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  • Craig A. Rolling (a1), Yuhong Yang (a2) and Dagmar Velez (a3)

Estimating a treatment’s effect on an outcome conditional on covariates is a primary goal of many empirical investigations. Accurate estimation of the treatment effect given covariates can enable the optimal treatment to be applied to each unit or guide the deployment of limited treatment resources for maximum program benefit. Applications of conditional treatment effect estimation are found in direct marketing, economic policy, and personalized medicine. When estimating conditional treatment effects, the typical practice is to select a statistical model or procedure based on sample data. However, combining estimates from the candidate procedures often provides a more accurate estimate than the selection of a single procedure. This article proposes a method of model combination that targets accurate estimation of the treatment effect conditional on covariates. We provide a risk bound for the resulting estimator under squared error loss and illustrate the method using data from a labor skills training program.

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*Address correspondence to Craig A. Rolling, Department of Epidemiology and Biostatistics, Saint Louis University, St. Louis, Missouri, USA; e-mail:
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The authors would like to thank Arthur Lewbel and four anonymous referees for their helpful comments and suggestions that have improved the article.

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Abadie, A. & Imbens, G.W. (2006) Large sample properties of matching estimators for average treatment effects. Econometrica 74, 235267.
Abadie, A. & Imbens, G.W. (2011) Bias-corrected matching estimators for average treatment effects. Journal of Business and Economic Statistics 29, 111.
Abrevaya, J., Hsu, Y.-C., & Lieli, R.P. (2015) Estimating conditional average treatment effects. Journal of Business & Economic Statistics 33, 485505.
Akaike, H. (1974) A new look at the statistical model identification. IEEE Transactions on Automatic Control 19, 716723.
Althauser, R.P. & Rubin, D. (1970) The computerized construction of a matched sample. American Journal of Sociology 76, 325346.
Bhattacharya, D. & Dupas, P. (2012) Inferring welfare maximizing treatment assignment under budget constraints. Journal of Econometrics 167, 168196.
Buckland, S., Burnham, K., & Augustin, N. (1997) Model selection: An integral part of inference. Biometrics 53, 603618.
Cai, T., Tian, L., Wong, P.H., & Wei, L. (2011) Analysis of randomized comparative clinical trial data for personalized treatment selections. Biostatistics 12, 270282.
Chang, M., Lee, S., & Whang, Y.-J. (2015) Nonparametric tests of conditional treatment effects with an application to single-sex schooling on academic achievements. The Econometrics Journal 18, 307346.
Claeskens, G. & Hjort, N.L. (2008) Minimizing average risk in regression models. Econometric Theory 24, 493527.
Claeskens, G., Magnus, J.R., Vasnev, A.L., & Wang, W. (2016) The forecast combination puzzle: A simple theoretical explanation. International Journal of Forecasting 32, 754762.
Cook, R.D. (1998) Regression Graphics. Wiley.
Cook, R.D. & Li, B. (2002) Dimension reduction for conditional mean in regression. The Annals of Statistics 30, 455474.
Dehejia, R.H. & Wahba, S. (1999) Causal effects in nonexperimental studies: Reevaluating the evaluation of training programs. Journal of the American Statistical Association 94, 10531062.
Green, D.P. & Kern, H.L. (2012) Modeling heterogeneous treatment effects in survey experiments with Bayesian additive regression trees. Public Opinion Quarterly 76, 491511.
Hirano, K. & Porter, J.R. (2009) Asymptotics for statistical treatment rules. Econometrica 77, 16831701.
Holland, P.W. (1986) Statistics and causal inference. Journal of the American Statistical Association 81, 945960.
Hsu, Y.-C. (2017) Consistent tests for conditional treatment effects. The Econometrics Journal 20, 122.
Imai, K. & Ratkovic, M. (2013) Estimating treatment effect heterogeneity in randomized program evaluation. The Annals of Applied Statistics 7, 443470.
Imbens, G.W. (2004) Nonparametric estimation of average treatment effects under exogeneity: A review. Review of Economics and Statistics 86, 429.
Imbens, G.W. & Wooldridge, J.M. (2009) Recent developments in the econometrics of program evaluation. Journal of Economic Literature 47, 586.
Kitagawa, T. & Tetenov, A. (2015) Who Should be Treated? Empirical Welfare Maximization Methods for Treatment Choice. Cemmap Working paper, CWP10/15.
LaLonde, R.J. (1986) Evaluating the econometric evaluations of training programs with experimental data. The American Economic Review 76, 604620.
Li, K.-C. (1991) Sliced inverse regression for dimension reduction. Journal of the American Statistical Association 86, 316327.
Li, K.-C., Lue, H.-H., & Chen, C.-H. (2000) Interactive tree-structured regression via principal Hessian directions. Journal of the American Statistical Association 95, 547560.
Li, L. (2007) Sparse sufficient dimension reduction. Biometrika 94, 603613.
Li, L. & Yin, X. (2008) Sliced inverse regression with regularizations. Biometrics 64, 124131.
Qian, M. & Murphy, S.A. (2011) Performance guarantees for individualized treatment rules. The Annals of Statistics 39, 11801210.
Qian, W., Rolling, C.A., Cheng, G., & Yang, Y. (2017) On the forecast combination puzzle. Preprint. Available at
Raftery, A.E. (1995) Bayesian model selection in social research. Sociological Methodology 25, 111163.
Rolling, C.A. & Yang, Y. (2014) Model selection for estimating treatment effects. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 76, 749769.
Schwarz, G. (1978) Estimating the dimension of a model. The Annals of Statistics 6, 461464.
Smith, J. & Wallis, K. (2009) A simple explanation of the forecast combination puzzle. Oxford Bulletin of Economics and Statistics 71, 331355.
Stoye, J. (2009) Minimax regret treatment choice with finite samples. Journal of Econometrics 151, 7081.
Taddy, M., Gardner, M., Chen, L., & Draper, D. (2016) A nonparametric Bayesian analysis of heterogenous treatment effects in digital experimentation. Journal of Business & Economic Statistics 34, 661672.
Wood, S.N. (2006) Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC.
Yang, Y. (2001) Adaptive regression by mixing. Journal of the American Statistical Association 96, 574588.
Yang, Y. (2003) Regression with multiple candidate models: Selecting or mixing? Statistica Sinica 13, 783809.
Yang, Y. (2004) Combining forecasting procedures: Some theoretical results. Econometric Theory 20, 176222.
Zhang, B. (2016) Empirical likelihood in causal inference. Econometric Reviews 35, 201231.
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Econometric Theory
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