Covariance Regression Models for Studying Treatment Effect Heterogeneity Across One or More Outcomes: Understanding How Treatments Shape Inequality

Abstract

Causal analyses typically focus on average treatment effects. Yet for substantive research on topics like inequality, interest extends to treatments’ distributional consequences. When individuals differ in their responses to treatment, three types of inequality may result. Treatment may shape inequalities between subgroups defined by pretreatment covariates, it may induce more inequality in one subgroup than another, or it may polarize people across multiple dimensions of well-being. We introduce a model, called a covariance regression, that captures all three types of inequality via the means, variances, and correlations between multiple outcomes. The model can test for heterogeneous treatment effects, quantify the heterogeneity, and explain its structure using covariates. Finding that a treatment creates inequalities could drive theoretical refinement and inform policy decisions (targeting groups where payoffs will be most predictable). We illustrate the utility of covariance regressions by analyzing the effects of sharing information about income inequality on redistributive preferences.