Regression Discontinuity Designs - Harvard University

Regression Discontinuity Designs

Kosuke Imai

Harvard University

STAT186/GOV2002 CAUSAL INFERENCE

Fall 2019

Kosuke Imai (Harvard) Regression Discontinuity Designs Stat186/Gov2002 Fall 2019 1 / 16

Observational Studies

In many cases, we cannot randomize the treatment assignmentethical constraintslogistical constraints

But, some important questions demand empirical evidence eventhough we cannot conduct randomized experiments!

Designing observational studies find a setting where crediblecausal inference is possibleKey = Knowledge of treatment assignment mechanism

Regression discontinuiety design (RD Design):1 Sharp RD Design: treatment assignment is based on a

deterministic rule2 Fuzzy RD Design: encouragement to receive treatment is based on

a deterministic rule

Originates from a study of the effect of scholarships on students’career plans (Thistlethwaite and Campbell. 1960. J. of Educ. Psychol)


Regression Discontinuity Design

Idea: Find an arbitrary cutpoint c which determines the treatmentassignment such that Ti = 1{Xi ≥ c}Close elections as RD design (Lee et al. 2004. Q. J. Econ):

be a continuous and smooth function of vote shares everywhere,except at the threshold that determines party membership. Thereis a large discontinuous jump in ADA scores at the 50 percentthreshold. Compare districts where the Democrat candidatebarely lost in period t (for example, vote share is 49.5 percent),with districts where the Democrat candidate barely won (forexample, vote share is 50.5 percent). If the regression disconti-nuity design is valid, the two groups of districts should appear exante similar in every respect—on average. The difference will bethat in one group, the Democrats will be the incumbent for thenext election (t � 1), and in the other it will be the Republicans.Districts where the Democrats are the incumbent party for elec-tion t � 1 elect representatives who have much higher ADAscores, compared with districts where the Republican candidate

FIGURE ITotal Effect of Initial Win on Future ADA Scores:

This figure plots ADA scores after the election at time t � 1 against theDemocrat vote share, time t. Each circle is the average ADA score within 0.01intervals of the Democrat vote share. Solid lines are fitted values from fourth-order polynomial regressions on either side of the discontinuity. Dotted lines arepointwise 95 percent confidence intervals. The discontinuity gap estimates

� 0�P*t �1D � P*t �1

R � 1�P*t �1D � P*t �1

R .

“Affect” “Elect”

828 QUARTERLY JOURNAL OF ECONOMICS


Identification

Estimand:E(Yi(1)− Yi(0) | Xi = c)

Assumption: E(Yi(t) | Xi = x) is continuous in x for t = 0,1deterministic rather than stochastic treatment assignmentviolation of the overlap assumption: 0 < Pr(Ti | Xi = x) < 1 for all xRD design is all about extrapolation

Regression modeling:

E(Yi(1) | Xi = c) = limx↓c

E(Yi(1) | Xi = x) = limx↓c

E(Yi | Xi = x)

E(Yi(0) | Xi = c) = limx↑c

E(Yi(0) | Xi = x) = limx↑c

E(Yi | Xi = x)

Advantage: internal validityDisadvantage: external validityMake sure nothing else is going on at Xi = c


Analysis Methods under the RD Design

Simple linear regression within a windowHow should we choose a window in a principled manner?How should we relax the functional form assumption?higher-order polynomial regression not robust to outliers

Local linear regression (same h for both sides): better behavior atthe boundary than other nonparametric regressions

(α̂+, β̂+) = argminα,β

n∑i=1

1{Xi > c}{Yi − α− (Xi − c)β}2 · K(

Xi − ch

)

(α̂−, β̂−) = argminα,β

n∑i=1

1{Xi < c}{Yi − α− (Xi − c)β}2 · K(

Xi − ch

)Weighted regression with a kernel function of one’s choice:

uniform kernel: K (u) = 12 1{|u| < 1}

triangular kernel: K (u) = (1− |u|)1{|u| < 1}


Optimal Bandwidth (Imbens and Kalyanaraman. 2012. Rev. Econ. Stud.)

