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To examine the effects of incumbency (running for re-election to a house seat you currently hold) in the 2018 US House of Representatives election, we run the following regression on our dataset of election winners: vsharei = 0 + 1Inci + 2Di + 3(share16i) + Ui ,(1) where the dependent variable, vshare, is the of the vote-share the winner of a given seat received in the 2018 election; Inc is a dummy variable with Inc = 1 if the winner was an incumbent and Inc = 0 if the winner was not the incumbent; D is a dummy variable indicating if the winner was a democrat or republican with D = 1 for Democrat and D = 0 for Republican (assume everyone in the sample is either a Democrat or Republican); (share16) is the share of the vote the winner's party received in the district in 2016. Assume E[Ui |Inci , Di ,(share16i)] = 0. The sample size is n = 435. It is large enough to assume all large sample approximations work well.

2 (a) How should we interpret 3?

(b) Suppose one wants to test if incumbency has any effect on the candidate's vote-share, given all else equal. Write down (i) the null and alternative hypotheses, (ii) the name of the test statistic, (iii) its distribution for large sample, (iv) how to get the critical value if the significance level is 5%. (Providing the exact critical value is not necessary.)

(c) Suppose one wants to test that the incumbency status and being a Democrat have no effect on vote-share (after accounting for 2016 vote-share). Write down (i) the null and alternative hypotheses, (ii) the name of the test statistic, (iii) how to get the critical value if the significance level is 5%. (Providing the exact critical value is not necessary.)

(d)Suppose that being a Democrat was an advantage in the 2018 elections and incumbents were more likely to be Republicans. Suppose we estimate the model without the D variable. Will the OLS estimate of 1, the coefficient of Inc, suffer from positive or negative omitted variable bias? Your argument should be based on the omitted variable bias formula. The regressor (share16) does not affect the the omitted variable bias analysis.

(e) Suppose we estimate the model given in Equation (1). Think of another variable that is not included in the regression equation (1) and argue that omitting that variable causes a bias in the estimation of 2.