We consider the problem of extrapolating treatment effects across
heterogeneous populations (``sites"/``contexts"). We consider an idealized
scenario in which the researcher observes cross-sectional data for a large
number of units across several ``experimental" sites in which an intervention
has already been implemented to a new ``target" site for which a baseline
survey of unit-specific, pre-treatment outcomes and relevant attributes is
available. We propose a transfer estimator that exploits cross-sectional
variation between individuals and sites to predict treatment outcomes using
baseline outcome data for the target location. We consider the problem of
obtaining a predictor of conditional average treatment effects at the target
site that is MSE optimal within a certain class and subject to data
constraints. Our approach is design-based in the sense that the performance of
the predictor is evaluated given the specific, finite selection of experimental
and target sites. Our approach is nonparametric, and our formal results concern
the construction of an optimal basis of predictors as well as convergence rates
for the estimated conditional average treatment effect relative to the
constrained-optimal population predictor for the target site. We illustrate our
approach using a combined data set of five multi-site randomized controlled
trials (RCTs) to evaluate the effect of conditional cash transfers on school
attendance