Choose the bandwidth by minimizing the MSE:

MSE = E[{(α̂+ − α̂−)− (α+ − α−)}2 | X]= E{(α̂+ − α+)

2 | X}+ E{(α̂− − α−)2 | X}−2 · E(α̂+ − α+ | X) · E(α̂− − α− | X)

= (Bias+ − Bias−)2 + Variance+ + Variance−

Bias and variance of local linear regression estimator at theboundary:

Bias = E(m̂(0) | X)−m(0), Variance = V(m̂(0) | X)

where m(x) = E(Yi | Xi = x), m̂(x) = α̂(x), and

(α̂(x), β̂(x)) = argminα,β

n∑i=1

(Yi − α− β(Xi − x))2 · K(

Xi − xh

)Refinements, e.g., bias correction (Calonico et al. 2014. Econometrica)


The “As-if Random” Assumption

RD design does NOT require the local randomization or “as-ifrandom” assumption within a window:

{Yi(1),Yi(0)}⊥⊥Ti | c0 ≤ Xi ≤ c1

The “as-if random” assumption implies zero slope of regressionlines difference-in-means within the windowThe assumption may be violated regardless of the window size

PL19CH20-Imai ARI 16 April 2016 8:58

a Democratic experience advantage b Share of total spending by Democratic candidate

–0.05 0.00 0.05

0.1

0.2

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0.7

0.8

Democratic margin

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Democratic margin

Prop

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Figure 1The problem of the local randomization assumption. Under the local randomization assumption, also called the as-if-randomassumption, the observations below and above the discontinuity threshold, a [−0.02, 0.02] window indicated by dotted lines in this case,are assumed to be identical on average. As a result, the estimated discontinuity is based on two flat lines with no slope (red dashed lines).In contrast, under the continuity assumption, the association with the forcing variable is not assumed to be absent (blue solid lines). Thetwo plots are based on the dataset on US House elections by Caughey & Sekhon (2011) using two pretreatment covariates: theexperience advantage of the Democratic candidate (a) and the proportion of total donations given to the Democrat (b). They show thatthe local randomization assumption can falsely discover a discontinuity (a) or overestimate one (b).

WHEN IS THE CONTINUITY ASSUMPTION VIOLATED?The discussion so far implies that imbalance in pretreatment covariates just below and above thethreshold does not necessarily imply the violation of the identification assumption for the RDdesign. Under the continuity assumption, such imbalance can exist so long as there is no discon-tinuous jump at the threshold. Lack of discontinuity in pretreatment covariates at the thresholdthen represents empirical evidence for the continuity of the expected potential outcomes so longas all pretreatment covariates relevant for the outcome of interest are measured and analyzed.

If covariate imbalance does not necessarily invalidate the RD design, what are the scenariosunder which the continuity assumption is violated? To answer this question, we must considerthe kind of sorting behavior that pushes would-be barely-losers up above the threshold or movespotential barely-winners down below the threshold.8 Such sorting will lead to a discontinuousjump at the threshold in the conditional expectation function of the potential outcomes. This inturn is likely to manifest as a discontinuous jump in pretreatment covariates, which are associatedwith the outcome.

Following the informative discussion of this issue by Eggers et al. (2015a), we consider two typesof sorting behavior. One is due to pre-election behavior or characteristics of candidates, whereasthe other is owing to postelection advantages in vote tallying, including the ability to engineerelectoral fraud. We argue that although the postelection sorting behavior clearly constitutes aviolation of the continuity assumption, the pre-election behavior may not. The occurrence ofelectoral fraud, for example, implies that a candidate who would have barely lost the election ends

8See Eggers et al. (2015b) for a discussion of sorting behavior when using population thresholds as the discontinuity cutoffs.

www.annualreviews.org • Regression Discontinuity Design 383

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Close Elections Controversy (de la Cuesta and Imai. 2016. Annu. Rev.

