above we plot Trump-Romney differences against the unemployment rate (2015) at the county level and by region of the United States. The pictured relationships between unemployment and Trump-Romney differences are in general positive, although the situation is a bit murky for counties in the West. However, even in this region of the United States, we observe a positive relationship between unemployment and Trump-Romney differences, albeit one that is not as seemingly pronounced as the relationship elsewhere.

There are other measures beyond race, education, and unemployment that one might conjecture were related to Trump’s success relative to Romney’s. Trump’s advantage in the 2016 presidential election may be better explained, for example, by a county’s income level, by the nature of its industrial problems, by resentment toward urbanites, and so forth. And, perhaps the explanatory power of these various social and economic rationalizations is not constant across the United States. Thus, to better understand the variation in Trump’s vote relative to Romney’s, the table below considers a variety of bivariate relationships that seek to account for Trump-Romney differences, and it reports simple $$R^2$$ values, broken down by various subsets of counties where all calculations are weighted by the two-party vote in 2016. Here the West Coast consists of California, Oregon, and Washington, we continue to rely on our earlier definition of the Rustbelt, and we use the pre-election definition of battleground states in the Wall Street Journal. These states consist of Arizona, Colorado, Florida, Georgia, Iowa, Michigan, North Carolina, New Hampshire, Nevada, Ohio, Pennsylvania, Virginia, and Wisconsin. In contrast, non-battleground states are known as safe.

What stands out in this table is the sheer explanatory power of education in the matter of Trump-Romney differences. To wit, across the United States the percent of a county’s population without a college degree explains 48% of Trump-Romney differences in vote share. Moreover, the explanatory power of education is extraordinarily high in the Midwest (where it explains 77% of the variance in Trump-Romney differences) and in particular in Rustbelt states (where it explains 79%). It is remarkable that this single variable explains nearly 80% of the vote share differences between Trump and Romney in these regions, and this quantity is much greater—between 25 and 30 points higher—than the next best explanatory variable in the table, namely, urban-rural differences.

The reason we have offered this table is not because we seek to engage in a debate over the relative importance of education, income, and race in determining Trump vote share. While the table suggests that education is certainly a key variable, it is hard, if not impossible, to use aggregate units like counties to determine precisely which of a set of highly correlated measures is most important for vote choice in a presidential election. Rather, the table serves as motivation for our inclusion of both county-level education and measures of economic performance in the regression models that appear below.

### Regression analysis of Trump-Romney and Democratic turnout differences

We have thus far argued that race, education, and economics are part of the story of voting in the 2016 presidential election, and the following two regression models build on the figures that we have presented above. In our first regression, we seek to explain differences at the county level between Trump and Romney vote shares, and for explanatory variables we rely on the types of county characteristics that we have elaborated on above. The regression, which we call the Trump vote share regression, is a coarse model of vote choice in the 2016 presidential election in part because it uses aggregate units that in some cases are rather large. We weight the Trump vote share regression by the two-party vote because counties with more presidential votes are more informative about Trump-Romney differences than are smaller counties, all things equal. There are certainly many ways to improve our linear specification of the extent to which Trump outperformed Romney, but we believe that what we offer below is a useful initial pass. Our weighted regression includes state fixed effects. Since states are sorted into regions like the Midwest and so forth, this means that our model controls for regions of the country as well.

We want to emphasize that the objective of our Trump vote share regression is not producing the definitive story of vote choice in the 2016 presidential election. Rather, what we are striving for is a basic model that is consistent with the plots we have discussed above and with academic literature that examines county level vote shares in presidential elections. We have developed our model inductively, and in this spirit it also includes variables that describe the religious composition of counties. Religion is often a part of scholarly election analyses, and, while religion did not appear to be a dominant issue in 2016, it certainly was present in Utah, where the religious contrast between Trump and Romney was pronounced. Our data on religious adherents per county are drawn from the 2010 U.S. Religion Census, Religious Congregations & Membership Study.

Our second regression focuses not on vote share per se but rather on total votes, specifically on the total number of Democratic votes. Many of Trump’s claims about voter fraud in 2016 implied that there would be a surge in Democratic turnout, presumably in specified locations in the United States, because Democrats would mobilize ineligible voters to cast illegal votes there. If this were to have happened, the number of Democratic votes would be greater in said locations than would be expected in an election without fraud. With this in mind, we model the difference between total Clinton votes and total Obama votes from 2012 as a percentage of the citizen voting age population (where citizen voting age population is taken from the American Community Survey conducted between 2010 and 2014 and therefore constant between 2012 and 2016). The meaning of this difference is premised on the idea that there is significant overlap between Clinton supporters and Obama supporters, and we call our Clinton and Obama vote difference regression, the Democratic turnout regression.

