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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SELECTION BIAS: Index Sufficient Methods 2

Much applied econometric activity is devoted to eliminating the mean effect of unobservables on estimates of functions like g0 and дг. However, the mean difference in unobservables is an essential component of the definition of the parameter of interest in evaluating social programs.20 In the traditional separable framework, the selection bias that arises from using a nonexperimental comparison group is
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SELECTION BIAS: Index Sufficient Methods

A major advantage of the method of randomized trials over the method of matching in evaluating programs is that randomization works for any choice of X. In the method of matching, there is the same uncertainty about which X to use as there is in the specification of conventional econometric models. Even if one set of X values satisfies condition (A-l), an augmented or reduced version of this set may not. Heckman, Ichimura and Todd (1997; first draft 1993) discuss tests that can be used to determine the appropriate choice of X variables. We discuss this problem in Section 4.3 below. Since nonparametric methods can be used to perform matching, the method does not, in principle, require that arbitrary functional forms be imposed to estimate program impacts.

Index Sufficient Methods and the Classical Econometric Selection Model

The traditional econometric approach to the selection problem adopts a more tightly-specified model relating outcomes to regressors X. This is in the spirit of much econometric work that builds models to estimate a variety of counterfactual states, rather than just the single counterfactual required to estimate the mean impact of treatment on the treated, the parameter of interest in most applications of the methods of matching or random assignment. In the simplest econometric approach, two functions are postulated: Yi = gi{X, U\) and Y0 = go{X, Uq), where Uo and U\ are unobservables. A selection equation is specified to determine which outcome is observed. Separability between X and (Un, b\) is

SELECTION BIAS: The Method of Matching 3

The analysis of Rosenbaum and Rubin (1983) assumes that P(X) is known rather than estimated. They do not present a distribution theory for the pointwise estimator (5) or averaged estimator (6). Heckman, Ichimura and Todd (1997, 1998; first drafts 1993) present the asymptotic distribution theory for the kernel matching estimator for the cases where P is known and where it is estimated.
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