Exact Bias Correction for Linear Adjustment of Randomized Controlled Trials

In an influential critique of empirical practice, Freedman \cite{freedman2008A,freedman2008B} showed that the linear regression estimator was biased for the analysis of randomized controlled trials under the randomization model. Under Freedman's assumptions, we derive exact closed-form bias corrections for the linear regression estimator with and without treatment-by-covariate interactions. We show that the limiting distribution of the bias corrected estimator is identical to the uncorrected estimator, implying that the asymptotic gains from adjustment can be attained without introducing any risk of bias. Taken together with results from Lin \cite{lin2013agnostic}, our results show that Freedman's theoretical arguments against the use of regression adjustment can be completely resolved with minor modifications to practice.