 Dependent variable: Trump-Romney share Democratic turnout (1) (2) % Unemployed 17.100*** -0.009 (3.622) (0.041) Log median household income -1.997*** 0.033*** (0.276) (0.003) % Employed in manufacturing -0.311 -0.033*** (0.973) (0.011) % Male 15.318*** 0.128*** (3.319) (0.038) % White -38.535*** 0.291*** (8.932) (0.101) % Black -12.172 0.120 (9.052) (0.103) % Hispanic -15.787* 0.268** (9.223) (0.105) % Asian 6.737 0.084 (9.414) (0.107) % No college degree -11.295 0.164 (10.150) (0.115) % White, no college degree 52.721*** -0.363*** (10.354) (0.117) % Black, no college degree 7.950 -0.175 (10.442) (0.118) % Hispanic, no college degree 11.319 -0.239** (10.504) (0.119) % Asian, no college degree -11.817 0.008 (11.343) (0.129) % Mormon -8.226*** 0.064*** (1.604) (0.018) % Evangelical Christian -5.953*** 0.034*** (0.516) (0.006) % Jewish 9.190*** -0.087** (3.069) (0.035) % Islamic -12.395*** 0.168*** (4.250) (0.048) % Foreign born citizen -1.851 0.073*** (1.901) (0.022) Observations 3,111 3,111 R2 0.895 0.751 Adjusted R2 0.893 0.746 Note: p<0.1; p<0.05; p<0.01 (Intercepts and state fixed effects not displayed)

Considering first the Trump vote share model, recall that this regression speaks to Trump-Romney vote share differences. What we see in the table of regression results (left column) is roughly consistent with our earlier plots and with others’ findings on vote choice and Trump’s coalition in the 2016 presidential election. Namely, counties facing economic pressure (via, for example, high unemployment and low median household income) had disproportionately large Trump vote share compared to Romney vote share; the same is true of counties that are heavily White and relatively uneducated. However, counties with large Evangelical, Islamic, or Mormon populations voted for Trump at lower rates than they had previously voted for Romney. This is not to say that these counties voted heavily in favor of Hillary Clinton, of course. This means only that, with Romney vote share as a baseline, said counties were less supportive of Trump, all things equal.

Comparing the Democratic turnout regression (right column) to the aforementioned Trump vote share model, one can see from the table above that almost all of the significant predictors of Trump’s advantage over Romney in vote share are also predictors of Obama’s advantage in total votes over Clinton. For example, in counties where the median household income is low and in counties where Whites lack college degrees, Republican vote share increased relative to 2012 and the number of Democratic votes decreased. These results could be a reflection of Obama supporters having failed to turn out for Clinton, but they may also reflect some Obama supporters having supported Trump in 2016. We cannot distinguish these explanations using our aggregate data.

## Step 2. Possible sources of voter fraud

Having now developed a rationalization of the 2016 presidential election, one that is captured by our two regression models, we now consider four possible sources of voter fraud. As will be clear, regression residuals—that is, variance not explained in our regression models by county demographics and state fixed effects—play an important role in what follows.

### Non-citizen voting

One presidential campaign-period allegation about voter fraud turned on the theory that non-citizens would turn out and vote in the presidential election. Beyond claims from Darrell Issa that his recent Congressional race featured “illegal, unregistered voters,” and general claims from Trump about “border states,” we are not aware of any compelling assertions that non-citizens played a meaningful role in the 2016 presidential contest.

The American Community Survey that we used to characterize citizen voting age population by county also provides us with counts of non-citizens of voting age by county and by race. Consider, then, the residuals from our Trump-Romney vote share regression. One can interpret these residuals as features of Trump-Romney differences that cannot be explained by typical county-level characteristics. We ask, do we see patterns in these residuals that seem to be related to the presence of non-citizens? When we say patterns here, we want to emphasize that our objective is searching for patterns in residuals that highlight the presence of systematic voter fraud on a dramatic scale—the type of fraud that has the potential to swing an election that is otherwise not particularly close.

We start by plotting residuals from both of our regression models against the fraction of a county’s voting age population that is composed of non-citizens. This is a standard exercise that one would carry out if concerned that an important variable—here, non-citizen population—had been omitted from a set of regression models. In these plots and those that follow, battleground counties are colored purple and non-battleground counties, black.

The residuals from the Trump vote share regression are plotted along the vertical axis in the left panel. These residuals can best be described as measures of the degree to which the Trump vote share regression misestimates Trump’s vote share compared to Romney’s. Counties with positive residuals are those where Trump’s vote share is unexpectedly high—see the figure for the the location of such residuals. In contrast, counties with negative residuals are those where Trump’s vote share is unexpectedly low. This is also labeled in the figure. Similarly, residuals from the Democratic turnout regression are plotted along the vertical axis in the right panel above. These residuals are measures of the degree to which the Democratic turnout regression model misestimates Clinton’s vote advantage over Obama. In the right panel, counties with positive residuals are those where Clinton’s total vote is unexpectedly high, and counties with negative residuals are those where Clinton’s total vote is unexpectedly low. In the context of Trump’s allegations, counties most likely to be affected by fraud are those where Trump’s vote share is unexpectedly low and Clinton’s vote, unexpectedly high. We do not replicate the annotations in our two panels (“Trump’s vote share unexpectedly high,” and so forth) that describe how to interpret our residuals, but the same interpretations apply in all future residual figures.

The average residual in a linear least squares regression is by construction zero (as long as the regression includes an intercept, and all of our regressions do), and the figures above accordingly have horizontal lines at zero. In the left panel, a blue smoother summarizes all points in the figure, and this smoother is essentially pegged at zero. Most importantly, the blue smoother is basically flat. This is not consistent with a surge of non-citizens voting for Hillary Clinton and in so doing driving down Trump’s share of the two-party vote in 2016. Had this form of voter fraud characterized the 2016 presidential election, then the blue smoother should have had a distinctly negative slope, indicating that, as a county’s population of non-citizens increased, our Trump vote share regression model consistently overestimated Trump support.

