Nonparametric Identification and Estimation with Independent, Discrete Instruments

Abstract

In a nonparametric instrumental regression model, we strengthen the conventional moment independence assumption towards full statistical independence between instrument and error term. This allows us to prove identification results and develop estimators for a structural function of interest when the instrument is discrete, and in particular binary. When the regressor of interest is also discrete with more mass points than the instrument, we state straightforward conditions under which the structural function is partially identified, and give modified assumptions which imply point identification. These stronger assumptions are shown to hold outside of a small set of conditional moments of the error term. Estimators for the identified set are given when the structural function is either partially or point identified. When the regressor is continuously distributed, we prove that if the instrument induces a sufficiently rich variation in the joint distribution of the regressor and error term then point identification of the structural function is still possible. This approach is relatively tractable, and under some standard conditions we demonstrate that our point identifying assumption holds on a topologically generic set of density functions for the joint distribution of regressor, error, and instrument. Our method also applies to a well-known nonparametric quantile regression framework, and we are able to state analogous point identification results in that context.