Identification and Estimation of Heterogeneous Treatment Effects under Non-compliance or Non-ignorable assignment

We provide sufficient conditions for the identification of the heterogeneous treatment effects, defined as the conditional expectation for the differences of potential outcomes given the untreated outcome, under the nonignorable treatment condition and availability of the information on the marginal distribution of the untreated outcome. These functions are useful both to identify the average treatment effects (ATE) and to determine the treatment assignment policy. The identification holds in the following two general setups prevalent in applied studies: (i) a randomized controlled trial with one-sided noncompliance and (ii) an observational study with nonignorable assignment with the information on the marginal distribution of the untreated outcome or its sample moments. To handle the setup with many integrals and missing values, we propose a (quasi-)Bayesian estimation method for HTE and ATE and examine its properties through simulation studies. We also apply the proposed method to the dataset obtained by the National Job Training Partnership Act Study.