Impossible Inference in Econometrics: Theory and Applications

TL;DR

This paper explores theoretical limitations in econometric hypothesis testing, demonstrating conditions under which tests have trivial power or confidence sets are unbounded, with implications for economic models with discontinuities and time-series.

Contribution

It introduces a unified framework for understanding impossible inference in econometrics, connecting various notions of indistinguishability and nearly unidentified parameters.

Findings

Impossibility arises when models are indistinguishable or nearly unidentified.

Weak topology simplifies proving impossibility and aids robust testing.

Applications include models with discontinuities and time-series.

Abstract

This paper studies models in which hypothesis tests have trivial power, that is, power smaller than size. This testing impossibility, or impossibility type A, arises when any alternative is not distinguishable from the null. We also study settings in which it is impossible to have almost surely bounded confidence sets 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 theoretical framework connects many existing publications on impossible inference that rely on different notions of topologies to show models are not distinguishable or nearly unidentified. We also derive both types of impossibility using the weak topology induced by convergence in distribution. Impossibility in the weak topology is often easier to prove, it is applicable for many widely-used tests, and it is useful for robust hypothesis testing. We conclude by demonstrating impossible inference in multiple economic applications of models with discontinuity and time-series models.