A Simple Model of Senate Elections

Without a President seeking re-election at the top of the ticket, the Democrats face a substantial challenge if they wish to win back the Senate in 2016.

While the Presidential race gets all the media and pundit attention, the battle for control of the Senate also looms large in this election year.  Republicans enter the election holding 54 of the 100 Senate seats, so a net Republican loss of just five seats would put the Senate back in Democratic hands.  The Democrats have the advantage that many more Republican seats, twenty-four, are at risk in the 2016 election compared to only ten held by Democrats.  This lopsided margin reflects the result of the 2010 off-year election when Republicans picked up six seats from the Democrats.  In principle, some of those Republican senators may be more vulnerable in a Presidential year with higher turnouts and more visibility.  The Democrats certainly believe they can retake the Senate this November.  The Party aggressively recruited candidates for the Senate elections and had secured bids from all but one of its top-tier candidate selections by early October of 2015.

The Democrats also have the advantage that the party of the incumbent President usually wins a slim majority of the Senate vote in “on-year” elections when a Presidential election also takes place but loses in “off-year” elections.


Unfortunately for the Democrats the relationship between Senate electoral success and type of election is not so simple.  If I divide up on-year elections into ones when the President ran for re-election and ones when, like the upcoming election, he did not, a very different pattern emerges.  The President’s party fared substantially worse in the five open-seat elections since 1946 than it did in elections with the President at the top of the ticket.  While open-seat years gave the President’s party a one-percent boost compared to off-years, that difference is not statistically significant.  What matters is whether the President is running or not.


The Democrats’ optimism is also based on the much larger number of Republican seats at risk in 2016.  I find some support for the notion that a Senate “Class” with a comparatively lopsided division of the vote in one election becomes more competitive six years later.  Statisticians call this phenomenon “regression toward the mean,” where observations that were outliers at one time show more average scores when measured again. But this effect is weak, and the division of the Senate vote in 2010, 53-47 percent Republican, was not as lopsided as the margin in terms of seats, 65-35 percent Republican.  All told the estimated “rebound” effect given the Republican 2010 landslide is just 0.6%, raising the expected Democratic vote from 47.1% to 47.7%.

Where else might the Democrats gain some relief?  Perhaps the generally positive state of the economy might provide some help.  Political scientists and economists have tested many different measures of economic conditions in models of voting for President and Congress.  One simple measure that has consistently proven significant is the change in personal income, and that proves true for Senate elections as well.


This chart adds the effects of the year-on-year percent change in real disposable personal income to our simple model.  While the effect of rising incomes is positive and statistically significant, it alone cannot overcome the substantial deficit facing the Democrats in an open-seat election year.  In the six years of the Obama Administration, personal income rose by at most 2.2 percent in a single year, 2012.  With a likely figure for annual income growth in 2016 at around two percent, we should expect the Democrats to win only about 48 percent of the Senate vote in 2016.

Winning a majority of the Senate vote is not a requirement for winning a majority of the contested seats.  In 2004, and most dramatically in 1982, the Republicans managed to win a majority of the seats with a minority of the votes cast.


The high “swing ratio” of 2.4 means that a change of one percent in the percentage of votes won translates on average to a 2.4 percent increase in the percentage of seats.  So even fairly small changes in the division of the vote can have much larger effects on the composition of the United States Senate.

I tried some other possible influences like the approval rating of the President and the size of the President’s margin in on-year elections.  I found no “coattails” effect for the Presidential vote either in years when the President is running or years when he not.  Presidential approval does matter, but only in off-year elections, so I did not include it in this discussion about 2016.  That finding is consistent with a conventional view that off-year elections reflect public opinion about the President’s performance in office.


Technical Appendix: Modelling Senatorial Elections

In an effort to examine how the 2016 Senatorial elections might turn out, I have been estimating some simple models of Senate elections using aggregate data from 1946 to 2014.[1]   For my dependent variable I have chosen to use the (logit of the) total vote for Senate candidates in the President’s party. This removes party labels from the analysis and treats the two parties symmetrically. I conduct some “regression experiments” of this measure using three types of predictors.

One type represents the electoral setting.  Is this an on-year or off-year election?  And, in on-years, is the incumbent President running for re-election?  Alone these two factors account for over twenty percent of the variance in incumbent support, with President re-election bids having by far the greatest impact.  This results for this model appears in the left-hand column of the table below.  The remaining columns add additional explaatory factors to the basic political environment.


