Causal Regression Models II: Unconfoundedness and Causal Unbiasedness

We consider regression models with discrete units and a discrete treatment variable. In this framework, individual and average causal effects as well as causal unbiasedness of conditional expected values E(Y | X = x) and of their differences were defined in a previous paper where it was also noted that a hypothesis of causal unbiasedness is not empirically testable outside the randomized experiment. Therefore, we study a stronger causality criterion which we call "unconfoundedness". To our knowledge, this is the weakest empirically testable condition implying causal unbiasedness of the conditional expected values E(Y | X = x). Unconfoundedness holds in randomized experiments, but it may hold in nonrandomized experiments, as well. We derive theorems about sufficient and necessary conditions, about sufficient conditions, and about necessary conditions for unconfoundedness. The latter identify the hypotheses to be tested in nonrandomized experiments when it comes to testing the weakest empirically testable sufficient condition for conditional expected values E(Y | X = x ) to be causally unbiased.