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I propose a treatment selection model that introduces unobservedheterogeneity in both choice sets and preferences to evaluate the averageeffects of a program offer. I show how to exploit the model structure to defineparameters capturing these effects and then computationally characterize theiridentified sets under instrumental variable variation in choice sets. Iillustrate these tools by analyzing the effects of providing an offer to theHead Start preschool program using data from the Head Start Impact Study. Ifind that such a policy affects a large number of children who take up theoffer, and that they subsequently have positive effects on test scores. Theseeffects arise from children who do not have any preschool as an outside option.A cost-benefit analysis reveals that the earning benefits associated with thetest score gains can be large and outweigh the net costs associated with offertake up.

Identifying the Effects of a Program Offer with an Application to Head Start