In defining selection bias, we have considered only the parallel universe of the treated group or the untreated group rather than including the parallel universe of the untreated group or the treated group. This makes causal inference theories unbalanced because they were developed on one side...

We provide novel bounds on average treatment effects (on the treated) that are valid under an unconfoundedness assumption. Our bounds are designed to be robust in challenging situations, for example, when the conditioning variables take on a large number of different values in the observed...