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

  • Jens Hainmueller (a1)

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.

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Abadie, A., and Imbens, G. 2007. Simple and bias-corrected matching estimators for average treatment effects. Working paper. Harvard University.
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.
DellaVigna, S., and Kaplan, E. 2007. The Fox News effect: Media bias and voting. Quarterly Journal of Economics 122: 1187–34.
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.
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 Unpublished manuscript, Department of Political Science, UC Berkeley.
Drake, C. 1993. Effects of misspecification of the propensity score on estimators of treatment effect. Biometrics 49: 1231–36.
Eggers, A., and Hainmueller, J. 2009. MPs for sale? Returns to office in postwar British politics. American Political Science Review 103: 513–33.
Erlander, S. 1977. Entropy in linear programs—an approach to planning. Report No. LiTH-MAT-R-77-3. Department of Mathematics, Linköping University, Sweden.
Erlander, S. 2004. Finite sample properties of propensity-score matching and weighting estimators. Review of Economics and Statistics 86: 7790.
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: 359407.
Graham, B. S., Pinto, C., and Egel, D. 2010. Inverse probability tilting for moment condition models with missing data. Working paper. New York University.
Gu, X., and Rosenbaum, P. 1993. Comparison of multivariate matching methods: Structures, distances, and algorithms. Journal of Computational and Graphical Statistics 2: 405–20.
Hahn, J. 1998. On the role of the propensity score in efficient semiparametric estimation of average treatment effects. Econometrica 66: 315–31.
Hansen, B. B., and Bowers, J. 2008. Covariate balance in simple, stratified and clustered comparative studies. Statistical Science 23: 219–36.
Hansen, L. 1982. Large sample properties of generalized method of moments estimators. Econometrica 50: 1029–54.
Hellerstein, J., and Imbens, G. 1999. Imposing moment restrictions from auxiliary data by weighting. The Review of Economics and Statistics 81: 114.
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.
Hirano, K., Imbens, G., and Ridder, G. 2003. Efficient estimation of average treatment effects using the estimated propensity score. Econometrica 71: 1161–89.
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: 199236.
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.
Iacus, S., King, G., and Porro, G. 2009. Causal inference without balance checking: Coarsened exact matching. Mimeo Harvard University.
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: 481502.
Imbens, G. 1997. One-step estimators for over-identified generalized method of moments models. The Review of Economic Studies 64: 359–83.
Imbens, G. 2004. Nonparametric estimation of average treatment effects under exogeneity: A review. Review of Economics and Statistics 86: 429.
Imbens, G., Spady, R., and Johnson, P. 1998. Information theoretic approaches to inference in moment condition models. Econometrica 66: 333–57.
Ireland, C., and Kullback, S. 1968. Contingency tables with given marginals. Biometrika 55: 179–88.
Kapur, J., and Kevsavan, H. 1992. Entropy optimization principles with applications. London: Academic Press.
Kitamura, Y., and Stutzer, M. 1997. An information-theoretic alternative to generalized method of moments estimation. Econometrica 65: 861–74.
Kullback, S. 1959. Information theory and statistics. New York: Wiley.
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.
LaLonde, R. J. 1986. Evaluating the econometric evaluations of training programs with experimental data. American Economic Review 76: 604–20.
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.
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.
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.
Owen, A. 2001. Empirical likelihood. Boca Raton, FL: Chapman & Hall.
Qin, J., and Lawless, J. 1994. Empirical likelihood and general estimating equations. Annals of Statistics 22: 300–25.
Qin, J., Zhang, B., and Leung, D. 2009. Empirical likelihood in missing data problems. Journal of the American Statistical Association 104: 1492–503.
Read, T., and Cressie, N. 1988. Goodness-of-fit statistics for discrete multivariate data. New York: Springer.
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.
Rosenbaum, P. R., and Rubin, D. B. 1983. The central role of the propensity score in observational studies for causal effects. Biometrika 70: 4155.
Rubin, D. 2006. Matched sampling for causal effects. Cambridge: Cambridge University Press.
Särndal, C. E., and Lundström, S. 2006. Estimation in surveys with nonresponse. New York: John Wiley & Sons, Ltd.
Schennach, S. 2007. Point estimation with exponentially tilted empirical likelihood. The Annals of Statistics 35: 634–72.
Sekhon, J. 2006. Alternative balance metrics for bias reduction in matching methods for causal inference. Unpublished manuscript, Department of of Political Science, UC Berkeley.
Sekhon, J. S. 2009. Opiates for the matches: Matching methods for causal inference. Annual Review of Political Science 12: 487–08.
Smith, J., and Todd, P. 2001. Reconciling conflicting evidence on the performance of propensity-score matching methods. American Economic Review 91: 112–18.
Zaslavsky, A. 1988. Representing local reweighting area adjustments by of households. Survey Methodology 14: 265–88.
Zhao, Z. 2004. Using matching to estimate treatment effects: Data requirements, matching metrics, and Monte Carlo evidence. Review of Economics and Statistics 86: 91107.
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Political Analysis
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