DS-UA 201: Causal Inference: Regression Discontinuity Part 2
University Center for Data Science
August 16, 2022
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Acknowledgement: Slides including material from DS-US 201 Fall 2021 offered by .
Overview of the Sharp RD Estimator
Our goal is to identify the local effect of assignment to treatment Di knowing that assignment is being driven by a running variable Xi and a cut-point c.
▶ Units with Xi above the cut-point receive treatment ▶ Units below the cut-point receive control
In this setting we can identify: E[Yi(1)−Yi(0)|Xi =c],
lim E[Yi|Xi =x]− lim E[Yi|Xi =x]. x→c+ x→c−
Overview of the Sharp RD Estimator
RDD Estimation strategy:
1. Subset the data to only observations that are within h of the
2. Fit one regression model of Yi on Xi above the cut-point and
another of Yi on Xi below the cut-point.
3. Use the models to predict the value of Yi at the cut-point.
4. The difference in predictions is the estimated treatment effect.
RDD Challenges:
▶ How do we choose a model?
▶ How do we choose h?
▶ Can we test the RDD assumptions?
▶ Imperfect treatment assignment at the threshold.
Illustration: Lee (2008) Election RDD
We’ll use the Lee (2008) election dataset to illustrate our results Research question: Does being an incumbent give you an advantage at the next election?
▶ Currently in power politicians have an easier time winning than runner ups for many different reasons
▶ identification problem: do they win because the people that previously elected them still like them or because they have an advantage?
Variables:
▶ X: Democratic margin of victory in time t ▶ Y : Democratic vote share in time t + 1 ▶ D: Victory in time t (margin > 0).
Design: Compare democrats that won by a small margin to democrats that lost by a small margin: similar voters but different incumbency outcomes.
Illustration: Lee (2008) Election RDD
# generate a treatment indicator
house$d <− as.integer(house$x > 0) # Subset to the close observations
house close <− subset(house, x>−.25&x<.25)
# Fit the regression model w/ interaction
rd reg <− lm robust(y ̃ d + x + d∗x, data=house close)
( Intercept ) 0.4509 0.4399 0.4618
d 0.0827 0.0663 0.0991
x 0.3665 0.2854 0.4476
d:x 0.0760 −0.0473 0.1993
Pr(>|t|) CI 80.82 0.00e+00 9.87 1.31e−22 8.86 1.37e−18 1.21 2.27e−01
Estimate Std. Error t value Lower CI Upper
0.00558 0.00838 0.04135 0.06288
1 2 3 4 5 6 7 8 9
Illustration: Lee (2008) Election RDD
#Add it to the plot
# Scatterplot w/ regression
bin scatter close reg <− ggplot(aes(x=x, y=y), data= house close ) +
stat summary bin(fun.y=’mean’, bins=50, size=2, geom=’point’) +
geom vline(xintercept=0, col=”red”, lty=2) +
geom smooth(data=subset(house close ,d==1), formula= y
̃ x, method=”lm robust”) +
geom smooth(data=subset(house close ,d==0), formula= y
̃ x, method=”lm robust”, col=”orange”) + xlab (”Democratic vote share margin of victory ,
Election t”) +
ylab (”Democratic vote share , Election t+1”) + theme bw()
Illustration: Lee (2008) Election RDD
−0.2 −0.1 0.0 0.1 0.2
Democratic vote share margin of victory, Election t
Democratic vote share, Election t+1
Model Choice in RDD
It is a popular idea to sometimes fit polynomial regressions to the data to estimate RDDs:
Y = α+D τ+X β +X2β +X3β +···+D (X λ +X2λ +X3λ +...)+ε , i ii1i2i3 ii1i2i3 i
This is usually advocated because:
▶ We need to predict the expected outcome well for the RDD to be valid
▶ The relationship between X and Y could be nonlinear
Illustration: Lee (2008) Election RDD
# Suppose we used a polynomial fit to the entire data
bin scatter close poly full <− ggplot(aes(x=x, y=y) ,
data=house) +
stat summary bin(fun.y=’mean’, bins=100,
size=2, geom=’point’) + geom vline(xintercept=0, col=”red”, lty=2) +
geom smooth(data=subset(house ,d==1),
formula= y ̃ x + I(xˆ2) + I(xˆ3),
method=”lm robust”) + geom smooth(data=subset(house ,d==0),
formula= y ̃ x + I(xˆ2) + I(xˆ3),
method=”lm robust” , col=”orange”) + xlab (”Democratic vote share margin of victory ,
Election t”) +
ylab (”Democratic vote share , Election t+1”) + theme bw()
3 4 5 6 7 8 9
Illustration: Lee (2008) Election RDD
−0.5 0.0 0.5 1.0
Democratic vote share margin of victory, Election t
●● ● ●● ● ●
Democratic vote share, Election t+1
Illustration: Lee (2008) Election RDD
# Polynomial fit to the entire dataset gives the most extreme estimate
rd reg poly full <− lm robust(y ˆ3)), data=subset(house))
> rd reg poly full
Estimate Std. Error
Lower CI Upper
̃ d∗(x + I(xˆ2) + I(x
( Intercept ) 0.4278 0.4149 0.4407
d 0.1115 0.0933 0.1297
x −0.0971 −0.2512 0.0569
I (xˆ2) −1.7177 −2.1725 −1.2630
I (xˆ3) −1.4636 −1.7986 −1.1286 d:x 0.4524
0.2492 0.6556
d : I ( x ˆ2) 1.9109
0.00659 0.00929 0.07859 0.23197 0.17089 0.10366 0.28731
1.3477 2.4741
value Pr(>|t|) CI 64.94 0.00e+00 12.00 7.61e−33 −1.24 2.17e−01 −7.41 1.47e−13 −8.56 1.33e−17
4.36 1.29e−05 6.65 3.14e−11
d:I(xˆ3) 1.2525 0.20437 6.13 9.37e−10
Caution with polynomials!
