Bayesian treatment effects
We continue with the selection setting discussed in the previous three chapters and apply Bayesian analysis. Bayesian augmentation of the kind proposed by Albert and Chib  in the probit setting (see chapter 5) can be extended to selection analysis of treatment effects (Li, Poirier, and Tobias ). An advantage of the approach is treatment effect distributions can be identified by bounding the unidentified parameter. As counterfactuals are not observed, the correlation between outcome errors is unidentified. However, Poirier and Tobias  and Li, Poirier, and Tobias  suggest using the positive definiteness of the variancecovariance matrix (for the selection equation error and the outcome equations’ errors) to bound the unidentified parameter.
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