  Causal inference in a program evaluation setting faces the problem of
external validity when the treatment effect in the target population is
different from the treatment effect identified from the population of which the
sample is representative. This paper focuses on a situation where such
discrepancy arises by a stratified sampling design based on the individual
treatment status and other characteristics. In such settings, the design
probability is known from the sampling design but the target population depends
on the underlying population share vector which is often unknown, and except
for special cases, the treatment effect parameters are not identified. In this
paper, we propose a method of constructing confidence sets that are valid for a
given range of population shares. When a benchmark population share vector and
a corresponding estimator of a treatment effect parameter are given, we develop
a method to discover the scope of external validity with familywise error rate
control. Finally, we derive an optimal sampling design which minimizes the
semiparametric efficiency bound given a population share associated with a
target population. We provide Monte Carlo simulation results and an empirical
application to demonstrate the usefulness of our proposals.
