Discretize by Bin in Stata Regression Continuity

0.1 Introduction

Regression discontinuity (RD) designs with multiple cutoffs or multiple scores are commonly encountered in empirical work in Education, Economics, Political Science, Public Policy, and many other disciplines. As a consequence, these specific settings have also received attention in the recent RD methodological literature (Papay-Willett-Murnane_2011_JoE; Reardon-Robinson_2012_JREE; Wong-Steiner-Cook_2012-JEBS; Keele-Titiunik_2015_PA; Keele-Titiunik-Zubizarreta_2015_JRSSA; Cattaneo-Keele-Titiunik-VazquezBare_2016_JOP; Cattaneo-Keele-Titiunik-VazquezBare_2019_wp, and references therein). In this article, we introduce the software package rdmulti, which includes three Stata commands (and analogous R functions) for the analysis of RD designs with multiple cutoffs or multiple scores.

The command rdmc applies to non-cumulative and cumulative multi-cutoff RD settings, following recent work in Cattaneo-Keele-Titiunik-VazquezBare_2016_JOP; Cattaneo-Keele-Titiunik-VazquezBare_2019_wp . Specifically, it calculates pooled and cutoff-specific RD treatment effects, employing local polynomial estimation and robust bias-corrected inference procedures. Post estimation and inference is allowed. The companion command rdmcplot offers RD plots for multi-cutoff settings. Finally, the command rdms concerns multi-score settings, covering in particular cumulative cutoffs and bivariate score contexts. It also calculates pooled and cutoff-specific RD treatment effects based on local polynomial methods, and allows for post-estimation estimation and inference. These commands employ the Stata (and R) package rdrobust for plotting, estimation, and inference; see Calonico-Cattaneo-Titiunik_2014_Stata; Calonico-Cattaneo-Titiunik_2015_R; Calonico-Cattaneo-Farrell-Titiunik_2017_Stata for software details. See also Cattaneo-Titiunik-VazquezBare_2017_JPAM for a comparison of RD methodologies, Cattaneo-Idrobo-Titiunik_2019_Book; Cattaneo-Idrobo-Titiunik_2020_Book and Cattaneo-Titiunik-VazquezBare_2020_Sage for a practical introductions to RD designs, and Cattaneo-Escanciano_2017_AIE for a recent edited volume with further references.

To streamline the presentation, this article employs only simulated data to showcase all three settings covered by the package rdmulti: non-cumulative multiple cutoffs, cumulative multiple cutoffs, and bivariate score settings. For further discussion and illustration employing real data sets see Cattaneo-Idrobo-Titiunik_2020_Book . The three settings covered by the package correspond, respectively, to (i) RD designs where different subgroups in the data are exposed to distinct but only one of the cutoff points (non-cumulative case), (ii) RD designs where units receive one single score and units are confronted to a sequence of ordered cutoffs points (cumulative case), and (iii) RD designs where units received two scores and there is a boundary on the plane determining the control and treatment areas. Well-known examples of each of these settings are:
Non-Cumulative Multiple Cutoffs: units in different groups (e.g., schools) receive an univariate score (e.g., test score) but the RD cutoff varies by group;
Cumulative Multiple Cutoffs: units receive the an univariate score (e.g., age) but different treatments are assigned at distinct score levels (e.g., at age 60 and at age 65);
Multiple Scores: units receive two scores (e.g., latitude and longitude) and treatment is assigned based on a boundary depending on both scores (e.g., geographic boundary).

We elaborate further on these cases in the upcoming sections, where we also give graphical representations of each case.

The Stata (and R) package rdmulti complements several recent software packages for RD designs. First, it explicitly relies on rdrobust (Calonico-Cattaneo-Titiunik_2014_Stata; Calonico-Cattaneo-Titiunik_2015_R; Calonico-Cattaneo-Farrell-Titiunik_2017_Stata) for implementation, and hence further extends its scope to the case of RD designs with multiple cutoffs or multiple scores. Second, while the package focuses on local polynomial methods, related methods employing local randomization ideas and implemented in the package rdlocrand can also be used in the contexts of multiple cutoffs and multiple scores (Cattaneo-Titiunik-VazquezBare_2016_Stata). Third, the package rddensity (Cattaneo-Jansson-Ma_2018_Stata) can also be used in multiple cutoffs or multiple scores settings for falsification purposes. Finally, see the package rdpower (Cattaneo-Titiunik-VazquezBare_2018_rdpower) for power calculations and sampling design methods, which can also be applied in the contexts discussed in this article.

