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Entropy Balancing for Causal Effects: A Multivariate Reweighting Method to Produce Balanced Samples in Observational Studies

Published online by Cambridge University Press:  04 January 2017

Jens Hainmueller
Jens Hainmueller*
Affiliation:
Department of Political Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139. e-mail: [email protected]
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Abstract

This paper proposes entropy balancing, a data preprocessing method to achieve covariate balance in observational studies with binary treatments. Entropy balancing relies on a maximum entropy reweighting scheme that calibrates unit weights so that the reweighted treatment and control group satisfy a potentially large set of prespecified balance conditions that incorporate information about known sample moments. Entropy balancing thereby exactly adjusts inequalities in representation with respect to the first, second, and possibly higher moments of the covariate distributions. These balance improvements can reduce model dependence for the subsequent estimation of treatment effects. The method assures that balance improves on all covariate moments included in the reweighting. It also obviates the need for continual balance checking and iterative searching over propensity score models that may stochastically balance the covariate moments. We demonstrate the use of entropy balancing with Monte Carlo simulations and empirical applications.

Type
Research Article
Information
Political Analysis , Volume 20 , Issue 1 , Winter 2012 , pp. 25 - 46
Copyright
Copyright © The Author 2011. Published by Oxford University Press on behalf of the Society for Political Methodology 

