Consider a partially linear approximation to equation,
where 9 represents the reduced-form causal parameter in equation, т(•) is a general mapping, and the residual e is meant to capture other factors potentially affecting infant health (e.g., maternal age).
Identifying the effect of education on infant health, 9, requires solving two difficult problems. The first problem is the endogeneity of schooling. For example, education at motherhood may be related to family background, which may be related to infant health through non-schooling mechanisms. The second problem is sample selection. Education may affect a woman’s decision to have children, leading to a selected sample of those observed giving birth. The standard regression discontinuity approach will, under continuity assumptions, circumvent the endogeneity problem. However, except in unusual circumstances, it will not circumvent the sample selection problem. Problems with both endogeneity and sample selection lead to two endogenous regressors, necessitating two instruments instead of one.
To describe our solution to these two identification problems, let us for the moment ignore sample selection issues and assume that mothers are a random sample of women. Empirically, we will show that schooling is a discontinuous function of day of birth. Consider the model
where R is an individual’s day of birth relative to the school entry date for the state in which the individual begins school, n(^) is a continuous function, D = 1(R > 0), and v is mean zero given R (cf., Porter 2003). For example, R = 0 for an individual born on the school entry date, and R = 5 for an individual born 5 days after the school entry date.12 In this notation, the parameter в measures the discontinuity in expected schooling at the school entry date, or the vertical gap at r = 0 in the conditional expectation of S given R = r. amoxil online
Equation may be thought of as the “first stage” equation in a simultaneous equation
where a = 9в and u is an error term.13 This equation may be derived by substituting equation into equation and taking the conditional expectation of Y given R.
Discontinuity in the conditional expectation of infant health at the school entry date iden tifies a+limrj_0 E[т(W)|R = r]—limrf0 E[т(W)|R = r]+limr^0 E[e|R = r]—limrf0 E[e|R = r]. This will be different from a generally. If the conditional distribution of W given R = r is continuous in r, then limr j_0 E[t(W)|R = r] — limr|0 E[т(W)|R = r] = 0.14 If the conditional distribution of e given R = r is continuous in r, then limrj_0 E[e|R = r] — limr|0 E[e|R = r] = 0. If both these conditions hold, then a is identified from the infant health discontinuity. In words, we assume (i) continuity of background characteristics in day of birth, and (ii) continuity of unobservable characteristics in day of birth. The first assumption is testable if background characteristics are observed. We demonstrate continuity of background characteristics in Section V, below. The second assumption is plausible on prior grounds and is partially corroborated by failure to reject tests of the first assumption.