Political Sci)

Sorting?1 Pre-election

behavior orcharacteristicsof candidates,e.g., resourceadvantages steep slope

2 Post-electionadvantages invote tallying,e.g., electionfraud sorting

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–1.5 –1.0 –0.5 0.0 0.5 1.0 1.5

a Difference-in-means in window

Estimated discontinuity(standard deviation unitsfor nonbinary measures)



Rep. # prev. terms

Inc.'s D1 nominate

Partisan swing

Rep. experience adv.

Rep. inc. in race

Dem. pres. margin

% Govt. worker

% Voter turnout

% Black

Rep.-held open seat

Dem.-held open seat

Dem. governor

% Foreign born

Open seat

Dem. sec. of state

Dem. inc. in race

Dem. win t − 1

Dem. experience adv.

Dem. margin t − 1

Dem. share t − 1

Dem. # prev. terms

% Urban

Dem. spending %

Dem. donation %

CQ rating

–1.5 –1.0 –0.5 0.0 0.5 1.0 1.5

b Linear regression in window

Inc.'s D1 nominate

Partisan swing

Rep. # prev. terms


Rep. inc. in race

Dem. governor

Dem. pres. margin

Rep.-held open seat

% Voter turnout

Open seat

Dem.-held open seat

% Black

% Foreign born

% Urban

Dem. sec. of state

Dem. # prev. terms

% Govt. worker

Dem. margin t − 1

Dem. share t − 1

Dem. spending %

Dem. donation %

Dem. inc. in race


Dem. win t − 1

CQ rating

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

c Local linear regression

Rep. # prev. terms


% Govt. worker

Partisan swing

Inc.'s D1 nominate

Rep. inc. in race

Dem. pres. margin

% Voter turnout

Rep.-held open seat

Dem. sec. of state

Dem. inc. in race

% Foreign born

Dem.-held open seat

Dem. margin t − 1

% Black

Open seat

Dem. governor

Dem. win t − 1


Dem. share t − 1

% Urban

CQ rating

Dem. # prev. terms

Dem. donation %

Dem. spending %

Figure 2Comparison of estimated discontinuities in pretreatment covariates across three methods. Solid and dashed lines in each panel represent 95% confidence intervals, notcorrected for multiplicity. (a) Filled blue circles represent estimates based on the difference-in-means estimator within the 2-percentage-point window on either side ofthe threshold; open red circles represent estimates within the one-half-percentage-point window. Panel (b) shows the estimates based on the linear regression in the samesets of windows. Panel (c) presents the estimates based on the local linear regression proposed by Calonico et al. (2014). Abbreviations: adv., advantage; CQ, CongressionalQuarterly; Dem., Democratic; govt., government; inc., incumbent; pres., president; prev., previous; Rep., Republican; sec., secretary; t, time period.

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–1.5 –1.0 –0.5 0.0 0.5 1.0 1.5

