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Regression Discontinuity Design
Basics
Two potential outcomes Yi(0) and Yi(1),
causal effect Yi(1) − Yi(0), binary treatment indicator Wi, covariate
Xi, and the observed outcome equal to:
At Xi = c incentives to participate change.
Two cases, Sharp Regression Discontinuity:
(SRD)
and Fuzzy Regression Discontinuity Design:
(FRD)
Outline
Introduction
 Basics
 Graphical Analyses
 Local Linear Regression
 Choosing the Bandwidth
 Variance Estimation
 Specification Checks

RD History





In the beginning there was Thislethwaite and Campbell (1960)
Only a few economists were involved initially (Goldberger, 1972)
Then RDD went into hibernation
It recently experienced a renaissance among economists (e.g.
Hahn, Todd and van der Klaauw, 2001; Jacob and Lefgren, 2002)
Tom Cook has written about this story


Recent applications in politics include analyses of the
incumbency effect (Lee, 2008), electoral competition on policy
(Lee, Moretti, and Butler 2004), the effect of gender on legislator
behavior (Rehavi nd), the value of a seat in the legisalture
(Eggers and Hainmueller 2009), the effect of mayoral party ID
on policy (Ferreria and Gyourko 2009).
Recent important theoretical work has dealt with identification
issues (Hahn, Todd, and Van Der Klaauw, 2001), optimal
estimation (Porter, 2003), tests for validity of the design
(McCrary, 2008), bandwidth selection (Imbens andKalyanaraman
2011).

General surveys include Lee and Lemieux (2009), Van Der
Klaauw (2008), and Imbens and Lemieux (2008).

Angrist & Pischke, “Mostly Harmless Econometrics”
Lee and Lemieux, 2010, "Regression Discontinuity Designs in
Economics." Journal of Economic Literature, 48(2), 281–355
Imbens and Lemieux (2008). “” Journal of Econometrics, 2008,
142(2)


Introduction



A Regression Discontinuity (RD) Design is a powerful and
widely applicable identification strategy.
Often access to, or incentives for participation in, a service or
program is assigned based on transparent rules with criteria
based on clear cutoff values, rather than on discretion of
administrators.
Comparisons of individuals that are similar but on different sides
of the cutoff point can be credible estimates of causal effects for
a specific subpopulation.
RD Logic
Many different rules work like this.
Examples:







Whether you pass a test
Whether you are eligible for a program
Who wins an election
Which school district you reside in
Whether some punishment strategy is enacted
Birth date for entering kindergarten
The key insight is that right around the cutoff we can think of
people slightly above as identical to people slightly below
Two cases of RD

Sharp Regression Discontinuity (SRD)

Fuzzy Regression Discontinuity Design (FRD)
This is what is referred to as a “Sharp Regression
Discontinuity”

There is also something called a “Fuzzy Regression
Discontinuity”


This occurs when rules are not strictly enforced
SRD Example: Lee (2007)



What is effect of incumbency on election outcomes? (More
specifically, what is the probability of a Democrat winning the
next election given that the last election was won by a
Democrat?)
This conjecture sounds plausible, but the success of incumbents
need not reflect a real electoral advantage. Incumbents - by
definition, candidates and parties who have shown they can win may simply be better at satisfying voters or getting the vote out.
Compare election outcomes in cases where previous election was
very close.
Fuzzy Regression Discontinuity
Examples (VanderKlaauw, 2002)
What is effect of financial aid offer on acceptance of college
admission.
College admissions office puts applicants in a few categories based
on numerical score.
Financial aid offer is highly correlated with category.
Compare individuals close to cutoff score.
RD Compared to an Experiment





RD is often described as a “close cousin” of a randomized
experiment or as a “local randomized experiment.”
Consider an experiment in which each participant is assigned a
randomly generated number, v, from a uniform distribution over
the range [0,1].
Units with v ≥ 2 assigned to treatment; units with v < 2 assigned
to control. This randomized experiment can be thought of as an
RD where the assignment variable X = v and the cutoff is at 2.
Because the assignment variable is random, the curves [Y(0)|X]
and E[Y(1)|X] are flat. The ATE can be computed as the
difference in the mean value of Y on either side of the cutoff.
Good for internal validity, not much external validity.
Figure 3. Randomized Experiment as a RD Design
SRD

Key assumption:
E[Y (0)|X = x] and E[Y (1)|X = x] are continuous in x.

