# Semiparametric estimation of heterogeneous treatment effects under the nonignorable assignment condition

## TL;DR

This paper introduces a semiparametric two-stage least squares estimator for heterogeneous treatment effects, addressing the ill-posed nature of the integral equations involved by using orthogonal series approximation to ensure stability.

## Contribution

The paper develops a novel semiparametric estimator for HTE that stabilizes solutions to ill-posed integral equations using orthogonal series, improving estimation under nonignorable assignment.

## Key findings

- Estimator performs well in simulation experiments.
- Addresses ill-posedness in integral equations for HTE.
- Provides a stable solution to a challenging estimation problem.

## Abstract

We propose a semiparametric two-stage least square estimator for the heterogeneous treatment effects (HTE). HTE is the solution to certain integral equation which belongs to the class of Fredholm integral equations of the first kind, which is known to be ill-posed problem. Naive semi/nonparametric methods do not provide stable solution to such problems. Then we propose to approximate the function of interest by orthogonal series under the constraint which makes the inverse mapping of integral to be continuous and eliminates the ill-posedness. We illustrate the performance of the proposed estimator through simulation experiments.