a Difference-in-means in window




Rep. # prev. terms

Inc.'s D1 nominate

Partisan swing


Rep. inc. in race

Dem. pres. margin

% Govt. worker

% Voter turnout

% Black

Rep.-held open seat

Dem.-held open seat

Dem. governor

% Foreign born

Open seat

Dem. sec. of state

Dem. inc. in race

Dem. win t − 1


Dem. margin t − 1

Dem. share t − 1

Dem. # prev. terms

% Urban

Dem. spending %

Dem. donation %

CQ rating

–1.5 –1.0 –0.5 0.0 0.5 1.0 1.5

b Linear regression in window

Inc.'s D1 nominate

Partisan swing

Rep. # prev. terms


Rep. inc. in race

Dem. governor

Dem. pres. margin

Rep.-held open seat

% Voter turnout

Open seat

Dem.-held open seat

% Black

% Foreign born

% Urban

Dem. sec. of state

Dem. # prev. terms

% Govt. worker

Dem. margin t − 1

Dem. share t − 1

Dem. spending %

Dem. donation %

Dem. inc. in race


Dem. win t − 1

CQ rating

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

c Local linear regression

Rep. # prev. terms


% Govt. worker

Partisan swing

Inc.'s D1 nominate

Rep. inc. in race

Dem. pres. margin

% Voter turnout

Rep.-held open seat

Dem. sec. of state

Dem. inc. in race

% Foreign born

Dem.-held open seat

Dem. margin t − 1

% Black

Open seat

Dem. governor

Dem. win t − 1


Dem. share t − 1

% Urban

CQ rating

Dem. # prev. terms

Dem. donation %

Dem. spending %

Figure 2Comparison of estimated discontinuities in pretreatment covariates across three methods. Solid and dashed lines in each panel represent 95% confidence intervals, notcorrected for multiplicity. (a) Filled blue circles represent estimates based on the difference-in-means estimator within the 2-percentage-point window on either side ofthe threshold; open red circles represent estimates within the one-half-percentage-point window. Panel (b) shows the estimates based on the linear regression in the samesets of windows. Panel (c) presents the estimates based on the local linear regression proposed by Calonico et al. (2014). Abbreviations: adv., advantage; CQ, CongressionalQuarterly; Dem., Democratic; govt., government; inc., incumbent; pres., president; prev., previous; Rep., Republican; sec., secretary; t, time period.

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Density Test of Sorting (McCrary. 2008. J. Econom.)

r ¼ X 1;X 2; . . . ;X J . The binsize and bandwidth were again chosen subjectively after using the automaticprocedure. Much more so than the vote share density, the roll call density exhibits very specific features nearthe cutoff point that are hard for any automatic procedure to identify.27

The figure strongly suggests that the underlying density function is discontinuous at 50%. Outcomes withina handful of votes of the cutoff are much more likely to be won than lost; the first-step histogram indicatesthat the passage of a roll call vote by 1–2 votes is 2.6 times more likely than the failure of a roll call vote by 1–2

ARTICLE IN PRESS

0

30

60

90

120

150

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1Democratic Margin

Fre

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0.00

0.20

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Fig. 4. Democratic vote share relative to cutoff: popular elections to the House of Representatives, 1900–1990.

Table 2Log discontinuity estimates

Popular elections Roll call votes

#0.060 0.521(0.108) (0.079)

N 16,917 35,052

Note: Standard errors in parentheses. See text for details.

0

50

100

150

200

250

300

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Percent Voting in Favor of Proposed Bill

Fre

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0.00

0.50

1.00

1.50

2.00

2.50

Den

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Est

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Fig. 5. Percent voting yeay: roll call votes, U.S. House of Representatives, 1857–2004.

27I use a binsize of b ¼ 0:003 and a bandwidth of h ¼ 0:03. The automatic procedure would select b ¼ 0:0025 and h ¼ 0:114.

J. McCrary / Journal of Econometrics 142 (2008) 698–714710

1 Create histogram with a selected bin size2 Fit local linear regression to bin midpoints to smooth the histogram3 Estimate the difference in the logged histogram height at the

thresholdKosuke Imai (Harvard) Regression Discontinuity Designs Stat186/Gov2002 Fall 2019 9 / 16

Placebo Test

V.C. Sensitivity to Alternative Measures of Voting Records

Our results so far are based on a particular voting index, theADA score. In this section we investigate whether our resultsgeneralize to other voting scores. We find that the findings do notchange when we use alternative interest groups scores, or othersummary measures of representatives’ voting records.

Table III is analogous to Table I, but instead of using ADAscores, it is based on two alternative measures of roll-call voting.The top panel is based on McCarty, Poole, and Rosenthal’s DW-NOMINATE scores. The bottom panel is based on the percent ofindividual roll-call votes cast that are in agreement with theDemocrat party leader. All the qualitative results obtained usingADA scores (Table I) hold up using these measures. When we usethe DW-NOMINATE scores, is �0.36, remarkably close to thecorresponding estimate of 1[Pt�1

D � Pt�1R ] in column (4), which

is �0.34. The estimates are negative here because, unlike ADAscores, higher Nominate scores correspond to a more conservativevoting record. When we use the measure “percent voting with theDemocrat leader,” is 0.13, almost indistinguishable from the

FIGURE VSpecification Test: Similarity of Historical Voting Patterns between Bare

Democrat and Republican DistrictsThe panel plots one time lagged ADA scores against the Democrat vote share.

