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(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(viDi = 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:

yD,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)]

TD,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


δvd,p

T |D,P


δ+


η
T |D,P


References

Agodini, R., and M. Dynarski (2004): “Are Experiments the Only Option? A Look at
Dropout Prevention Programs,”
Review of Economics and Statistics, 86(1), 180-194.

Alderman, H., J. Behrman, V. Lavy, and R. Menon (1997): “Child Nutrition, Child
Health, And School Enrollment: A Longitudinal Analysis,” World Bank Policy Research
Working Paper No. 1700.

Angrist, J. (1990): “Lifetime earnings and the vietnam era draft lottery: evidence from social
security administrative records,”
American Economic Review, 80, 313-335.

Angrist, J., and J. Hahn (2004): “When to Control for Covariates? Panel Asymptotics for
Estimates of Treatment Effects,”
Review of Economics and Statistics, 86(1), 58-72.

Angrist, J. D., and A. B. Krueger (1999): “Empirical Strategies in Labor Economics,”
in
Handbook of Labor Economics, ed. by O. Ashenfelter, and D. Card. Elsevier, Amsterdam,
North Holland.

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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