  This paper presents a new estimator of the intercept of a linear regression
model in cases where the outcome varaible is observed subject to a selection
rule. The intercept is often in this context of inherent interest; for example,
in a program evaluation context, the difference between the intercepts in
outcome equations for participants and non-participants can be interpreted as
the difference in average outcomes of participants and their counterfactual
average outcomes if they had chosen not to participate. The new estimator can
under mild conditions exhibit a rate of convergence in probability equal to
$n^{-p/(2p+1)}$, where $p\ge 2$ is an integer that indexes the strength of
certain smoothness assumptions. This rate of convergence is shown in this
context to be the optimal rate of convergence for estimation of the intercept
parameter in terms of a minimax criterion. The new estimator, unlike other
proposals in the literature, is under mild conditions consistent and
asymptotically normal with a rate of convergence that is the same regardless of
the degree to which selection depends on unobservables in the outcome equation.
Simulation evidence and an empirical example are included.