Right away we see that when a President is running for re-election, his co-partisans in the Senate have a much greater chance of winning.  Because these are measured as logits, values below zero correspond to a percentage value below fifty, while positive logits imply values above fifty percent.  Without the President running, the model has a slight negative prediction equal to the constant term.  In Presidential re-election years that negative value turns positive being the sum of the constant (-0.08) and the effect for relection years (0.14).  By this model the Democrats in the Senate will be short that extra boost that comes from having an incumbent seeking re-election.[2]

One reason the Democrats are optimistic about their chances to retake the Senate in 2016 is that these seats were last contested in the Republican wave election of 2010.  This year those seats will be fought in the context of a Presidential election with its greater visibility and higher turnout.  I have measured this effect by including the vote from the election held six years prior.  In principle, we should expect a negative effect, as “regression toward the mean” sets in.  Republicans perhaps won by unexpectedly larger margins in 2010 so their margins should fall closer to the average this time around.

Adding the prior vote for each Senatorial “class” improves the predictive power of this simple model slightly, but the coefficient itself fails to reach significance.  It has the expected negative sign, however, and will prove much more significant in further reformulations.

The third column adds the effect of presidential approval, a common predictor in models of voting for the House.  For the Senate it turns out to have a more subtle effect.  Presidential approval has the expected positive effect on votes for the incumbent’s Senators, but only in off-year elections.  A long literature in political science has examined off-year elections espousing a variety of theories to explain the President’s usual losses.  I generally adhere to the “referendum” school of thought on off-years, that they give the public a chance to express their approval or disapproval of a President mid-way through his term.  That presidential approval matters not in years when a Presidential election is being held reinforces my belief in the referendum explanation for off-year voting.

The last explanatory factor is the year-on-year percent change in real disposable personal income.  Political scientists and economists have included pretty much every economic variable that might affect election outcomes in their models of presidential and congressional voting, but the one factor that often proves significant is personal income. Adding it to the model increased “explained” variance by over ten percent.

Here is a chart showing the expected national vote for the Senate Democrats as a function of the size of the increase in personal income heading into the election. They do get a small positive compensation from having lost the popular vote for these seats six years before.  However, without the re-election boost, even reaching an absurdly high four percent growth real income would not push the expected vote for the Democrats over the 50 percent line.  In the best years of the Obama Administration, real income growth reached slightly over two percent, which would give the Democratic candidates for Senate about 48 percent of the vote.


I admit there are many shortcomings to this analysis.  First, I only account for 45 percent of the variance in Senatorial voting, and this only at the national level.  Senatorial campaigns are played out in states, where local forces can exert a major role.  With only 33 or 34 seats up in each election, idiosyncratic factors can swing a few decisive states.

If, as the model predicts, the Democrats should expect to win about 48 percent of the popular vote for Senate, they can nevertheless still win back the Senate.  They might follow the path taken by the Republicans in 2004 and most dramatically in the first off-year election under Ronald Reagan in 1982.  In both those years the Republicans won a majority of the contested Senate seats with a minority of the popular vote.


The Rhythm of Senate Elections

From reading media reports of the 2014 election results you might believe the nation has experienced a political change of cataclysmic proportions. Republicans won 23 of the 36 states where a senatorial election was held, enough to give them control of the Senate for the next two years. Yet we need look back only to another strong Republican year, 2010, to see nearly identical results. In that year the Republicans took 24 of the 37 states where elections were held.

Historically the parties’ shares of Senate elections have swung back and forth quite substantially with the last decade appearing unusually unstable.  Here are the results for Senate elections back to 1936:



The Democrats reached their peak in the 1964 Johnson landslide, though this election merely confirmed the Democrats dominance of the Senate “class” elected six years during the 1958 Eisenhower recession.  Republicans have won about two-thirds of the seats in half-a-dozen elections over the same period of time.  The 2014 result is quite similar to the Republican margins in 2010, 2002, 1980, 1952, and 1946.

If you look carefully at this graph, you’ll see a certain rhythm in these results, one created by the six-year length of a Senatorial term and the power of incumbency.  In fact, if we slide the graph forward six years and superimpose the results, we get this:


Now we see how the partisan split in a Senate “class” helps explain the variation from election to election.  Because incumbents have an advantage when it comes time for re-election, the partisan composition of a Senate class tends to repeat at six-year intervals.  The Republican edge in 1946 was replicated six years later when Eisenhower won the White House. Six years after that a major political shift occurs.  The largely Republican class of 1946 and 1952 was replaced with a largely Democratic class during the Democrats sweep of the 1958 off-year elections. The Republicans’ victories in 1980 constituted a similar shift for their party.

Of course the partisanship of the class facing re-election just sets the stage on which each year’s set of electoral forces plays out.  We would expect that factors like broader trends in partisanship and the state of the economy might also influence the outcome of Senate elections.  I turn to those influences in the next article.