Even though polynomials are often used in RDD analysis, there are problems with them (Gelman and Imbens 2016)
▶ Noisy estimates
▶ Sensitivity to model specification and degree of polynomial ▶ Poor coverage of confidence intervals.
Intuition: Polynomial fit is closer to the datapoints, therefore:
▶ It might exaggerate the gap at the threshold due to noise
▶ Variance estimates will be smaller, therefore CIs will also be smaller
Suggestion: Either use linear, quadratic at most regressions, or other smooth functions.
Bandwith Choice in RDD
Another problem in RDD analysis is that of choosing the bandwith, h.
Choosing a bandwidth (h) for a local linear regression is a classic bias-variance trade-off.
▶ A metric for adjudicating between bias vs. variance is Mean Squared Error (MSE)
MSE(τˆ) = E[(τˆ − τ)2] = Bias(τˆ)2 + Var(τˆ)
▶ Larger choices of h lead to smaller variance but larger bias ▶ Smaller choices of h lead to larger variance but smaller bias
Illustration: Lee (2008) Election RDD
−0.2 −0.1 0.0 0.1 0.2
Democratic vote share margin of victory, Election t
▶ Bandwidth: 0.1
▶ Estimate: 0.06, 95% CI: [0.03, 0.08]
Democratic vote share, Election t+1
Illustration: Lee (2008) Election RDD
−0.2 −0.1 0.0 0.1 0.2
Democratic vote share margin of victory, Election t
▶ Bandwidth: 0.05
▶ Estimate: 0.07, 95% CI: [0.03, 0.1]
Democratic vote share, Election t+1
Choosing a bandwidth
Strategies for choosing bandwidth:
▶ Plot estimates for a lot of bandwidths – assess “robustness”
to bandwidth choice.
▶ Cross-validation: Randomly split the data into training and test sets – fit models of different bandwidths on training, compare predictive accuracy on the test for values of Xi close to the discontinuity (MSE)
▶ Optimal Bandwidth Selection: Imbens-Kalyanaraman (2008) develop a data-driven method for choosing h for local-linear regression.
General intuition: Smaller samples ⇝ larger bandwidth choices ⇝ more dependence on the underlying model.
Overview of Fuzzy RDD
Last lecture, we looked at the case where treatment was perfectly determined by the running variable Xi
▶ Units with Xi above the cut-point c have Di = 1
▶ Units with Xi below the cut-point c have Di = 0
What if not all units above the cut-point receive treatment?
What if not all units below the cut-point receive control?
Key idea: There is still a discontinuity at c in the probability of receiving treatment.
▶ The discontinuity is due an imperfectly applied rule
▶ For example: Van der Klauuw (2002) looks at the effect of financial aid on college enrollment, knowing that the aid assignment function for the university being studied incorporated cut-offs based on a GPA/SAT score index.
Visualizing Fuzzy RDD
Figure taken from Van der Klauuw (2002) “Estimating the Effect of Financial Aid Offers on College Enrollment: A Regression-Discontinuity Approach”
Fuzzy RDD Setup
▶ Running/forcing variable Xi with cut-off c. ▶ Zi : Indicator for being above the cut-off
▶ Di : Actual receipt of treatment.
▶ Yi: Outcome.
Fuzzy RDD Assumptions
Instead of assuming that the treatment assignment “jumps” from 0 to 1 at the cut-point c, the “fuzzy” RD design assumes that the probability of treatment is discontinuous at the threshold
Discontinuous Propensity of Treatment
lim Pr(Di =1|Xi =x)̸= lim Pr(Di =1|Xi =x) x→c+ x→c−
In other words, there is a subset of units, who would take the treatment if they were above the discontinuity and control if below.
