Regression-Adjusted Estimators in Randomized Experiments with a Diverging Number of Predictors

• Activity Number: 34 - Linear Models for Large or Complex Data
• Type: Contributed
• Date/Time: Sunday, July 29, 2018 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Lihua Lei* and Peng Ding
• Companies: UC Berkeley and UC Berkeley
• Keywords: causal inference; randomized trial; Neyman-Rubin model; regression-adjusted estimator; bias correction; finite population analysis

Abstract

Extending R. A. Fisher and D. A. Freedman's results on the analysis of covariance in completely randomized experiments, Lin (2013) studied asymptotic performance of least squares adjusted estimates for average treatment effect in randomized trials under the Neyman-Rubin model. His result holds for a fixed dimension p of the covariates with the sample size n approaching infinity. In this article we consider a more practical asymptotic regime where the dimension of covariates is diverging. We show that Lin's estimator is consistent when p = o(n) and asymptotically normal when p = o(n^{1/2}), but it can have a non-vanishing asymptotic bias when p is larger than O(n^{1/2}). Therefore, it is crucial to conduct bias correction, and we propose a bias-corrected estimator that is asymptotically normal when p = o(n^{2/3}), even if the linear model is misspecified. Although this asymptotic regime appeared in the classical high dimensional regression literature, previous results hold under independent sampling of the units or a correctly specified regression model. Our results are model-free and require novel analytic tools in finite population analysis.