# Posterior Average Effects

## TL;DR

This paper introduces posterior average effects (PAE), a method for estimating population averages of unobservables using sample-conditioned posterior means, justified by minimax error properties and applied to economic data.

## Contribution

It develops the theoretical foundation for PAE, demonstrating their minimax optimality under misspecification and introducing a measure of informativeness for posterior conditioning.

## Key findings

- PAE have minimum worst-case specification error under misspecification.
- A measure of informativeness quantifies the robustness of PAE.
- Empirical applications include neighborhood effects and income dynamics.

## Abstract

Economists are often interested in estimating averages with respect to distributions of unobservables, such as moments of individual fixed-effects, or average partial effects in discrete choice models. For such quantities, we propose and study posterior average effects (PAE), where the average is computed conditional on the sample, in the spirit of empirical Bayes and shrinkage methods. While the usefulness of shrinkage for prediction is well-understood, a justification of posterior conditioning to estimate population averages is currently lacking. We show that PAE have minimum worst-case specification error under various forms of misspecification of the parametric distribution of unobservables. In addition, we introduce a measure of informativeness of the posterior conditioning, which quantifies the worst-case specification error of PAE relative to parametric model-based estimators. As illustrations, we report PAE estimates of distributions of neighborhood effects in the US, and of permanent and transitory components in a model of income dynamics.