Regression for causal inference: incumbency and voting#

Part 4 · Stage 11 · 📈 Regression Foundations · Lesson 093 of 144 · advanced

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Important

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Does holding office help you win?#

The incumbency advantage is the extra vote share a candidate receives merely by being the sitting representative. It cannot be randomised — nobody assigns incumbency by coin flip — so the question is observational, and regression must carry the causal weight. Stage 7’s warnings apply in full.

Why the naive measures fail#

Two classical estimators look reasonable and are biased. Sophomore surge compares a legislator’s vote share in their second election (as incumbent) with their first (as challenger). Retirement slump compares a party’s share before and after its incumbent retires. Each conflates the incumbency effect with the fact that incumbents are not a random sample: they are the candidates who won, in districts that favoured them. Gelman and King showed the bias directly — and showed that the information in these comparisons, placed in a regression framework, yields an unbiased estimate.

The model#

Model district \(i\)’s vote share \(v_{it}\) in election year \(t\), controlling for the district’s previous vote share (a proxy for partisan strength) and party, with an indicator \(I_{it}\) for whether an incumbent is running:

\[v_{it} = \beta_0 + \beta_1 \, v_{i,t-1} + \beta_2 \, P_{it} + \underbrace{\gamma}_{\text{incumbency}} I_{it} + \epsilon_{it} .\]

The coefficient \(\gamma\) is the estimand. Conditioning on the lagged vote is what removes the selection: districts where incumbents run are compared with similar districts holding open seats.

import pymc as pm
with pm.Model():
    b = pm.Normal("b", 0, 0.5, shape=3)
    gamma = pm.Normal("gamma", 0, 0.1)               # incumbency effect, on vote-share scale
    mu = b[0] + b[1] * v_lag + b[2] * party + gamma * incumbent
    pm.Normal("v", mu, pm.HalfNormal("sigma", 0.1), observed=v)
    idata = pm.sample()
# posterior for gamma, by decade, shows the advantage growing over the century

What makes it causal, and what does not#

The regression is the causal claim only under the no-unmeasured-confounding assumption of Stage 7: that, given the lagged vote and party, whether an incumbent runs is as good as random. That is an assumption about why incumbents retire — health, scandal, redistricting, or anticipated defeat — and the last of these breaks it. If incumbents strategically retire when they expect to lose, retirement is informative about \(\epsilon_{it}\), and \(\gamma\) is biased upward.

So the honest report has three parts: the estimate, the assumption it rests on, and a sensitivity analysis for how strong a confounder would need to be to erase it. Regression supplies the arithmetic; the design supplies the license.

Hint