Robust Likelihood Ratio Tests for Incomplete Economic Models

Abstract

Economic models with multiple equilibria, self-selection, or weak behavioral restrictions often make set-valued predictions and therefore do not imply a unique likelihood. This paper develops robust likelihood-ratio tests for structural hypotheses in such incomplete models. We evaluate tests by their power guarantee, defined as the smallest rejection probability over all selection mechanisms compatible with the alternative, while requiring uniform size control over all null selections. Using the Huber--Strassen theory of least favorable pairs, we construct finite-sample minimax likelihood-ratio tests. The main result shows that, in repeated experiments, the least favorable pair is the product of the single-experiment least favorable pairs whenever the latent variables are independent across experiments. This product structure holds even though unrestricted selection may induce arbitrary heterogeneity and dependence in the observed outcomes. It reduces a high-dimensional robust testing problem to single-experiment calculations and yields exact finite-sample critical values and Gaussian approximations. For directed one-sided alternatives, we provide conditions under which conditioning on a selection-invariant statistic delivers exact conditional uniformly most powerful (UMP) tests with respect to the power guarantee. In the examples, these optimal tests are simple and interpretable, using selection-invariant features of the data that are directly tied to the hypothesis of interest. Monte Carlo experiments in entry-game and Roy-model designs illustrate size control, power, and the role of robust testability.