A.2 Derivation of the Probability Limit of the Wald Estimator Using
D × P as an IV
We begin by computing the following conditional expectations:
E(yij |Di = 1, Pij = 1) = α + β + γ + δE (Tij |Di = 1, Pij = 1)
+ η + E(vi|Di = 1)
E(yij |Di = 1, Pij = 0) = α + β + δE(Tij |Di = 1, Pij = 0) + E(vi |Di = 1)
E(yij |Di = 0,Pij = 1) = α + γ + 6E(Tij |Di = 0,Pj = 1) + E(vi∣Di = 0)
E(yij|Di = 0, Pij = 0) = α + δE(Tij |Di = 0, Pij = 0) + E(vi|Di = 0)
We will also need to compute:
E(Tij|Di = 1|Pij = 1)
E (Tij |Di = 1|Pij = 0)
E(Tij|Di = 0|Pij = 1)
E (Tij |Di = 0|Pij = 0)
We can now construct difference-in-difference estimators for the effect of D and P on consump-
tion, as well as on the dose variable:
∆y∣D,P = [E(yij|Di = 1,Pij = 1) - E(yij|Di = 1,Pij = 0)]
- [E(yij |Di = 0, Pij = 1) - E(yij |Di = 0, Pij = 0)]
∆T∣D,P = [E(Tij|Di = 1, Pij = 1) - E(Tij|Di = 1, Pij = 0)]
- [E(Tij |Di = 0, Pij = 1) - E(Tij |Di = 0, Pij = 0)]
Taking the ratio of these two estimators produces a Wald estimator with probability limit,
~
διv
δv∖d,p
∆T |D,P
δ+
η
∆T |D,P
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