Under this assumption,
The estimand is the difference of two regression functions at a
point. Extrapolation is unavoidable.
FRD
What do we look at in the FRD case: ratio of discontinuities in
regression function of outcome and treatment:
(FRD)
RD Compared to an Experiment





Note that there are several ways in which an RD differs from a
randomized experiment in actuality
First, it may be possible for units to alter their assignment to
treatment by manipulating the forcing variable in a way that is
not possible when it is assigned at random by the investigator
Second, the functional form need not be flat (or linear or
monotonic) and may not even be known.
not possible when it is assigned at random by the investigator.
Third, only the sample near to the cutoff points are useful.
Manipulation

If individuals have control over the assignment variable, then we
should expect them to sort into (out of) treatment if treatment is
desirable (undesirable)


Think of a means‐tested income support program, or an election
Those just above the threshold will be a mixture of those who
would passed and those who barely failed without manipulation.

If individuals have precise control over the assignment variable,
we would expect the density of X to be zero just below the
threshold but positive just above the threshold (assuming the
treatment is desirable).



McCrary (2008) provides a formal test for maniupulaiton of the
assignment variable in an RD. The idea is that the marginal density
of X should be continuous without manipulation and hence we
look for discontinuities in the density around the threshold.
How precise must the manipulation must be in order to threaten
the RD design? See Lee and Lemieux (2010).
This means that when you run an RD you must know something
about the mechanism generating the assignment variable and
how susceptible it could be to manipulation.
Example: Hysterectomy cases in Taiwan




Each year about 25000 women have hysterectomy in Taiwan.
NHI records show that one out of five women have
hysterectomy in her life in Taiwan.
Each year about 6-7000 labor enrollees have hysterectomy, with
NT130-140k per case. In total, LI paid NT 1 billion for disability
benefits each year.
There is no clear associations between the number of
hysterectomy and the economic conditions
28
Disability insurance

Three social programs:

Government Employees Insurance (GEI): workers in public
sector, school teachers

Labor Insurance (LI): workers in the private sector

Farmers’ Insurance (FI): farmers

Women under age 45, undergoing a hysterectomy (surgical
removal of uterus，子宮切除術) or an oophorectomy (surgical
removal of both ovaries ，卵巢切除術)

Disability benefit = 6 months of insured salary for GEI, 5.5
months for LI and FI
29
Hazard by quarter
Total and partial hysterectomies
Labor
3
2
1
0
0
1
2
3
hazard (of 1,000)
4
4
Public
-20
-10
0
10
quarters from age 45
20
-20
-10
20
3
2
1
0
0
1
2
3
hazard (of 1,000)
4
Uninsured
4
Farmer
0
10
quarters from age 45
-20
-10
0
10
quarters from age 45
20
-20
-10
0
10
quarters from age 45
20
30
Hazard by quarter
Total and partial hysterectomies
1
2
3
4
Hysterectomy
-20
-10
0
quarters from age 45
Public
Farmer
10
Labor
Uninsured
20
31
Hazard by quarter
Partial oophorectomies
.75
.5
0
.25
0
.25
.5
.75
hazard (of 1,000)
1
Labor
1
Public
-20
-10
0
10
quarters from age 45
20
-20
-10
20
.75
.5
0
.25
0
.25
.5
.75
hazard (of 1,000)
1
Uninsured
1
Farmer
0
10
quarters from age 45
-20
-10
0
10
quarters from age 45
20
-20
-10
0
10
quarters from age 45
20
32
The size of the discontinuity at the cutoff is the size of the effect.
Here we see a discontinuity between the regression lines at the
cutoff, which would lead us to conclude that the treatment worked.
But this conclusion would be wrong because we modeled these data
with a linear model when the underlying relationship was nonlinear
Here we see a discontinuity that suggests a treatment effect.
However, these data are again modeled incorrectly, with a linear
model that contains no interaction terms, producing an
artifactualdiscontinuity at the cutoff…
Sample Size Implications

Because of the substantial multi-collinearity that exists between
its forcing variable and treatment indicator, an RDD requires 3
to 4 times as many sample members as a corresponding
randomized experiment
External Validity
The estimatand has little external validity. It is at best valid for a
population defined by the cutoff value c, and by the subpopulation
that is affected at that value.
Figure 2. Nonlinear RD


