Cattaneo-Drukker-Holland_2013_Stata
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Cattaneo-Drukker-Holland_2013_Stata
<. OwMcrutKUIreThe State Journal (2013)13. Number 3. pp. 4ÍXT 450Estimation of multivalued treatment effects under conditional independenceMatias D. C Cattaneo-Drukker-Holland_2013_StataCattaneo Department of Economics University of Michigan Ann Arbor. MI cattaneo'^umich.eduDavid M. Drukker StataCorp College Station. TX ddrukkerostata.comAshley I). Holland Department of Science and Mathematics Cedarville University Cedarville, OH ahollandocedarville.eduAbstract. 1'his article discu Cattaneo-Drukker-Holland_2013_Statasses the popanns command, which implements two semi parametric estimators for multivalued treatment effects discussed in Cattaneo (2010, Journal of EcCattaneo-Drukker-Holland_2013_Stata
onometrics 1-55: 138 154). The first is a properly reweighted inverse-probability weighted estimator, and the second is an cfficient-influcnco functio<. OwMcrutKUIreThe State Journal (2013)13. Number 3. pp. 4ÍXT 450Estimation of multivalued treatment effects under conditional independenceMatias D. C Cattaneo-Drukker-Holland_2013_Stataoutcome distributions, allowing for multiple, discrete treatment levels. These estimators are then used to estimate a variety of multivalued treatment effects. We discuss pre- and postestimation approaches that can In' used in conjunction with our main implementation. We illustrate the program and p Cattaneo-Drukker-Holland_2013_Statarovide a simulation study assessing the finite-sample performance of the inference procedures.Keywords: stO3O3, poparms, bill, inverse-probability weiCattaneo-Drukker-Holland_2013_Stata
ghting, treatment effects, semiparametric estimation, unconfoundedness, generalized propensity scone, multivalued treatment effects1 IntroductionThis <. OwMcrutKUIreThe State Journal (2013)13. Number 3. pp. 4ÍXT 450Estimation of multivalued treatment effects under conditional independenceMatias D. C Cattaneo-Drukker-Holland_2013_Statality, that is. under the selection-on-ol>servables and common support assumptions. In particular, this command implements the two flexible, semiparametric-efficient estimation procedures proposed in Cattaneo (2010) to conduct joint inference on mean and quant ile treatment effects. For recent review Cattaneo-Drukker-Holland_2013_Statas on the vast literature of treatment effects, see, among others. Heckman and Vytlacil (2007), Imbens and Wooldridge (2009), and Wooldridge (2010) inCattaneo-Drukker-Holland_2013_Stata
economics: Morgan and Winship (2007) in sociology; and van der Limn and Robins (2003) and Tsiatis (2006) in biostatistics.Many multivalued treatment e<. OwMcrutKUIreThe State Journal (2013)13. Number 3. pp. 4ÍXT 450Estimation of multivalued treatment effects under conditional independenceMatias D. C Cattaneo-Drukker-Holland_2013_StataataCorp LPstO3O3408Multivalued treatment effectsment. These distributions are called the potential-outcome distributions and are identifiable from the oliserved data under the selection-on-observables or unconfoundcdness assumption. Under this assumption, Cattanco (2010) derives the large-sample pro Cattaneo-Drukker-Holland_2013_Stataperties of inverse-probability weighted (ll’W) estimators and efficient-influence-function (EIF) estimators for the means, quantiles, and other featurCattaneo-Drukker-Holland_2013_Stata
es of the potential-outcome distributions when the treatment variable can have multiple distinct values. Using either of these estimators, which are s<. OwMcrutKUIreThe State Journal (2013)13. Number 3. pp. 4ÍXT 450Estimation of multivalued treatment effects under conditional independenceMatias D. C Cattaneo-Drukker-Holland_2013_Statavalid inference procedures for multivalued treatment effects.In this article, we describe the new poparts command, which implements these IPW and EIF estimators to estimate the menus and quantiles of each potential-outcome distribution as well as the associated standard-error estimators. Different c Cattaneo-Drukker-Holland_2013_Stataontrasts of these estimated parameters are then used to produce semiparametric-efficient estimators with valid standard-error estimators for average aCattaneo-Drukker-Holland_2013_Stata
nd quantile multivalued treatment effects. These procedures require implementing nonparametric series estimators to flexibly approximate certain nonpa<. OwMcrutKUIreThe State Journal (2013)13. Number 3. pp. 4ÍXT 450Estimation of multivalued treatment effects under conditional independenceMatias D. C Cattaneo-Drukker-Holland_2013_Statased methods crucially rely on the select ion-on-observables assumption to identify and estimate the parameters of the potential-outcome distributions. This assumption maintains that after one controls for observed covariates, the potentialoutcome distributions lire independent of the treatment level Cattaneo-Drukker-Holland_2013_Stata, and therefore rules out that some unoltservable factor correlated with treatment assignment affects the potential-outcome dist ributions. This assumCattaneo-Drukker-Holland_2013_Stata