We want to emphasize that the blue smoother in the left panel, above, is evidence of the lack of a pattern in regression residuals. The fact that the blue line is basically flat does not tell us that literally zero non-citizens cast votes in the 2016 presidential election. What it does say, however, is that there does not seem to be evidence of systematic presidential election voting by non-citizens across the United States.

One might conjecture that voter fraud via non-citizen voting should only be a problem in battleground states, that is, in states that were anticipated to be reasonably close. After all, the potential payoff of fraud in battleground states is presumably much greater than in safer states, all things equal. We have already noted that counties in so-called battleground states are colored purple, and in the left panel, above, a purple smoother summarizes these counties in particular. Restricting attention to battleground states, we see either no evidence of a relationship between Trump-Romney residuals and battleground non-citizens (i.e., the purple line is flat), or evidence that, as the fraction of non-citizens in battleground counties increases, there is a very small increase in Trump-Romney regression residuals (i.e., the purple line is sloped up). Neither of these interpretations is consistent with a surge of non-citizens voting for Hillary Clinton and in so doing driving down Trump’s vote share in 2016.

The right panel above is similar to the left, except that the former describes residuals from our second regression model, the Democratic turnout regression. Where residuals are large and positive, counties received significantly more Democratic votes than our regression model would have anticipated; in contrast, where residuals are large and negative, counties received significantly fewer Democratic votes. If non-citizen voting were systematic across the United States and if mobilized non-citizens were supportive of Clinton, one would expect our Democratic turnout model to underestimate consistently the number of Democratic votes in counties with larger non-citizen populations. Hence, in the right panel, if this type of fraud were occurring, there should be a significant negative correlation between a county’s non-citizen population and its Democratic turnout residual. As we can see from the panel, Democratic turnout residuals appear evenly distributed around zero, and no correlation with non-citizens appears to exist. This is true for all counties pictured in the right panel as well as battleground counties.

Trump’s previously cited concerns about non-citizen voting mentioned the “border,” and from this we infer that Trump was particularly focused on Hispanic non-citizen voting (as opposed to Canadian citizens voting) in the presidential election. A stance against the Mexican border is consistent with Trump’s campaign rhetoric, which was known to be anti-Hispanic and anti-Mexican in particular. To this end, we now consider residuals from our two regressions, focusing attention specifically on Hispanic non-citizens.

If it were the case that ineligible Hispanics voted in the 2016 presidential election, then among other things Democrats would receive additional support as the non-citizen Hispanic population in a county increased. With this in mind, consider the left hand panel, above. If Trump’s allegations about non-citizen Hispanics were accurate, we would expect there to be a significant negative relationship between the non-citizen Hispanic population and the degree to which our model overestimates Trump-Romney differences in the share of the two-party vote. In other words, if non-citizen Hispanics turned out at the polls and voted Democratic, they would consequently reduce Trump’s vote share beyond that which would be expected given Romney’s support in the county as well as other underlying county characteristics. This would result in negative residuals in counties with larger non-citizen Hispanic populations. Considering all counties (blue smoother) as well as just those counties in battleground states (purple smoother), we see no relationship between county voting in the 2016 presidential election and the presence of Hispanic non-citizens. This is not consistent with Trump’s campaign-period allegations.

Considering the right hand panel, above, if Trump’s allegations of illegal Hispanic fraud were valid, we would expect there to be a significant positive relationship between the non-citizen Hispanic population and the degree to which our model overestimates Clinton’s vote advantage over Obama. There is a very slight upward slope in the right panel’s residual smoothers, but it is barely present if at all.

What if we focus attention only on counties that share a border with Mexico? One might conjecture that these 23 counties are uniquely susceptible to non-citizen voting, and some of the counties (colored purple as always) are located in battleground states. Most Mexican border counties are relatively small in terms of citizen voting age population, and this is evident in the sizes of the circles in the figures above. In the sense of voter fraud as articulated by Trump, neither panel appears problematic. The largest counties have residuals that are close to zero, and there does not appear to be anything systematic with respect to the purple battleground counties. One weak regularity in the left hand panel, above, is that most residuals are positive, meaning that they are associated with excessive Trump-Romney differences. This is inconsistent with the theory that non-citizens traveled to the United States from Mexico and cast votes disproportionately against Trump. And on the right, we see a number of negative residuals, which is consistent with fewer Democratic votes in 2016 than in 2012, all things equal. Again, this is not what one would have expected in light of Trump’s claims about non-citizen voter fraud in 2016.

### Ballots cast in the names of deceased voters

Beyond non-citizens, another voter fraud claim made during the course of the 2016 presidential election campaign was that some voters would cast ballots using the registration records of deceased individuals. Former New York mayor Rudy Guiliani asserted that such ballots would likely be cast for “Democrats rather than Republicans,” and here we consider whether there is any evidence in favor of Giuliani’s claims.

It is certainly true that refined voter data, well beyond the sort of broad, county-level measures on which we rely here, would be necessary to investigate comprehensively the claim that large scale fraud was perpetrated in the 2016 presidential election via registration records of deceased individuals. For example, if one were concerned about this possibility, a natural research design might mandate sampling randomly from voter records across the United States (or in a specific area of interest) and then checking records that were used in the 2016 General Election against official death records. This sort of a project is extensive and certainly could not be executed quickly in the aftermath of a presidential election, and hence we ask, how might we use county-level variables to assess the claim that deceased voting registration records were fraudulently used in the 2016 presidential race?

