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Regression discontinuity (RD) estimator
Syntax
rd [varlist] [if] [in] [weight] [, options]
where varlist has the form outcomevar [treatmentvar] assignmentvar
+---------+
----+ Weights +------------------------------------------------------------
aweights, fweights, and pweights are allowed; see help weights. Under
Stata versions 9.2 or before (using locpoly to construct local regression
estimates) aweights and pweights will be converted to fweights
automatically and the data expanded. If this would exceed system memory
limits, error r(901) will be issued; in this case, the user is advised to
round weights. In any case, the validity of bootstrapped standard errors
will depend on the expanded data correctly representing sampling
variability, which may require rounding or replacing weight variables.
Under Stata versions 10 or later (using lpoly to construct local regression
estimates), all weights will be treated as aweights.
bs [, options]: rd varlist [if] [in] [weight] [, options]
+----------------------------+
----+ Table of Further Contents +-----------------------------------------
General description of estimator
Examples
Detailed syntax
Description of options
Remarks and saved results
References
Acknowledgements
Citation of rd
Author information
+-------------+
----+ Description +--------------------------------------------------------
rd implements a set of regression-discontinuity estimation methods that are
thought to have very good internal validity, for estimating the causal effect of
some explanatory variable (called the treatment variable) for a particular
subpopulation, under some often plausible assumptions. In this sense, it is much
like an experimental design, except that levels of the treatment variable are not
assigned randomly by the researcher. Instead, there is a jump in the conditional
mean of the treatment variable at a known cutoff in another variable, called the
assignment variable, which is perfectly observed, and this allows us to estimate
the effect of treatment as if it were randomly assigned in the neighborhood of
the known cutoff.
rd is an alternative to various regression techniques that purport to allow
causal inference (e.g. panel methods such as xtreg), instrumental variables (IV)
and other IV-type methods (see the ivreg2 help file and references therein), and
matching estimators (see the psmatch2 and nnmatch help files and references
therein). The rd approach is in fact an IV model with one exogenous variable
excluded from the regression (excluded instrument), an indicator for the
assignment variable above the cutoff, and one endogenous regressor (the treatment
variable).
rd estimates local linear or kernel regression models on both sides of the
cutoff, using a triangle kernel. Estimates are sensitive to the choice of
bandwidth, so by default several estimates are constructed using different
bandwidths. In practice, rd uses kernel-weighted suest (or ivreg if suest fails)
to estimate the local linear regressions and reports analytic SE based on the
regressions.
Further discussion of rd appears in Nichols (2007).
+----------+
----+ Examples +-----------------------------------------------------------
In the simplest case, assignment to treatment depends on a variable Z being above
a cutoff Z0. Frequently, Z is defined so that Z0=0. In this case, treatment is 1
for Z>=0 and 0 for Z<0, and we estimate local linear regressions on both sides of
the cutoff to obtain estimates of the outcome at Z=0. The difference between the
two estimates (for the samples where Z>=0 and where Z<0) is the estimated effect
of treatment.
For example, having a Democratic representative in the US Congress may be
considered a treatment applied to a Congressional district, and the assignment
variable Z is the vote share garnered by the Democratic candidate. At Z=50%, the
probability of treatment=1 jumps from zero to one. Suppose we are interested in
the effect a Democratic representative has on the federal spending within a
Congressional district. rd estimates local linear regressions on both sides of
the cutoff like so:
ssc inst rd, replace
net get rd
use votex
rd lne d, gr mbw(100)
rd lne d, gr mbw(100) line(`"xla(-.2 "Repub" 0 .3 "Democ", noticks)"')
rd lne d, gr ddens
rd lne d, mbw(25(25)300) bdep ox
rd lne d, x(pop-vet)
rd lne d, mbw(100) bin binvar(bins) scopt(mcol(black))
In a fuzzy RD design, the conditional mean of treatment jumps at the cutoff, and
that jump forms the denominator of a Local Wald Estimator. The numerator is the
jump in the outcome, and both are reported along with their ratio. The sharp RD
design is a special case of the fuzzy RD design, since the denominator in the
sharp case is just one.
g byte ranwin=cond(uniform()<.1,1-win,win)
rd lne ranwin d, mbw(100)
The default bandwidth from Imbens and Kalyanaraman (2009) is designed to minimize
MSE, or squared bias plus variance, in a sharp RD design. Note that a smaller
bandwidth tends to produce lower bias and higher variance. The optimal bandwidth
will tend to be larger for a fuzzy design due to the additional variance arising
from the estimation of the jump in the conditional mean of treatment.
Unfortunately, a larger bandwidth also leads to additional bias, which will be
greater if the curvature of the response function is greater (meaning that a
linear regression over a larger range is a poorer approximation). The increase
in squared bias due to dividing by the estimated jump in the conditional mean of
treatment (using observations away from the discontinuity) can easily dominate
the increase in variance and lead to the optimal bandwidth in a fuzzy design to
be smaller than in the sharp design. No clear guidance is offered; conducting
simulations using plausible generating functions for your specific application
are highly recommended. The rd option bdep facilitates visualizing the
dependence of the estimate on bandwidth.
There are also a varitey of alternative implementations on
{browse:https://sites.google.com/a/umich.edu/cattaneo/software}{the website of
Matias Cattaneo}.
rd lne ranwin d, mbw(25(25)300) bdep ox
+-----------------------------+
----+ Detailed Syntax and Options +----------------------------------------
There should be two or three variables specified after the rd command; if two are
specified, a sharp RD design is assumed, where the treatment variable jumps from
zero to one at the cutoff. If no variables are specified after the rd command,
the estimates table is displayed.
rd outcomevar [treatmentvar] assignmentvar [if] [in] [weight] [, options]
+-----------------+
----+ Options summary +----------------------------------------------------
mbw(numlist) specifies a list of multiples for bandwidths, in percentage terms.