ption is strong and may not be valid in some applications, although it is popular and frequently used in empirical work. While it IS testable in some <. OwMcrutKUIreThe State Journal (2013)13. Number 3. pp. 4ÍXT 450Estimation of multivalued treatment effects under conditional independenceMatias D. C Cattaneo-Drukker-Holland_2013_Stata in this article lead to more efficient estimators than the usual parametric estimators. We further discuss this assumption and its implications below.In the remainder of the article, we discuss the implemented methods with notation ami formality, an example, the syntax of the poparms command, and t Cattaneo-Drukker-Holland_2013_Statahe methods and formulas implemented in the command.2 Setup, parameters, and estimators2.1 Model and samplingWe consider a standard cross-sectional setCattaneo-Drukker-Holland_2013_Stata
ting where we observe a random sample of size n from a large population hi which each individual has been assigned one of./ + 1 possible treatment lev<. OwMcrutKUIreThe State Journal (2013)13. Number 3. pp. 4ÍXT 450Estimation of multivalued treatment effects under conditional independenceMatias D. C Cattaneo-Drukker-Holland_2013_Statates the treatment level administered, and Xi is a kt X 1 vector of covariates. We also introduce the indicator variables (j) = l(«'j = j), which lake the value 1 if unit t received treatment j and the value 0 otherwise, l(-) denotes the indicator function, theM. Cat.taneo. D. Dmkker, and .4. Holland Cattaneo-Drukker-Holland_2013_Stata41 wolxscrvcd vectors z,, i 1.2,... ,n. arc independent and identically distributed draws of the vector 7. = (y, w, x')', and d(j) = I (wi = j).lb desCattaneo-Drukker-Holland_2013_Stata
cribe the cstimands and estimators OÍ interest, we use the classical polenliul-ontcome framework in the context of multivalued treatment effects. This<. OwMcrutKUIreThe State Journal (2013)13. Number 3. pp. 4ÍXT 450Estimation of multivalued treatment effects under conditional independenceMatias D. C Cattaneo-Drukker-Holland_2013_Statae observed outcome variable is given by■V. = d,(o)y,(0) + d,(I )y,(I) + - ■. 4- d,(.7)y,(.7)where (y,(<>),<;,(I),..., >a(-7)}' is an independent, and identically distributed draw from {p(0), y(l), . -.. id-7)1' for each individual i 1,2,....n in the sample. The distribution of each y(j) is the distr Cattaneo-Drukker-Holland_2013_Stataibution of the outcome variable that would occur if individuals were given treatment level j; it is known as the potential outcome distribution of treCattaneo-Drukker-Holland_2013_Stata
at ment level j. Many treatment effects of interest reduce to contrasts between parameters of these distributions. Because it is central to parameter <. OwMcrutKUIreThe State Journal (2013)13. Number 3. pp. 4ÍXT 450Estimation of multivalued treatment effects under conditional independenceMatias D. C Cattaneo-Drukker-Holland_2013_Statanalysis.Only one of the J + 1 possible potential outcomes can be oliservcd for each individual in the sample Iwcause each individual can receive only one treatment level. Holland (1986) termed this sit nation the fundamental problem of causal inference. Rom this perspective, estimating the parameter Cattaneo-Drukker-Holland_2013_Statas of the potential-outcome distribution is a missingdata problem l>ecause we can see only one outeome per individual. The observed y are draw’s from dCattaneo-Drukker-Holland_2013_Stata
istribution of y(j) conditional on tư = j. and hence, we need further assumptions to identify the unconditional distribution of y( j) from the ol^ervc<. OwMcrutKUIreThe State Journal (2013)13. Number 3. pp. 4ÍXT 450Estimation of multivalued treatment effects under conditional independenceMatias D. C Cattaneo-Drukker-Holland_2013_Statand it allows ILS to recover the parameters of the unobserved unconditional dist ribution from the observer! conditional distribution.Assumption 1. For all j = 0.1,..., J:(a)(Sclcction-on-observablcs) y(j) -11 d(j)|x.(b)(A'o-cmptv-ccll) 0 < Anin < Pj(x) with p/x) LJ(u' j|x).Assumption 1(a) implies th Cattaneo-Drukker-Holland_2013_Stataat the distribution of each potential outcome y(j) is independent, of the random treatment d(j), conditional on the covariates X. This as sumption hasCattaneo-Drukker-Holland_2013_Stata
a long history; sec, among many others, Heckman. Ichimura, and Todd (1998), Imbcus (2001). Heckman and Vytlacil (2007). Imbcus and Wooldridge (2009),<. OwMcrutKUIreThe State Journal (2013)13. Number 3. pp. 4ÍXT 450Estimation of multivalued treatment effects under conditional independenceMatias D. C Cattaneo-Drukker-Holland_2013_Statae observable char acteristics, treatment, assignment should be independent of the potential outcome. This assumption, although weaker than plain random assignment, is indeed strong because it rules out the presence of observed characteristics that could affect both treatment and outcomes. Nonetheles Cattaneo-Drukker-Holland_2013_Statas, in some empirical contexts, this assumption is reasonable and often imposed to estimate treatment effects.410Multivalued treatment effectsAssumptioCattaneo-Drukker-Holland_2013_Stata
n 1(b) says that for every possible X in the population, there is a strictly positive probability that someone with that covariate pattern could he asGọi ngay
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