Suppose that a large set of individuals in 2016 cast presidential votes using registration records associated with deceased registered voters. If this were to have happened, then we posit that counties with the greatest number of deaths in the recent past should have unusually small Trump-Romney differences and Democratic turnout bumps as well. This assumes that the number of deaths local to a county constrains the use of fraudulent votes based on registration records of deceased individuals. With this supposition in mind, we gathered data from the Centers for Disease Control and Prevention (CDC) on county deaths, and we summed total county deaths from 1999 though 2014, the last year that we could gather these data. The two figures below plot residuals from the Trump-Romney vote share regression (left panel) and residuals from the Democratic vote share regression (right panel) against the ratio of county deaths to citizen voting age population. The two scatter plots do not include some small counties for which the CDC did not publish mortality data.

What is evident in the plots runs counter to Trump’s claims about widespread voter fraud via records of deceased individuals, conditional on our operationalization of deaths per county. For example, we see that, as the number of previously deceased individuals in a county increases, Trump-Romney residuals increase. Hence, Trump’s vote share is actually greater than expected in counties with larger deceased populations. This is not consistent with the idea that fraudulent voting via records of deceased voters occurred systematically in 2016 in a way that depressed Trump’s vote share. Similarly, as the number of deceased individuals per county increases, Democratic turnout residuals decrease. This is also not consistent with the idea that fraudulent voting via deceased voter records systematically occurred in 2016 in a way that led to Democratic turnout surges. The patterns in both panels above might be evidence of a specification error in our regression models, but we can say with confidence that this error is not one that appears to be consistent with anti-Trump voter fraud via voting in the name of deceased individuals.

### Unusual regression residuals

Thus far, we have not been able to identify voter fraud consistent with theories of non-citizen voting or voting by individuals in the name of those deceased. Trump’s vote share relative to Romney’s does not decrease beyond expectation as the non-citizen population or the deceased population increases, and the same is true of the normalized difference in the change in Clinton votes compared to Obama votes. However, one may be concerned that fraud was a significant feature of the 2016 election, just not systematically across counties with many non-citizens or previous deaths. Instead, one could argue hypothetically, perhaps fraud is isolated to a handful of counties. In this case, we might in principle be able to identify fraudulent outcomes by identifying counties where Trump vote share differences deviate considerably from expected quantities, and a similar statement applies to Democratic turnout differences.

That voter fraud occurred sporadically across the United States is not, we want to emphasize, consistent with our reading of Trump’s voter fraud allegations, those made during his presidential campaign and those promulgated in November, 2016. Indeed, what Trump alleged throughout the 2016 campaign period and beyond the election is that fraud was systematic. Indeed, our research project is based explicitly on Trump’s allegations and not on the matter of detecting any sort of voter fraud that might have occurred in and around November, 2016.

Nonetheless, one might ask about the unusual residuals from our two regressions, and one might argue that these residuals will focus attention on areas of the country that are possibly problematic. We have already divided counties across the country into two groups, battleground and safe, corresponding to whether the counties are in so-called battleground states or not. We now plot the distribution of residuals for each set of counties in the box plots that appear below. As in a typical box plot, bars extend from the lower bound of the interquartile range (IQR) to the upper bound, and each circle beyond the bar represents outlying county residuals that are beyond the IQR. The circles lie beyond three times the IQR can be considered extreme outliers and are labeled with the county name and state. These extreme outliers are counties where Trump-Romney differences (left panel) and Democratic turnout differences (right panel) deviated significantly from expectations, conditional on our regression models. If Democrats were gaining votes through fraudulent voting, we would expect to see unusual residuals associated with counties in which Democrats received far more votes than would be expected. And, because of the incentive to commit voter fraud in locations that might affect the outcome of the presidential election, we might expect more outliers in counties within battleground states than in safe states.

The outliers in the box plots above do not seem consistent with voter fraud. First, there are more unusual residuals (which are proxies for unexpected outcomes) among counties in safe states than in battleground states, contrary to the theory that fraud is more likely to occur where it would have the greatest impact. Second, there are seeming outliers in both directions. Some counties have a greater Trump vote share than expected while other counties have a smaller Trump vote share than expected. The same occurs with Democratic turnout. Third, most of these outlying counties are very small. For example, the largest outliers in the left panel—Perry County, Poinsett County, and Leslie County—have only between ten and twenty thousand residents. If one were to perpetrate voter fraud in a presidential election, targeting small counties would be a puzzling approach.

The subject of identifying outliers in a set of observations is a complicated one, and our research project is not intended to engage the question, what is the most unusual county in the United States regarding Trump’s vote share, for example. That is an interesting question but one that is not necessarily related to our objective, which is testing Trump’s allegations of voter fraud. We do not believe that the two boxplots, above, are consistent with sporadic, albeit meaningful, anti-Trump voter fraud in the United States, and we look forward to research, complementary to this report, that digs deeply into the question of whether there are one or two counties in the country that have Trump vote share so low so as to suggest something nefarious.

### Time required to report results

During the campaign, Donald Trump regularly invoked the idea that the 2016 presidential election was going to be “rigged,” that is, decided before a single ballot was cast, and here we briefly consider what this might mean and whether we can uncover any evidence of a predetermined election. The definition of rigged is certainly broad and thus somewhat difficult to pin down, and our analysis in this section—which focuses on possible irregularities in timing of election returns—captures only one element of a potentially rigged vote. Given that Trump won the 2016 presidential election, it is perhaps difficult to envision a conspiracy of election officials directed against him, but here we consider this possibility nonetheless.

