Regression Discontinuity Designs in Stata
Matias D. Cattaneo
University of Michigan
July 30, 2015
Overview
Main goal: learn about treatment e¤ect of policy or intervention.
If treatment randomization available, easy to estimate treatment e¤ects.
If treatment randomization not available, turn to observational studies.
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Instrum ental variables.
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Selection on observables.
Regression discontinuity (RD) designs.
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Sim ple and ob jective. Requires little inform ation, if design available.
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M ight b e viewed as a “lo cal” random ized trial.
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Easy to falsify, easy to interpret.
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C areful: very local!
Overview of RD packages
https://sites.google.com/site/rdpackages
rdrobust package: estimation, inference and graphical presentation using local
polynomials, partitioning, and spacings estimators.
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rdrobust: RD inference (p oint estim ation and CI; classic, bias-corrected, robust).
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rdbwselect: bandwidth or window selection (IK, CV, CCT).
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rdplot: plots data (with “optim al” blo ck length).
rddensity package: discontinuity in density test at cuto¤ (a.k.a. manipulation testing)
using novel local polynomial density estimator.
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rddensity: m anipulation testing using lo cal p olynom ial density estim ation.
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rdbwdensity: bandwidth or window selection.
rdlocrand package: covariate balance, binomial tests, randomization inference
methods (window selection & inference).
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rdrandinf: inference using random ization inference m etho ds.
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rdwinselect: falsi…cation testing and window selection.
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rdsensitivity: treatm ent e¤ect m o dels over grid of windows, CI inversion.
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rdrbounds: Rosenbaum b ounds.
Randomized Control Trials
Notation: (Yi (0); Yi (1); Xi ), i = 1; 2; : : : ; n.
Treatment: Ti 2 f0; 1g,
Ti independent of (Yi (0); Yi (1); Xi ).
Data: (Yi ; Ti ; Xi ), i = 1; 2; : : : ; n, with
(
Yi (0)
Yi =
Yi (1)
if Ti = 0
if Ti = 1
Average Treatment E¤ect:
ATE
= E[Yi (1)
Experimental Design.
Yi (0)] = E[Yi jT = 1]
E[Yi jT = 0]
Sharp RD design
Notation: (Yi (0); Yi (1); Xi ), i = 1; 2; : : : ; n,
Treatment: Ti 2 f0; 1g,
Ti = 1(Xi
Xi continuous
x).
Data: (Yi ; Ti ; Xi ), i = 1; 2; : : : ; n, with
(
Yi (0)
Yi =
Yi (1)
if Ti = 0
if Ti = 1
Average Treatment E¤ect at the cuto¤:
SRD
= E[Yi (1)
Yi (0)jXi = x] = lim E[Yi jXi = x]
x#x
lim E[Yi jXi = x]
x"x
Quasi-Experimental Design: “local randomization” (more later)
3
2
1
0
−1
−2
Outcome variable (Y)
τ0
−0.6
−0.4
−0.2
0.0
0.2
Assignment variable (R)
0.4
0.6
3
2
1
0
−1
−2
Outcome variable (Y)
τ0
−0.6
−0.4
−0.2
0.0
0.2
Assignment variable (R)
0.4
0.6
3
2
1
0
−1
−2
Outcome variable (Y)
τ0
−0.6
−0.4
−0.2
0.0
0.2
Assignment variable (R)
0.4
0.6
6
4
2
0
−2
−4
Local Random Assignment
−6
Outcome variable (Y)
τ0
−0.6
−0.4
−0.2
0.0
0.2
Assignment variable (R)
0.4
0.6
Empirical Illustration: Cattaneo, Frandsen & Titiunik (2015, JCI)
Problem: incumbency advantage (U.S. senate).
Data:
Yi = election outcome.
Ti = whether incumbent.
Xi = vote share previous election
(x = 0).
Zi = covariates (demvoteshlag1, demvoteshlag2, dopen, etc.).
Potential outcomes:
Yi (0) = election outcome if had not been incumbent.
Yi (1) = election outcome if had been incumbent.
Causal Inference:
Yi (0) 6= Yi jTi = 0
and
Yi (1) 6= Yi jTi = 1
Graphical and Falsi…cation Methods
Always plot data: main advantage of RD designs!
Plot regression functions to assess treatment e¤ect and validity.
Plot density of Xi for assessing validity; test for continuity at cuto¤ and elsewhere.
Important: use also estimators that do not “smooth-out” data.
