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Regression Discontinuity Design

Marcelo Coca Perraillon

University of Chicago

April 24 & 29, 2013

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Introduction

Basics

Method developed to estimate treatment effects in non-experimentalsettings

Provides causal estimates of treatment effects

Good internal validity; some assumptions can be empirically verified

Treatment effects are local (LATE)

Limits external validity

Easy to estimate

First application: Thistlethwaite and Campbell (1960)

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Introduction Thistlethwaite and Campbell

Thistlethwaite and Campbell

They studied the impact of merit awards on future academic outcomes

Awards allocated based on test scores

If a person had a score greater than c, the cutoff point, then she

received the awardSimple way of analyzing: compare those who received the award tothose who didnt. Why is this wrong?

Confounding: factors that influence the test score are also related tofuture academic outcomes (income, parents education, motivation)

Thistlethwaite and Campbell noted that they could compareindividuals just above and just below the cutoff point

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Introduction Validity

Validity

Simple idea: assignment mechanism is completely knownWe know that the probability of treatment jumps to 1 if test score >c

Assumption is that individuals cannot manipulate with precision theirassignment variable (think about the SAT)

Key word: precision. Consequence: comparable individuals near cutoffpoint

If treated and untreated individuals are similar near the cutoff pointthen data can be analyzed as if it were a (conditionally) randomizedexperiment

If this is true, then background characteristics should be similar nearc(can be checked empirically)

The estimated treatment effect applies to those near the cutoff point(external validity)

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Introd ction Graphical Example


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Introduction Graphical Example

No effect

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Estimation


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Estimation

Estimation: Parametric

Simplest case is linear relationship between Y and X

Yi =0+ 1Ti+3Xi+i

Ti= 1 if subject i received treatment and Ti= 0 otherwise. You can

also write this as Ti =1(Xi>c) orTi= [Xi>c]

Xis the assignment variable (sometimes called forcing or runningvariable)

Usually centered at cutoff point

Yi=0+1Ti+3(Xi c) +i. Treatment effect is given by 1.

E[Y|T = 1,X =c] =0+ 1 and E[Y|T = 0,X =c] =0.

E[Y|T = 1,X =c] E[Y|T = 0,X =c] =1.

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Estimation Centering


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Estimation Centering

Reminder on centering

Centering changes the interpretation of the intercept:

Y = 0+ 1(Age 65) +2Edu

= 0+ 1Age 165 +2Edu

= (0 165) +1Age+2Edu

Compare to:Y =0+ 1Age+2Edu

1=1, 2=2, but 0= (0 165)

Useful with interactions:

Y =0+ 1Age+2Edu+3Age Edu

Compare to:

Y =0+ 1(Age 65) + 2(Edu 12) + 3(Age 65) (Edu 12)

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Estimation Extrapolation


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Extrapolation

Note that the estimation of treatment effect in RDD depends onextrapolation

To the left of cutoff point only non-treated observationsTo the right of cutoff point only treated observations

What is the treatment effect at X= 130? Just plug in:

E[Y|T,X= 130] =0+ 1T+3(130 140)

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p

Extrapolation...

Dashed lines are extrapolations

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Counterfactuals

The extrapolation is a counterfactual or potential outcomeEach person ihas two potential outcomes (Rubins causalframework).

Yi(1) denotes the outcome of person i if in the treated group

Yi(0) denotes the outcome of person i if in the non-treated groupCausal effect of treatment for person i is Yi(1) Yi(0)

Average treatment effect is E[Yi(1) Yi(0)]

Only one potential outcome is observed. In randomized experiments,

one group provides the conterfactual for the other because they arecomparable (exchangeable)

Exchangeability (epi). Also called selection on observables or nounmeasured confounders

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Counterfactuals, II

In RDD the counterfactuals are conditional on Xas in a conditionallyrandomized trial (think severity)

We are interested in the treatment effect at X =c:E[Yi(1) Yi(0)|Xi =c]

Treatment effect is limxcE[Yi|Xi=x] limxcE[Yi|Xi=x]

Estimation possible because of the continuity ofE[Yi(1)|X] andE[Yi(0)|X]

Since the estimation of the treatment effect is based on extrapolationbecause of lack of overlap, the functional relationship between X andYmust be correctly specified

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Estimation Functional form


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Need to model relationship between X and Y correctly

What if nonlinear? Could result in a biased treatment effect if one

assumes a linear model.

