Counterfactual worlds: Characterizing the identifying power of incomplete models with conditional and marginal independence restrictions

We study a generalization of the treatment e¤ect model in which an observed discrete classi…er indicates in which one of a set of counterfactual processes a decision maker is observed. The other observed outcomes are delivered by the particular counterfactual process in which the decision maker is found. Models of the counterfactual processes can be incomplete in the sense that even with knowledge of the values of observed exogenous and unobserved variables they may not deliver a unique value of the endogenous outcomes. We study the identifying power of models of this sort that incorporate (i) conditional independence restrictions under which unobserved variables and the classi…er variable are stochastically independent conditional on some of the observed exogenous variables and (ii) marginal independence restrictions under which unobservable variables and a subset of the exogenous variables are independently distributed. We use random set theory methods to characterize the identifying power of these models for fundamental structural relationships and probability distributions and for interesting functionals of these objects, some of which may be point identi…ed. In one example of an application, we observe the entry decisions of …rms that can choose which of a number of markets to enter and we observe various endogenous outcomes delivered in the markets they choose to enter.