Some Impossibility Results for Inference With Cluster Dependence with Large Clusters

TL;DR

This paper demonstrates fundamental limitations in statistical inference with large cluster dependence, showing that without multiple large clusters, consistent mean discrimination and variance estimation are impossible.

Contribution

It establishes new impossibility results for inference under cluster dependence, highlighting the necessity of multiple large clusters for consistent estimation.

Findings

Single large cluster prevents consistent mean discrimination.

At least two large clusters are needed for $\

Long run variance is not consistently estimable with at least one large cluster.

Abstract

This paper focuses on a setting with observations having a cluster dependence structure and presents two main impossibility results. First, we show that when there is only one large cluster, i.e., the researcher does not have any knowledge on the dependence structure of the observations, it is not possible to consistently discriminate the mean. When within-cluster observations satisfy the uniform central limit theorem, we also show that a sufficient condition for consistent $\sqrt{n}$-discrimination of the mean is that we have at least two large clusters. This result shows some limitations for inference when we lack information on the dependence structure of observations. Our second result provides a necessary and sufficient condition for the cluster structure that the long run variance is consistently estimable. Our result implies that when there is at least one large cluster, the long run variance is not consistently estimable.