Impossible Inference in Econometrics: Theory and Applications to Regression Discontinuity, Bunching, and Exogeneity Tests

This paper presents necessary and sufficient conditions for tests to have trivial power. By inverting these impractical tests, we demonstrate that the bounded confidence regions have error probability equal to one. This theoretical framework establishes a connection among many existing impossibility results in econometrics, those using the total variation metric and those using the L\'{e}vy-Prokhorov metric (convergence in distribution). In particular, the theory establishes conditions under which the two types of impossibility exist in econometric models. We then apply our theory to Regression Discontinuity Design models and exogeneity tests based on bunching.