Potential outcomes and finite-population inference for M-estimators

When a sample is drawn from or coincides with a finite population, the uncertainty of the coefficient estimators is often reported assuming the population is effectively infinite. The recent literature on finite-population inference instead derives an alternative asymptotic variance of the ordinary least squares estimator. Here, I extend the results to the more general setting of M-estimators and also find that the usual robust 'sandwich' estimator is conservative. The proposed asymptotic variance of M-estimators accounts for two sources of variation. In addition to the usual sampling-based uncertainty arising from (possibly) not observing the entire population, there is also design-based uncertainty, which is usually ignored in the common inference method, resulting from lack of knowledge of the counterfactuals. Under this alternative framework, we can obtain smaller standard errors of M-estimators when the population is treated as finite.