How much of an incumbency advantage do winners in U.S. House elections enjoy? This is the topic of a paper by David S. Lee (2008) "Randomized experiments from nonrandom selection in U.S. House elections," Journal of Econometrics, 142(2), 675-697. Lee uses a regression discontinuity approach to estimate the effect. There are 435 Congressional districts in the United States and elections are held every 2 years. Representatives serve a term of 2 years. We employ a subset of Lee's data. The data file rddhouse_small has 1200 observations. See the rddhouse_small.def for data details. The data file rddhouse is larger. The forcing variable is SHARE, which is the Democratic share of the votes in a election in year \(t\) minus 0.50, so that SHARE is the Democratic margin of victory. The outcome of interest is the Democratic share of the vote in the next election, SHARENEXT.

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

b. The dummy variable \(D=1\) if SHARE \(>0\) and \(D=0\) if SHARE \(<0\). Estimate the regression model with SHARENEXT as dependent variable, and SHARE, D, and SHARE \(\times D\) as explanatory variables. Interpret the magnitudes, signs, and significance of the coefficients of \(D\) and \(S H A R E \times D\). Graph the fitted value from this regression against SHARE.

c. The variable \(B I N\) is the center of an interval of width 0.005 , starting at -0.25 . There are 100 bins between -0.25 and 0.25. Define a "narrow" win or loss as being an election where the margin of victory, or loss, is within the interval -0.005 to 0.005. Calculate the sample means of SHARENEXT when \(B I N=-0.0025\) and when \(B I N=0.0025\). 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 in Appendix C.7.2, Case 1. 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 SHARE2, SHARE3, and SHARE4 are SHARE raised to the second, third, and fourth power, respectively. Estimate the regression model with SHARENEXT as dependent variable, with explanatory variables SHARE and its powers, \(D\) and \(D\) times SHARE and its powers. Interpret the magnitudes, signs, and significance of the coefficients of \(D\), and \(D\) times SHARE.

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

g. Estimate the regression with SHARENEXT as dependent variable with explanatory variables SHARE and its powers, for the observations when \(D=0\). Reestimate the regression for the observations when \(D=1\). Compare these results to those in (e).

h. The variable \(B I N\) in part (c) was created using the equation \(B I N=S H A R E-\bmod (S H A R E, 0.005)+\) 0.0025 , where "mod" is the "modulus operator," a common software function. In particular, \(\bmod (x, y)=x-y \times\) floor \((x / y)\) where the operator "floor" rounds the argument down to the next integer. Explain how this operator works in this application to create "bins" of width 0.005.