Additive Nonparametric Regression in the Presence of Endogenous Regressors

Abstract

In this article we consider nonparametric estimation of a structural equation model under full additivity constraint. We propose estimators for both the conditional mean and gradient which are consistent, asymptotically normal, oracle efficient, and free from the curse of dimensionality. Monte Carlo simulations support the asymptotic developments. We employ a partially linear extension of our model to study the relationship between child care and cognitive outcomes. Some of our (average) results are consistent with the literature (e.g., negative returns to child care when mothers have higher levels of education). However, as our estimators allow for heterogeneity both across and within groups, we are able to contradict many findings in the literature (e.g., we do not find any significant differences in returns between boys and girls or for formal versus informal child care). Supplementary materials for this article are available online.

KEY WORDS:

• Additive regression
• Endogeneity
• Generated regressors
• Oracle estimation
• Structural equation

SUPPLEMENTARY MATERIALS

The supplementary appendix gives the proofs of Lemmas B.1–B.5 in Appendix B.

ACKNOWLEDGMENTS

The authors thank two anonymous referees, the Joint Editors Rong Chen, Peter Brummund, David Jacho-Chavez, Chris Parmeter, and Anton Schick for useful comments and suggestions. They also thank participants in talks given at the State University of New York at Albany, the University of Alabama, the University of North Carolina at Greensboro, New York Camp Econometrics (Bolton, NY), the annual meeting of the Midwest Econometric Group (Bloomington, IN), and the Western Economic Association International annual conference (Seattle, WA). L. Su acknowledges support from the Singapore Ministry of Education for Academic Research Fund under grant number MOE2012-T2-2-021. The R code used in the article can be obtained from the authors upon request.