# **Appendix C:** Estimation of Treatment Effects on Children

This appendix provides an overview of the empirical issues involved in estimating treatment effects on child outcomes when the intervention under investigation targets prospective mothers and starts prior to their pregnancies. As the intervention may have altered the composition of children subsequently observed, we suggest a simple sequence of assumptions and steps in an attempt to move from the total reduced-form effect of the intervention to a more standard causal effect on the children actually born.

The natural experimental literature has recognized that maternal selection and differential mortality represent plausible causal pathways when analyzing child outcomes. Using data from the US, Buckles and Hungerman (2013) show that nearly half of the well-documented effects of season of birth on later-life outcomes can be explained by variation in the types of women giving birth across seasons. Aaronson, Lange and Mazumder (2014) show extensive margin selection also contributes to the quantity-quality tradeoff for children. This phenomenon is well documented in the developed world, and yet Currie and Vogl (2012) suggest that these extensive margin effects are likely to be more pronounced in the developing world where differential mortality, as well as differential fertility, is an operative channel. It has now become standard in the natural experiments literature to test for the presence of selection effects when studying child outcomes (see, for example, Almond 2006; Black et al. 2014; Adhvaryu et al. 2016). In many cases in this literature, the obvious selection effect suggests that the simple casual effects are lower bounds, but we present a context in which selection and treatment effects go in the same direction. It is therefore critical to attempt to control for the selection mechanism as a mediator, in order to isolate how much of the observed effect may have a simple causal interpretation.

To use counterfactual outcomes to represent impacts on subsequent children, we must define the universe as being the _potential children_ : all those who might exist under either the treated or control state.<sup>38</sup> Outcome measured for potential child _i_ at time _t_ is<sup>_Y_</sup> _it_<sup>. This outcome is</sup> only observed if the child is born and survives, a binary outcome denoted by _Sit_  1. The probability that woman _i_ has a surviving child at time _t_ , as a function of her baseline characteristics, can be written as _Sit_  _S_ ( _X i_ ) , where _X i_ is a set of pre-treatment maternal characteristics. For children who are born, we observe _Yit_  _Yit_ ( _xit_ , _ait_ ; _X i_ , _Ait_ , _Zit_ | _Sit_  1) where _Ait_ is the age of the mother at the time at which child data is collected,<sup>_x_</sup> _it_<sup>are child-level</sup> determinants of the outcome,<sup>_a_</sup> _it_<sup>is child age, and</sup> _Zit_ gives attributes of the father.

Critically, in a study tracking potential mothers, _X i_ is observed for all respondents (regardless of whether or not they had children) and hence can be used to predict fertility within the universe of these potential mothers. Controlling for selection into motherhood thus resembles standard attrition adjustment, and requires a (weaker) selection on observables assumption. Controlling for factors that are observed only among extant children, however, must be done on the intensive margin, and so resembles mediation analysis as in Baron and Kenny (1986), which

> 38 This study tracks female respondents and their descendants, so it is concerned with the potential children of a fixed set of women. We do not attempt to capture all potential children born to the fathers in this study, and to do so would have needed to specify a group of males at baseline and tracked them. The control for father characteristics in this context therefore sits behind a layer of female selection, and so the assumptions needed to control for father type are stronger than those needed to control for mother type.