Counterfactual Identification and Latent Space Enumeration in Discrete Outcome Models

Abstract This article provides a unified framework for studying the identification of counterfactual parameters in a general class of discrete outcome models, allowing for endogenous regressors and multidimensional latent variables, all without parametric distributional assumptions. Our main theoretical result is that, when the covariates are discrete, the infinite-dimensional latent variable distribution can be replaced with a finite-dimensional version that is equivalent from an identification perspective. The finite-dimensional latent variable distribution is constructed in practice by enumerating regions of the latent variable space with a new and efficient cell enumeration algorithm for hyperplane arrangements. We then show that bounds on a certain class of counterfactual parameters can be computed by solving a sequence of linear programming problems, and show how the researcher can introduce additional assumptions as constraints in the linear programmes. Finally, we apply the method to a mobile phone choice example with heterogeneous choice sets, and to an airline entry game example.