Difference-in-Means Estimator

Its unbiasedness comes from the randomization itself, not from a model of the outcomes. Because assignment is independent of the potential outcomes, the treated group is, in expectation, a representative sample of the whole population's treated potential outcomes, and likewise for controls. The expected treated mean therefore equals the average treated potential outcome and the expected control mean equals the average control potential outcome, so their difference has expectation equal to the average treatment effect. No assumption about functional form or error distribution is needed.

Regressing the outcome on a treatment indicator with no covariates yields exactly the difference in means as the coefficient on treatment, so the two are numerically identical in the simplest case. They diverge once covariates are added: regression adjustment can improve precision but introduces the risk of bias in finite samples, which design-based corrections such as Lin's covariate-adjusted estimator address. The difference-in-means estimator with robust or randomization-based standard errors remains the assumption-light benchmark.

The true sampling variance of the difference in means depends on the variances of both potential outcomes and on their covariance across units. That covariance can never be observed, because each unit reveals only one potential outcome. Neyman's estimator effectively assumes the least favorable case for that unidentified term, which makes the resulting variance an upper bound on the true variance under constant treatment effects. As a result, confidence intervals built from it tend to be wider than necessary, erring toward caution rather than overconfidence.