On nonparametric additive error models with discrete regressors

Abstract

This thesis contributes to the literature on nonparametric additive error models with discrete explanatory variables. Although nonparametric methods have become very popular in recent decades, research on the impact of the discreteness of regressors is sparse. Main interest is in an unknown nonparametric conditional mean function in the presence of endogenous explanatory variables. Under endogeneity, the identifying power of the model depends on the number of support points of the discrete instrument relative to that of the regressor. Under non-parametric identification failure, we show that some linear functionals of the conditional mean function are point-identified, while some are completely unconstrained. A test for point identification is suggested. Observing that the simple nonparametric model can be interpreted as a linear regression, new approaches to testing for exogeneity of the regressor(s) are proposed. For the point-identified case, the test is an adapted version of the familiar Durbin-Wu-Hausman approach. This extends the work of Blundell and Horowitz (2007) to the case of discrete regressors and instruments. For the partially identified case, the Durbin-Wu-Hausman approach is not available, and the test statistic is derived from a constrained minimization problem. In this second case, the asymptotic null distribution is non-standard, and a simple device is suggested to compute the critical values in practical applications. Both tests are shown to be consistent, and a simulation study reveals that both have satisfactory finite-sample properties. The practicability of the suggested testing procedures is illustrated in applications to the modelling of returns to education.

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ORCID for Grant Hillier: orcid.org/0000-0003-3261-5766

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