RD Plots (Calonico, Cattaneo & Titiunik, JASA):
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Two ingredients: (i) Sm o othed global p olynom ial …t & (ii) binned discontinuous
lo cal-m eans …t.
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Two goals: (i) detention of discontinuities, & (ii) representation of variability.
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Two tuning param eters:
F
G lo b a l p o ly n o m ia l d e g re e (kn ).
F
L o c a tio n (E S o r Q S ) a n d nu m b e r o f b in s (Jn ).
Manipulation Tests & Covariate Balance and Placebo Tests
Density tests near cuto¤:
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Idea: distribution of running variable should b e sim ilar at either side of cuto¤.
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M etho d 1: Histogram s & Binom ial count test.
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M etho d 2: Density Estim ator at b oundary.
F
P re -b in n e d lo c a l p o ly n o m ia l m e th o d – M c C ra ry (2 0 0 8 ).
F
N e w tu n in g -p a ra m e te r-fre e m e th o d – C a tta n e o , J a n sso n a n d M a (2 0 1 5 ).
Placebo tests on pre-determined/exogenous covariates.
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Idea: zero RD treatm ent e¤ect for pre-determ ined/exogenous covariates.
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M ethods: global p olynom ial, lo cal p olynom ial, random ization-based.
Placebo tests on outcomes.
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Idea: zero RD treatm ent e¤ect for outcom e at values other than cuto¤.
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M ethods: global p olynom ial, lo cal p olynom ial, random ization-based.
Estimation and Inference Methods
Global polynomial approach (not recommended).
Robust local polynomial inference methods.
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Bandwidth selection.
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Bias-correction.
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Con…dence intervals.
Local randomization and randomization inference methods.
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W indow selection.
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Estim ation and Inference m etho ds.
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Falsi…cation, sensitivity and related m etho ds
Conventional Local-polynomial Approach
Idea: approximate regression functions for control and treatment units locally.
“Local-linear” estimator (w/ weights K( )):
hn
Yi =
I
+ (Xi
Xi < x :
x)
x
+"
Yi =
;i
Treatm ent e¤ect (at the cuto¤): ^ SRD = ^ +
+
Xi
+ (Xi
hn :
x)
+
+ "+;i
^
Can be estimated using linear models (w/ weights K( )):
Yi =
+
SRD
Ti + (Xi
x)
1
+ Ti (Xi
x)
1
+ "i ,
Once hn chosen, inference is “standard”: weighted linear models.
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Details com ing up next.
hn
Xi
hn
Conventional Local-polynomial Approach
How to choose hn ?
Imbens & Kalyanaraman (2012, ReStud): “optimal” plug-in,
^ IK = C
^IK n
h
1=5
Calonico, Cattaneo & Titiunik (2014, ECMA): re…nement of IK
^ CCT = C
^CCT n
h
1=5
Ludwig & Miller (2007, QJE): cross-validation,
^ CV = arg min
h
h
n
X
w(Xi ) (Yi
^ 1 (Xi ; h))2
i=1
Key idea: trade-o¤ bias and variance of ^SRD (hn ). Heuristically:
" Bias(^SRD )
=)
^
#h
and
" Var(^SRD )
=)
^
"h
Local-Polynomial Methods: Bandwidth Selection
Two main methods: plug-in & cross-validation. Both MSE-optimal in some sense.
Imbens & Kalyanaraman (2012, ReStud): propose MSE-optimal rule,
1=5
n
hMSE = CMSE
1=5
CMSE = C(K)
Var(^SRD )
Bias(^SRD )2
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IK im plem entation: …rst-generation plug-in rule.
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CCT im plem entation: second-generation plug-in rule.
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They di¤er in the way Var(^ SRD ) and Bias(^ SRD ) are estim ated.
Imbens & Kalyanaraman (2012, ReStud): discuss cross-validation approach,
^ CV = arg min CV (h) ,
h
CV (h) =
h>0
n
X
1(X
;[ ]
Xi
X+;[ ] ) (Yi
^ (Xi ; h))2 ,
i=1
where
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^ +;p (x; h) and ^
2 (0; 1), X
;[ ]
;p (x; h)
are lo cal p olynom ials estim ates.
and X+;[
Our im plem entation uses
]
denote
-th quantile of fXi : Xi < xg and fXi : Xi
= 0:5; but this is a tuning param eter!
xg.