Because the assignment variable is random, the curves [Y(0)|X]
and E[Y(1)|X] are flat. The ATE can be computed as the
difference in the mean value of Y on either side of the cutoff.
Because the functions are flat everywhere, the “optimal
bandwidth” is to use all the data
How to Estimate SRD Model

Compare means
Approach #1: Compare means




In the data, we never observe E[Y(0)|X=c], that is there are no
units at the cutoff that don’t get the treatment, but in principle it
can be approximated arbitrarily well by E[Y(0)|X=c‐ε].
Therefore we estimate: E[Y|X=c+ε]-E[Y|X=c‐ε]
This is the difference in means for those just above and below
the cutoff.
This is a nonparametric approach. A great virtue is that it does
not depend on correct specification of functional forms.
Figure 2. Nonlinear RD
OLS (with Polynomials)




The original RD design (Thistlewaite and Campbell 1960) was
implemented by OLS.
where τ is the causal effect of interest and η is an error term.
This regression distinguishes the nonlinear and discontinuous
jump from the smooth linear function.
OLS with one linear term in X is seldom used anymore because
the functional form assumptions are very strong.



Suppose the underlying functions are nonlinear and maybe
unknown. In particular, suppose you want to estimate
where f(X) is a smooth nonlinear function of X.
Perhaps the simplest f(X) is with in X way to approximate via
OLS polynomials X. Common practice is to fit different
polynomial functions on each side of the cutoff by including
interactions between T and X.
Modeling f(X) with a pth‐order polynomial in this way leads to


Centering X at the cutoff prior to running the regression ensures
that the coefficient on T is the treatment effect.
Common practice, for whatever reason, seems to use a 4th order
polynomial, though you should be sure that your results are
robust to other specifications (more on this below).


OLS with polynomials is a particularly simple way of allowing a
flexible functional form in X. A drawback is that it provides
global estimates of the regression function that use data far from
the cutoff.
There many are other ways, but the RD setup poses a couple of
problems for standard nonparametric smoothers.


We are interested in the estimate of a function at a boundary point.
(For why this is a problem, see HTV or Imbens and Lemieux.)
Standard nonparametric kernel regression does not work well here
(converging rate too slow)
Local Linear Regression




Instead of locally fitting a constant function (e.g., the mean), fit
linear regressions to observations within some bandwidth of the
cutoff
A rectangular kernel seems to work best (see Imbens and
Lemieux), but optimal bandwidth selection is an open question
A serious discussion of local linear regression is beyond the
scope of this lecture. See, for example, Fan and Gijbels (1996)
But, really, we’re just talking about running regressions on data
near the cutoff.
Local Linear Regression
We are interested in the value of a regression function at the
boundary of the support. Standard kernel regression
does not work well for that case (slower convergence rates)
Better rates are obtained by using local linear regression. First
The value of left hand limit μl(c) is then estimated as
Similarly for right hand side. Not much gained by using a
nonuniform kernel.
Alternatively one can estimate the average effect directly in a single
regression,
thus solving
which will numerically yield the same estimate of τSRD.
This interpretation extends easily to the inclusion of covariates.
Estimation for the FRD Case
Do local linear regression for both the outcome and the treatment
indicator, on both sides,
and similarly
and
. Then the FRD estimator is
Alternatively, define the vector of covariates
Then we can write
(TSLS)
Then estimating τ based on the regression function (TSLS) by TwoStage-Least-Squares methods, using
Wi as the endogenous regressor,
the indicator 1{Xi ≥ c} as the excluded instrument
Vi as the set of exogenous variables
This is numerically identical to
before (because of uniform
kernel)
Can add other covariates in straightfoward manner.
Choosing the Bandwidth (Ludwig & Miller)
We wish to take into account that (i) we are interested in the
regression function at the boundary of the support, and (ii) that we
are interested in the regression function at x = c.
Define
and
as the solutions to
Define
Define
to be δ quantile of the empirical distribution of X for
the subsample with Xi < c, and let
be δ quantile of the
empirical distribution of X for the subsample with Xi ≥ c.
Now we use the cross-validation criterion
for, say δ = 1/2, with the corresponding cross-validation choice for
the binwidth
Bandwidth for FRD Design

Calculate optimal bandwidth separately for both regression
functions and choose smallest.

