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)
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.
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.
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.
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
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
Imbens, Guido, and Karthik Kalyanaraman. 2009. "Optimal Bandwidth Choice for the Regression Discontinuity Estimator." NBER WP 14726.
I would like to thank Justin McCrary for helpful discussions. Any errors are my own.
The optimal bandwidth calculations are from Fuji, Imbens, and Kalyanaraman (2009), available at http://www.economics.harvard.edu/faculty/imbens/software_imbens.
rd is not an official Stata command. It is a free contribution to the research community, like a paper. Please cite it as such:
Nichols, Austin. 2011. rd 2.0: Revised Stata module for regression discontinuity estimation. http://ideas.repec.org/c/boc/bocode/s456888.html
Author
Austin Nichols Urban Institute Washington, DC, USA austinnichols@gmail.com
Also see
Manual: [U] 23 Estimation and post-estimation commands [R] bootstrap [R] lpoly in Stata 10, else locpoly (findit locpoly to install) [R] ivregress in Stata 10, else [R] ivreg [R] regress [XT] xtreg
On-line: help for (if installed) rd_obs (prior version of rd), ivreg2, overid, ivendog, ivhettest, ivreset, xtivreg2, xtoverid, ranktest, condivreg; psmatch2, nnmatch.