Nonparametric Identification Using Instrumental Variables: Sufficient Conditions for Completeness

Econometric Theory

Hu, Yingyao, Ji-Liang Shiu

Abstract

This paper provides sufficient conditions for the nonparametric identification of the regression function m (·) in a regression model with an endogenous regressor x and an instrumental variable z. It has been shown that the identification of the regression function from the conditional expectation of the dependent variable on the instrument relies on the completeness of the distribution of the endogenous regressor conditional on the instrument, i.e., f(x|z). We show that (1) if the relative deviation of the conditional density f(x|zk) from a known complete sequence of function is less than a constance determined by the complete sequence for some distinct sequence {zk : k = 1, 2, 3, ...} converging to z0, then f(x|z) itself is complete, and (2) if the conditional density f(x|z) can form a linearly independent sequence {f(·|zk) : k = 1, 2, ...} for some distinct sequence {zk : k = 1, 2, 3, ...} converging to z0 and its relative deviation from a known complete sequence of function under some norm is finite then f(x|z) itself is complete. We use this general result to provide specific sufficient conditions for completeness in three different specifications of the relationship between the endogenous regressor x and the instrumental variable z.

Keywords: Nonparametric identification; Instrumental variable completeness; Endogeneity 

Read more: https://doi.org/10.1017/S0266466617000251