▶ Unobserved confounders could be affecting treatment take up!
Fuzzy RDD is IV
The Fuzzy RD set-up is equivalent to an instrumental variables design where the instrumental variable is the indicator for being above or below the cut-off.
Standard IV assumptions apply:
▶ Exogeneity of Xi (within the area around the discontinuity) – same as in the sharp RD design (“local randomization”)
▶ Exclusion restriction (being slightly above the discontinuity only affects Yi through its effect on Di )
▶ Monotonicity (being above the discontinuity does not increase treatment propensity for some and decrease it for others).
Treating a fuzzy RD design as a “sharp” RD design gives us an intent-to-treat effect (what is the effect of being slightly above vs. below the discontinuity)
Fuzzy RDD is IV
Estimation is straightforward using classic 2SLS framework w/ (Xi − c) as a covariate in both regressions.
▶ Again, let Zi = ∞(Xi ≥ c) – the indicator for being above the threshold c.
First stage:
Di =δ0 +δ1Zi(Xi −c)+δ2(1−Zi)(Xi −c)+ρZi +ηi
Second stage:
Yi =β0+β1Zi(Xi −c)+β2(1−Zi)(Xi −c)+τDˆi +εi
Approach is equivalent to a Wald-type ratio estimator: the ratio of the sharp RD estimate over the estimated first-stage effect of the discontinuity on probability of treatment.
Fuzzy RDD is IV
As in IV, a Fuzzy RD effect is a local effect on compliers.
The Fuzzy RD identifies the Local Average Treatment Effect among compliers who would be induced to take the treatment if their Xi was slightly above the cut-off and take control if Xi were slightly below.
▶ We’re adopting the “local randomization” interpretation of RDD – Xi is as-good-as-random within the vicinity of the cut-point c.
▶ Because treatment assignment is not deterministic, only some units would actually change Di if they were above the cut-point vs. below. These are our “compliers”
The subset of units for which we estimate the ATE is even smaller! ▶ What can we say about populations of interest?ß
Example: Bleemer and Mehta (2020)
Question: Does majoring in Economics boost graduates’ wages?
▶ Major choice is endogenous!
▶ Bleemer and Mehta (2020) leverage the fact that UC Santa Cruz’s Economics department implemented a minimum GPA requirement in 2008 for students to be permitted to declare.
▶ Students had to earn a 2.8 GPA in Econ 1 and 2 to declare. But policy was imperfectly implemented. Grades are relatively noisy, hard for students to manipulate to get exactly above the threshold.
▶ Students just above the threshold were about 36pp more likely to declare.
▶ Being above the threshold raised post-graduation wages by about $8,000 annually
▶ This generated an IV estimate of about a $22,000 effect on annual early-career wages!
Example: Bleemer and Mehta (2020)
Assessing RD assumptions
How can we assess whether our “local randomization” assumption is plausible?
▶ Units slightly above the discontinuity differ on pre-treatment covariates from those slightly below the discontinuity – imbalance.
▶ Units are able to selectively manipulate their score to land just above or just below the cut-point (essentially a kind selection effect on an unobservable)
Solutions:
▶ Balance tests (is there a covariate that also exhibits a discontinuity around c?)
▶ Placebo tests (does the discontinuity have an “effect” around some fake discontinuity)
▶ Density tests (is the density of Xi discontinuous in the area of the threshold)?
Density tests
If units are not able to manipulate their Xi , then the density of Xi around the discontinuity should be continuous. (McCrary 2008)
▶ We shouldn’t expect surprisingly more or less units above versus below the cut-off.
Intuition:
▶ Construct a histogram of the running variable (with bins
selected to not overlap at the discontinuity)
▶ Smooth the histogram by fitting a local linear regression of the histogram heights on the bin mid-points
▶ Test for the difference in the smoothed histogram near the discontinuity
Implemented in the rdd package. 1
rdd :: DCdensity(house$x, cutpoint=0)
Density tests
● ●● ●●● ●●
●●●● ● ●●●●●
● ● ●● ● ●● ●●●
−0.5 0.0 0.5
0.0 0.2 0.4 0.6 0.8 1.0
Today we looked at four issues in RDD designs:
1. Model and polynomial choices: try to use simple models.
2. Bandwidth selection: show estimates for many bandwidths.
3. Imperfect treatment assignment at the threshold: fuzzy RDD as IV.
4. Assumption checking for RDD: Placebo and density tests.
Last 4 lectures: Special topics.
▶ Sensitivity analysis
▶ Causal inference and ML
▶ Causal inference case studies ▶ Course review
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