Bayesian Nonparametric Instrumental Variables Regression Based on Penalized Splines and Dirichlet Process Mixtures

Abstract

We propose a Bayesian nonparametric instrumental variable approach under additive separability that allows us to correct for endogeneity bias in regression models where the covariate effects enter with unknown functional form. Bias correction relies on a simultaneous equations specification 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 effects 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 effects, including data-driven smoothing parameter selection. In addition, improved robustness properties are achieved due to the flexible error distribution specification. Both these features are challenging in the classical framework, making the Bayesian one advantageous. In simulations, we investigate small sample properties and an investigation of the effect of class size on student performance in Israel provides an illustration of the proposed approach which is implemented in an R package bayesIV. Supplementary materials for this article are available online.

KEY WORDS:

• Endogeneity
• Markov chain Monte Carlo methods
• Simultaneous credible bands

ACKNOWLEDGMENTS

This work was partially supported by grant DE2009-0030 from the Spanish Ministry of Science and Innovation, grant MTM2011-28285-C02-01 from the Spanish Ministry of Economy and Competitiveness, and by the AECID (Agencia Española de Cooperación Internacional) grant system. The authors thank Daniel Miles Touya for suggestions with respect to the application.