Lecture Notes on Propensity Score Matching

1 Lecture Notes on Propensity Score MatchingJin-Lung LinThis Lecture note is intended solely for teaching. Some parts of the Notes are taken from varioussources listed below and no originality is IntroductionA specific question: Is takingmath lessons after schoolhelpful in improving Score ? ? ( : (2008) ? ,41,97-148)A first attempt to answer this question would be computing the difference between the scores ofthose who took the after school lessons and those who don so doing, one assumes that all the students are similar and are randomly selected to take afterschool reality, these are two different groups with different characteristics that would affect the learningand scoring ability.

2 In other words, there exist sample selection bias that seriously affects thevalidity of the basic concepts:Treatment = (D= 1); (D= 0)Y(1) : ;Y(0) : ATE (Average Treatment Effect) , ? ATT (Average Treatment Effect on the Treated): , , ? ATU (Average Treatment Effect on the Untreated): , , ?General questions: Is the treatment (for whatever) effective?Fact: some people receive question:What would have happened to those who, in fact, did receive treatment,if they had not received treatment (or the converse)?

3 In short, participants differ from nonparticipants and creates theselection bias. To minimize the1bias, we need to find a large group of nonparticipants those individuals who are similar to theparticipants in all relevant treatment 1: The counterfactual frameworkPotential outcomesGroupY(1)Y(0)Treatment effect (D=1)ObservableE(Y(1)|D= 1)CounterfactualE(Y(0)|D= 1)Control group (D=0)CounterfactualE(Y(1)|D= 0)ObservableE(Y(0)|D= 0)2 Matching basicsRoy-Rubin modelmain pillars: individual, treatment and potential outcome.

4 For binary treatment, treatment indicatorDi= 1if individualireceives treatment0if individualidoes not receive treatmentYi(Di)is the potential outcome for individuali,i= 1, ,N. Treatment effect i=Yi(1) Yi(0)Only one ofYi(1),Yi(0)is observed and the other unobservable outcome is calledcounterfactualoutcome. It is impossible to estimate ifor eachiand we could only estimate the average treatmenteffect. i=Yi(1) Yi(0) AT E=E( ) =E(Y(1) Y(0)) AT T=E( |D= 1) =E(Y(1)|D= 1) E(Y(0)|D= 1) AT U=E( |D= 0) =E(Y(1)|D= 0) E(Y(0)|D= 0)Population average treatment effect (ATE), AT E, answers the questionWhat is the expected effectof the outcome if individuals in the population were randomly assigned to treatment?

5 AT Eis notinteresting because it includes the effects on persons not the other hand, AT T, average effect of the treated is defined as the difference between ex-pected outcome values with and without treatment for those who actually participate in determines the realized gross gain from the programme and can be compared with its (Y(0)|D= 1)is counterfactual (unobserved) andE(Y(0)|D= 0)is usually not a good exists selection (Y(1)|D= 1) E(Y(0)|D= 0) = AT T+E(Y(0)|D= 1) E(Y(0)|D= 0) AT Tis only identified if the selection bias,E(Y(0)|D= 1) E(Y(0)|D= 0) = 03 Furthermore, letP(D= 1) = , then AT E=E( ) =E(Y(1) Y(0))= [ E(Y(1)|D= 1) + (1 )E(Y(1)|D= 0)] [ E(Y(0)|D= 1) + (1 )E(Y(0)|D= 0)]= [E(Y(1)|D= 1) E(Y(0)|D= 1)] + (1 )[E(Y(1)|D= 0) E(Y(0)|D= 0)]= E( |D= 1) + (1 )E( |D= 0)= ATT+ (1 )ATUR egards to previous example, ?

6 Yes, E[Y(1)|D= 0] =E[Y(1)|D= 1],No,E[Y(1)|D= 0] E[Y(1)|D= 1]is the baseline bias. treatment ?Yes, E[Y(0)|D= 1] =E[Y(0)|D= 0]No,E[Y(0)|D= 1] E[Y(0)|D= 0]is the differential effect biasThen,E[Y(1)|D= 1] E[Y(0)|D= 0] =E( ) + [E(Y(0)|D= 1) E(Y(0)|D= 0)]+ (1 )[E( |D= 1) E( |D= 0)]Naive Estimate = average causal effect + baseline bias + differential effect biasFundamental assumptions: Unconfoundedness and Common SupportAssumption 1:Unconfoundedness:Y(0),Y(1) D|XGiven a set of observable covariates,X, which is not affected by treatment, potential outcomesare independent of treatment assignment.

7 This implies that all variables that influence treatmentassignment and potential outcomes simultaneously have to be observed by the researchers. Un-confoundedness is also called selection on observable or conditional 2:Overlap:0< P(D= 1|X)<1 Persons with the sameXvalues have a positive probability of being participants and 3:Unconfoundedness for controls:Y(0) D|XAssumption 4:Weak overlap:P(D= 1|X)< : as the dimension ofXincreases, the unconfoundedness is difficult to hold. Rosenbaumand Rubin (1983) suggested using balancing scoreb(X).

8 The Propensity Score ,P(D= 1|X) =P(X), the probability for an individual to participate in a treatment given his observed covariatesX, is one balancing given the Propensity Score :Y(0),Y(1) D|P(X)Estimation strategy P SMAT T=EP(X)|D=1(E(Y(1)|D= 1,P(X)) E(Y(0)|D= 1,P(X))PSM estimator is the mean difference in outcomes over the common support, appropriately weightedby the Propensity Score distribution of Implementation of Propensity Score Estimating the Propensity scoreTwo choices:1. Model to be used for the estimation2.)

9 Variables to be included in this modelModel choice - Binary Treatment logit model probit model linear probability modelModel choice - Multiple treatments multinominal probit model multinominal logit model Series of binomial model linear probability modelvariable choice Omitting important variables can seriously increase bias in the Only variables that influence simultaneously the participation decision and the outcome vari-able should be included. Only variables unaffected by participation should be included in the model.

10 Participants and nonparticipants should stem from the same source (dataset). Should avoid including too many variables as execrates the support problem and increasesthe Steps of Implementation PSMStep 0: Decide between PSM and CVM (covariate Matching )Step 1: Propensity Score estimationStep 2: Choose Matching algorithmStep 3: Check overlap/common supportStep 4: Matching quality/effect estimationStep 5: sensitivity Matching algorithmDistance measures1. Exact:Mij= 0ifXi=Xj ifXi6=Xj2. Mahalanobis:Mij= (Xi Xj) 1(Xi Xj)where is the covariance matrix ofXin the full control Propensity Score :Mij=|ei ej|4.