Finite Sample Inference in Incomplete Models

We propose confidence regions for the parameters of incomplete models with exact coverage of the true parameter in finite samples. Our confidence region inverts a test, which generalizes Monte Carlo tests to incomplete models. The test statistic is a discrete analogue of a new optimal-transport formulation of the structural model. Both test statistic and critical values rely on simulation draws from the distribution of latent variables and are computed using solutions to discrete optimal transport, hence linear programming problems. We also propose a fast preliminary search in the parameter space with an alternative, more conservative yet consistent test, based on a parameter-free critical value. We compare size and power of our procedure with competing proposals in simulations based on a regression with interval valued regressors. Finally, we apply our methodology to the model of airline entry and price competition in Ciliberto et al. [2021].