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Estimation Flexible specification


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Other specifications

More general: Yi=0+ 1Ti+3f(Xi c) +i

If (Xi c) = Xi then Yi=0+ 1Ti+3f(Xi) +i

Most common form for f(Xi) are polynomials

Polynomials of order p:Yi=0+1Ti+2Xi+3Xi

2+4Xi

3+ +p+1Xi

p+i

More flexibility with interactions

2nd degree with interactions:

Yi=0+1Ti+3Xi+4Xi2

+5Xi Ti+6Xi2

Ti+i

Question: Why not controlling for other covariates?

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Estimation Flexible specification


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Third degree polynomial. Actual model second degree polynomial (seeStata do file).

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Example


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Real dataset

Data from Lee, Moretti, Buttler (2004)

Forcing variable is Democratic vote share. If share >50 thenDemocratic candidate is elected

Outcome is a liberal voting score from the Americans for DemocraticAction (ADA)

Do candidates who are elected in close elections tend to moderatetheir congressional voting?

Data: http://harrisschool.uchicago.edu/Blogs/EITM/wp-content/uploads/2011/06/EITM-RD-Examples.zip

Some of the code is old; syntax no longer works

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Example
http://harrisschool.uchicago.edu/Blogs/EITM/wp-content/uploads/2011/06/EITM-RD-Examples.ziphttp://harrisschool.uchicago.edu/Blogs/EITM/wp-content/uploads/2011/06/EITM-RD-Examples.ziphttp://harrisschool.uchicago.edu/Blogs/EITM/wp-content/uploads/2011/06/EITM-RD-Examples.ziphttp://harrisschool.uchicago.edu/Blogs/EITM/wp-content/uploads/2011/06/EITM-RD-Examples.zip


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Graph a bit messy

scatter score demvoteshare, msize(tiny) xline(0.5) ///

xtitle("Democrat vote share") ytitle("ADA score")

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Example


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Always a good idea to add some jittering

With the jitter option, it is easier to see where is the mass

scatter score demvoteshare, msize(tiny) xline(0.5) ///xtitle("Democrat vote share") ytitle("ADA score") jitter(5)

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Example


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Could add linear trend

scatter score demvoteshare, msize(tiny) xline(0.5) xtitle("Democrat vote share") //

ytitle("ADA score") || lfit score demvoteshare if democrat ==1, color(red) || ///

lfit score demvoteshare if democrat ==0, color(red) legend(off)

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Example


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Better something more flexible

What about a polynomial of degree 5?

gen demvoteshare2 = demvoteshare^2




qui reg score demvoteshare demvoteshare2 demvoteshare3 demvoteshare4 ///demvoteshare5 democrat

qui predict scorehat

scatter score demvoteshare, msize(tiny) xline(0.5) ///

xtitle("Democrat vote share") ///

ytitle("ADA score") || ///

line scorehat demvoteshare if democrat ==1, sort color(red) || ///

line scorehat demvoteshare if democrat ==0, sort color(red) legend(off)

graph export lee3.png, replace

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Example


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Example


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Regression results

Center the forcing variable and use polynomials and interactions

gen x_c = demvoteshare - 0.5gen x2_c = x_c^2gen x3_c = x_c^3gen x4_c = x_c^4gen x5_c = x_c^5

reg score i.democrat##(c.x_c c.x2_c c.x3_c c.x4_c c.x5_c)

Source | SS df MS Number of obs = 13577

-------------+------------------------------ F( 11, 13565) = 1058.32Model | 6677033.81 11 607003.073 Prob > F = 0.0000Residual | 7780253.69 13565 573.553534 R-squared = 0.4618

-------------+------------------------------ Adj R-squared = 0.4614Total | 14457287.5 13576 1064.91511 Root MSE = 23.949

---------------------------------------------------------------------------------score | Coef. Std. Err. t P>|t| [95% Conf. Interval]

----------------+----------------------------------------------------------------

