SELECTION BIAS: Index Sufficient Methods 3

Index sufficiency is only a necessary condition for applying the classical index sufficient selection model in a nonparametric or semiparametric setting. As noted by Heckman (1990a), it is also necessary to know a point or interval of P where E(Uq \ P{X)} D = 0) = 0. Unless this condition is satisfied, it is not possible to use the index-sufficient selection model to construct the required counterfactual. Thus in order to implement this method, it is necessary (a) that such a point or interval exists and (b) that it is possible to discover it.
The traditional selection-correction method parameterizes the bias function B(P(Z)) and eliminates bias by estimating B(P(Z)) along with the other parameters of the model.25 Heckman and Robb (1985, 1986) term the dependence between Uq and D operating through the v “selection on unobservables” while the dependence between Uq and D operating through dependence between Z and Uq is termed “selection on observables”. In their framework, the method of matching assumes selection on observables, because conditioning on Z controls the dependence between D and U0, producing a counterpart to (4) for the residuals: E(Uq | Z, D = 1) = E(U0 \ Z,D = 0). When selection is on unobservables, it is impossible to condition on v and eliminate the selection bias. Thus the choice of an appropriate econometric model critically depends on the properties of the data on which it is applied.
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The classical before-after estimator compares the outcomes of participants after they participate in the program with their outcomes before they participate. With the difference-in-differences estimator, common time and age trends are eliminated by subtracting the before-after change in nonparticipant outcomes from the before-after change for participant outcomes. This method can be generalized to include regressors.26 The simplest application of the method does not condition on X and forms simple averages over the treatment and comparison groups.