Optimal Stratification in Randomized Experiments

Abstract

This paper shows that stratifying on the conditional expectation of the outcome given baseline variables is optimal in matched-pair randomized experiments. The assignment minimizes the variance of the post-treatment difference in mean outcomes between treatment and controls. Optimal pairing depends only on predicted values of outcomes for experimental units, where the predicted values are the conditional expectations. After randomization, both frequentist inference and randomization inference depend only on the actual strata chosen and not on estimated predicted values. This gives experimenters a way to use big data (possibly more covariates than the number of experimental units) ex-ante while maintaining simple post-experiment inference techniques. Optimizing the randomization with respect to one outcome allows researchers to credibly signal the outcome of interest prior to the experiment. Inference can be conducted in the standard way by regressing the outcome on treatment and strata indicators. We illustrate the application of the methodology by running simulations based on a set of field experiments. We find that optimal designs have mean squared errors 23% less than randomized designs, on average. In one case, mean squared error is 43% less than randomized designs. Date: January 10, 2014. I am grateful to my Ph.D. advisers Gary Chamberlain, Ed Glaeser, Guido Imbens, and Larry Katz for their generous guidance. I thank Don Rubin, Max Kasy, Claudia Goldin, Stefano DellaVigna, Roland Fryer, Michael Kremer, Sendhil Mullainathan, Jose Montiel Olea, Raj Chetty, Rick Hornbeck, Nathaniel Hilger, and Silvia Robles for helpful comments. I am grateful to seminar participants at the Harvard Labor, Development, and Econometrics workshops and at the MIT Development workshop. I acknowledge support from an Education Innovation Lab Research Fellowship. 1 2 THOMAS BARRIOS DEPARTMENT OF ECONOMICS, HARVARD UNIVERSITY

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