Partial identification by extending subdistributions

Abstract

I show that sharp identified sets in a large class of econometric models can be characterized by solving linear systems of equations. These linear systems determine whether, for a given value of a parameter of interest, there exists an admissible joint dis- tribution of unobservables that can generate the distribution of the observed variables. The joint distribution of unobservables is not required to satisfy any parametric re- strictions, but can (if desired) be assumed to satisfy a variety of location, shape and/or conditional independence restrictions. To prove sharpness of the characterization, I generalize a classic result in copula theory concerning the extendibility of subcopulas to show that related objects—termed subdistributions—can be extended to proper dis- tribution functions. I describe this characterization argument as partial identification by extending subdistributions, or PIES. One particularly attractive feature of PIES is that it focuses directly on the sharp identified set for a parameter of interest, such as an average treatment effect, without needing to construct the identified set for the entire model. I apply PIES to univariate and bivariate bivariate response models. A notable product of the analysis is a method for characterizing the sharp identified set for the average treatment effect in Manski’s (1975; 1985; 1988) semiparametric binary response model.