If the 2016 presidential election were indeed rigged, then perhaps the election officials who ran it were biased against Trump. This might lead officials to tamper with election returns to secure a Clinton victory from the start or to adjust nefariously election returns in response to returns from other counties/states. This might mean, for instance, that results in early-reporting counties might be different than late-reporting counties, all things equal, and that county results might dramatically change at the last minute or as a winner emerges.

Having said this, we associate with each county in the United States the number of minutes between the county’s first report of election returns and its last. Our report times are based on the database of election returns maintained by the Associated Press. Overall, these durations are roughly trimodal insofar as some counties reported all of their results at once (so that the number of minutes between first and last reporting is literally zero) or in very a short period, some counties took a few hours before their results were finalized, and some counties took several days/weeks. To avoid compressing observations in the first two terciles of durations (intuitively speaking, in the very fast and medium counties), we break our durations into three approximately equally-sized groups.

These groups are displayed in the above figure, which has one panel for each tercile. The bottom panel displays times for all counties, and the vertical axis in each of the four panels is the Trump-Romney difference that appears in a variety of contexts in this report. The horizontal axes in the three tercile panels differ because each panel incorporates durations of different scales. As we have done before, counties in the plots above are shaded by color with purple denoting counties in battleground states. Each panel contains two linear trend lines, one for battleground and one for all counties.

The pictured relationships between election outcomes (vertical axis) and times required for complete reporting (horizontal axis) are relatively flat, and there are not sizable differences between battleground and safe counties. There is, though, an upward trend among battleground states in the third tercile of durations; this reflects results in New Hampshire and Michigan.

If election officials had conspired to rig the 2016 election against Trump, we might expect to observe large differences in durations between pro-Clinton counties and pro-Trump counties. Why? Because, perhaps, the act of rigging of counties votes slows down results reporting, implying that pro-Clinton counties should have long durations. Or, perhaps extensive rigging speeds up reporting since, after all, rigged votes do not need to be counted. In this latter situation case, pro-Clinton counties should have short durations. Since we see in the plots, above, little evidence of a systematic relationship between election outcomes and durations (that is, pro-Clinton counties are associated neither with unusually short durations nor unusually long ones), we conclude that our duration data are not consistent with a rigged elections against Trump, assuming that such rigging would be manifested through either very short or very long durations in rigged counties.

Suppose that one were concerned that anti-Trump election officials deliberately tried to manipulate vote totals when it became apparent that Clinton was at risk of losing the 2016 presidential election. If this were to have happened, then we would have observed the following. First, later reporting counties would have been disproportionately pro-Clinton; and, these counties would have been pivotal counties in pivotal states. With this in mind, consider the cluster of reporting times around 10,000 minutes. These counties represent Utah, easily won by Trump, and Washington, easily won by Clinton. Among late-reporting Utah counties, Trump won by 153,543 votes; among late reporting Washington counties, Trump lost by 481,404 votes. In both states, late reporting counties were not pivotal and thus had no effect on the final outcome of the Electoral College.

The cluster of counties reporting results at around 18,000 minutes includes just three states: California, Michigan and Wisconsin. Results in California are historically slow as the state allows for voters to mail in ballots up to the day of the election and also allows for provisional ballots that must be adjudicated. The outcome in Michigan has appeared relatively clear for weeks, but the state endeavored to avoid reaching a re-count threshold by conducting an exhaustive tabulation before certifying. Wisconsin is currently in the process of reporting updated results. In all cases, there is no evidence of results suddenly changing during the extended tabulation period. California went, as it has for the last several decades, to the Democratic presidential candidate, in this case, Clinton. Although Michigan results were not certified until November 28, 2016, the long delay in this state is not compatible with anti-Trump voter fraud: Trump won Michigan and the results from this state were not in the Electoral College. Though it is in theory possible that fraud could have occurred in Michigan, election officials there have known for weeks that Trump is the President-elect, which left them few incentives to tamper with results in a pro-Clinton direction, late in November.

Flips by precincts reporting
Precincts reporting Battleground flips Safe flips
<= 25% 25 33
> 25% and <= %50 23 33
> 50% and <= 75% 16 37
> 75% 24 60

To explore potentially unexpected county-level changes that took place during vote tabulation processes starting on the evening of November 8, we consider changes in the leader reported by a county as a function of the percentage of precincts counted in the county. A county can have zero changes if, for example, Trump were ahead of Clinton in every report made by the county. On the other hand, if Trump were ahead on the first report, Clinton in the second, and Trump in the third and final, then the county had two so-called leader flips. If, say, a subset of county election officials were rigging the election in favor of Clinton, then perhaps we might expect to see county vote reports suddenly changing, as corrupted election officials made their presence known. We note that leader flips often occur naturally as returns come in from different types of precincts, urban versus rural for example.

Of 15,796 Associated Press observations from 3,111 counties in our dataset, we find only 251 instances where the winner of a county flipped from Trump to Clinton or from Clinton to Trump. These flips are summarized in the table above. Of the 251 flips, the majority, 163, took place in safe states. In the change from 75% to 100% of precincts reporting, 84 flips occurred—24 in battleground and 60 in safe states—representing a net gain of 41,388 votes for Trump (33,165 votes in battleground states). In battleground states, 12 flips were from Clinton to Trump. This is, of course, the opposite of what we would expect if the results were rigged against Trump.

