Modeling structural equations with endogenous regressors and heterogeneity through derivative constraints

Abstract

In this paper, I present a general modeling framework for nonparametric models with endogenous regressors and heterogeneity. I show that many existing models in the literature can be derived from a structural equation with unobserved het- erogeneity by imposing constancy assumptions on the first and second deriva- tives. I consider a less restrictive model that imposes constancy assumptions on the second partial derivative of the structural equation. Assuming the existence of suitable instrumental variables, I provide identification results and show that the model can be estimated using a generalized control function approach. I consider an application to the estimation of the returns to education in Chile, exploiting variation across regions and cohorts in educational infrastructure and compul- sory schooling laws. Using penalized spline functions to approximate the com- ponents of the average structural function, I find that the local average returns to schooling are highly nonlinear and typically underestimated by flexible models that ignore the endogeneity of schooling. I also find evidence of credential effects for high school and college graduates, and limited evidence of comparative ad- vantage bias in the returns to certain levels of education. Keywords. Nonparametric regression, endogenous regressors, control function, endogenous treatment, returns to schooling. JEL classification. C14, C21, C31, J31.