Regression Discontinuity

Today

Regression Discontinuity
- One more tool for learning about causal effects
- Example of a design: like experiment or IV
- A situation in which data produced in a way which allows learning causal effect
- See Angrist and Pischke Ch 4
Method to learn from this data
- Extends event study idea to case where treatment determined by variable other than time
- Works well with nonparametric regression

Regression Discontinuity: Situation

Interested in causal effect of a binary (Yes/No) treatment variable \(D\) on outcome \(Y\)
E.g. effect of politician winning one election on chance of winning next election
- Called “incumbency advantage” in poli sci
Don’t have random assignment
- Can’t just compare treated and nontreated units
Sometimes have treatment assigned by a deterministic rule
- Variable \(X\), called running variable or assignment variable, determines treatment
- Units with \(X\geq c\) get treatment \(D=1\)
- Units with \(X<c\) don’t \(D=0\)
  - \(D=1\{X\geq c\}\)
- Candidate who gets >50% of votes win election, <50% lose

Problem

Running variable \(X\) may have direct effect on outcome
Or be correlated with other variable having direct effect
- E.g. candidates who win election probably different from those who lose in lots of ways that make them likely to win next election too
\(X\) also deterministically related to treatment,
- Conditioning on \(X\) not quite what we want to do

Solution

Use fact that assignment rule is discontinuous at \(c\)
Direct effect of \(X\) likely to be continuous
- Politicians who get 49.9% of vote probably not much different from those who got 50.1%
- But big change in treatment
  - 50.1% means win, 49.9% means lose
So compare units just on left and right side of \(c\)
So long as no other things jump at \(c\), any discontinuous change in \(Y\) exactly at \(c\) must be due to \(D\)
\(X\) is not randomly assigned, but whether \(X\) is just below or above \(c\) is effectively random

Regression Discontinuity Estimate

Estimate is difference in means “just above” and “just below” c \[\tau^{RD}:=\underset{e\searrow 0}{\lim}E[Y|X=c+e]-\underset{e\searrow 0}{\lim}E[Y|X=c-e]\]
Estimates average treatment effect when \(X\) is at \(c\)
\(\tau^{RD}=E[Y^1-Y^0|X=c]\) in potential outcomes notation
So long as \(E[Y^1|X]\) and \(E[Y^0|X]\) continuous
- No jumps in anything except treatment at \(c\)
Interpretation: weighted average treatment effect at \(c\) with weighting proportional to probability \(X_i=c\)
- E.g., effect of just winning election for legislators likely to be in close race

Visual Proof (Lee and Lemieux 2008, Fig 2)

Potential and Observed Outcomes

Examples of Discontinuities

Lee (2008): effect of winning last election on winning next
- Discontinuity at 50% vote share
Effect of admission to a school on education outcomes
Students who score above certain number admitted into school, below that number not
Effect of legal access to alcohol on health
- Illegal if age<21, legal if age>=21
Many welfare programs given only to those with income below a threshold

Regression Discontinuity: Simple Estimate

Model effect of \(D\) and \(X\) on \(Y\) by a regression \[ Y = b_0+\tau D + \beta_1X + u\]
Since \(D=1(X>c)\), this is same as \[ Y = b_0+\tau 1(X>c) + \beta_1X + u\]
Accounts for effect of \(X\), if linear and \(D\) additive
Very restrictive form
- May have interaction of \(X\) with treatment \[Y = b_0+\tau D + \beta_1(X-c) + \beta_2 D(X-c)+ u\]
- Different slope above and below c
Nonlinearity of effect of \(X\)
- Can add polynomials in \(X\), interacted with \(D\)
Need a correct model of effect of \(X\) and \(D\)
OLS regression provides estimate of jump in \(Y\) at threshold from \(\hat{\tau}\) size of jump at \(c\)

When does it work?

