Analyzing Electoral Advantage of Incumbency in U.S. House Elections Using Regression Discontinuity Design

Lee (2008) uses a regression discontinuity design (RDD) to study whether the incumbent party in a district has an electoral advantage in the United States House of Representatives. He is interested in studying whether winning an election has a causal influence on the probability that the candidate will run for office again and eventually win the next election.
The dataset election.dta contains data on vote shares from 1946 to 1998. The main vote share variable is the Democratic vote share minus the vote share of the strongest opponent (which in most cases is a Republican nominee). The Democrat wins the election when this variable “Democratic vote share margin of victory” (difshare in the dataset) crosses the zero threshold and loses the election otherwise. Two measures of the success of the party in the subsequent election are used: the probability that the party’s candidate will both become the party’s nominee and win the election (mmyoutcomenext in the dataset) and the probability that the party’s candidate will become the nominee in the election (mrunagain in the dataset).
He considers the following RDD model:

𝑣𝑖𝑡+1 = 𝛼𝑤𝑖𝑡 + 𝛽𝑣𝑖𝑡 + 𝛾𝑑𝑖𝑡+1 + 𝑒𝑖𝑡+1

The dependent variable 𝑣𝑖𝑡+1 is a measure of success of the Democratic candidate in district
𝑖 in election year t+1 (either mmyoutcomenext or mrunagain in the dataset). The variable
𝑣𝑖𝑡 is the vote share margin of victory for the Democratic candidate in district 𝑖 in election year t (difshare in the dataset). The vector 𝑤𝑖𝑡 is a set of characteristics of the candidate in year t. 𝑑𝑖𝑡+1 is an indicator variable equal to 1 if the Democrats are the incumbent party during the electoral race in year t+1. This is a deterministic function of whether the Democrats won election t:

𝑑𝑖𝑡+1
= {1 𝑖𝑓 𝑣𝑖𝑡 > 0
0 𝑖𝑓 𝑣𝑖𝑡 ≤ 0

NOTE: for all regressions in this question, you should use robust standard errors with no clustering.

Suppose that you run an OLS regression of the probability that the Democratic candidate wins the election in year t+1 on the vote share margin of victory of the Democratic candidate in year t. Would this regression identify the causal effect of incumbency on the vote share? Explain. (8 marks)

Under what identification assumption does RDD lead to a valid estimate of the causal effect of incumbency on the vote share? Explain in detail. (8 marks)

Estimate the RDD model for the probability of a Democrat both running in and winning election t+1 (mmyoutcomenext in the dataset). You should regress this variable on a fourth-order polynomial in the democratic vote share margin of victory, separately for each side of the threshold. Replicate figure 2 (a) in the paper showing a scatter plot of the data and the fitted line of this RDD model. Does the incumbent candidate appear to have an electoral advantage? Explain. (6 marks)

Repeat the analysis in part c) for the probability that the Democrat remains the nominee for the party in election t+1 (mrunagain in the dataset). Replicate figure 3 (a) in the paper showing a scatter plot of the data and the fitted line of this RDD model. Does the incumbent candidate appear to have an electoral advantage? Explain. (6 marks)

Now estimate the RDD model separately for two dependent variables that have already been determined as of election t: the average number of terms the candidate has served in Congress (mofficeexp in the dataset) and the average number of times he has been a nominee (melectexp in the dataset). You should run two separate regressions of each of these variables on a fourth-order polynomial in the democratic vote share margin of victory, separately for each side of the threshold. Replicate figures 2 (b) and 3 (b) in the paper showing a scatter plot of the data and the fitted line of these RDD models. How do these results inform the validity of the identification assumption that you discussed in part b)? Explain in detail. (12 marks)

Now repeat the analysis in parts c) and d) adding the pre-determined variables mofficeexp and melectexp as controls in the regressions. Generate the two figures again using these estimates. Do the results change significantly when you control for these pre-determined variables? Explain in detail. (10 marks)

References:
Lee, David S. (2008), “Randomized experiments from non-random selection in U.S. House elections”, Journal of Econometrics, Vol. 142, pp. 675–697.