Minimizing bias in selection on observables estimators when unconfoundness fails

We characterize the bias of propensity score based estimators of common average treatment effect parameters in the case of selection on unobservables. We then propose a new minimum biased estimator of the average treatment effect. We assess the finite sample performance of our estimator using simulated data, as well as a timely application examining the causal effect of the School Breakfast Program on childhood obesity. We find our new estimator to be quite advantageous in many situations, even when selection is only on observables.