Need whether \(X\) is above or below value to be “effectively” random
Need well-specified control for effects of \(X\)
- If effect of X nonlinear, could mistake for effect of treatment
No jumps in other covariates at \(c\)
- They can change, but density has to be continuous
- Candidates who get 50.1% of vote only slightly more skilled politicians than those who get 49.9%
No discontinuous effect of treatment assignment on \(X\)
- Test score example: people study harder to get higher score
- This is fine, so long as people with score \(c-0.01\) and score \(c+0.01\) studied basically the same amount
- Not fine if passing associated with big jump in studying
  - People with \(c+0.01\) studied, \(c-0.01\) didn’t
- Concern in welfare case: people accept jobs earning cap - 1 cent to also get benefits

Accounting for underlying trend

As graphic for proof makes clear, don’t know anything about shape of causal effects away from jump point
Really only want Conditional Expectation Function at this point
Linear regression provides this if correctly specified
Otherwise biased, from trying to fit points far away from jump
Solution
- Nonparametric Regression!
Run linear regression only in narrow neighborhood to the left and to the right of \(c\)
Almost the same as estimate from last class
- Due to discontinuity, window should only contain data on left or data on right, not both

Local Linear Estimate

\[(\hat{\beta}^{-}_0(x),\hat{\beta}{-}_1(x))=\underset{(\beta_0,\beta_1)}{\arg\min}\sum_{i=1}^{n}1\{X_i<c\}K(\frac{X_i-x}{h})(Y_i-\beta_0-\beta_1(X_i-x))^2\]

\[(\hat{\beta}^{+}_0(x),\hat{\beta}{+}_1(x))=\underset{(\beta_0,\beta_1)}{\arg\min}\sum_{i=1}^{n}1\{X_i\geq c\}K(\frac{X_i-x}{h})(Y_i-\beta_0-\beta_1(X_i-x))^2\]

\[\hat{\tau}^{RD}=\hat{\beta}^{+}_0(c)-\hat{\beta}^{-}_0(c)\]

Difference in local linear estimates at point
Local linear tends to work better than local constant at boundary
- Accounts for slope up or down at the point

Choosing a Bandwidth: Considerations

Since we care about function only at point of discontinuity, optimize error at this point instead of average error over all points
Small \(h\) lowers bias, big \(h\) lowers variance
MSE optimal rate still gives \(h=cn^{-1/5}\) for some \(c\)
Can estimate optimal \(c\)
- Imbens & Kalyanaraman give procedure, implemented by default in most RD software
- Or use CV on local error
- R libraries rdd, rdestimate, or rdrobust
In practice: try a bunch of bandwidths, report all results

Lee Data w/ OLS, different slope on each side: 11.8% incumbency advantage (Code)

#Load RD package containing Lee (2008) incumbency data 
library(rddtools)
data(house) #Lee data
# Specify x, y, and cutpoint = 0 Dem-Rep vote share
house_rdd<-rdd_data(y=house$y,x=house$x,cutpoint=0)
# Run OLS RD estimate, linear 
# with different slope on each side
reg_para <- rdd_reg_lm(rdd_object=house_rdd)
plot(reg_para,xlab="Dem-Rep Vote Share, t", 
     ylab="Dem-Rep Vote Share, t+1", 
     main="Parametric RD")

Lee Data w/ OLS, different slope on each side: 11.8% incumbency advantage

Lee Data Local Linear, IK BW=0.29, 8.0% incumbency advantage (Code)

#Estimate MSE- optimal bandwidth on Lee 
# data by Imbens-Kalyanaraman procedure
LEEbw<-rdd_bw_ik(house_rdd)
# Run local linear RD estimate, 
# linear with different slope on each side
LEEnp<-rdd_reg_np(rdd_object=house_rdd,bw=LEEbw)
plot(LEEnp,xlab="Dem-Rep Vote Share, t", 
     ylab="Dem-Rep Vote Share, t+1", 
     main="Nonparametric RD")

Lee Data Local Linear, IK BW=0.29, 8.0% incumbency advantage

Confidence intervals and hypothesis tests

If estimated by OLS, use standard formulas
- Doesn’t account for bias if specification wrong
If estimated locally by nonparametric regression, formulas exist
- Scale SEs by \(\sqrt{nh}\), not \(\sqrt{n}\), so intervals wider
When functional form of regression not exact, bias exists even in large samples
Confidence intervels centered on \(\hat{\tau}^{RD}\) not centered on truth
2 solutions
Undersmoothing: Choose bandwidth smaller than optimal
- Bias small enough to be ignored when building (wider) confidence intervals
- Coverage/size still won’t be exact, but error no bigger than usual from using CLT for normal approx
Or use some formula robust to bias: see library rdrobust

