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