# Iterative Estimation of Nonparametric Regressions with Continuous Endogenous Variables and Discrete Instruments

## TL;DR

This paper introduces an iterative method for estimating nonparametric regression models with continuous endogenous variables using only discrete instruments, addressing a challenging nonlinear integral equation problem.

## Contribution

It proposes a simple iterative estimation procedure for such models and analyzes its asymptotic properties, especially with binary instruments.

## Key findings

- The method effectively estimates the regression function.
- Asymptotic properties are established.
- Application to returns to education demonstrates practical utility.

## Abstract

We consider a nonparametric regression model with continuous endogenous independent variables when only discrete instruments are available that are independent of the error term. Although this framework is very relevant for applied research, its implementation is challenging, as the regression function becomes the solution to a nonlinear integral equation. We propose a simple iterative procedure to estimate such models and showcase some of its asymptotic properties. In a simulation experiment, we detail its implementation in the case when the instrumental variable is binary. We conclude with an empirical application to returns to education.