Y5

5. (LATE aggregation) Suppose Z1, Z2 and Z3 are three different instruments. All of them are dummies. Let B1, B2 and B3 be the LATE using each instrument for a binary treatment D. Always-takers, never- takers and compliers are different across instruments. Now suppose the always-takers under Z, are compliers under Z2, whereas never-takers under Z, are compliers under Z3. Suppose all the three assumptions for LATE theorem hold for every Z;, j - 1, 2, 3. i) Prove that although we don't know who are the compliers, the proportion of compliers under Z, j = 1, 2, 3 is identified by Pr(D = 1|Z, =1) - Pr(D = 1\\Z, = 0) ii) Derive ATE as a function of (81, (2, 83) and conditional probabilities of D given Z1, Z2 or Z3