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.
AbadieAlberto. 2002. “Bootstrap Tests for Distributional Treatment Effects in Instrumental Variable Models.” Journal of the American Statistical Association97: 284–92.
2.
AllisonPaul. 1978. “Measures of Inequality.” American Sociological Review43: 865–80.
3.
AtheySusanImbensGuido. 2016. “Recursive Partitioning for Heterogeneous Causal Effects.” Proceedings of the National Academy of Sciences113: 7353–60.
4.
AlwinDuane. 2007. Margins of Error: A Study of Reliability in Survey Measurement. New York: Wiley-Interscience.
5.
BaldassarriDeliaGelmanAndrew. 2008. “Partisans without Constraint: Political Polarization and Trends in American Public Opinion.” American Journal of Sociology114: 408–46.
6.
BanerjeeOnureenaGhaouiLaurent Eld’AspremontAlexandre. 2008. “Model Selection Through Sparse Maximum Likelihood Estimation for Multivariate Gaussian or Binary Data.” Journal of Machine Learning Research9: 485–16.
7.
BarnettW. StevenJungKwangheeYaroszDonaldThomasJessicaHornbeckAmyStechukRobertBurnsSusan. 2008. “Educational Effects of the Tools of the Mind Curriculum: A Randomized Trial.” Early Childhood Research Quarterly23: 299–13.
8.
BitlerMarianneGelbachJonahHoynesHilary. 2017. “Can Variation in Subgroups’ Average Treatment Effects Explain Treatment Effect Heterogeneity? Evidence from a Social Experiment.” Review of Economics and Statistics99: 683–97.
9.
BoikRobert. 2002. “Spectral Models for Covariance Matrices.” Biometrika89: 159–82.
10.
BrandJennieThomasJuli Simon. 2013. “Causal Effect Heterogeneity.” Pp. 189–213 in Handbook on Causal Analysis for Social Research, edited by MorganStephen. New York: Springer.
BrickelPeterLevinaElizaveta. 2008. “Regularized Estimation of Large Covariance Matrices.” Annals of Statistics36: 199–227.
13.
BrykAnthonyRaudenbushStephen. 1988. “Heterogeneity of Variance in Experimental Studies: A Challenge to Conventional Interpretations.” Psychological Bulletin104: 396–404.
14.
CookSamanthaGelmanAndrewRubinDonald. 2006. “Validation of Software for Bayesian Models Using Posterior Quantiles.” Journal of Computational and Graphical Statistics15: 675–92.
15.
CrumpRichardJoseph HotzV.ImbensGuidoMitnikOscar. 2008. “Nonparametric Tests for Treatment Effect Heterogeneity.” Review of Economics and Statistics90: 389–405.
16.
DehejiaRajeev. 2005. “Program Evaluation as a Decision Problem.” Journal of Econometrics125: 141–73.
17.
DingPengFellerAviMiratrixLuke. 2018. “Decomposing Treatment Effect Variation.” Journal of the American Statistical Association. doi:10.1080/01621459.2017.1407322
18.
DingPengFellerAviMiratrixLuke. 2016. “Randomization Inference for Treatment Effect Variation.” Journal of the Royal Statistical Society, Series B78: 655–71.
19.
EmbrechtsPaulMcNeilAlexanderStraumannDaniel. 2002. “Correlation and Dependency in Risk Management: Properties and Pitfalls.” Pp. 176–223 in Risk Management: Value at Risk and Beyond, edited by DempsterM. A. H.. Cambridge: Cambridge University Press.
20.
FurrerReinhardBengtssonThomas. 2007. “Estimation of High-dimensional Prior and Posterior Covariance Matrices in Kalman Filter Variants.” Journal of Multivariate Analysis98: 227–55.
21.
GelmanAndrew. 2004. “Treatment Effects in before-after Data.” Pp. 195–202 in Applied Bayesian Modeling and Causal Inference from Incomplete Data Perspectives, edited by GelmanAndrewMengXiao-Li. London: Wiley.
GelmanAndrewHillJennifer. 2007. Data Analysis Using Regression and Multilevel/Hierarchical Models. New York: Cambridge University Press.
24.
HeckmanJamesUrzuaSergioVytlacilEdward. 2006. “Understanding Instrumental Variables in Models with Essential Heterogeneity.” Review of Economics and Statistics88: 389–432.
25.
HeckmanJamesVytlacilEdward. 1999. “Local Instrumental Variables and Latent Variable Models for Identifying and Bounding Treatment Effects.” Proceedings of the National Academy of Sciences96: 4730–34.