The rest of the article is organized as follows. Section 0.2 gives a brief overview of the methods implemented in the package rdmulti, and also provides further references. Sections 0.3, 0.4 and 0.5 discuss the syntax of the commands rdmc, rdmcplot and rdms, respectively. Section 0.6 gives numerical illustrations, and Section 0.7 concludes. The latest version of this software, as well other software and materials useful for the analysis of RD designs, can be found at:

https://sites.google.com/site/rdpackages/

0.2 Overview of Methods

In this section we briefly describe the main ideas and methods used in the package rdmulti. For further methodological details see Keele-Titiunik_2015_PA , Cattaneo-Keele-Titiunik-VazquezBare_2016_JOP; Cattaneo-Keele-Titiunik-VazquezBare_2019_wp , Cattaneo-Idrobo-Titiunik_2020_Book , and references therein. All estimation and inference procedures employ rdplots (Calonico-Cattaneo-Titiunik_2015_JASA) as well as local polynomial point estimation and robust bias correction inference methods (Calonico-Cattaneo-Titiunik_2014_ECMA; Calonico-Cattaneo-Farrell_2018_JASA; Calonico-Cattaneo-Farrell-Titiunik_2019_RESTAT; Calonico-Cattaneo-Farrell_2020_ECTJ; Calonico-Cattaneo-Farrell_2019_wp-optcer).

0.2.1 Non-cumulative Multiple Cutoffs

In this case, individuals have a running variable

and a vector of potential outcomes

. Each individual faces a cutoff with . Unlike in a standard single-cutoff RD design,

is a random variable. In a sharp design, individuals are treated when their running variable exceeds their corresponding cutoff,

. A key feature of this design is that the variable partitions the population, that is, each unit faces one and only one value of . As the notation suggests, the potential outcomes for each individual are the same regardless of the specific cutoff they are exposed to; see Cattaneo-Keele-Titiunik-VazquezBare_2016_JOP; Cattaneo-Keele-Titiunik-VazquezBare_2019_wp for more discussion. Finally, we only consider finite multiple cutoffs because this is the most natural setting for empirical work: in practice, continuous cutoff are discretized for estimation and inference, as discussed and illustrated below.

Under regularity conditions, which include smoothness of conditional expectations among other things (see aforementioned references for details), the cutoff-specific treatment effects, , are identified by:

The pooled RD estimate is obtained by recentering the running variable, , thus normalizing the cutoff at zero:

All these parameters can be readily estimated using local polynomial methods (see Cattaneo-Idrobo-Titiunik_2019_Book, for a practical introduction), conditioning on cutoffs when appropriate. In other words, RD methods can by applied to each cutoff separately, in addition to pooling the data. Therefore, the rdmulti package implements bandwidth selection, estimation and inference based on local polynomial methods using the rdrobust command, described in Calonico-Cattaneo-Titiunik_2014_Stata; Calonico-Cattaneo-Titiunik_2015_R; Calonico-Cattaneo-Farrell-Titiunik_2017_Stata . Specifically, the command rdmc allows for multi-cutoffs RD designs.

For the pooled parameter , the weights are estimated using the fact that ; see Cattaneo-Keele-Titiunik-VazquezBare_2016_JOP for further details. Then, given a bandwidth ,

When not specified by the user, the rdmc command uses the bandwidth selected by rdrobust when estimating the pooled effect to estimate the weights.

0.2.2 Cumulative Multiple Cutoffs

In an RD setting with cumulative cutoffs, individuals receive different treatments (or different dosages of a treatment) for different ranges of the running variable. For instance, individuals receive treatment 1 if , treatment 2 if , and so on, until the last treatment value at . Denote the values of these treatments as , so that the treatment variable is now . Under standard regularity conditions, we have:

Since, unlike the case with multiple non-cumulative cutoffs, the population is not partitioned, each observation can be used to estimate two different (but contiguous on the score dimension) treatment effects. For example, units receiving treatment dosage are used as "treated" (i.e. above the cutoff ) when estimating and as "controls" when estimating (i.e. below the cutoff ). As a result, cutoff-specific estimators may not be independent, although the dependence disappears asymptotically as long as the bandwidths around each cutoff decrease with the sample size. On the other hand, bandwidths can be chosen to be non-overlapping to ensure that observations are used only once.

Once the data has been assigned to each cutoff under analysis, local polynomial methods can also be applied cutoff by cutoff in the cumulative multiple cutoffs case. We illustrate this approach below; for further discussion see Cattaneo-Idrobo-Titiunik_2020_Book , and the references therein.

0.2.3 Multiple Scores

In a multi-score RD design, treatment is assigned based on multiple running variables and some function determining a treatment "region" or "area". We focus on the case with two running variables, , which is by far the most common case in empirical work. This case occurs naturally when, for instance, a treatment is assigned based on scores in two different exams (such as language and mathematics), or based on geographic location (e.g., latitude and longitude).

