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NONPARAMETRIC IDENTIFICATION USING INSTRUMENTAL VARIABLES: SUFFICIENT CONDITIONS FOR COMPLETENESS

Published online by Cambridge University Press:  19 June 2017

Yingyao Hu  and
Ji-Liang Shiu
Yingyao Hu*
Affiliation:
Johns Hopkins University
Ji-Liang Shiu
Affiliation:
Jinan University
*
*Address correspondence to Yingyao Hu, Department of Economics, Johns Hopkins University, 3400 N. Charles Street, Baltimore, MD 21218, USA; e-mail: yhu@jhu.edu.
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Abstract

This paper provides sufficient conditions for the nonparametric identification of the regression function $m\left( \cdot \right)$ 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., $f\left( {x|z} \right)$. We show that (1) if the deviation of the conditional density $f\left( {x|{z_k}} \right)$ from a known complete sequence of functions is less than a sequence of values determined by the complete sequence in some distinct sequence $\left\{ {{z_k}:k = 1,2,3, \ldots } \right\}$ converging to ${z_0}$, then $f\left( {x|z} \right)$ itself is complete, and (2) if the conditional density $f\left( {x|z} \right)$ can form a linearly independent sequence $\{ f( \cdot |{z_k}):k = 1,2, \ldots \}$ in some distinct sequence $\left\{ {{z_k}:k = 1,2,3, \ldots } \right\}$ converging to ${z_0}$ and its relative deviation from a known complete sequence of functions under some norm is finite then $f\left( {x|z} \right)$ itself is complete. We use these general results to provide specific sufficient conditions for completeness in three different specifications of the relationship between the endogenous regressor x and the instrumental variable $z.$

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Type
ARTICLES
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Econometric Theory , Volume 34 , Issue 3 , June 2018 , pp. 659 - 693
Copyright
Copyright © Cambridge University Press 2017 

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Footnotes

We thank Peter C.B. Phillips, two associate editors, and two anonymous referees for their time and helpful comments. We also thank the participants at the International Economic Association 16th World Congress, Beijing, China, and the 2013 Annual Meeting of American Economic Association, San Diego, USA, for various suggestions regarding this paper. The usual disclaimer applies.

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