# Asymptotic Theory for Clustered Samples

## TL;DR

This paper develops a comprehensive asymptotic distribution theory for large samples of clustered data, accommodating heterogeneity and unbounded cluster sizes, and extends classical results to complex clustered sampling scenarios.

## Contribution

It generalizes classical asymptotic results to clustered data with heterogeneous and unbounded clusters, providing a unified framework for various estimators.

## Key findings

- Unified asymptotic distribution theory for clustered data
- Conditions that include classical i.i.d. results as special cases
- Applicable to linear, 2SLS, nonlinear MLE, and GMM estimators

## Abstract

We provide a complete asymptotic distribution theory for clustered data with a large number of independent groups, generalizing the classic laws of large numbers, uniform laws, central limit theory, and clustered covariance matrix estimation. Our theory allows for clustered observations with heterogeneous and unbounded cluster sizes. Our conditions cleanly nest the classical results for i.n.i.d. observations, in the sense that our conditions specialize to the classical conditions under independent sampling. We use this theory to develop a full asymptotic distribution theory for estimation based on linear least-squares, 2SLS, nonlinear MLE, and nonlinear GMM.