Information Bounds and Impossibility Theorems for Simultaneous Discrete Response Models

This paper considers information bounds for semiparametric models with endogenous regressors, under different support conditions on the outcome variable as well as the endogenous regressor. Our base case is the model where both the outcome variable and the endogenous regressor are binary. Our first result is that under stated conditions, the information for the coefficient on the endogenous variable is 0, implying the impossibility of estimating the coefficient at the parametric rate. This impossibility result extends to to other parameters of interest such as the average treatment effect (ATE). An immediate implication of this result is the need to conduct alternative methods for inference, such as that suggested in Khan and Tamer(2010) or Abrevaya, Hausman, and Khan(2010). Extending from the base model by allowing the support of the endogenous variable to be more rich, we find that the bound can be positive, both for the regression coefficient as well as the average structural function (ASF), suggesting the existence of regular estimators converging at the parametric rate. Finally, while our first set of results applies to triangular discrete systems, we then consider non-triangular systems, of which a leading example is a simultaneous game (e.g. Tamer(2003) and Ardillas-Lopez(2010)). For these models, under stated conditions we attain analogous impossibility theorems for parameters of interest, suggesting the need for alternative inference procedures here as well. JEL Classification: C14, C25, C13.