We consider estimation of policy relevant treatment effects in a data-rich environment where there may be many more control variables available than there are observations. In addition to allowing many control variables, the setting we consider allows heterogeneous treatment effects, endeogenous receipt of treatment, and function-valued outcomes. To make information inference possible, we assume that reduced form predictive relationships are approximately sparse. That is, we require that the relationsihp between the covariates and the outcome, treatment status, and instrument status can be captured up to a small approximation error using a small number of controls whose identities are unknown to the researcher. This condition allows estimation and inference for a wide variety of treatment parameters to process after selection of an appropriate set of control variables formed by selecting controls separately for each reduced form relationship and then appropriately combining this set of reduced form predictive models and associated selected controls. We provide conditions under which post-selection inferences is uniformly valid across a wide-range of models and show that a key condition underlying uniform validity of post-selection inference allowing for imperfect model selection is the use of approximately unbiased estimating equations. We illustrate the use of the proposed treatment effect estimation methods with an application to estimating the effect of 401(k) participation on accumulated assets.