This type of assignment defines a continuum of treatment effects over the boundary of the treatment region, denoted by . For instance, if treatment is assigned to students scoring below 50 in language and mathematics, the treatment boundary is . For each point , the treatment effect at that point is given by

and under regularity conditions,

where and denote the control and treatment areas, respectively, and is a metric.

Since estimating a whole curve of treatment effects may not be feasible in practice, it is common to define a set of boundary points of interest at which to estimate the RD treatment effects. In the previous example, for instance, three points of interest on the boundary determining treatment assignment could be . On the other hand, the pooled RD estimand requires defining some measure of distance to the cutoff, such as the perpendicular (Euclidean) distance. This distance can be seen as the recentered running variable , which allows defining the pooled estimand as in Equation 2.

0.3 rdmc syntax

This section describes the syntax of the command rdmc, which estimates the pooled and cutoff-specific RD effects using rdrobust.

0.3.1 Syntax

rdmc depvar runvar , cvar(cutoff_var) pooled_opt(string)
derivvar(string) pvar(string) qvar(string) hvar(string)
hrightvar(string) bvar(string) brightvar(string) rhovar(string)
covsvar(string) kernelvar(string) weightsvar(string)
bwselectvar(string) vcevar(string) level(#) scaleparvar(string)
scaleregulvar(string) fuzzy(string) plot graph_opt(string) verbose depvar is the dependent variable.

runvar is the running variable (a.k.a. score or forcing variable).

c var(cutoff_var) specifies the numeric variable cutoff_var that indicates the cutoff faced by each unit in the sample.

pooled_opt(string) specifies the options to be passed to rdrobust to calculate pooled estimate. See help rdrobust for details.

deriv var(string) a variable of length equal to the number of different cutoffs that specifies the order of the derivative for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

p var(string) a variable of length equal to the number of different cutoffs that specifies the order of the polynomials for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

q var(string) a variable of length equal to the number of different cutoffs that specifies the order of the polynomials for bias estimation for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

h var(string) a variable of length equal to the number of different cutoffs that specifies the bandwidths for rdrobust to calculate cutoff-specific estimates. When hrightvar is specified, hvar indicates the bandwidth to the left of the cutoff. When hrightvar is not specified, the same bandwidths are used at each side. See help rdrobust for details.

hright var(string) a variable of length equal to the number of different cutoffs that specifies the bandwidths to the right of the cutoff for rdrobust to calculate cutoff-specific estimates. When hrightvar is not specified, the same bandwidths in hvar are used at each side. See help rdrobust for details.

b var(string) a variable of length equal to the number of different cutoffs that specifies the bandwidths for the bias for rdrobust to calculate cutoff-specific estimates. When brightvar is specified, bvar indicates the bandwidth to the left of the cutoff. When brightvar is not specified, the same bandwidths are used at each side. See help rdrobust for details.

bright var(string) a variable of length equal to the number of different cutoffs that specifies the bandwidths to the right of the cutoff for rdrobust to calculate cutoff-specific estimates. When brightvar is not specified, the same bandwidths in bvar are used at each side. See help rdrobust for details.

rho var(string) a variable of length equal to the number of different cutoffs that specifies the value of rho for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

covs var(string) a variable of length equal to the number of different cutoffs that specifies the covariates for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

kernel var(string) a variable of length equal to the number of different cutoffs that specifies the kernels for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

weights var(string) a variable of length equal to the number of different cutoffs that specifies the weights for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

bwselect var(string) a variable of length equal to the number of different cutoffs that specifies the bandwidth selection method for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

vce var(string)

a variable of length equal to the number of different cutoffs that specifies the variance-covariance matrix estimation method for

rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

scalepar var(string) a variable of length equal to the number of different cutoffs that specifies the value of scalepar for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

scaleregul var(string) a variable of length equal to the number of different cutoffs that specifies the value of scaleregul for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

fuzzy(string) indicates a fuzzy design. See help rdrobust for details.

plot plots the pooled and cutoff-specific estimates and the weights given by the pooled estimate to each cutoff-specific estimate.

graph_opt(string) options to be passed to the graph when plot is specified.

verbose displays the output from rdrobust to calculate pooled estimand.

0.4 rdmcplot syntax

This section describes the syntax of the command rdmcplot , which plots the regression functions for each of the groups facing each cutoff using rdplot .

