# Identification of Regression Models with a Misclassified and Endogenous Binary Regressor

## TL;DR

This paper establishes conditions under which regression functions with misclassified and endogenous binary regressors are identifiable, using instrumental variables and covariates, and proposes a mixture-based framework for treatment effect identification.

## Contribution

It provides new identification results for nonparametric regression models with misclassified and endogenous binary regressors, incorporating instrumental variables and covariates, and introduces a mixture-based approach for treatment effects.

## Key findings

- Regression function is nonparametrically identified under specified conditions.
- Instrumental variable corrects endogeneity, covariate corrects misclassification.
- Treatment effects can be identified using the proposed mixture framework.

## Abstract

We study identification in nonparametric regression models with a misclassified and endogenous binary regressor when an instrument is correlated with misclassification error. We show that the regression function is nonparametrically identified if one binary instrument variable and one binary covariate satisfy the following conditions. The instrumental variable corrects endogeneity; the instrumental variable must be correlated with the unobserved true underlying binary variable, must be uncorrelated with the error term in the outcome equation, but is allowed to be correlated with the misclassification error. The covariate corrects misclassification; this variable can be one of the regressors in the outcome equation, must be correlated with the unobserved true underlying binary variable, and must be uncorrelated with the misclassification error. We also propose a mixture-based framework for modeling unobserved heterogeneous treatment effects with a misclassified and endogenous binary regressor and show that treatment effects can be identified if the true treatment effect is related to an observed regressor and another observable variable.