Heterogeneous Treatment Effects for Networks, Panels, and other Outcome Matrices

TL;DR

This paper introduces a novel spectral approach to analyze treatment effects in experiments where outcomes are represented as matrices, such as networks or panels, using eigenvalues to bound and compare effects.

Contribution

It proposes a new empirical strategy based on eigenvalues of outcome matrices, extending the Fréchet-Hoeffding bounds to matrix settings for treatment effect analysis.

Findings

Bounded the distribution of treatment effects using eigenvalues.

Introduced spectral treatment effects as eigenvalue differences.

Provided a matrix analog of quantile treatment effects.

Abstract

We are interested in the distribution of treatment effects for an experiment where units are randomized to a treatment but outcomes are measured for pairs of units. For example, we might measure risk sharing links between households enrolled in a microfinance program, employment relationships between workers and firms exposed to a trade shock, or bids from bidders to items assigned to an auction format. Such a double randomized experimental design may be appropriate when there are social interactions, market externalities, or other spillovers across units assigned to the same treatment. Or it may describe a natural or quasi experiment given to the researcher. In this paper, we propose a new empirical strategy that compares the eigenvalues of the outcome matrices associated with each treatment. Our proposal is based on a new matrix analog of the Fr\'echet-Hoeffding bounds that play a key role in the standard theory. We first use this result to bound the distribution of treatment effects. We then propose a new matrix analog of quantile treatment effects that is given by a difference in the eigenvalues. We call this analog spectral treatment effects.