Conventional Approach to RD
“Local-linear” estimator (w/ weights K( )):
hn
Yi =
I
Xi < x :
+ (Xi
x)
x
+"
Treatm ent e¤ect (at the cuto¤): ^ SRD = ^ +
Construct usual t-test. For H0 :
SRD
Yi =
;i
+
+ (Xi
hn :
x)
^
= 0,
^SRD
^+ ^
T^(hn ) = p
= q
^n
^
^
V
V+;n + V
d
;n
95% Con…dence interval:
^ n) =
I(h
Xi
^SRD
1:96
q
^n
V
N (0; 1)
+
+ "+;i
Bias-Correction Approach to RD
Note well: for usual t-test,
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^SRD
T^(hMSE ) = p
^n
V
d
N (B; 1)
6=
B>0
N (0; 1),
Bias B in RD estim ator captures “curvature” of regression functions.
^ n = 0:5 h
^ IK ?
Undersmoothing/“Small Bias” Approach: Choose “smaller” hn ... Perhaps h
=) Not clear guidance & power loss!
Bias-correction Approach:
^n
^SRD B
T^bc (hn ; bn ) = p
^n
V
=) 95% Con…dence Interval:
I^bc (hn ; bn ) =
d
h
^ n
How to choose bn ? Same ideas as before... ^bn = C
N (0; 1)
^SRD
1=7
^n
B
1:96
p
^n
V
i
Robust Bias-Correction Approach to RD
Recall:
^ SRD
T^ (hn ) = p
^n
V
I
d
N (0; 1)
and
^ SRD
bc
T^ (hn ; bn ) =
p
^n
B
^n
V
d
N (0; 1)
^ n is constructed to estim ate leading bias B.
B
Robust approach:
^n
^n
^SRD B
Bn B
^SRD Bn
T^bc (hn ; bn ) = p
+ p
= p
^n
^n
^n
V
V
V
| {z } | {z }
d
N (0;1)
d
N (0; )
d
N (0; 1)
Robust bias-corrected t-test:
^n
^n
^SRD B
^SRD B
= q
T^rbc (hn ; bn ) = p
^n + W
^n
bc
^
V
Vn
=) 95% Con…dence Interval:
I^rbc (hn ; bn ) =
^SRD
^n
B
1:96
q
bc
^n
V
,
bc
^n
^n + W
^n
V
=V
Local-Polynomial Methods: Robust Inference
Approach 1: Undersmoothing/“Small Bias”.
^ n) =
I(h
^SRD
q
^n
V
1:96
Approach 2: Bias correction (not recommended).
I^bc (hn ; bn ) =
^SRD
^n
B
1:96
q
^n
V
Approach 3: Robust Bias correction.
I^rbc (hn ; bn ) =
^SRD
^n
B
1:96
q
^n + W
^n
V
Local-randomization approach and …nite-sample inference
Popular approach: local-polynomial methods.
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Approxim ates regression function and relies on continuity assum ptions.
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Requires: choosing weights, bandwidth and p olynom ial order.
Alternative approach: local-randomization + randomization-inference
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Gives an alternative that can b e used as a robustness check.
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K ey assum ption: exists window W = [ hn ; hn ] around cuto¤ ( hn < x < hn ) where
Ti indep endent of (Yi (0); Yi (1))
(for all Xi 2 W )
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In words: treatm ent is random ly assigned within W .
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Go o d news: if plausible, then RCT ideas/m etho ds apply.
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Not-so-good news: m ost plausible for very sm all windows (very few observations).
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One solution: em ploy sm all window but use random ization-inference m etho ds.
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Requires: choosing random ization rule, window and statistic.
Local-randomization approach and …nite-sample inference
Recall key assumption: exists W = [ hn ; hn ] around cuto¤ ( hn < x < hn ) where
Ti independent of (Yi (0); Yi (1))
(for all Xi 2 W )
How to choose window?
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Use balance tests on pre-determ ined/exogenous covariates.
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Very intuitive, easy to im plem ent.
How to conduct inference? Use randomization-inference methods.
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Cho ose statistic of interest. E.g., t-stat for di¤erence-in-m eans.
2
Cho ose random ization rule. E.g., numb er of treatm ents and controls given.
3
Com pute …nite-sam ple distribution of statistics by p ermuting treatm ent assignm ents.
Local-randomization approach and …nite-sample inference
Do not forget to validate & falsify the empirical strategy.
1
Plot data to m ake sure lo cal-random ization is plausible.
2
Conduct placeb o tests.
(e.g., use pre-intervention outcom es or other covariates not used select W )
3
Do sensitivity analysis.
See Cattaneo, Frandsen and Titiunik (2015) for introduction.
See Cattaneo, Titiunik and Vazquez-Bare (2015) for further results and
implementation.