0.4.1 Syntax

rdmcplot depvar runvar , c var( cutoff_var ) h var( string )
p var( string ) nbins var( string ) nbinsright var( string )
covs var( string ) binselect var( string ) scale var( string )
scaleright var( string ) kernel var( string ) weights var( string )
h var( string ) hright var( string ) support var( string )
supportright var( string ) nobins nopoly nodraw depvar is the dependent variable.

runvar is the running variable (a.k.a. score or forcing variable).

c var( cutoff_var ) specifies the numeric variable cutoff_var that indicates the cutoff faced by each unit in the sample.

p var( string ) a variable of length equal to the number of different cutoffs that specifies the order of the polynomials for rdplot . See help rdplot for details.

nbins var( string ) a variable of length equal to the number of different cutoffs that specifies the number of bins for rdplot . When nbinsrightvar is specified, nbinsvar indicates the number of bins to the left of the cutoff. When nbinsrightvar is not specified, the same number of bins is used at each side. See help rdplot for details.

nbinsright var( string ) a variable of length equal to the number of different cutoffs that specifies the number of bins to the right of the cutoff for rdplot . When nbinsrightvar is not specified, the number of bins in nbinsvar used at each side. See help rdplot for details.

covs var( string ) a variable of length equal to the number of different cutoffs that specifies the covariates for rdplot . See help rdplot for details.

binselect var( string ) a variable of length equal to the number of different cutoffs that specifies the bin selection method for rdplot . See help rdplot for details.

scale var( string ) a variable of length equal to the number of different cutoffs that specifies the scale for rdplot . When scalerightvar is specified, scalevar indicates the scale to the left of the cutoff. When scalerightvar is not specified, the same scale is used at each side. See help rdplot for details.

scaleright var( string ) a variable of length equal to the number of different cutoffs that specifies the scale to the right of the cutoff for rdplot . When scalerightvar is not specified, the scale in scalevar is used at each side. See help rdplot for details.

kernel var( string ) a variable of length equal to the number of different cutoffs that specifies the kernels for rdplot . See help rdplot for details.

weights var( string ) a variable of length equal to the number of different cutoffs that specifies the weights for rdplot . See help rdplot for details.

h var( string ) a variable of length equal to the number of different cutoffs that specifies the bandwidths for rdplot . When hrightvar is specified, hvar indicates the bandwidth to the left of the cutoff. When hrightvar is not specified, the same bandwidth is used at each side. See help rdplot for details.

hright var( string ) a variable of length equal to the number of different cutoffs that specifies the bandwidth to the right of the cutoff for rdplot . When hrightvar is not specified, the bandwidth in hvar is used at each side. See help rdplot for details.

support var( string ) a variable of length equal to the number of different cutoffs that specifies the support for rdplot . When supportrightvar is specified, supportvar indicates the support to the left of the cutoff. When supportrightvar is not specified, the same support is used at each side. See help rdplot for details.

supportright var( string ) a variable of length equal to the number of different cutoffs that specifies the support to the right of the cutoff for rdplot . When supportrightvar is not specified, the support in supportvar is used at each side. See help rdplot for details.

nobins omits the bins plot.

nopoly omits the polynomial curve plot.

nodraw omits the plot.

0.5 rdms syntax

This section describes the syntax of the command rdms , which analyzes RD designs with cumulative cutoffs or two running variables.

0.5.1 Syntax

rdms depvar runvar1 [ runvar2 treatvar ] , c var( cutoff_var1
[cutoff_var2]) range( range1 [range2] ) xnorm( string )
pooled_opt( string ) deriv var( string ) p var( string ) q var( string )
h var( string ) hr ightvar( string ) b var( string ) br ightvar( string )
rho var( string ) covs var( string ) kernel var( string ) weights var( string )
bwselect var( string ) vce var( string ) level( # ) scalepar var( string )
scaleregul var( string ) fuzzy( string ) plot graph_opt( string ) depvar is the dependent variable.

runvar1 is the running variable (a.k.a. score or forcing variable) in a cumulative cutoffs setting.

runvar2 if specified, is the second running variable (a.k.a. score or forcing variable) in a two-score setting.

treatvar if specified, is the treatment indicator in a two-score setting.

c var( cutoff_var1 [cutoff_var2] ) specifies the numeric variable cutoff_var1 that indicates the cutoff faced by each unit in the sample in a cumulative cutoff setting, or the two running variables cutoff_var1 and cutoff_var2 in a two-score RD design.

range( range1 [range2] ) specifies the range of the running variable to be used for estimation around each cutoff. Specifying only one variable implies using the same range at each side of the cutoff.

xnorm( string ) specifies the normalized running variable to estimate pooled effect.

