On the relationship between prediction intervals, tests of sharp nulls and inference on realized treatment effects in settings with few treated units

TL;DR

This paper explores how inference methods for small treated groups can be adapted to heterogeneous treatment effects, revealing deep connections and conditions under which these methods remain valid.

Contribution

It demonstrates that methods based on treatment effect homogeneity can be extended to heterogeneous effects, providing theoretical insights and justifications.

Findings

Inference methods for sharp nulls, realized effects, and prediction intervals are interconnected.

Methods under homogeneity assumptions can be valid for heterogeneous effects under certain conditions.

Theoretical results clarify when existing methods remain applicable in more complex settings.

Abstract

We study how inference methods for settings with few treated units that rely on treatment effect homogeneity extend to alternative inferential targets when treatment effects are heterogeneous -- namely, tests of sharp null hypotheses, inference on realized treatment effects, and prediction intervals. We show that inference methods for these alternative targets are deeply interconnected: they are either equivalent or become equivalent under additional assumptions. Our results show that methods designed under treatment effect homogeneity can remain valid for these alternative targets when treatment effects are stochastic, offering new theoretical justifications and insights on their applicability.