1.democrat | 47.73325 2.042906 23.37 0.000 43.72887 51.73763x_c | - 28.79765 74.09167 -0.39 0.698 -174.0276 116.4323x2_c | - 1138.232 1144.497 -0.99 0.320 -3381.605 1105.141x3_c | - 10681.29 7137.315 -1.50 0.135 -24671.42 3308.839x4_c | - 33490.23 18844.92 -1.78 0.076 -70428.88 3448.424x5_c | - 32873.77 17212.08 -1.91 0.056 -66611.83 864.302

|democrat#c.x_c |

1 | - 5.793828 97.79088 -0.06 0.953 -197.4775 185.8899|

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Example


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Could restrict to a window

Run a flexible regression like a polynomial with interactions(stratified) but dont use observations away from the cutoff. Choose abandwidth around X = 0.5. Lee et al (2004) used 0.4 to 0.6.

reg score demvoteshare demvoteshare2 if democrat ==1 & ///

(demvoteshare>.40 & demvoteshare.40 & demvoteshare


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Example


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Limit to window, 2nd degree polynomial

reg score i.democrat##(c.x_c c.x2_c) if (demvoteshare>.40 & demvoteshare F = 0.0000Residual | 2104043.2 4626 454.829918 R-squared = 0.5549

-------------+------------------------------ Adj R-squared = 0.5544Total | 4726805.22 4631 1020.6878 Root MSE = 21.327

---------------------------------------------------------------------------------score | Coef. Std. Err. t P>|t| [95% Conf. Interval]

----------------+----------------------------------------------------------------1.democrat | 45.9283 1.892566 24.27 0.000 42.21797 49.63863

x_c | 38.63988 60.77525 0.64 0.525 -80.5086 157.7884x2_c | 295.1723 594.3159 0.50 0.619 -869.9704 1460.315

|democrat#c.x_c |

1 | 6.507415 88.51418 0.07 0.941 -167.0226 180.0374|democrat#c.x2_c |

1 | - 744.0247 862.0435 -0.86 0.388 -2434.041 945.9916|

_cons | 17.71198 1.310861 13.51 0.000 15.14207 20.28189---------------------------------------------------------------------------------

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Nonparametric


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Nonparametric methods

Paper by Hahn, Todd, and Van der Klaauw (2001) clarifiedassumptions about RDD and framed estimation as a nonparametricproblem

Emphasized using local polynomial regression

Nonparametric methods means a lot of things in statistics

In the context of RDD, the idea is to estimate a model that does notassume a functional form for the relationship between Y and X. Themodel is something like Yi=f(Xi) +i

A very basic method: calculate E[Y] for each bin on X (think of ahistogram)

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Nonparametric

Stata has a command to do just that:


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Stata has a command to do just that: cmogramAfter installing the command (ssc install cmogram) type help cmogram. Lotsof useful optionsCommon way to show RDD data. See for example Figure II of Almond et al.

(2010). To recreate something like Figure 1 of Lee et al (2004):cmogram score demvoteshare, cut(.5) scatter line(.5) qfit

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Nonparametric

C li d LOWESS fi


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Compare to linear and LOWESS fits

cmogram score demvoteshare, cut(.5) scatter line(.5) lfit

cmogram score demvoteshare, cut(.5) scatter line(.5) lowess

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Nonparametric Local polynomials

L l l i l i


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Local polynomial regression

Hahn, Todd, and Van der Klaauw (2001) showed that one-side Kernelestimation (like LOWESS) may have poor properties because thepoint of interest is at a boundary

Proposed to use instead a local linear nonparametric regression

Statas lpoly command estimates kernel-weighted local polynomialregression

Think of it as a weighted regression restricted to a window (hencelocal). The Kernel provides the weights

A rectangular Kernel would give the same result as taking E[Y] at agiven bin on X. The triangular Kernel gives more importance toobservations close to the center

Method sensitive to choice of bandwidth (window)

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L l i i hi h d


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Local regression is a smoothing method

Kernel-weighted local polynomial regression is a smoothing method

lpoly score demvoteshare if democrat == 0, nograph kernel(triangle) gen(x0 sdem0) bwidth(0.1)lpoly score demvoteshare if democrat == 1, nograph kernel(triangle) gen(x1 sdem1) bwidth(0.1)

