Suppose Z, Z2 and Z3 are three different instruments. All of them are dummies. Let B,...

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Suppose Z₁, Z2 and Z3 are three different instruments. All of them are dummies. Let B₁, B2 and 33 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 Z₁ are compliers under Z2, whereas never-takers under Z₁ are compliers under Z3. Suppose all the three assumptions for LATE theorem hold for every Z₁, j = 1, 2, 3. i) Prove that although we don't know who are the compliers, the proportion of compliers under Zj, j = 1, 2, 3 is identified by Pr(D = 1|Z, = 1) - Pr(D = 1|Z, = 0). ii) Derive ATE as a function of (B1, B2, B3) and conditional probabilities of D given Z₁, Z₂ or Z3. Suppose Z₁, Z2 and Z3 are three different instruments. All of them are dummies. Let B₁, B2 and 33 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 Z₁ are compliers under Z2, whereas never-takers under Z₁ are compliers under Z3. Suppose all the three assumptions for LATE theorem hold for every Z₁, j = 1, 2, 3. i) Prove that although we don't know who are the compliers, the proportion of compliers under Zj, j = 1, 2, 3 is identified by Pr(D = 1|Z, = 1) - Pr(D = 1|Z, = 0). ii) Derive ATE as a function of (B1, B2, B3) and conditional probabilities of D given Z₁, Z₂ or Z3.

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