pooled_opt( string ) specifies the options to be passed to rdrobust to calculate pooled estimate. See help rdrobust for details.

deriv var( string ) a variable of length equal to the number of different cutoffs that specifies the order of the derivative for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

p var( string ) a variable of length equal to the number of different cutoffs that specifies the order of the polynomials for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

q var( string ) a variable of length equal to the number of different cutoffs that specifies the order of the polynomials for bias estimation for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

h var( string ) a variable of length equal to the number of different cutoffs that specifies the bandwidths for rdrobust to calculate cutoff-specific estimates. When hrightvar is specified, hvar indicates the bandwidth to the left of the cutoff. When hrightvar is not specified, the same bandwidths are used at each side. See help rdrobust for details.

hright var( string ) a variable of length equal to the number of different cutoffs that specifies the bandwidths to the right of the cutoff for rdrobust to calculate cutoff-specific estimates. When hrightvar is not specified, the same bandwidths in hvar are used at each side. See help rdrobust for details.

b var( string ) a variable of length equal to the number of different cutoffs that specifies the bandwidths for the bias for rdrobust to calculate cutoff-specific estimates. When brightvar is specified, bvar indicates the bandwidth to the left of the cutoff. When brightvar is not specified, the same bandwidths are used at each side. See help rdrobust for details.

bright var( string ) a variable of length equal to the number of different cutoffs that specifies the bandwidths to the right of the cutoff for rdrobust to calculate cutoff-specific estimates. When brightvar is not specified, the same bandwidths in bvar are used at each side. See help rdrobust for details.

rho var( string ) a variable of length equal to the number of different cutoffs that specifies the value of rho for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

covs var( string ) a variable of length equal to the number of different cutoffs that specifies the covariates for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

kernel var( string ) a variable of length equal to the number of different cutoffs that specifies the kernels for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

weights var( string ) a variable of length equal to the number of different cutoffs that specifies the weights for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

bwselect var( string ) a variable of length equal to the number of different cutoffs that specifies the bandwidth selection method for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

vce var( string ) a variable of length equal to the number of different cutoffs that specifies the variance-covariance matrix estimation method for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

level( # ) specifies the confidence levels for confidence intervals. See help rdrobust for details.

scalepar var( string ) a variable of length equal to the number of different cutoffs that specifies the value of scalepar for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

scaleregul var( string ) a variable of length equal to the number of different cutoffs that specifies the value of scaleregul for rdrobust to calculate cutoff-specific estimates. See help rdrobust for details.

fuzzy( string ) indicates a fuzzy design. See help rdrobust for details.

plot plots the pooled and cutoff-specific estimates and the weights given by the pooled estimate to each cutoff-specific estimate.

graph_opt( string ) options to be passed to the graph when plot is specified.

0.6 Illustration of Methods

0.6.1 Non-cumulative Multiple Cutoffs

We begin by illustrating rdmc using a simulated dataset, simdata_multic . In this dataset, y is the outcome variable, x is the running variable and c is a variable indicating the cutoff that each unit in the sample faces. As shown below, there are four different cutoffs, each with the sample sample size.

[auto]. use simdata_multic, clear . sum Variable Obs Mean Std. Dev. Min Max 13 57 c 2,000 49.5 16.50413 33 66 x 2,000 50.79875 28.95934 .0184725 99.97507 t 2,000 .516 .4998689 0 1 y 2,000 1728.135 545.0856 540.0849 3015.232 . tab c c Freq. Percent Cum. 12 35 33 1,000 50.00 50.00 66 1,000 50.00 100.00 12 35 Total 2,000 100.00

The basic syntax for rdmc is the following:

The output shows the cutoff-specific estimate at each cutoff, together with the corresponding robust bias-corrected p-value, 95 percent robust confidence interval and sample size at each cutoff, and two "global" estimates. The first one is a weighted average of the cutoff specific estimates using the estimated weights described in Section 0.2 . These estimated weights are shown in the last column. The second one is the pooled estimate obtained by normalizing the running variable. While these two estimators converge to the same population parameter, they can differ in finite samples as seen in this example. In this example, the effect is statistically insignificant at the lowest cutoff, with evidence of a positive treatment effect at the remaining three cutoffs. Given the distribution of the running variable and the sample size at each cutoff, the weighted and pooled estimates give approximately the same weight to each cutoff.

All the results in the above display are calculated using rdrobust . The user can specify options for rdrobust to calculate both the pooled estimates using the option pooled_opt . For instance, the syntax below specifies a bandwidth of 20 and a local quadratic polynomial for the pooled estimand. By default, rdmc omits the output from rdrobust when estimating the effects. The output from the pooled effect estimation can be displayed using the option verbose , which we use below to show how the options are passed to rdrobust .