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T t t ff t


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Treatment effect

Were interested in getting the treatment at X = 0.5

gen forat = 0.5 in 1

lpoly score demvoteshare if democrat == 0, nograph kernel(triangle) gen(sdem0) ///

at(forat) bwidth(0.1)

lpoly score demvoteshare if democrat == 1, nograph kernel(triangle) gen(sdem1) ///

at(forat) bwidth(0.1)gen dif = sdem1 - sdem0

list sdem1 sdem0 dif in 1/1

+----------------------------------+

| sdem1 sdem0 dif |

|----------------------------------|

1. | 64.395204 16.908821 47.48639 |+----------------------------------+

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Diff t i d


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Different windows

What happens when we change the bandwidth?

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Nonparametric


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Nonparametric

Several methods to choose optimal windows: trade off between biasand variance

In practical applications, you may want to check balance around thatwindow

Standard error of treatment effect can be bootstrapped but there areother alternatives

Could add other variables to nonparametric methods

See Stata do file for examples using commands rd obs and rdrobust

rd obs is old and buggy. rdrobust is new and promising butpreliminary

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Using rdrobust


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Using rdrobust

. rdrobust score demvoteshare, c(0.5) all bwselect(IK)

Sharp RD Estimates using Local Polynomial Regression

Cutoff c = .5 | Left of c Right of c Number of obs = 13577----------------------+---------------------- Rho (h/b) = 0.770

Number of obs | 3535 3318 NN Matches = 3Order Loc. Poly. (p) | 1 1 BW Type = IK

Order Bias (q) | 2 2 Kernel Type = TriangularBW Loc. Poly. (h) | 0.152 0.152

BW Bias (b) | 0.197 0.197

--------------------------------------------------------------------------------------| Loc. Poly. Robust [Robust

score | Coef. Std. Err. z P>|z| 95% Conf. Interval]----------------------+---------------------------------------------------------------

demvoteshare | 47.171 1.262 36.9043 0.000 44.1 49.047108--------------------------------------------------------------------------------------

All Estimates. Outcome: score. Running Variable: demvoteshare.

--------------------------------------------------------------------------------------Method | Coef. Std. Err. z P>|z| [95% Conf. Interval]

----------------------+---------------------------------------------------------------Conventional | 47.171 .98131 48.0692 0.000 45.247 49.093991

Bias-Corrected | 46.574 .98131 47.4608 0.000 44.65 48.496943Robust | 46.574 1.262 36.9043 0.000 44.1 49.047108

--------------------------------------------------------------------------------------

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Conclusion

Parametric or non parametric?


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Parametric or non-parametric?

When would parametric or non-parametric or window size matter?

Small effectRelationship between Y and Xdifferent away from cutoffFunctional form not well captured by polynomials (or other functional

form). Splines may work too.Easier to work with traditional methods (parametric)

Could add random effects, robust standard errors, clustering SEs

In practical applications a regression with polynomials usually works

wellIf conclusions are different, worry. A lot.

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Almond et al (2010)

Marginal returns to medical care


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Marginal returns to medical care

Big picture: is spending more money on health care worth it (in termsof health gained)?

Actual research: is spending more money on low-weight newbornsworth it in terms of mortality reductions? Compare marginal costs

(dollars) to marginal benefits (mortality transformed into dollars).

On jargon: In economics marginal = additional. So compareadditional spending to additional benefit

RDD part used to estimate marginal benefits

Forcing variable is newborn weight. Cutoff point c= 1, 500 grams(almost 3 lbs)

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Almond et al (2010) Estimation

Estimating equation


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Estimating equation

Their model is:

Yi=0+ 1VLBWi+2VLBWi (gi 1500)+

3(1 VLBWi)(gi 1500) +t+s+X

i +i (1)

Change notation so VLBW =Tand (gi 1500) = Xand after doingsome algebra the model is:

Y =0+ 1T+3X+ (2 3)T X+ (t+s+X) +

(t+s+X) are covariates

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Almond et al (2010) Estimation...

Covariates


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Covariates

They compared means of covariates above and beyond cutoff point

They found some differences (large sample) so they include covariatesin the model

They did a RDD-type analysis on covariates to see if they weresmooth (no jump at cutoff)

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