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Abstract:
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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. In this article 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 $\kappa \log p\rightarrow 0$ and asymptotically normal when $\kappa p \rightarrow 0$ under mild moment conditions, where $\kappa$ is the maximum leverage score of the covariate matrix. In addition, we propose a bias-corrected estimator that is consistent when $\kappa \log p\rightarrow 0$ and is asymptotically normal, with the same variance in the fixed-p regime, when $\kappa^2 p\log p \rightarrow 0$. In the favorable case where leverage scores are all close together, the latter condition reduces to $p = o(n^{2/3}/(\log n)^{1/3})$. Similar to Lin [2013], our results hold without any model specification. This work is closed related to Professor Brown's work on sampling theory, inference on misspecified model and causal inference.
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