Net Trump vote change in battleground flipped counties
State County Number of flips Net Trump gain
Arizona Apache County 1 -6833
Florida Pinellas County 1 12455
Iowa Bremer County 1 2139
Iowa Clinton County 1 1217
Iowa Des Moines County 1 1325
Iowa Dubuque County 1 2068
Iowa Jefferson County 1 203
Iowa Muscatine County 1 1906
Iowa Warren County 1 4053
Michigan Marquette County 1 -1473
Michigan Muskegon County 1 -2229
New Hampshire Hillsborough County 2 -382
Pennsylvania Berks County 1 15855
Pennsylvania Centre County 1 -1588
Pennsylvania Erie County 1 4838
Pennsylvania Northampton County 1 5543
Virginia Montgomery County 1 -1747
Virginia Staunton city 2 -83
Virginia Suffolk city 1 -1679
Wisconsin Douglas County 1 -1883
Wisconsin Sauk County 2 -540

Of the 24 flips in battleground states, ten occurred in states with a small margin of victory. In two states—Florida and Pennsylvania—flips increased Trump’s lead. In Arizona, Michigan, New Hampshire, Virginia, and Wisconsin, flips decreased Trump’s margin but never by more than 6,833 votes. In all of these cases, changes in votes associated with flips for Trump were not sufficient to sway a state-level election.

Perhaps the last opportunity to rig the 2016 election outcome occurred when counties reported final vote totals. Looking at just the last two reports from each county (in counties with more than two reports), there were flips in just three counties—Hillsborough County in New Hampshire, Dubuque County in Iowa, and Sauk County in Wisconsin—with these flips decreasing Donald Trump’s margin by 856 votes. This number of votes is insufficient to have swayed the election in either New Hampshire or Iowa. Sauk County reported updated results on November 28, 2016, and here the total number of votes decreased for Trump and Clinton but the Trump victory margin actually increased.

### Voting technology

Electronic voting machines are consistently identified as potential sources of voter fraud. Some assert that these voting machines allow nefarious groups and/or individuals to alter vote choices, conspiracy theories speculate that voting machines change tabulations, and some observers worry that these machines “flip” voter selections during the process of voting (here we are using the word “flip” to refer to a change in the candidate selected by a voter as opposed to changes in the winner reported by a given county). Investigations of flipping largely show that the phenomenon occurs because of voter error or touchscreen miscalibration. Nevertheless, given the putative association between electronic voting and threats of voter fraud, and in light of some concerns about voting in Michigan, Pennsylvania, and Wisconsin, we briefly investigate whether counties with electronic voting machines had unusual results in the 2016 presidential contest.

The above figures describe regression residuals from our overall election models grouped by voting technology. We divide counties based on whether they used digital voting, optical scan technology, or hand counted paper ballots. If voting method varied within a county, we say that the county used digital voting if only digital voting took place there, and we say that the county used hand counted paper ballots if such a method were used at all. We code any combination of digital voting and optical scan technology as optical scan technology. Our data on voting technology were gathered from the Verified Voting Foundation on August 14, 2016. According to our rules for assigning voting technology to counties, in the 2016 presidential election there were 112 digital counties, 2,614 optical counties, and 417 hand counted paper counties

If there were voter fraud directed at counties using digital voting technology, then we would expect to see unusual Trump-Romney and Democratic turnout regression residuals in the counties that used these technologies. We do not see this. For each of our two regressions, the three associated boxplots (one per type of voting technology) are quite similar, and in fact there appear to be a disproportionate number of unusual residuals associated with counties that used paper ballots. In other words, we do not see any evidence that voting technology, and in particular purely digital voting, is associated with unusual regression results in the 2016 presidential election.

Another perspective on the issue of voting technology can be gleaned by comparing by county the votes that Trump received in the 2016 General Election with the votes he received in the 2016 primary. We can make this sort of a comparison in the 2494 counties in states that featured presidential primaries prior to the 2016 General Election, and the above figure plots Trump General Election versus Trump primary votes (both on a log scale) against each other. The figure has three panels, one for each type of voting technology that we noted above, and each panel has a 45-degree line. It should be clear that comparing Trump primary votes across states is not ideal because the set of Republican primary candidates was not constant in time. This will have the feature of adding noise to any comparison of Trump primary and General Election vote totals.

Two features of the above plots are notable. First, the three clouds of points pictured lie above associated 45-degree lines. This means, simply put, that Trump did well in the General Election where he did well in the primary. At some level, this is a trivial statement and in ordinary circumstances might not merit being mentioned. On the other hand, viewed from the perspective of voter fraud, it is not trivial. If, for example, we observed in the General Election Trump doing significantly worse in counties were he performed well in the primary, that might be cause for concern—the election was “rigged”—particularly if the extent of such a hypothetical drop in Trump performance were associated with voting technology. Second, and looking across the panels, the three clouds of points across do not appear appreciably different from one another. This statement holds even though our naive comparison of Trump votes with voting technology does not control for the fact that the distribution of different vote tabulating mechanisms across the country is not uniform. Prima facie, the panels above are not consistent with the idea that voting technology was an important feature in the 2016 presidential election.

