We introduce a new framework for characterizing identified sets of structural
and counterfactual parameters in econometric models. By reformulating the
identification problem as a set membership question, we leverage the separating
hyperplane theorem in the space of observed probability measures to
characterize the identified set through the zeros of a discrepancy function
with an adversarial game interpretation. The set can be a singleton, resulting
in point identification. A feature of many econometric models, with or without
distributional assumptions on the error terms, is that the probability measure
of observed variables can be expressed as a linear transformation of the
probability measure of latent variables. This structure provides a unifying
framework and facilitates computation and inference via linear programming. We
demonstrate the versatility of our approach by applying it to nonlinear panel
models with fixed effects, with parametric and nonparametric error
distributions, and across various exogeneity restrictions, including strict and
sequential.