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References

Abadie, A., and Imbens, G. 2007. Simple and bias-corrected matching estimators for average treatment effects. Working paper. Harvard University.Google Scholar
Brookhart, M., Schneeweiss, S., Rothman, K., Glynn, R., Avorn, J., and Sturmer, T. 2006. Variable selection for propensity score models. American Journal of Epidemiology 163: 1149–56.CrossRefGoogle ScholarPubMed
DellaVigna, S., and Kaplan, E. 2007. The Fox News effect: Media bias and voting. Quarterly Journal of Economics 122: 1187–34.Google Scholar
Deming, W., and Stephan, F. 1940. On the least squares adjustment of a sampled frequency table when the expected marginal totals are known. The Annals of Mathematical Statistics 11: 427–44.CrossRefGoogle Scholar
Diamond, A. J., and Sekhon, J. 2006. Genetic matching for causal effects: A general multivariate matching method for achieving balance in observational studies. Unpublished manuscript, Department of Political Science, UC Berkeley.Google Scholar
Drake, C. 1993. Effects of misspecification of the propensity score on estimators of treatment effect. Biometrics 49: 1231–36.CrossRefGoogle Scholar
Eggers, A., and Hainmueller, J. 2009. MPs for sale? Returns to office in postwar British politics. American Political Science Review 103: 513–33.CrossRefGoogle Scholar
Erlander, S. 1977. Entropy in linear programs—an approach to planning. Report No. LiTH-MAT-R-77-3. Department of Mathematics, Linköping University, Sweden.Google Scholar
Erlander, S. 2004. Finite sample properties of propensity-score matching and weighting estimators. Review of Economics and Statistics 86: 77–90.Google Scholar
Frölich, M. 2007. Propensity score matching without conditional independence assumption with an application to the gender wage gap in the United Kingdom. The Econometrics Journal 10: 359–407.CrossRefGoogle Scholar
Graham, B. S., Pinto, C., and Egel, D. 2010. Inverse probability tilting for moment condition models with missing data. Working paper. New York University.Google Scholar
Gu, X., and Rosenbaum, P. 1993. Comparison of multivariate matching methods: Structures, distances, and algorithms. Journal of Computational and Graphical Statistics 2: 405–20.Google Scholar
Hahn, J. 1998. On the role of the propensity score in efficient semiparametric estimation of average treatment effects. Econometrica 66: 315–31.CrossRefGoogle Scholar
Hansen, B. B., and Bowers, J. 2008. Covariate balance in simple, stratified and clustered comparative studies. Statistical Science 23: 219–36.CrossRefGoogle Scholar
Hansen, L. 1982. Large sample properties of generalized method of moments estimators. Econometrica 50: 1029–54.Google Scholar
Hellerstein, J., and Imbens, G. 1999. Imposing moment restrictions from auxiliary data by weighting. The Review of Economics and Statistics 81: 1–14.CrossRefGoogle Scholar
Hirano, K., and Imbens, G. 2001. Estimation of causal effects using propensity score weighting: An application of data on right hear catherization. Health Services and Outcomes Research Methodology 2: 259–78.CrossRefGoogle Scholar
Hirano, K., Imbens, G., and Ridder, G. 2003. Efficient estimation of average treatment effects using the estimated propensity score. Econometrica 71: 1161–89.CrossRefGoogle Scholar
Ho, D., Imai, K., King, G., and Stuart, E. 2007. Matching as nonparametric preprocessing for reducing model dependence in parametric causal inference. Political Analysis 15: 199–236.CrossRefGoogle Scholar
Horvitz, D., and Thompson, D. 1952. A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association 47: 663–85.CrossRefGoogle Scholar
Iacus, S., King, G., and Porro, G. 2009. Causal inference without balance checking: Coarsened exact matching. Mimeo Harvard University.Google Scholar
Imai, K., King, G., and Stuart, E. 2008. Misunderstandings among experimentalists and observationalists: Balance test fallacies in causal inference. Journal of the Royal Statistical Society, Series A 171: 481–502.CrossRefGoogle Scholar
Imbens, G. 1997. One-step estimators for over-identified generalized method of moments models. The Review of Economic Studies 64: 359–83.CrossRefGoogle Scholar
Imbens, G. 2004. Nonparametric estimation of average treatment effects under exogeneity: A review. Review of Economics and Statistics 86: 4–29.CrossRefGoogle Scholar
Imbens, G., Spady, R., and Johnson, P. 1998. Information theoretic approaches to inference in moment condition models. Econometrica 66: 333–57.Google Scholar
Ireland, C., and Kullback, S. 1968. Contingency tables with given marginals. Biometrika 55: 179–88.CrossRefGoogle ScholarPubMed
Kapur, J., and Kevsavan, H. 1992. Entropy optimization principles with applications. London: Academic Press.CrossRefGoogle Scholar
Kitamura, Y., and Stutzer, M. 1997. An information-theoretic alternative to generalized method of moments estimation. Econometrica 65: 861–74.CrossRefGoogle Scholar
Kullback, S. 1959. Information theory and statistics. New York: Wiley.Google Scholar
Ladd, J., and Lenz, G. 2009. Exploiting a rare communication shift to document the persuasive power of the news media. American Journal of Political Science 53: 394–10.CrossRefGoogle Scholar
LaLonde, R. J. 1986. Evaluating the econometric evaluations of training programs with experimental data. American Economic Review 76: 604–20.Google Scholar
Mattos, R., and Veiga, A. 2004. Entropy optimization: Computer implementation of the maxent and minexent principles. Working paper. Universidade Federal de Juiz de Fora, Brazil.Google Scholar
McCaffrey, D., Ridgeway, G., and Morral, A. 2004. Propensity score estimation with boosted regression for evaluating adolescent substance abuse treatment. Psychological Methods 9: 403–25.CrossRefGoogle Scholar
Oh, H. L., and Scheuren, F. J. 1978. Multivariate ratio raking estimation in the 1973 exact match study. Proceedings of the Section on Survey Research Methods XXV: 716–22.Google Scholar
Owen, A. 2001. Empirical likelihood. Boca Raton, FL: Chapman & Hall.Google Scholar
Qin, J., and Lawless, J. 1994. Empirical likelihood and general estimating equations. Annals of Statistics 22: 300–25.CrossRefGoogle Scholar
Qin, J., Zhang, B., and Leung, D. 2009. Empirical likelihood in missing data problems. Journal of the American Statistical Association 104: 1492–503.CrossRefGoogle Scholar
Read, T., and Cressie, N. 1988. Goodness-of-fit statistics for discrete multivariate data. New York: Springer.CrossRefGoogle Scholar
Robins, J., Rotnitzky, A., and Zhao, L. 1995. Analysis of semiparametric regression models for repeated outcomes in the presence of missing data. Journal of the American Statistical Association 90: 106–21.Google Scholar
Rosenbaum, P. R., and Rubin, D. B. 1983. The central role of the propensity score in observational studies for causal effects. Biometrika 70: 41–55.CrossRefGoogle Scholar
Rubin, D. 2006. Matched sampling for causal effects. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
Särndal, C. E., and Lundström, S. 2006. Estimation in surveys with nonresponse. New York: John Wiley & Sons, Ltd.Google Scholar
Schennach, S. 2007. Point estimation with exponentially tilted empirical likelihood. The Annals of Statistics 35: 634–72.CrossRefGoogle Scholar
Sekhon, J. 2006. Alternative balance metrics for bias reduction in matching methods for causal inference. Unpublished manuscript, Department of of Political Science, UC Berkeley.Google Scholar
Sekhon, J. S. 2009. Opiates for the matches: Matching methods for causal inference. Annual Review of Political Science 12: 487–08.CrossRefGoogle Scholar
Smith, J., and Todd, P. 2001. Reconciling conflicting evidence on the performance of propensity-score matching methods. American Economic Review 91: 112–18.Google Scholar
Zaslavsky, A. 1988. Representing local reweighting area adjustments by of households. Survey Methodology 14: 265–88.Google Scholar
Zhao, Z. 2004. Using matching to estimate treatment effects: Data requirements, matching metrics, and Monte Carlo evidence. Review of Economics and Statistics 86: 91–107.CrossRefGoogle Scholar
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