We already noted that three states, Michigan, Pennsylvania, and Wisconsin in particular have attracted a fair bit of attention post-election on account of allegations that voting technology may have played a role in the vote outcomes there. These states were all won by Trump. Green Party presidential candidate Jill Stein is supporting efforts for a recount in the three states, and in Wisconsin state officials are apparently preparing for such an event. Voting technology and concerns about Russian hacking of the 2016 election are at the forefront of recount efforts in the aforementioned states, and we now briefly consider Michigan, Pennsylvania, and Wisconsin.

In terms of voting technology, all voters in Michigan use optical scan voting, but voters in Wisconsin and Pennsylvania use varied technologies. We already noted that vote tabulating technologies can vary within counties as well as across counties, and in fact this is the case in Wisconsin (variance is by town). To the best of our knowledge, a compilation of town-level election returns for the 2016 presidential election has not yet been released by the Wisconsin Elections Commissions.

Notwithstanding within county variance in voting technology, the above figure describes the relationship between (log) Trump primary vote and (log) Trump General Election vote in our three states of interest, and different colors in the three panels denote alternative voting technologies. Technology uniformity in Michigan is evident insofar as all of the dots in this state’s panel are green, indicating the presence of optical scan voting. Most importantly, the patterns we see in the figure above are reminiscent of those observed earlier. Namely, we observe clouds of points above respective 45-degree lines—indicating that Trump did well in the General Election in those places where he did well in the primary—and we do not observe striking differences by voting technology. We have not controlled for local factors associated with the distribution of voting technology; even so, nothing strikes us as inherently problematic in the Trump vote distributions in Michigan, Pennsylvania, and Wisconsin.

### Trump’s post-election claim regarding California, New Hampshire, and Virginia

Trump alleged on November 27, 2016, via Twitter that there was “serious voter fraud” votes in three states—California, New Hampshire, and Virginia, all of which supported Clinton in the 2016 presidential election. We are not sure what led Trump to highlight these three states in particular, but four hours prior to naming them Trump referred, again via Twitter, to “millions of people who voted illegally.” With these two tweets as background, we presume that Trump is concerned about illegal voters in California, New Hampshire, and Virginia, and thus we consider the following figure, which plots (the log of) Obama votes from 2012 against (the log of) Clinton votes from 2016.

The plot shows all counties for which we have data, and counties in California, New Hampshire, and Virginia are colored differently than counties from other states. As is evident in the plot, the counties from Trump’s three states of interest look no different than counties from elsewhere. That is, where Obama did well in 2012, Clinton tended to do well in 2016 (although one can see from the figure that Clinton performed disproportionately poorly among relatively small counties with low numbers of Obama voters). Insofar as, to the best of our knowledge, there were no claims of massive voter fraud in California, New Hampshire, and Virginia in 2012, it follows that the the numbers of votes in 2016 for Clinton in California, New Hampshire, and Virginia are inconsistent with Trump’s claims about “millions” of illegal voters in the 2016 presidential election, whom we presume would have voted for Clinton if they were to have existed.

On the subject of California, New Hampshire, and Virginia, we should also note that, throughout this report, we have explored various statistics that describe outlying or unusual counties. We have not seen clusters of counties from these states, and this casts additional aspersions on the assertion that there was significant voter fraud there in the 2016 presidential election.

## Conclusion

We motivated this report by describing the extent of voter fraud allegations that surrounded the recently-concluded 2016 presidential campaign. Donald Trump, the Republican candidate for president and current president-elect, was responsible for many of these allegations, all of which, Trump averred, would be directed at him. Here we have posed the question, is there anything in county-level election returns that is consistent with the widespread fraud allegations put forth by Trump and those allied with him? We believe that the answer is, no. We have explored in particular fraud allegations that involve non-citizens and deceased individuals, and we have not uncovered any evidence that there was a widespread, anti-Trump fraud effort that relied on either of these sources.

We also considered Trump’s assertion that the 2016 presidential election would be “rigged.” Keeping in mind that this charge is general, we operationalized it by considering patterns in the way that election returns were released starting in the evening of November 8, 2016. We do not find any unusual patterns in result timing, and we do not find that some counties switched from Trump to Clinton (or vice versa) in a way that appears suspicious. There are certainly many ways that one could argue that an election is rigged, but we are comfortable dismissing the idea that election officials conspired against Trump by nefariously manipulating the timing of results.

Lastly, we considered recent charges about problematic voting technology in Michigan, Pennsylvania, and Wisconsin. As of this report’s writing, there is a push from presidential candidate Jill Stein of the Green Party for a recount in these three states, and Stein and her supporters are citing evidence of unusual votes in them, correlated in some fashion with voting technology. Although the status of recounts remains uncertain, the election data we have brought to bear on Michigan, Pennsylvania, and Wisconsin do not look problematic. This does not mean that recounts, should they occur, will recover pre-recount results exactly. However, based on our comparisons of Trump’s General Election and primary vote support, we are skeptical that a recount will uncover any evidence of voter fraud.

Given the close proximity of the 2016 General Election, there are many types of election data that we do not (yet) have, and this limits the types of voter fraud that our analysis can engage.

