Bayesian Nonparametric Instrumental Variable Regression based on Penalized Splines and Dirichlet Process Mixtures

We propose a Bayesian nonparametric instrumental variable approach that allows us to correct for endogeneity bias in regression models where the covariate eff ects enter with unknown functional form. Bias correction relies on a simultaneous equations speci cation with flexible modeling of the joint error distribution implemented via a Dirichlet process mixture prior. Both the structural and instrumental variable equation are specified in terms of additive predictors comprising penalized splines for nonlinear eff ects of continuous covariates. Inference is fully Bayesian, employing efficient Markov Chain Monte Carlo simulation techniques. The resulting posterior samples do not only provide us with point estimates, but allow us to construct simultaneous credible bands for the nonparametric e ffects, including data-driven smoothing parameter selection. In addition, improved robustness properties are achieved due to the flexible error distribution speci fication. Both these features are extremely challenging in the classical framework, making the Bayesian one advantageous. In simulations, we investigate small sample properties and an investigation of the eff ect of class size on student performance in Israel provides an illustration of the proposed approach which is implemented in an R package bayesIV.