2.1 Identification of the PE

To identify the PE, formally defined in Section C of the Supplementary Material, the data must include more than one election-year and exhibit variation in the party who controls the assembly.Footnote ⁵ Assume that $D_{it}$ is randomized; our main strategy is to simply control for $M_{it}$ in the regression of $Y_{it}$ on $D_{it}$ . Note that $M_{it}$ depends only on $D_{it}$ and on which party has the majority in the assembly. It is therefore sufficient to assume that the overall majority is determined at the national level (and not at the district level) and to control for time fixed effects to safely include $M_{it}$ in the regression without introducing a selection bias. The assumption is more likely to hold (i) when the number of districts n is large, and thus small is the probability that the outcome in one district determines the overall majority, and (ii) the smaller the fraction of districts that never changes political color, because in that case the control of the assembly would be determined only by the outcome in the few contestable districts. Both (i) and (ii) are testable. Finally, note that Albouy (Reference Albouy2013) already makes the same assumption with the aim to identify $M_{it}$ , but he does not discuss the importance of controlling for $M_{it}$ in order to identify the PE, which is our focus.

In reality $D_{it}$ is not randomized and thus researchers rely on the RDD CE. In this design, Calonico et al. (Reference Calonico, Cattaneo, Farrell and Titiunik2019) recommend including controls, which is crucial in our identification strategy, only to improve precision and after checking that such controls are balanced at the threshold. This recommendation is based on the presumption that covariates imbalance might suggest that the potential outcome function is not continuous at the threshold, so that the crucial identifying assumption is violated. Furthermore, the authors add that covariates can be included to restore identification if the researchers are willing to impose additional assumptions. In our case, we are aware that $M_{it}$ might not be balanced at the threshold, and that the outcome might be a function of it. In fact, as elaborated above, we propose to include $M_{it}$ in the regression under the additional assumption that assembly control is determined at the national level.

Finally, note that if our argument does not convince the reader on the viability of controlling for $M_{it}$ , it is always possible to balance the sample in terms of years with democratic/republican control, so that the correlation between $M_{it}$ and $D_{it}$ is negligible and is not necessary to include majority status. In practice, one may selectively drop years or, more efficiently, use post-stratification (Miratrix, Sekhon, and Yu Reference Miratrix, Sekhon and Yu2013), that is, re-weight the sample such that observations under the two types of years have equal weight.

4.1 Roll-Call Voting and Incumbency Advantage 1946–1994

We replicate the analysis in Lee, Moretti, and Butler (Reference Lee, Moretti and Butler2004) using the original dataset, which includes results for the U.S. House in the period 1946–1994, and voting scores of representatives on a right–left scale 0–100 based on roll-call votes. In this sample, there is only one republican-controlled year. The authors use a RDD CE to estimate the PE on three outcomes: contemporaneous policy stance $RC_{it}$ , policy stance in the next term $RC_{it+1}$ , and the treatment in the next term $D_{it+1}$ (incumbency advantage). Results, reported in Table 1, show that including majority status considerably affects the estimate of the coefficient on $D_{it}$ for all outcomes.

Table 1 Replication of Lee, Moretti, and Butler (Reference Lee, Moretti and Butler2004)

Note: Linear model estimated with OLS without controlling for the margin of victory. Robust standard errors in parenthesis. Bandwidth = $2$ percentage points.

Despite some differences, the qualitative conclusion in Lee, Moretti, and Butler (Reference Lee, Moretti and Butler2004) is robust to this replication. Nevertheless this exercise shows that the PE changes more than one would expect in a valid RDD CE when we control for majority status.

4.2 Roll-Call Voting 1947–2008

We extend the dataset in the previous section until 2008, obtaining a sample with 23 terms under democratic control and 8 under republican control. The estimation is conducted separately on subsamples that feature a different ratio of observations from democratic- and republican-controlled years, resulting in different covariance between $D_{it}$ and $M_{it}$ . For simplicity, we only focus on the PE on contemporaneous roll-call voting $RC_{it}$ . Table 2 reports the results. In the most balanced period 1982–2004, the correlation between $D_{it}$ and $M_{it}$ is close to zero. As expected, the coefficient on $D_{it}$ is the same (approximately $56$ ) irrespective of whether we control for majority status. The coefficient on $M_{it}$ is approximately $-5$ , suggesting that majority members have on average a less liberal stance compared to opposition members, holding party constant. Results from the other subsamples are broadly consistent with what predicted theoretically in Section 2: relative to 1982–2004, the coefficient on $D_{it}$ in the model without $M_{it}$ is lower the more democratic years (positive covariance), and higher the more republican years (negative covariance). Furthermore, in all partially unbalanced subsamples controlling for majority status yields a coefficient on $D_{it}$ closer to $56$ , relative to the model without $M_{it}$ . Introducing time fixed effects makes little difference. The results confirm our theoretical insights which, however, has a limited quantitative relevance in this application, due to the moderate effect of majority status on roll-call voting.

Table 2 Roll-call voting.

Note: Linear model estimated with OLS controlling linearly for the margin of victory on each side of the threshold. Standard errors clustered at the electoral district. Bandwidth = 0.183 selected using the method by Calonico, Cattaneo, and Titiunik (Reference Calonico, Cattaneo and Titiunik2014).

4.3 Electoral Financing 1979–2006

We estimate the effect of a victory of the democratic party in a district on the campaign funds raised by the incumbent party in the next election.Footnote ⁸ Since most incumbents seek reelection, this is almost equivalent to testing whether democratic members raise more funds than their republican colleagues to finance their reelection campaign. This could happen if members of one party are on average more able to attract funds, or if donors have a partisan bias. The analysis is interesting in light of Cox and Magar (Reference Cox and Magar1999), who find that majority status yields an advantage in terms of campaign financing. The outcome is the amount of campaign funds (in thousands of 1990 dollars) raised in a district from non-investor donors by the party that won the previous election.

As before, in the balanced subsample (1978–2004) the coefficient on $D_{it}$ is the same (approximately $-133$ ) irrespective of whether we control for majority status (see Table 3). Moreover, here the coefficient on $M_{it}$ is sizable ( $80$ ), and thus its omission makes for very large difference in the estimate of the coefficient on $D_{it}$ in unbalanced subsamples: $-51$ in 1978–1992 versus $-205$ in 1994–2004. As before, controlling for majority status makes the estimate of the coefficient on $D_{it}$ more similar across subsamples.

Table 3 Campaign financing.

Note: Linear model estimated with OLS controlling linearly for the margin of victory on each side of the threshold. Standard errors clustered at the electoral district in parenthesis. Bandwidth = 0.09 selected using the method by Calonico, Cattaneo, and Titiunik (Reference Calonico, Cattaneo and Titiunik2014).