• Absentee ballots cast per county. If we had national data on absentee ballots, we could in principle investigate whether there were surges in absentee ballots cast in select counties. Some people believe that ballots cast by mail are disproportionately prone to fraud, and we could investigate whether rates of absentee ballots by county are inexplicably correlated with Clinton support, as Trump might aver.
• Provisional ballots cast per county. Provisional ballots often indicate the presence of voter registration problems, and as such spikes in provisional ballot counts may be indicative of deficiencies in voter registration databases. In principle, one might be concerned about attempted voter fraud in locales that had spikes in provisional ballots.
• Precinct and town-level data. Our analysis here relies on county-level election returns. This has the advantage of allowing for national coverage, something that is very important. However, counties are large, and this is a limitation of our research. More refined data will help us better understand the 2016 election, and over time we expect that precinct and town-level data will become increasingly available. With these sorts of data, we may be able to refine some of our tests for the types of voter fraud cited by Trump.

Our conceptualization here of voter fraud has focused on widespread fraud, almost exclusively in the vein of Donald Trump’s campaign-period allegations, and our conclusions pertain only to this sort of fraud. Although we have not found evidence of massive fraud in the 2016 presidential election, it does not follow that there was literally zero fraud in this race. Research directed at uncovering all instances of fraud will require more refined data than are presently available, and detailed studies of fraud will be complementary to ours. We look forward to developing this report in the future and to other research projects on fraud that scholars, election officials, and journalists pursue.

## Acknowledgements

The authors thank Hollye Swinehart and Morgan Waterman for research assistance, Rich Houseal of Research Services, Church of the Nazarene Global Ministry Center, for forwarding to us the Religious Congregations & Membership Study, and seminar participants at the Freie Universität Berlin for helpful comments. This academic research project is non-partisan and not affiliated with any of the candidates in the 2016 General Election. None of the authors has received external funding for the work described here.

## Variable descriptions and terms

• Trump-Romney difference. Trump’s share of the 2016 two-party presidential vote minus Romney’s share of the 2012 two-party presidential vote. Source: the 2016 presidential vote is from Associated Press Election Services and the 2012 presidential vote is from Dave Leip’s Atlas of United States Presidential Elections.
• Democratic turnout difference. The total number of votes for Clinton in 2016 minus the total number of votes for Obama in 2012 divided by the total citizen voting age population.
• Citizen Voting Age Population. The population that is a U.S. citizen and 18 years or older, averaged between 2010 and 2014. Source: 2010-2014 5-year American Community Survey.
• Non-Citizen Voting Age Population. The population that is 18 years or older minus the citizen voting age population. source: 2010-2014 5-year American Community Survey.
• % White. Percent of the citizen voting age population that identify as White alone and not Hispanic. Source: 2010-2014 5-year American Community Survey.
• % Black. Percent of the citizen voting age population that identify as African American alone. Source: 2010-2014 5-year American Community Survey.
• % Hispanic. Percent of the citizen voting age population that identify as Hispanic. Source: 2010-2014 5-year American Community Survey.
• % Asian. Percent of the citizen voting age population that identify as Asian alone. Source: 2010-2014 5-year American Community Survey.
• % No College Degree. Percent of adults over 25 without a bachelor’s degree. Source: 2010-2014 5-year American Community Survey.
• % White no college degree. Percent of adults over 25 that are without a bachelor’s degree and identify as White alone and not Hispanic. Source: 2010-2014 5-year American Community Survey.
• Unemployment Rate, 2015. Annual average percent of the labor force that is unemployed in 2015. Source: Labor Force Data by County, 2015 Annual Averages, Local Area Unemployment Statistics, Bureau of Labor Statistics.
• Log Unemployment Rate, 2015. The natural log of the unemployment rate in 2015
• State Adjusted Unemployment Rate, 2015. The difference between each county’s unemployment rate and that of the average county within the same state
• Change in Unemployment Rate, 2012 - 2015. The difference between the annual average percent of the labor force that is unemployed in 2015 and the annual average percent of the labor force that is unemployed in 2012. Source: Labor Force Data by County, 2015 Annual Averages and 2012 Annual Averages.
• % Employed in Manufacturing.Percent of the civilian employed population 16 years and over that is employed in manufacturing jobs. Source: 2010-2014 5-year American Community Survey.
• % Employed in Agriculture, Forestry, Fishing, Mining. Percent of the civilian employed population 16 years and over that is employed in agriculture, forestry, fishing and hunting, and mining jobs. Source: 2010-2014 5-year American Community Survey.
• % Employed in Construction. Percent of the civilian employed population 16 years and over that is employed in construction jobs. Source: 2010-2014 5-year American Community Survey.
• Log Med HH Income, 2014. Natural log of the median household income within a county for 2014. Source: Small Area Income and Poverty Estimates (SAIPE), U.S. Census.
• Change in Log Med HH Income, 2012 - 2014. The difference between the natural log of the median household income within a county for 2014 and the natural log of the median household income within a county for 2012. Source: Small Area Income and Poverty Estimates (SAIPE), U.S. Census.
• Med HH Income as Pct of Statewide Median, 2014. The median household income within a county for 2014 as a percent of the state’s median household income. Source: Small Area Income and Poverty Estimates (SAIPE), U.S. Census.
• Rural-Urban Continuum, 9pt Scale. A nine-point rating from heavily populated metro counties (1) to non-metro counties with small urban populations (9). USDA Rural-Urban Continuum codes
• % Urban, 2010. Percent of a county’s total population that resides in urban areas or clusters. Source: Decennial Census, 2010.
• Battleground. States where electors may go to either party: Arizona, Colorado, Florida, Georgia, Iowa, Michigan, North Carolina, New Hampshire, Nevada, Ohio, Pennsylvania, Virginia, and Wisconsin. Source: Wall Street Journal.