Estimation of nonseparable models with censored dependent variables and endogenous regressors

ABSTRACT

In this article we develop a nonparametric estimator for the local average response of a censored dependent variable to endogenous regressors in a nonseparable model where the unobservable error term is not restricted to be scalar and where the nonseparable function need not be monotone in the unobservables. We formalize the identification argument put forward in Altonji, Ichimura, and Otsu (2012), construct a nonparametric estimator, characterize its asymptotic property, and conduct a Monte Carlo investigation to study its small sample properties. Identification is constructive and is achieved through a control function approach. We show that the estimator is consistent and asymptotically normally distributed. The Monte Carlo results are encouraging.

KEYWORDS:

• Average derivatives
• censored dependent variables
• endogeneity
• nonparametric estimation
• nonseparable models

JEL CLASSIFICATION:

• C14
• C24
• C26

Acknowledgments

The authors would like to thank Javier Hidalgo for helpful comments.

Notes

¹It is possible to allow both L(⋅) and H(⋅) to depend on additional observed variables, for example $L\left(\tilde{X}\right)$ and $H\left(\tilde{X}\right)$ where $\tilde{X}$ contains X as a subvector, without affecting the proceeding results. We restrict attention only to X for ease of exposition.

²In this article, we employ the Nadaraya–Watson kernel estimator to construct $\widehat{\beta }\left(x\right)$ because it simplifies the theoretical analysis below. It is also possible to use the formula in (6) and estimate the right-hand side by local linear or polynomial estimators as in AIO. It is known that local polynomial fitting has some desirable properties, such as an absence of boundary effects and minimax efficiency (see, Section 3.2 of Fan and Gijbels, 1996). On the other hand, to estimate the conditional probabilities G_M, G_H, and G_L, local polynomial estimators are not constrained to lie between 0 and 1 (Hall, Wolff, and Yao, 1999). Furthermore, the formula in (6) involves the conditional density dP(v|X = x,I_M(X,U) = 1), and its local polynomial fitting may require an additional bandwidth parameter for the dependent variable (Fan, Yao, and Tong, 1996). A full comparison of different estimation methods is beyond the scope of this article.