Quantitative Economics: Mar, 2013, Volume 4, Issue 1

Modeling structural equations with endogenous regressors and heterogeneity through derivative constraints

Tomás Rau

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.

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