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.

Comparison groups produced assuming (A-l) is valid differ from the control groups produced by a random experiment in an important way. Randomization equates the distributions of characteristics in the treatment and control groups. Without randomization, the distributions of characteristics in the treatment and comparison groups axe not necessarily equated even if (A-l) is satisfied. The supports of the distributions of X may be different in the two groups and the shapes of the distributions may be different over regions of common support. Because counterparts to participants cannot always be found in the comparison group, estimators based on (A-l) or equation (7) do not necessarily identify treatment impacts for all values of X among program participants, unless the impacts do not depend on X.