Time t and t � 1 refer to congressional sessions. Each point is the average laggedADA score within intervals of 0.01 in Democrat vote share. The continuous line isfrom a fourth-order polynomial in vote share fitted separately for points above andbelow the 50 percent threshold. The dotted line is the 95 percent confidenceinterval.

838 QUARTERLY JOURNAL OF ECONOMICS

What is a good placebo?1 expected not to have any effect2 closely related to outcome of interest

Lagged outcome future cannot affect pastInterpretation: failure to reject the null 6= the null is correct


Fuzzy RD Design (Hahn et al. 2001. Econometrica)

Sharp regression discontinuity design: Ti = 1{Xi ≥ c}What happens if we have noncompliance?Forcing variable as an instrument: Zi = 1{Xi ≥ c}Potential outcomes: Ti(z) and Yi(z, t)Assumptions

1 Monotonicity: Ti(1) ≥ Ti(0)2 Exclusion restriction: Yi(0, t) = Yi(1, t)3 E(Ti(z) | Xi = x) and E(Yi(z,Ti(z)) | Xi = x) are continuous in x

Estimand:

E(Yi(1,Ti(1))− Yi(0,Ti(0)) | Complier ,Xi = c)

Estimator:

limx↓c E(Yi | Xi = x)− limx↑c E(Yi | Xi = x)limx↓c E(Ti | Xi = x)− limx↑c E(Ti | Xi = x)

Disadvantage: external validityKosuke Imai (Harvard) Regression Discontinuity Designs Stat186/Gov2002 Fall 2019 11 / 16

Class Size Effect (Angrist and Lavy. 1999. Q. J. Econ)

Effect of class-size on student test scoresMaimonides’ Rule: Maximum class size = 40

f (z) =z⌊ z−1

40

⌋+ 1

0 50 100 150 200

1015

2025

3035

40

Enrollment Count

Cla

ss S

ize

Maimonides RuleActual class size


Empirical Analysis

Yi : class average verbal test scoreTi : classsizeWindow size: wConstruction of forcing variable:

Xi =

40− Zi if 40− w/2 ≤ Zi ≤ 40 + w/280− Zi if 80− w/2 ≤ Zi ≤ 80 + w/2

......

Linear models (cluster standard errors by schools):

Ti = δ1 + α1 × I{Xi ≥ 0}+ β1Xi + γ1Xi × I{Xi ≥ 0}+ ε1i

Yi = δ2 + α2 × I{Xi ≥ 0}+ β2Xi + γ2Xi × I{Xi ≥ 0}+ ε2i

where α̂1 = −7.90 (s.e. = 1.90) and α̂2 = −0.056 (s.e. = 2.08)Two-stage least squares estimate: est. = 0.007 (s.e. = 0.261)


Interrupted Time Series Design

Time as the forcing variablePossibility of multiple events at the same timeMust model time trend: seasonality, etc.

Effect of the “stand your ground” bill in Florida on homicide

(Humphreys, Gasparrini, and Wiebe. 2017. JAMA)

Use of other states difference-in-differences designs


Geographical RD Design (Keele and Titiunik. 2015. Political Anal.)

