Pitfalls of First-Stage Diagnostics and Valid Inference in IV Regression with Heavy Tailed Variables

Abstract

This paper investigates instrumental variable (IV) estimation in linear instrumental variable regressions with heavy-tailed distributions. We allow for the possibility of weak instruments in the conventional sense, meaning that the first-stage regression coefficients may converge to zero as the sample size increases. Within this framework, we generalize the notion of weak instruments by defining vague instruments as those for which the first-stage F-statistic remains stochastically bounded. Notably, under certain heavy-tailed distributions, instruments can be vague even when the first-stage coefficients are constant. We show that the IV estimator can remain consistent even if the IV is vague. However, standard inference procedures—such as the IV t-test and weak-instrument robust methods—may fail to provide valid inference regardless of the vagueness of the instruments. To address this, we propose a valid inference procedure based on the second-stage F-statistic, incorporating a directional transformation of the instruments. We establish that the proposed test controls the size correctly and achieves consistency, provided the instruments are not severely vague.