How much of an incumbency advantage do winners in U.S. Senate elections enjoy? This issue is examined by Matias D. Cattaneo, Brigham R. Frandsen and Rocío Titiunik (2015) "Randomization Inference in the Regression Discontinuity Design: An Application to Party Advantages in the U.S. Senate, Journal of Causal Inference, 3(1): 1-24.32 As they describe (p. 11): "Term length in the U.S. Senate is 6 years and there are 100 seats. These Senate seats are divided into three classes of roughly equal size (Class I, Class II, and Class III), and every 2 years only the seats in one class are up for election. As a result, the terms are staggered: In every general election, which occurs every 2 years, only one-third of Senate seats are up for election. Each state elects two senators in different classes to serve a 6-year term in popular statewide elections. Since its two senators belong to different classes, each state has Senate elections separated by alternating 2-year and 4-year intervals." We employ a subset of their data, contained in the file rddsenate. See rddsenate.def for data details. The forcing variable is MARGIN, which is the Democratic share of the votes in an election in year \(t\) minus 50: it is the Democratic margin of victory. The outcome of interest is the Democratic share of the vote in the next election for that Senate seat, VOTE.

a. Create a scatter plot with MARGIN on the horizontal axis and VOTE on the vertical axis. Does there appear to be a positive relationship, an inverse relationship, or no relationship?

b. The dummy variable \(D=1\) if MARGIN \(>0\) and \(D=0\) if \(M A R G I N<0\). Estimate the regression model with VOTE as dependent variable, and MARGIN, \(D\), and MARGIN \(\times D\) as explanatory variables. Interpret the magnitudes, signs, and significance of the coefficients of \(D\) and MARGIN \(\times D\). Graph the fitted value from this regression against MARGIN.

c. The variable \(B I N\) is the center of an interval of width 5, starting at -97.5 and ending at 102.5. Define a "narrow" win or loss as being an election where the margin of victory, or loss, is within the interval -2.5 to 2.5. Calculate the sample means of \(V O T E\) when \(B I N=-2.5\) and when \(B I N=2.5\). Is the difference in means an estimate of the value of incumbency? Explain how.

d. Treat the two groups created in (c) as two populations. Carry out a test of the difference between the two population means using the test: Using a two-tail test and the \(5 \%\) level of significance, do we reject the equality of the two population means, or not?

e. The variables MARGIN2, MARGIN3, and MARGIN4 are MARGIN raised to the second, third, and fourth powers, respectively. Estimate the regression model with VOTE as dependent variable, with explanatory variables MARGIN and its powers, \(D\) and \(D\) times MARGIN and its powers. Interpret the magnitudes, signs, and significance of the coefficients of \(D\) and \(D\) times MARGIN.

f. Graph the fitted value from the regression in (e) against MARGIN. Is the fitted line similar to the one in (b)?

g. How would the results of (e) compare to the regression with VOTE as dependent variable with explanatory variables MARGIN and its powers, for the observations when \(D=0\). What if the regression was estimated for the observations when \(D=1\) ?