Calculate optimal bandwidth only for outcome and use that for
both regression functions.
Typically the regression function for the treatment indicator is
flatter than the regression function for the outcome away from the
discontinuity point (completely flat in the SRD case). So using same
criterion would lead to larger bandwidth for estimation of
regression function for treatment indicator. In practice it is easier to
use the same bandwidth, and so to avoid bias, use the bandwidth
from criterion for SRD design or smallest.
Variance Estimation
The asymptotic covar of
and
Finally, the asymptotic distribution has the form
is
This asymptotic distribution is a special case of that in HTV, using
the rectangular kernel, and with h = N−δ, for 1/5 < δ <2/5 (so that
the asymptotic bias can be ignored).
Can use plug in estimators for components of variance.
TSLS Variance for FRD Design
The second estimator for the asymptotic variance of
interpretation of the as a TSLS estimator.
exploits the
The variance estimator is equal to the robust variance for TSLS
based on the subsample of observations with c−h ≤ Xi ≤ c+h, using
the indicator 1{Xi ≥ c} as the excluded instrument, the treatment
Wi as the endogenous regressor and the Vi as the exogenous
covariates.
Concerns about Validity
Two main conceptual concerns in the application of RD designs,
sharp or fuzzy.
 Other Changes


Possibility of other changes at the same cutoff value of the
covariate. Such changes may affect the outcome, and these effects
may be attributed erroneously to the treatment of interest.
Manipulation of Forcing Variable

The second concern is that of manipulation of the covariate value.
Specification Checks





Discontinuities in average Covariates
Discontinuity in the Distribution of the Forcing Variable
Discontinuities in Average Outcomes at Other Values
Sensitivity to Bandwidth Choice
RD Designs with Misspecification
Discontinuities in Average Covariates
Test the null hypothesis of a zero average effect on pseudo
outcomes known not to be affected by the treatment.
Such variables includes covariates that are by definition not affected
by the treatment. Such tests are familiar from settings with
identification based on unconfoundedness assumptions.
Although not required for the validity of the design, in most cases,
the reason for the discontinuity in the probability of the treatment
does not suggest a discontinuity in the average value of covariates.
If we find such a discontinuity, it typically casts doubt on the
assumptions underlying the RD design.
A Discontinuity in the Distribution of the Forcing
Variable
McCrary (2007) suggests testing the null hypothesis of continuity of
the density of the covariate that underlies the assignment at the
discontinuity point, against the alternative of a jump in the density
function at that point.
Again, in principle, the design does not require continuity of the
density of X at c, but a discontinuity is suggestive of violations of
the no-manipulation assumption.
If in fact individuals partly manage to manipulate the value of X in
order to be on one side of the boundary rather than the other, one
might expect to see a discontinuity in this density at the
discontinuity point.
Discontinuities in Average Outcomes at Other Values
Taking the subsample with Xi < c we can test for a jump in the
conditional mean of the outcome at the median of the forcing
variable.
To implement the test, use the same method for selecting the
binwidth as before. Also estimate the standard errors of the jump
and use this to test the hypothesis of a zero jump.
Repeat this using the subsample to the right of the cutoff point with
Xi ≥ c. Now estimate the jump in the regression function and at
and test whether it is equal to zero.
Sensitivity to Bandwidth Choice
One should investigate the sensitivity of the inferences to this
choice, for example, by including results for bandwidths twice (or
four times) and half (or a quarter of) the size of the originally
chosen bandwidth.
Obviously, such bandwidth choices affect both estimates and
standard errors, but if the results are critically dependent on a
particular bandwidth choice, they are clearly less credible than if
they are robust to such variation in bandwidths.
RD Designs with Misspecification
Lee and Card (2007) study the case where the forcing variable
variable X is discrete. In practice this is of course always true. This
implies that ultimately one relies for identification on functional
form assumptions for the regression function μ(x).
They consider a parametric specification for the regression function
that does not fully saturate the model and interpret the deviation
between the true conditional expectation and the estimated
regression function as random specification error that introduces a
group structure on the standard errors.
Lee and Card then show how to incorporate this group structure
into the standard errors for the estimated treatment effect. Within
the local linear regression framework discussed in the current paper
one can calculate the Lee-Card standard errors and compare them
to the conventional ones.
Figure 4. Density of Assignment Variable Conditional on
W = w, U = u
Figure 5. Treatment, Observables, and Unobservables in
Four Research Designs
Figure 9. Winning the Next Election, Bandwidth of 0.02
(50 bins)
Figure 10. Winning the Next Election, Bandwidth of 0.01
(100 bins)
Figure 11. Winning the Next Election, Bandwidth of
0.005 (200 bins)
Figure 12. Cross-Validation Function for Choosing the
Bandwidth in a RD Graph: Winning the Next Election
Figure 13. Cross-Validation Function for Choosing
Bandwidth in a RD Graph: Share of Vote at Next
Election
Figure 14. Cross-Validation Function for Local Linear
Regression: Share of Vote at Next Election
Figure 15. Cross-Validation Function for Local Linear
Regression: Winning the Next Election
Figure 16. Density of the Forcing Variable (Vote Share in
Previous Election)
Figure 17. Discontinuity in Baseline Covariate (Share of
Vote in Prior Election)
Figure 18. Local Linear Regression with Varying
Bandwidth: Share of Vote at Next Election
Estimation Basics 1