[auto]. rdmc y x, c(c) pooled_opt(h(20) p(2)) verbose Sharp RD estimates using local polynomial regression. Cutoff c = 0 Left of c Right of c Number of obs = 2000 19 22 BW type = Manual Number of obs 968 1032 Kernel = Triangular Eff. Number of obs 409 416 VCE method = NN Order est. (p) 2 2 Order bias (q) 3 3 BW est. (h) 20.000 20.000 BW bias (b) 20.000 20.000 rho (h/b) 1.000 1.000 Outcome: y. Running variable: __000002. 19 60 Method Coef. Std. Err. z P¿—z— [95% Conf. Interval] 19 60 Conventional 437.04 129.8 3.3671 0.001 182.643 691.441 Robust - - 3.0118 0.003 185.618 877.381 19 60 Cutoff-specific RD estimation with robust bias-corrected inference 12 67 Cutoff Coef. P¿—z— [95% Conf. Int.] hl hr Nh Weight 12 67 33 484.83 0.00 421.18 552.53 14.66 14.66 289 0.540 66 297.98 0.00 220.35 362.27 11.95 11.95 246 0.460 12 67 Weighted 398.91 0.00 348.74 445.14 . . 535 . Pooled 437.04 0.00 185.62 877.38 20.00 20.00 825 . 12 67

The following syntax shows how to manually change options for the cutoff-specific estimates.

[auto]. gen double h = 11 in 1 (1,999 missing values generated) . replace h = 10 in 2 (1 real change made) . rdmc y x, c(c) h(h) Cutoff-specific RD estimation with robust bias-corrected inference 12 67 Cutoff Coef. P¿—z— [95% Conf. Int.] hl hr Nh Weight 12 67 33 495.43 0.00 368.13 563.21 11.00 11.00 207 0.498 66 303.77 0.00 220.40 403.32 10.00 10.00 209 0.502 12 67 Weighted 399.14 0.00 321.56 455.23 . . 416 . Pooled 436.40 0.00 179.34 676.63 13.68 13.68 550 . 12 67

The cutoff-specific options are passed in a similar fashion, defining a new variable that indicates the options for each cutoff in its values. For instance, the following syntax indicates different bandwidth selection methods at each cutoff:

[auto]. gen bwselect = "msetwo" in 1 (1,999 missing values generated) . replace bwselect = "certwo" in 2 (1 real change made) . rdmc y x, c(c) bwselect(bwselect) Cutoff-specific RD estimation with robust bias-corrected inference 12 67 Cutoff Coef. P¿—z— [95% Conf. Int.] hl hr Nh Weight 12 67 33 481.14 0.00 418.03 545.23 15.03 16.91 318 0.572 66 298.84 0.00 227.70 367.26 14.85 7.95 238 0.428 12 67 Weighted 403.10 0.00 355.73 449.87 . . 556 . Pooled 436.40 0.00 179.34 676.63 13.68 13.68 550 . 12 67

The rdmc command saves the bias-corrected estimates and variances in the matrices e(b) and e(V) , which allows for post-estimation testing using lincom or test . For instance, to test whether the effects at the two cutoffs are the same, type:

[auto]. rdmc y x, c(c) Cutoff-specific RD estimation with robust bias-corrected inference 12 67 Cutoff Coef. P¿—z— [95% Conf. Int.] hl hr Nh Weight 12 67 33 484.83 0.00 421.18 552.53 14.66 14.66 289 0.540 66 297.98 0.00 220.35 362.27 11.95 11.95 246 0.460 12 67 Weighted 398.91 0.00 348.74 445.14 . . 535 . Pooled 436.40 0.00 179.34 676.63 13.68 13.68 550 . 12 67 . matlist e(b) c1 c2 weighted pooled 13 44 y1 486.8578 291.3082 396.9415 427.9832 . lincom c1-c2 ( 1) c1 - c2 = 0 13 64 Coef. Std. Err. z P¿—z— [95% Conf. Interval] 13 64 (1) 195.5496 49.3309 3.96 0.000 98.86279 292.2364 13 64

The rdmcplot command jointly plots the estimated regression functions at each cutoff. The output from rdmcplot is shown in Figure 1 . The basic syntax is the following:

[auto]. rdmcplot y x, c(c)

The rdmcplot includes all the options available for rdplot . For example, the plot can be restricted to a bandwidth using the option h() and to use a polynomial of a specified order using the option p() , as shown below. This option allows the user to plot the linear fit and estimated treatment effects at each cutoff.

