Semiparametric modeling and estimation of instrumental variable models

We apply Bayesian methods to a model involving a binary nonrandom treatment intake variable and an instrumental variable in which the functional forms of some of the covariates in both the treatment intake and outcome distributions are unknown. Continuous and binary response variables are considered. Under the assumption that the functional form is additive in the covariates, we develop efficient Markov chain Monte Carlo-based approaches for summarizing the posterior distribution and for comparing various alternative models via marginal likelihoods and Bayes factors. We show in a simulation experiment that the methods are capable of recovering the unknown functions and are sensitive neither to the sample size nor to the degree of confounding as measured by the correlation between the errors in the treatment and response equations. In the binary response case, however, estimation of the average treatment effect requires larger sample sizes, especially when the degree of confounding is high. The methods are applied to an example dealing with the effect on wages of more than 12 years of education.