  This paper concerns robust inference on average treatment effects following
model selection. In the selection on observables framework, we show how to
construct confidence intervals based on a doubly-robust estimator that are
robust to model selection errors and prove that they are valid uniformly over a
large class of treatment effect models. The class allows for multivalued
treatments with heterogeneous effects (in observables), general
heteroskedasticity, and selection amongst (possibly) more covariates than
observations. Our estimator attains the semiparametric efficiency bound under
appropriate conditions. Precise conditions are given for any model selector to
yield these results, and we show how to combine data-driven selection with
economic theory. For implementation, we give a specific proposal for selection
based on the group lasso, which is particularly well-suited to treatment
effects data, and derive new results for high-dimensional, sparse multinomial
logistic regression. A simulation study shows our estimator performs very well
in finite samples over a wide range of models. Revisiting the National
Supported Work demonstration data, our method yields accurate estimates and
tight confidence intervals.
