If the effect of a treatment is heterogeneous and the decision to take-up is based on individual characteristics unobservable to the econometrician, then we are in the presence of "essential heterogeneity". This concept was coined by Heckman and he has been working in the issue for more than 2 decades, producing various papers with co-authors (including one of my professors from Chile), that are generally obscure and not easy to follow for applied researchers.
Martin Ravallion has a new paper that can be a good starting point to understand the problem. He shows that under essential heterogeneity the use of a randomized treatment assignment as an IV for take-up can be worse than the naive OLS specification in estimating the causal effect of a program. As a nice extension, he allows the unobserved heterogeneous characteristics to also affect the covariates.
Martin Ravallion has a new paper that can be a good starting point to understand the problem. He shows that under essential heterogeneity the use of a randomized treatment assignment as an IV for take-up can be worse than the naive OLS specification in estimating the causal effect of a program. As a nice extension, he allows the unobserved heterogeneous characteristics to also affect the covariates.
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