“Fuzzy” Regression Discontinuity

RD provides “essentially” random assignment of a treatment
Sometimes above and below cutoff units only treated with some probability
Instead of \(D=1\{X\geq c\}\) have \(Pr(D=1|X\geq c)>Pr(D=1|X< c)\)
Sharp jump in probability, but not perfect assignment
E.g. Test score > c gives admission to school
- But not everyone chooses to accept
Formally identical to Instrumental Variables
- Discontinuity in \(X\) acts as instrument for \(D\)

Fuzzy RDD as Instrumental Variables Estimate

Recall Univariate IV: \(\tau^{IV}=\frac{\rho}{\phi}\)
Reduced form \(\rho\) causal effect of instrument on outcome \(Y\)
First stage \(\phi\) causal effect of instrument on treatment \(D\)
Both can be estimated by RD \[\tau^{Fuzzy}=\frac{\underset{e\searrow 0}{\lim}E[Y|X=c+e]-\underset{e\searrow 0}{\lim}E[Y|X=c-e]}{\underset{e\searrow 0}{\lim}E[D|X=c+e]-\underset{e\searrow 0}{\lim}E[D|X=c-e]}\]
Use OLS or local linear RD estimator for both
Since direct effects continuous, jump near \(c\) exogenous
Relevance holds if jump increases probability of treatment \(D\)
Can interpret as LATE if monotonicity holds
- Nobody induced to not take treatment by being over threshhold

Extensions

Regression kink design
- Slope changes suddenly, instead of level
- If slope of \(Y\) changes at same point, attribute to kink
Multivariate regression discontinuity
- Sudden change is function of multiple variables
- E.g., at the border of a state, law changes suddenly
- Estimate change in level at border
Adding controls
- Like with experiments, RD has regression representation
- If variables not correlated with jump in running variable, not needed to find causal effect
- Might get more precise estimates

Ensuring assumptions hold

Big requirement is that nothing else jumps at \(c\)
Can we test this? Yes, for some observable variables
- (Obviously can’t test for jumps in unobservables)
Issue if some individuals choose \(X\) to be just on one side
Problem if those individuals not identical on other characteristics affecting outcome
Not a problem if choice has smooth noise, so some randomly end up on either side
Density of \(X\) should not jump at \(c\)
Moreover, density of any observable covariates \(W\) also should not change at \(X=c\)

Density Estimation (Code)

#Run McCrary test 
# Estimates density left and right of cutoff
# null is no difference in density
LEEtest<-dens_test(LEEnp,plot=FALSE)
pvalue<-LEEtest[2] #get p value 
#Run McCrary test and display plot
dens_test(LEEnp)

Density Estimation

Estimate density locally by kernel density estimator
For each \(x\), use proportion of \(X_i\)’s in \(h\) neighborhood of \(x\) \[\hat{f}(x)=\frac{1}{n_x}\sum_{i=1}^{n}1(x-\frac{h}{2}<X_i<x+\frac{h}{2})\]
Compare Histogram
- Uses proportion over fixed, not rolling neighborhood
Use jump in density of \(X\) at c as test for manipulation
- McCrary test: if jump at cutoff, reject exogeneity
- Implemented in all RD packages
Also test density of other covariates when \(X\) just above/below
Lee (2008) study looks fine on density difference
- Cannot reject density same on left/right (p=0.1952357)
But does poorly on covariate balance
- Winners and losers in close races very different
- If race is close, put in a lot of effort to win

Lee Data: Density Above and Below Cutoff

Summary

Sometimes a treatment is assigned when a variable just passes a boundary
Near this point, assignment effectively random
- And can test to make sure
Can use difference in outcome just above and just below boundary to estimate causal effect
Use nonparametric (local) regression to focus just on this boundary area

Next Time

Maximum Likelihood and Discrete Choice Estimation
See Wooldridge Ch 17.1