# Identification and Estimation of Categorical Random Coefficient Models

## TL;DR

This paper introduces a linear categorical random coefficient model with parametric distributions, providing a GMM estimation method and demonstrating its application in analyzing heterogeneity in returns to education.

## Contribution

It develops a novel model for categorical random coefficients with a moment-based identification and estimation approach, and applies it to real-world educational data.

## Key findings

- Estimates of random coefficient moments are reasonably accurate with large samples.
- Heterogeneity in returns to education varies by gender and educational level.
- Rising heterogeneity is driven mainly by increased returns for postsecondary education.

## Abstract

This paper proposes a linear categorical random coefficient model, in which the random coefficients follow parametric categorical distributions. The distributional parameters are identified based on a linear recurrence structure of moments of the random coefficients. A Generalized Method of Moments estimation procedure is proposed also employed by Peter Schmidt and his coauthors to address heterogeneity in time effects in panel data models. Using Monte Carlo simulations, we find that moments of the random coefficients can be estimated reasonably accurately, but large samples are required for estimation of the parameters of the underlying categorical distribution. The utility of the proposed estimator is illustrated by estimating the distribution of returns to education in the U.S. by gender and educational levels. We find that rising heterogeneity between educational groups is mainly due to the increasing returns to education for those with postsecondary education, whereas within group heterogeneity has been rising mostly in the case of individuals with high school or less education.