Econometrics

a) Suppose that we are interested in the causal effea of a binary treatment D on an outcome Y.Assume that s(y°lo = 1,10 = E(Y°ID = 0.1!) where V4 d = 1.0 are the potential outcomes max is aset ot‘ observed pre-treatment covamtes Let 1? the OLS estimator offi in the following regressionmodel:Y;= a+pog+x§6+£hShow that ph’mw) = [9(1″1 – Y‘olD = 1) if either one crboth of those two conditions are satisfied:1. 503‘ – yle = 1.2:) = 80:1 – Y,°lo = 1),2. Y.” – Y.-° = 5‘” l D,- = 1 (Effect homogeneity among the treated).(Hint use the observational rule Y; = 0;?) + (1 – DOV? and the plim ot‘ the OLS coefficient of a binaryamiable.)b) Rm a small Monte Carlo simulation study in which compare OLS, a simple regession adjustment andinverse probability weighting (running two separate regression for treated and non-treated) in those twoscenarios:(Effect ha’nogmdty) In this scenario the true model isY‘= a+fio;+X.6+8.01′ = 10‘7“” U > 0),where I (~) is the indicator fimoticn.(Effect heterogeneity) In this Scenario the true model isK=d+figog+Z¢8+eeD; = ((Zflr +0 >0),Where 2 is a binary indicator andN(a. b2) in:I – 1 and 0‘ -1fl; ~N(c,d2) ifz‘ = 0 undo,- =10 if D, = 0Choose c as a and only discus what c = a impliesExamples and hintsDraw two random variablesA and B. For example choose A~N(1, 1) and B~N(2,1), then3‘: D:”Z“A+D“(1-ZJ‘B.Note that the true ATT is given by8(A) Pr(D = II?! = 1) Pr(Z = 1) + 8(8) Pr(D = 1IZ = 0) Pr(Z = 0)BUM D = 1)Pr(D – 1)Hint 1: va~N(0.1), then itis easyto compute Pr(D = 1IZ = 1) and Pr(D = llZ = 0) Earn 0;I(Z;rr+ v > 0), e. g , Pr(D =1IZ = 1) = Pr(v > -Z;rrlZ =1).Hint 2: Pr(D = 1) = Pr(D = IIZ = 1) Pr(Z = 1) + Pr(D = 1lZ = 0) Pr(Z = 0).