Answer Happy

Throughout the questions below, we consider the following set-up. Suppose that for each individual i, the random variables Y (1) and Y{(O) represent potential outcomes for treated state (state 1) and control state (state 0). The variable D; takes values from {1,0} indicating the treatment status of individual i and Xi is a vector of observed covariates. The econome- trician observes the i.id. random vectors {(Yị,Di,X;)}}=1, where Y; =D;Y{(1)+(1 – D;)Y[0). For simplicity, we will assume that Y{(1) and Y{() take values in [0,1]. (1) We make the following assumption: E[Y/OD; = 1] = E[Y{(0)|D; = 0). (a) Show that the average treatment effect on the treated: = AT T = E[Y (1)– YOD; = 1] is identified from the distribution of (Y;,D;). (b) Propose a consistent estimator of AT T based on the identification result in (a). (You need to prove its consistency.) ?