[auto]. gen p = 1 in 1/2 (1,998 missing values generated) . rdmcplot y x, c(c) h(h) p(p)

0.6.2 Cumulative Multiple Cutoffs

We know illustrate the use of rdms for cumulative cutoffs using the simulated dataset simdata_cumul . In this dataset, the running variable ranges from 0 to 100, and units with running variable below receive a certain treatment level whereas units with running variable above receive another treatment level . In this setting, the cutoffs are indicated as a variable in the dataset, where each row indicates a cutoff.

[auto]. use simdata_cumul, clear . sum Variable Obs Mean Std. Dev. Min Max 13 57 x 1,000 50.46639 28.69369 .0413166 99.8783 y 1,000 1508.638 488.2752 658.4198 2480.568 c 2 49.5 23.33452 33 66 . tab c c Freq. Percent Cum. 12 35 33 1 50.00 50.00 66 1 50.00 100.00 12 35 Total 2 100.00

The syntax for cumulative cutoffs is similar to rdmc . The user specifies the outcome variable, the running variable and the cutoffs as follows:

[auto]. rdms y x, c(c) Cutoff-specific RD estimation with robust bias-corrected inference 15 64 Cutoff Coef. P¿—z— [95% Conf. Int.] hl hr Nh 15 64 33 395.492 0.000 363.76 423.86 15.11 15.11 286 66 342.872 0.000 315.95 373.96 12.22 12.22 265 15 64

Options like the bandwidth, polynomial order and kernel for each cutoff-specific effect can be specified by creating variables as shown below.

[auto]. gen double h = 11 in 1 (999 missing values generated) . replace h = 8 in 2 (1 real change made) . gen kernel = "uniform" in 1 (999 missing values generated) . replace kernel = "triangular" in 2 variable kernel was str7 now str10 (1 real change made) . rdms y x, c(c) h(h) Cutoff-specific RD estimation with robust bias-corrected inference 15 64 Cutoff Coef. P¿—z— [95% Conf. Int.] hl hr Nh 15 64 33 394.701 0.000 339.45 439.30 11.00 11.00 215 66 342.505 0.000 301.56 375.95 8.00 8.00 166 15 64

Without further information, the rdms command could be using any observation above the cutoff 33 to estimate the effect of the first treatment level . This implies that some observations in the range are used. But these observations receive the second treatment level, . This feature can result in inconsistent estimators for . To avoid this problem, the user can specify the range of observations to be used around each cutoff. In this case, we can restrict the range at the first cutoff (33) to go from 0 to 65.5, to ensure that no observations above 66 are used, and the range at the second cutoff (66) to go from 33.5 to 100. This can be done as follows.

[auto]. gen double range_l = 0 in 1 (999 missing values generated) . gen double range_r = 65.5 in 1 (999 missing values generated) . replace range_l = 33.5 in 2 (1 real change made) . replace range_r = 100 in 2 (1 real change made) . rdms y x, c(c) range(range_l range_r) Cutoff-specific RD estimation with robust bias-corrected inference 15 64 Cutoff Coef. P¿—z— [95% Conf. Int.] hl hr Nh 15 64 33 394.698 0.000 356.12 430.45 10.96 10.96 214 66 342.180 0.000 312.20 372.04 11.18 11.18 246 15 64

The pooled estimate can be obtained using rdmc . For this, we need to assign each unit in the sample a value for the cutoff. One possibility is to assign each unit to the closest cutoff. For this, we generate a variable named cutoff that equals 33 for units with score below 49.5 (the middle point between 33 and 66), and equals 66 for units above 49.5.

[auto]. gen double cutoff = c[1]*(x¡=49.5) + c[2]*(x¿49.5) . rdmc y x, c(cutoff) Cutoff-specific RD estimation with robust bias-corrected inference 12 67 Cutoff Coef. P¿—z— [95% Conf. Int.] hl hr Nh Weight 12 67 33 389.53 0.00 332.94 443.69 6.26 6.26 119 0.531 66 341.02 0.00 300.39 377.33 5.04 5.04 105 0.469 12 67 Weighted 366.79 0.00 330.63 399.64 . . 224 . Pooled 363.97 0.00 180.11 551.78 8.14 8.14 333 . 12 67

Finally, we can use the variable cutoff to plot the regression functions using rdmcplot , shown in Figure 3

[auto]. rdmcplot y x, c(cutoff)

0.6.3 Multiple Scores

We now illustrate the use of rdms to analyze RD designs with two running variables using the simulated dataset simdata_multis . In this dataset, there are two running variables, x1 and x2 , ranging between 0 and 100, and units receive the treatment when and . We look at three cutoffs on the boundary: (25,50), (50,50) and (50,25).