Approach #1: Compare means




In the data, we never observe E[Y(0)|X=c], that is there are no
units at the cutoff that don’t get the treatment, but in principle it
can be approximated arbitrarily well by E[Y(0)|X=c‐ε].
Therefore we estimate: E[Y|X=c+ε]-E[Y|X=c‐ε]
This is the difference in means for those just above and below the
cutoff.
This is a nonparametric approach. A great virtue is that it does not
depend on correct specification of functional forms.
Estimation Basics 2

Approach #2: OLS (with Polynomials)





The original RD design (Thistlewaite and Campbell 1960) was
implemented by OLS.
where τ is the causal effect of interest and η is an error term.
This regression distinguishes the nonlinear and discontinuous jump
from the smooth linear function.
OLS with one linear term in X is seldom used anymore because the
functional form assumptions are very strong.
What are they?
Estimation Basics 3

Suppose the underlying functions are nonlinear and maybe
unknown. In particular, suppose you want to estimate

where f(X) is a smooth nonlinear function of X.
Perhaps the simplest f(X) is with in X way to approximate via
OLS polynomials X. Common practice is to fit different
polynomial functions on each side of the cutoff by including
interactions between T and X.
Modeling f(X) with a pth‐order polynomial in this way leads to




Centering X at the cutoff prior to running the regression ensures
that the coefficient on T is the treatment effect.
Common practice, for whatever reason, seems to use a 4th order
polynomial, though you should be sure that your results are
robust to other specifications (more on this below).
Estimation Basics 4


OLS with polynomials is a particularly simple way of allowing a
flexible functional form in X. A drawback is that it provides
global estimates of the regression function that use data far from
the cutoff.
There many are other ways, but the RD setup poses a couple of
problems for standard nonparametric smoothers.


We are interested in the estimate of a function at a boundary point.
(For why this is a problem, see HTV or Imbens and Lemieux.)
Standard nonparametric kernel regression does not work well here

This leads to Approach #3: Local Linear Regression




Instead of locally fitting a constant function (e.g., the mean), fit
linear regressions to observations within some bandwidth of the
cutoff
A rectangular kernel seems to work best (see Imbens and
Lemieux), but optimal bandwidth selection is an open question
A serious discussion of local linear regression is beyond the scope
of this lecture. See, for example, Fan and Gijbels (1996)
But, really, we’re just talking about running regressions on data
near the cutoff.
Manipulation

If individuals have control over the assignment variable, then we
should expect them to sort into (out of) treatment if treatment is
desirable (undesirable)


Think of a means‐tested income support program, or an election
Those just above the threshold will be a mixture of those who
would passed and those who barely failed without manipulation.

If individuals have precise control over the assignment variable,
we would expect the density of X to be zero just below the
threshold but positive just above the threshold (assuming the
treatment is desirable).



McCrary (2008) provides a formal test for maniupulaiton of the
assignment variable in an RD. The idea is that the marginal density
of X should be continuous without manipulation and hence we
look for discontinuities in the density around the threshold.
How precise must the manipulation must be in order to threaten
the RD design? See Lee and Lemieux (2010).
This means that when you run an RD you must know something
about the mechanism generating the assignment variable and
how susceptible it could be to manipulation.
Example of Manipulation


An income support program in which those earning under
$14,000 qualify for support
Simulated data from McCrary 2008