A Note on Close Elections and Regression Analysis of the Party Incumbency Advantage

Peter M. Aronow; David R. Mayhew; Winston Lin

doi:10.1515/spp-2014-0003

Article

A Note on Close Elections and Regression Analysis of the Party Incumbency Advantage

Peter M. Aronow , David R. Mayhew and Winston Lin

Published/Copyright: October 17, 2014

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From the journal Statistics, Politics and Policy Volume 5 Issue 1-2

Abstract

Much research has recently been devoted to understanding the effects of party incumbency following close elections, typically using a regression discontinuity design. Researchers have demonstrated that close elections in the US House of Representatives may systematically favor certain types of candidates, and that a research design that focuses on close elections may therefore be inappropriate for estimation of the incumbency advantage. We demonstrate that any issues raised with the study of close elections may be equally applicable to the ordinary least squares analysis of electoral data, even when the sample contains all elections. When vote share is included as part of a covariate control strategy, the estimate produced by an ordinary least squares regression that includes all elections either exactly reproduces or approximates the regression discontinuity estimate.

Corresponding author: Peter M. Aronow, Assistant Professor of Political Science, Yale University, 77 Prospect Street, New Haven, CT, 06520, USA, e-mail: peter.aronow@yale.edu

Acknowledgments

Thanks to Allison Carnegie, Don Green, Eitan Hersh, Seth Hill, Cyrus Samii, and Jas Sekhon for helpful comments.

Appendix

A Proof of Proposition 1

Let X_i0t=I[–J–1≤DemVot_it/k≤–J+1]. By Wooldridge (2002: p. 579),

β^OLS=(N−1∑j=0J∑it∈U˜jDemWinit2..)−1(N−1∑j=0J∑it∈U˜jDemWinit..DVi(t+1)..),

where DemWinit..≡DemWinit−∑j=0J[Xijt|U˜j|−1∑i′t′∈U˜jDemWini′t′],DVi(t+1)..≡DVi(t+1)−∑j=0J[Xijt|U˜j|−1∑i′t′∈U˜jDVi′(t′+1)] and U˜j≡{it:Xijt=1}.

When DemWin_it=0 or DemWin_it=1 for all it∈U˜j,DemẄin_it=0. Therefore, β^OLS=(∑i∈UbDemWinit2)−1..(∑i∈UbDemWinit..DVi(t+1)..), the least squares estimator as applied only to observations in the interval covering the discontinuity (noting that Ub=U˜J/2). Then, by simple algebraic manipulations, β^OLS=∑it∈UbDemWinitDVi(t+1)∑it∈UbDemWinit−∑it∈Ub(1−DemWinit)DVi(t+1)∑it∈Ub(1−DemWinit)=τ^RD.

B Proof of Proposition 2

Assume

E [DV|DemVot]=g(DemVot)I [DemVot<0]+f(DemVot)I [DemVot≥0]

where g() is continuous on [–1, 0) and f() is continuous on [0, 1]. Let α=limx↑0g(x) and β=f(0)–α. Note that we have not assumed that β has a causal interpretation. Rewriting the conditional expectation function,

E[DV|DemVot]=α+βI [DemVot≥0]+h(DemVot),

where

h(x)=[g(x)−α]I [x<0]+[f(x)−(α+β)]I [x≥0].

Our assumptions about g() and f() imply that h() is continuous on [–1, 1] and h(0)=0. By the Weierstrass Approximation Theorem, for any ϵ>0, there exist some J* and γ1*,…,γJ** such that |h(x)−∑j=1J*γj*xj|<ϵ for all x∈[–1, 1]. We assume that J in (2) is sufficiently large to ensure that there exist γ₁,…, γ_J such that |h(x)−∑j=1Jγjxj|≈0 for all x∈[–1, 1]. Then

E[DV|DemVot]≈α+βI [DemVot≥0]+∑j=1JγjDemVotj,

E[u|DemVot]≈0 and therefore E [β^OLS2]≈β. If the sample is drawn i.i.d. from a population U with Supp(DemVot)⊇[–k, k], where k>0, then for large enough N, we have β^OLS2≈β.

Under the same conditions, as N→∞, Nk→∞ and k→0, τ^RD≈β. We abbreviate the proof as it closely follows from the above. Under condition 2, ∑j=1JγjXijt may be represented as a constant B-spline, and consistency for β is assured under the usual conditions for RD or B-splines (see, e.g., Lee and Lemieux 2010; Imbens and Kalyanaraman 2012). Therefore, under the stated conditions, β^OLS2≈τ^RD.

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Published Online: 2014-10-17

Published in Print: 2014-12-1

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A Note on Close Elections and Regression Analysis of the Party Incumbency Advantage

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