Evaluating the Impact of Health Programmes

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(y_ij |D_i = 1, P_ij = 0)  =  α + β + δE(T_ij |D_i = 1, P_ij = 0) + E(v_i |D_i = 1)

E(yij |Di = 0,Pij = 1)  =  α + γ + 6E(T_ij |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 (T_ij |D_i = 1|P_ij = 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(T_ij |D_i = 0, P_ij = 1) - E(T_ij |D_i = 0, P_ij = 0)]

Taking the ratio of these two estimators produces a Wald estimator with probability limit,

References

Agodini, R., and M. Dynarski (2004): “Are Experiments the Only Option? A Look at
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Alderman, H., J. Behrman, V. Lavy, and R. Menon (1997): “Child Nutrition, Child
Health, And School Enrollment: A Longitudinal Analysis,” World Bank Policy Research
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Angrist, J. (1990): “Lifetime earnings and the vietnam era draft lottery: evidence from social
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Angrist, J., and J. Hahn (2004): “When to Control for Covariates? Panel Asymptotics for
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Angrist, J. D., and A. B. Krueger (1999): “Empirical Strategies in Labor Economics,”
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Attanasio, O., E. Battistin, E. Fitzsimons, A. Mesnard, and M. Vera-Hernandez
(2005): “How Effective Are Conditional Cash Transfers Evidence From Colombia,” Institute
For Fiscal Studies, Briefing Note No. 54.

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