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