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MPs For Sale? Estimating Returns to Office in Post-War British Politics
Unformatted Document Text:  strategically sort around the threshold. In fact, Lee (2008) shows that as long as the vote share includes some random component with a continuous density, treatment status is randomized at the threshold of winning. It is important to notice that while the RD design is likely to have a very high degree of internal validity, we pay a price in terms of decreased external validity and also efficiency. Most importantly, τ RDD is a local average treatment effect that is informative for marginal candidates close to the threshold. Unless additional homogeneity assumptions are intro- duced, the effect estimate is not informative about the effect of gaining office for candidates very far away from the threshold of winning. Figure 5 presents the graphical results from the RD design for Conservative candidates. Log wealth is plotted against the vote share margin (Z i ). The dotted vertical line at zero indicates the threshold separating MPs (to the right of the threshold, denoted by squares) and unsuccessful candidates (to the left of the threshold, denoted by circles). The solid lines represent the conditional expectation functions of wealth given the vote share margin approximated using a locally weighted polynomial regression fitted to both sides of the threshold; pointwise .95 confidence bounds are indicated by dashed lines. Recall that the effect of office on wealth in the RD design is the difference of two regression functions at the threshold. The graph indicates a discontinuity of about 0.7, which is very close to the matching results and again suggests that Conservative candidates roughly doubled their wealth by winning office. Given the weak assumptions underlying this estimate, we have strong confidence in this finding. It is unlikely that this difference in wealth could be driven by unobserved differences between the two groups of candidates, given the local random assignment around the threshold of winning (formal balance tests are provided below). Another notable feature in this graph is that the conditional expectation of wealth is not steeply increasing in the vote share margin over most of the support of the vote since W does not vary conditional on Z, but the overlap assumption is violated because the probability ofassignment is either P r(W i = 1|Z i > 0) = 1 or P r(W i = 1|Z i < 0) = 0 depending on whether a candidate scores below or above the threshold. By construction, there are no units at the threshold for which Y i (0) can be observed, but we can simply use units very close to the threshold to infer this conditional expectationof the outcome. The amount of extrapolation required becomes arbitrarily small in large samples. See(Imbens & Lemieux 2007) for an excellent discussion. 15

Authors: Eggers, Andy. and Hainmueller, Jens.
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strategically sort around the threshold. In fact, Lee (2008) shows that as long as the vote
share includes some random component with a continuous density, treatment status is
randomized at the threshold of winning.
It is important to notice that while the RD design is likely to have a very high degree of
internal validity, we pay a price in terms of decreased external validity and also efficiency.
Most importantly, τ
RDD
is a local average treatment effect that is informative for marginal
candidates close to the threshold. Unless additional homogeneity assumptions are intro-
duced, the effect estimate is not informative about the effect of gaining office for candidates
very far away from the threshold of winning.
Figure 5 presents the graphical results from the RD design for Conservative candidates.
Log wealth is plotted against the vote share margin (Z
i
). The dotted vertical line at zero
indicates the threshold separating MPs (to the right of the threshold, denoted by squares)
and unsuccessful candidates (to the left of the threshold, denoted by circles). The solid
lines represent the conditional expectation functions of wealth given the vote share margin
approximated using a locally weighted polynomial regression fitted to both sides of the
threshold; pointwise .95 confidence bounds are indicated by dashed lines. Recall that the
effect of office on wealth in the RD design is the difference of two regression functions at
the threshold. The graph indicates a discontinuity of about 0.7, which is very close to the
matching results and again suggests that Conservative candidates roughly doubled their
wealth by winning office. Given the weak assumptions underlying this estimate, we have
strong confidence in this finding. It is unlikely that this difference in wealth could be driven
by unobserved differences between the two groups of candidates, given the local random
assignment around the threshold of winning (formal balance tests are provided below).
Another notable feature in this graph is that the conditional expectation of wealth
is not steeply increasing in the vote share margin over most of the support of the vote
since W does not vary conditional on Z, but the overlap assumption is violated because the probability of
assignment is either P r(W
i
= 1|Z
i
> 0) = 1 or P r(W
i
= 1|Z
i
< 0) = 0 depending on whether a candidate
scores below or above the threshold. By construction, there are no units at the threshold for which Y
i
(0)
can be observed, but we can simply use units very close to the threshold to infer this conditional expectation
of the outcome. The amount of extrapolation required becomes arbitrarily small in large samples. See
(Imbens & Lemieux 2007) for an excellent discussion.
15


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