[auto]. use simdata_multis, clear . sum Variable Obs Mean Std. Dev. Min Max 13 57 x1 1,000 50.22881 28.87868 .6323666 99.94879 x2 1,000 50.63572 29.1905 .0775479 99.78458 t 1,000 .223 .4164666 0 1 y 1,000 728.5048 205.5627 329.4558 1372.777 c1 3 41.66667 14.43376 25 50 13 57 c2 3 41.66667 14.43376 25 50 . list c1 c2 in 1/3 9 c1 c2 9 1. 25 50 2. 50 50 3. 50 25 9

The following code provides a simple visualization of this setting:

[auto]. gen xaux = 50 in 1/50 (950 missing values generated) . gen yaux = _n in 1/50 (950 missing values generated) . twoway (scatter x2 x1 if t==0, msize(small) mfcolor(white) msymbol(X)) /// ¿ (scatter x2 x1 if t==1, msize(small) mfcolor(white) msymbol(T)) /// ¿ (function y = 50, range(0 50) lcolor(black) lwidth(medthick)) /// ¿ (line yaux xaux, lcolor(black) lwidth(medthick)) /// ¿ (scatteri 50 25, msize(large) mcolor(black)) /// ¿ (scatteri 50 50, msize(large) mcolor(black)) /// ¿ (scatteri 25 50, msize(large) mcolor(black)), /// ¿ text(25 25 "Treated", size(vlarge)) /// ¿ text(60 60 "Control", size(vlarge)) /// ¿ legend(off)

The basic syntax is the following:

[auto]. rdms y x1 x2 t, c(c1 c2) Cutoff-specific RD estimation with robust bias-corrected inference 15 64 Cutoff Coef. P¿—z— [95% Conf. Int.] hl hr Nh 15 64 (25,50) 243.842 0.111 -50.93 491.18 11.12 11.12 42 (50,50) 578.691 0.000 410.83 764.88 13.83 13.83 47 (50,25) 722.444 0.000 451.49 1060.15 10.83 10.83 38 15 64

Information to estimate each cutoff-specific estimate can be provided as illustrated before. For instance, to specify cutoff-specific bandwidths:

[auto]. gen double h = 15 in 1 (999 missing values generated) . replace h = 13 in 2 (1 real change made) . replace h = 17 in 3 (1 real change made) . rdms y x1 x2 t, c(c1 c2) h(h) Cutoff-specific RD estimation with robust bias-corrected inference 15 64 Cutoff Coef. P¿—z— [95% Conf. Int.] hl hr Nh 15 64 (25,50) 336.121 0.233 -119.35 491.36 15.00 15.00 87 (50,50) 583.047 0.000 501.94 1101.24 13.00 13.00 42 (50,25) 620.692 0.000 464.92 1159.99 17.00 17.00 86 15 64

Finally, the xnorm option allows the user to specify the normalized running variable to calculate a pooled estimate. In this case, we define the normalized running variable as the closest perpendicular distance to the boundary defined by the treatment assignment.

[auto]. gen double aux1 = abs(.5 - x1) . gen double aux2 = abs(.5 - x2) . egen xnorm = rowmin(aux1 aux2) . replace xnorm = xnorm*(2*t-1) (777 real changes made) . rdms y x1 x2 t, c(c1 c2) xnorm(xnorm) Cutoff-specific RD estimation with robust bias-corrected inference 15 64 Cutoff Coef. P¿—z— [95% Conf. Int.] hl hr Nh 15 64 (25,50) 243.842 0.111 -50.93 491.18 11.12 11.12 42 (50,50) 578.691 0.000 410.83 764.88 13.83 13.83 47 (50,25) 722.444 0.000 451.49 1060.15 10.83 10.83 38 15 64 Pooled 484.940 0.000 437.39 548.09 12.60 12.60 240 15 64

0.7 Conclusion

We introduced the Stata package rdmulti to anlyze RD designs with multiple cutoffs or scores. A companion R function with the same syntax and capabilities is also provided.

0.8 Acknowledgments

We thank Sebastian Calonico and Nicolas Idrobo for helpful comments and discussions. The authors gratefully acknowledge financial support from the National Science Foundation through grant SES-1357561.

References

About the Authors

Matias D. Cattaneo is a Professor at the Department of Operations Research and Financial Engineering, Princeton University.

Rocío Titiunik is a Professor of Political Science at Princeton University.

Gonzalo Vazquez-Bare is an Assistant Professor of Economics at the University of California, Santa Barbara.

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Source: https://deepai.org/publication/analysis-of-regression-discontinuity-designs-with-multiple-cutoffs-or-multiple-scores