26.
HeckmanJamesVytlacilEdward. 2001. “Policy-relevant Treatment Effects.” American Economic Review91: 107–11.
27.
HeteyRebecca C.EberhardtJennifer L.. 2014. “Racial Disparities in Incarceration Increase Acceptance of Punitive Policies.” Psychological Science25: 1949–54.
ImaiKosukeKeeleLukeYamamotoTeppei. 2010. “Identification, Inference and Sensitivity Analysis for Causal Mediation Effects.” Statistical Science25: 51–71.
30.
ImaiKosukeRatkovicMarc. 2013. “Estimating Treatment Effect Heterogeneity in Randomized Program Evaluation.” Annals of Applied Statistics7: 443–70.
31.
KingGaryRobertsMolly. 2014. “How Robust Standard Errors Expose Methodological Problems They Do Not Fix, and What to Do About It.” Political Analysis23: 159–79.
KuziemkoIlyanaNortonMichael I.SaezEmmanuelStantchevaStefanie. 2015. “How Elastic Are Preferences for Redistribution? Evidence from Randomized Survey Experiments.” American Economic Review105: 1478–1508.
34.
LohWei-YinHebXuManMichael. 2015. “A Regression Tree Approach to Identifying Subgroups with Differential Treatment Effects.” Statistics in Medicine34: 1818–33.
35.
LudwigJensDuncanGregGennetianLisaKatzLawrenceKesslerRonaldKlingJeffreySanbonmatsuLisa. 2012. “Neighborhood Effects on the Long-term Well-being of Low-income Adults.” Science337: 1505–10.
36.
MacCallumRobertAustinJames. 2000. “Applications of Structural Equation Modeling in Psychological Research.” Annual Review of Psychology51: 201–26.
37.
MacKinnonJames. 2012. “Thirty Years of Heteroskedasticity-robust Inference.” Pp. 437–59 in Recent Advances and Future Directions in Causality, Prediction, and Specification Analysis, edited by ChenXiaohongSwansonNorman. New York: Springer.
38.
MatteiAlessandraLiFanMealliFabrizia. 2013. “Exploiting Multiple Outcomes in Bayesian Principle Stratification Analysis with Application to a Job Training Program.” Annals of Applied Statistics7: 2336–60.
39.
McCullaghPeterNelderJohn. 1989. Generalized Linear Models. 2nd ed. New York: Chapman & Hall/CRC.
40.
MorganStephenWinshipChristopher. 2015. Counterfactuals and Causal Inference. 2nd ed. New York: Cambridge University Press.
41.
MutzDiana. 2018. “Status Threat, Not Economic Hardship, Explains the 2016 Presidential Vote.” Proceedings of the National Academy of Sciences. doi:10.1073/pnas.1718155115
42.
NiuXiaoyueHoffPeter. 2014. “Simultaneous Mean and Covariance Modeling of Chronic Kidney Disease.” Retrieved December 27, 2017 (http://arxiv.org/abs/1311.2610v4).
43.
PearlJudea. 2017. “Detecting Latent Heterogeneity.” Sociological Methods and Research46: 370–89.
44.
PourahmadiMohsen. 2011. “Covariance Estimation: The GLM and Regularization Perspectives.” Statistical Science26: 369–87.
45.
RaudenbushStephenBloomHoward. 2015. “Learning about and from a Distribution of Program Impacts Using Multisite Trials.” American Journal of Evaluation36: 475–99.
46.
RothmanAdamLevinaElizavetaJiZhu. 2010. “A New Approach to Cholesky-based Covariance Regularization in High Dimensions.” Biometrika97: 539–50.
47.
RubinDonald. 1974. “Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies.” Journal of Educational Psychology66: 688–701.
48.
Stan Development Team. 2015. Stan: A C++ Library for Probability and Sampling, Version 2.7.0. Retrieved December 27, 2017 (http://mc-stan.org).
VuoloMike. 2017. “Copula Models for Sociology: Measures of Dependence and Probabilities for Joint Distributions.” Sociological Methods and Research46: 604–48.
51.
WesternBruceBloomeDeirdre. 2009. “Variance Function Regressions for Studying Inequality.” Sociological Methodology39: 293–326.
52.
WoodWendy. 2000. “Attitude Change: Persuasion and Social Influence.” Annual Review of Psychology51: 539–70.
ZellnerArnold. 1962. “An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias.” Journal of the American Statistical Association57: 348–68.
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
