Treatment Effects I Whitney Newey Fall 2007 Cite as: Whitney Newey, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Treatment effects about how outcome of interest (ea rnings) is affected treatment (job training) program. Like structural model, outcome of interest is the left hand side variable, treatment is a right-hand side variable. Binary (endogenous) right-hand side variable with heterogenous coefficients. Have terminology all their own. i is individual, Di ∈ {0, 1} is treatment indicator (Di =1is enrollment in training) Yi0 potential outcome occurs when not treated (Di =0), Yi1 potential outcome when treated (Di =1). Observed outcome will be Yi = DiYi1+(1− Di)Yi0. Yi0 and Yi1 are counterfacutals; both not observed. Cite as: Whitney Newey, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Yi = DiYi1+(1− Di)Yi0. Treatment effect: βi = Yi1− Yi0. Not identifiable; only one of Yi1 and Yi0 are observed. Individual heterogenity in effect of treatment. Some objects may be identified. Average treatment effect: def AT E = E[βi]. Average treatment effect on treated: def TT = E[βi|Di =1]. Local average treatment effect and other effects described below. Cite as: Whitney Newey, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Yi = DiYi1+(1− Di)Yi0. Random coefficient interpretation. Yi = Yi0+(Yi1− Yi0)Di = αi + βiDi, αi = Yi0,βi = Yi1− Yi0. Treatment effect βi is the coefficient of Di and the constant αi and slope βi may vary over individuals. ATE is average of slope over entire population. TT is average of slope over treated subset where Di =1. Intellectual history: βi = Yi1− Yi0 is "counterfactual," Rubin (70’s). Econometricians know as "movement along a curve," Wright (1928). Here consider identification and estimation of various effects. Cite as: Whitney Newey, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Constant Treatment Effects Constant treatment effects: βi = β¯AT E = TT = β¯. For α¯ = E[αi] and εi = α, αi − ¯Yi =¯α +¯βDi + εi. Simple linear model with additive disturbance and constant coefficients. General model is linear model with additive disturbance but random slope coe ffi- cient. Note equivalence between random αi and a constant plus disturbance αi =¯α +εi. Cite as: Whitney Newey, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Instrument Zi will identify β¯and α¯ in usual way. Zi uncorrelated with εi and correlated with Di,thatis 0= Cov(Zi,εi)=Cov(Zi,αi)=Cov(Zi,Yi0), Cov(Zi,Di) =0.6Then ¯β = Cov(Zi,Yi)/Cov(Zi,Di). Estimate in the usual way. Nothing new here but terminology. Constant treatment effect too strong. Will not hold for effect of training, schooling, etc. βi will vary over individuals. Cite as: Whitney Newey, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Random Assignment Random assignment means Di not related to individual characteristics. Assume E[Yi0|Di]=E[Yi0], The mean of outcome without treatment does not depend on treatment.. Equivalently E[αi|Di]=E[αi]. Slightly more general than independence. To see what happens under this assumption note first that 0,Di =0,E[βi|Di]Di =( E[βi|Di =1],Di =1 = E[βi|Di =1]Di. Then E[Yi|Di]= E[αi + βiDi|Di]=E[αi]+E[βi|Di]Di = E[αi]+E[βi|Di =1]Di. Cite as: Whitney Newey, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].E[Yi|Di]=E[αi]+E[βi|Di =1]Di. Simple linear regression, constant is E[αi] slope is TT = E[βi|Di =1]. Assume also that E[Yi1|Di]=E[Yi1], Then we find that AT E = TT,since E[βi|Di =1]=E[Yi1|Di =1]− E[Yi0|Di =1] = E[Yi1] − E[Yi0]=E[βi]. Summarizing, if E[Yi0|Di]= E[Yi0] then by usual formula for OLS regression with constant and dummy variable, Cov(Di,Yi)TT = Var(Di)= E[Yi|Di =1]− E[Yi|Di =0]. If in addition E[Yi1|Di]=E[Yi1] then AT E = E[Yi|Di =1]− E[Yi|Di =0]. Cite as: Whitney Newey, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].Discussion Random assignment too strong for many applications. Individuals may choose whether to accept the treatment or not. May drop out of training programs May opt out of medical treatment. If decisions related to (αi,βi) then (αi,βi) and Di not independent. Di may be correlated with both αi and βi. Two approaches: a) Instrumental variables (IV). b) Selection on observables. Cite as: Whitney Newey, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].IV Identification of Treatment Effects Two cases: Dummy instrument and continuous instrument. Dummy Instruments Zi∈ {0, 1}, 0 < Pr(Zi=1)=P<1. (Why?) Assume throughout that that mean independence holds, as in E[αi|Zi]=E[Yi0|Zi]=E[Yi0]=E[αi]. Cite as: Whitney Newey, course materials for 14.385 Nonlinear Econometric Analysis, Fall 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].]Wald formula for IV limit: By usual least squares formula for slope when right-hand side variable is a dummy, Cov(Zi,Yi) Cov(Zi,Yi)/V ar(Zi) E[Yi|Zi =1]− E[Yi|Zi =0] = = Cov(Zi,Di) Cov(Zi,Di)/V ar(Zi) E[Di|Zi =1]− E[Di|Zi =0] Plugging in Yi = αi + βiDi, and using mean independence of αi we find Cov(Zi,Yi) E[αi|Zi =1]− E[αi|Zi =0]+E[βiDi|Zi =1]− E[βiDi|Zi =0 = Cov(Zi,Di) E[Di|Zi =1]− E[Di|Zi =0] E[βiDi|Zi =1]− E[βiDi|Zi =0] = . E[Di|Zi =1]− E[Di|Zi =0] In