Impossible Inference in Econometrics: Theory and Applications.

This paper studies models in which hypothesis tests have trivial power, thatis, power smaller than size. This testing impossibility, or impossibility typeA, arises when any alternative is not distinguishable from the null. We alsostudy settings where it is impossible to have almost surely bounded confidencesets for a parameter of interest. This second type of impossibility (type B)occurs under a condition weaker than the condition for type A impossibility:the parameter of interest must be nearly unidentified. Our theoreticalframework connects many existing papers on impossible inference that rely ondifferent notions of topologies to show models are not distinguishable ornearly unidentified. We also derive both types of impossibility using the weaktopology induced by convergence in distribution. Impossibility in the weaktopology is often easier to prove, it is applicable for many widely-used tests,and it is useful for robust hypothesis testing. We conclude by demonstratingimpossible inference in multiple economic applications of models withdiscontinuity and time series models.