Part 2: Monte Carlo Exercises

Consider the following potential outcomes model: Y0 =2+U0

Y1 =3+U1 V = 1 + UV

with (U0, U1, UV ) N(0, ), and corresponding to the following variance- covariance elements: V (U0) = 1, V (U1) = 1, V (UV ) = 1, Cov(U0, U1) = 0.5, Cov(U0, UV ) = 0.3, and Cov(U1, UV ) = 0.7.

a) For this model, compute the ATE.

b) Write computer code to simulate this model for 10,000 draws from the

(U0, U1, UV ) distribution.

c)AssumeD=1ifV 0,andD=0ifV <0. Giventhismodel for treated and control groups, using your simulation draws, compute the ATE, ATET, and ATEU, the corresponding OLS estimates for each of these parameters, and the exact bias. Report this in one Table. (Exact bias is the estimate minus the truth)

d) Continuing c), plot the distribution of treatment effects (), and the distribution of treatment effects for the treated and untreated (3 separate graphs).

e) Now assume D is randomly assigned. For this model, compute the OLSestimates and exact bias for each of the ATE, ATET, and ATEU parameters. Include these results in the same table as in c).

f) Using the random assignment model for D, compute QT E(q) at 5 quan- tiles: q = {0.1, 0.25, 0.5, 0.75, 0.9}. In a Table, report these QTE quantiles, the corresponding treatment effect quantiles (q), and the exact bias.