Extending R. A. Fisher and D. A. Freedman's results on the analysis of covariance, Lin [2013] proposed an ordinary least squares adjusted estimator of the average treatment effect in completely randomized experiments. We further study its statistical properties under the potential outcomes model in the asymptotic regimes allowing for a diverging number of covariates. We show that Lin [2013]'s estimator is consistent when @math and asymptotically normal when @math under mild moment conditions, where @math is the maximum leverage score of the covariate matrix. In the favorable case where leverage scores are all close together, his estimator is consistent when @math and is asymptotically normal when @math . In addition, we propose a bias-corrected estimator that is consistent when @math and is asymptotically normal, with the same variance in the fixed- @math regime, when @math . In the favorable case, the latter condition reduces to @math . Similar to Lin [2013], our results hold for non-random potential outcomes and covariates without any model specification. Our analysis requires novel analytic tools for sampling without replacement, which complement and potentially enrich the theory in other areas such as survey sampling, matrix sketching, and transductive learning.
Theoretical analyses under the finite-population randomization model are challenging due to the lack of probability tools. The closest work to ours is @cite_8 , which allows @math to grow with @math and potentially exceed @math . However, they assume that the potential outcomes have sparse linear representations based on the covariates, and require @math where @math is a measure of sparsity. Under additional regularities conditions, they show that @math is consistent and asymptotically normal with @math being the LASSO coefficients of the covariates. Although the LASSO-adjusted estimator can handle ultra-high dimensional case where @math , it has three limitations. First, the requirement @math is stringent. For instance, the PAC-man dataset considered by @cite_8 has @math and @math , so the condition reads @math , which implicitly imposes a strong sparse modelling assumption.
@cite_8
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We provide a principled way for investigators to analyze randomized experiments when the number of covariates is large. Investigators often use linear multivariate regression to analyze randomized experiments instead of simply reporting the difference of means between treatment and control groups. Their aim is to reduce the variance of the estimated treatment effect by adjusting for covariates. If there are a large number of covariates relative to the number of observations, regression may perform poorly because of overfitting. In such cases, the least absolute shrinkage and selection operator (Lasso) may be helpful. We study the resulting Lasso-based treatment effect estimator under the Neyman–Rubin model of randomized experiments. We present theoretical conditions that guarantee that the estimator is more efficient than the simple difference-of-means estimator, and we provide a conservative estimator of the asymptotic variance, which can yield tighter confidence intervals than the difference-of-means estimator. Simulation and data examples show that Lasso-based adjustment can be advantageous even when the number of covariates is less than the number of observations. Specifically, a variant using Lasso for selection and ordinary least squares (OLS) for estimation performs particularly well, and it chooses a smoothing parameter based on combined performance of Lasso and OLS.