The default is "100 50 200" (i.e. half and twice the requested bandwidth) and
100 is always included in the list, regardless of whether it is specified.
z0(real) specifies the cutoff Z0 in assignmentvar Z.
strineq specifies that mean treatment differs at Z0 from all Z>Z0 (e.g. treatment
is 1 for Z>0 and 0 for Z<=0); the default assumption is that mean treatment
differs at Z0 from all Z<Z0 (e.g. treatment is 1 for Z>=0 and 0 for Z<0).
x(varlist) requests estimates of jumps in control variables varlist.
ddens requests a computation of a discontinuity in the density of Z. This is
computed in a relatively ad hoc way, and should be redone using McCrary's
test described at http://www.econ.berkeley.edu/~jmccrary/DCdensity/.
s(stubname) requests that estimates be saved as new variables beginning with
stubname.
graph requests that local linear regression graphs for each bandwidth be
produced.
noscatter suppresses the scatterplot on those graphs.
cluster(varlist) requests standard errors robust to clustering on distinct
combinations of varlist (e.g. stratum psu).
scopt(string) supplies an option list to the scatter plot.
lineopt(string) supplies an option list to the overlaid line plots.
n(real) specifies the number of points at which to calculate local linear
regressions. The default is to calculate the regressions at 50 points above
the cutoff, with equal steps in the grid, and to use equal steps below the
cutoff, with the number of points determined by the step size.
bwidth(real) allows specification of a bandwidth for local linear regressions.
The default is to use the estimated optimal bandwidth for a "sharp" design as
given by Imbens and Kalyanaraman (2009). The optimal bandwidth minimizes
MSE, or squared bias plus variance, where a smaller bandwidth tends to
produce lower bias and higher variance. Note that the optimal bandwidth will
often tend to be larger for a fuzzy design, due to the additional variance
that arises from the estimation of the jump in the conditional mean of
treatment.
bdep requests a graph of estimates versus bendwidths.
bingraph requests a graph of binned means instead of a scatterplot, in bins
defined by binvar.
binvar(varname) specifies the variable across which binned means should be
calculated.
oxline adds a vertical line at the default bandwidth.
kernel(rectangle) requests the use of a rectangle (uniform) kernel. The default
is a triangle (edge) kernel.
covar(varlist) adds covariates to Local Wald Estimation, which is generally a
Very Bad Idea. It is possible that covariates could reduce residual variance
and improve efficiency, but estimation error in their coefficients could also
reduce efficiency, and any violations of the assumptions that such covariates
are exogenous and have a linear impact on mean treatment and outcomes could
greatly increase bias.
+---------------------------+
----+ Remarks and saved results +------------------------------------------
To facilitate bootstrapping, rd saves the following results in e():
Scalars
e(N) Number of observations used in estimation
e(w) Bandwidth in base model; other bandwidths are reported in e.g.
e(w50) for the 50% multiple.
Macros
e(cmd) rd
e(rdversion) Version number of rd
e(depvar) Name of dependent variable
Matrices
e(b) Coefficient vector of estimated jumps in variables at
different percentage bandwidth multiples
Functions
e(sample) Marks estimation sample
References
Many references appear in
Nichols, Austin. 2007. Causal Inference with Observational Data. Stata
Journal 7(4): 507-541.
but the interested reader is directed also to
Imbens, Guido and Thomas Lemieux. 2007. "Regression Discontinuity Designs:
A Guide to Practice." NBER Working Paper 13039.
McCrary, Justin. 2007. "Manipulation of the Running Variable in the
Regression Discontinuity Design: A Density Test." NBER Technical
Working Paper 334.
Shadish, William R., Thomas D. Cook, and Donald T. Campbell. 2002.
Experimental and Quasi-Experimental Designs for Generalized Causal
Inference. Boston: Houghton Mifflin.
Fuji, Daisuke, Guido Imbens, and Karthik Kalyanaraman. 2009. "Notes for
Matlab and Stata Regression Discontinuity Software."
http://www.economics.harvard.edu/faculty/imbens/software_imbens
{phang}Imbens, Guido, and Karthik Kalyanaraman. 2009. "Optimal
Bandwidth Choice for the Regression Discontinuity Estimator." NBER WP
14726. Acknowledgements {p}I would like to thank Justin McCrary for
helpful discussions. Any errors are my own.{p_end} {p}The optimal
bandwidth calculations are from Fuji, Imbens, and Kalyanaraman (2009),
available at
http://www.economics.harvard.edu/faculty/imbens/software_imbens.{p_end}
Citation of rd {p}rd is not an official Stata command. It is a free
contribution to the research community, like a paper. Please cite it as
such: {p_end} {phang}Nichols, Austin. 2011. rd 2.0: Revised Stata
module for regression discontinuity estimation.
http://ideas.repec.org/c/boc/bocode/s456888.html{p_end} Author Austin
Nichols Principal Scientist, Abt Associates, Bethesda MD
austinnichols@gmail.com Also see {p 1 14}Manual: [U] 23 Estimation
and post-estimation commands{p_end} {p 10 14}[R] bootstrap{p_end} {p
10 14}[R] lpoly in Stata 10, else locpoly (findit locpoly to
install){p_end} {p 10 14}[R] ivregress in Stata 10, else [R]
ivreg{p_end} {p 10 14}[R] regress{p_end} {p 10 14}[XT] xtreg{p_end} {p
1 10}On-line: help for (if installed) rd_obs (prior version of rd),
ivreg2, overid, ivendog, ivhettest, ivreset, xtivreg2, xtoverid,
ranktest, condivreg; psmatch2, nnmatch. {p_end}
(remote file ends)
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