A Framework for Eliciting, Incorporating, and Disciplining Identification Beliefs in Linear Models

TL;DR

This paper introduces a Bayesian framework that helps researchers formalize, incorporate, and ensure consistency of their beliefs about instrument validity, treatment endogeneity, and measurement error in linear models for causal inference.

Contribution

It characterizes the joint restrictions among beliefs about instrument invalidity, endogeneity, and measurement error, and provides a formal method to incorporate these into linear model estimation.

Findings

The framework ensures mutual coherence of beliefs and data constraints.

Application to empirical microeconomics examples demonstrates practical utility.

Formalizes the relationship between beliefs and model restrictions.

Abstract

To estimate causal effects from observational data, an applied researcher must impose beliefs. The instrumental variables exclusion restriction, for example, represents the belief that the instrument has no direct effect on the outcome of interest. Yet beliefs about instrument validity do not exist in isolation. Applied researchers often discuss the likely direction of selection and the potential for measurement error in their articles but lack formal tools for incorporating this information into their analyses. Failing to use all relevant information not only leaves money on the table; it runs the risk of leading to a contradiction in which one holds mutually incompatible beliefs about the problem at hand. To address these issues, we first characterize the joint restrictions relating instrument invalidity, treatment endogeneity, and non-differential measurement error in a workhorse linear model, showing how beliefs over these three dimensions are mutually constrained by each other and the data. Using this information, we propose a Bayesian framework to help researchers elicit their beliefs, incorporate them into estimation, and ensure their mutual coherence. We conclude by illustrating our framework in a number of examples drawn from the empirical microeconomics literature.