RD in two dimensions don’t use distance as forcing variableExample: media markets in Princeton, New Jersey

6.3 Compound Treatment Reduction

Before calculating each unit’s distance to each of the boundary points, we discuss how to minimize

the compound treatment assumption in this application. Figure 4 contains a map of the state of

New Jersey along with the location of the boundary between the New York and Philadelphia media

markets. We could calculate distances between voters and points along the entire media market

boundary and compare voters who are near each other along this border. However, it is first

important to reduce the incidence of compound treatments as much as possible. To do that, we

examined the boundaries of four different administrative units: U.S. congressional districts, state

senate districts, state house districts, and school districts. We found that for many parts of the

media market boundary, the boundaries of at least one of these units overlapped perfectly with the

media market boundary. In other words, in various boundary segments, not only did the media

market change at the boundary, but so did the school and/or the legislative districts. This is not

entirely surprising since, as we discussed above, the media market boundary in this area (and in

most of the United States) follows county boundaries.The overlap between media and county boundaries means that we cannot escape the problem of

compound treatments entirely, but we can minimize it by restricting our analysis to those segments

Fig. 4 Boundary between Philadelphia and New York City media markets. The dashed line represents

the boundary between the Philadelphia, PA, media market (located southwest of the boundary) and theNew York City, NY, media market (located northeast of the boundary), which divides the state ofNew Jersey. Area of detail is where legislative districts and school district are constant on both sides ofmedia market boundary. Figure 5 contains a detailed map of this area.

Geographic Boundaries as Regression Discontinuities 143

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along the media market boundary where voters are in identical legislative districts and schooldistricts. Despite the length of the media market boundary, we found only one short segmentalong the border where both legislative district or school district boundaries did not also followthe county–media market boundary. The area of detail in Fig. 4 marks this boundary segment.Figure 5 contains a detailed map of this area. The area marked with gray hash lines is WestWindor-Plainsboro school district, which is split in two by the county–media market boundary.This school district also lies within a single U.S. House, state house, and state senate district.We restrict our analysis to residents in this school district since we can more plausibly assumethat areas on either side of this segment of the media market border are comparable—though wetest this assumption empirically below. Thus, while we cannot avoid compound treatments in theapplication, by finding an area where school and legislative districts are constant, we hope toincrease the plausibility of the compound treatment irrelevance assumption.

It is also worth noting that by holding units such as legislative and school districts constantin order to reduce compound treatments, we are making our analysis conditional on distance.That is, by removing compound treatments, we are already restricting our analysis to areaswithin a specific distance of the border. We suspect that, in many applications of the GRDdesign, addressing the compound treatment problem will prompt researchers to indirectly conditionon distance to the boundary and will be a useful first step to identify areas that are similar along thediscontinuity of interest.

New York-Philadelphia Media Market Boundary

New York Media Market

Philadelphia Media Market

Treated Area of Analysis

Control Area of Analysis

Montgomery Township School District

Princeton School District

Hopewell Valley School District

Frankling Township School District

South Brunswick School District

Cranbury Township School District

East Windsor School District

Robbinsville Township School DistrictMilestone Township School Distric

West Windsor-Plainsboro School District

Upper Freehold School District

Lawrence Township School District

Fig. 5 Detail of the boundary between Philadelphia and New York City media markets. Area marked with

gray hash lines indicates the West Windsor-Plainsboro school district, which straddles the media marketboundary. Empirical analysis is confined to the West Windsor-Plainsboro school district only, where legis-lative districts are also the same on both sides of the border. Treated area is southwest of media market

boundary, inside Philadelphia media market, where volume of political ads is high; control area is northeastof media market boundary, inside New York City media market, where volume of ads is zero.

Luke J. Keele and Rocı́o Titiunik144

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Summary

Observational studies treatment assignment is not random“Design” observational studies for credible causal inference

Sharp regression discontinuity (RD) designs:deterministic (rather than stochastic) treatment assignment rulecontinuity assumption no sortingdoes not require “as-if random” assumptionlimited external validity extrapolation required for generalizationincumbency effects controversy (Eggers et al. 2015. Am. J. Political Sci)Importance of placebo tests

Fuzzy RD designs: noncomplianceOther RD designs: interrupted time series, geographical boundary

Suggested readings:ANGRIST AND PISCHKE, Chapter 6IMBENS AND LEMIEUX. (2008). J. of Econom


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