Bimodal t-ratios: The Surprising Impact of Thick Tails on Statistical Inference
Carlo Fiorio, Vassilis Hajivassiliou, and Peter Phillips
This paper studies the distribution of the classical t-ratio with data gnerated from distributions with no finite moments and shows how classical testing is affected by bimodality. A key condition in generating bimodality is independence of the observations in the underlying data generating process (DGP). The paper highlights the strikingly different implications of lack of correlation versus statistical independence in DGPs with infinite moments and shows how standard inference can be invalidated in such cases, thereby pointing to the need for adapting estimation and inference procedures to the special problems induced by thick-tailed (TT) distributions. The paper presents theoretical results for the Cauchy case and develops a new distribution termed the ``double Pareto,'' which allows the thickness of the tails and the existence of moments to be determined parametrically. It also investigates the relative importance of tail thickness in case of finite moments by using TT distributions truncated on a compact support, showing that bimodality can persist even in such cases. Simulation results highlight the dangers of relying on naive testing in the face of TT distributions. Novel density estimation kernel methods are employed, given that our theoretical results yield cases that exhibit density discontinuities.
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