The Statistics of Return Forecasting: Heavy Tails, Estimation, and the Limits of Predictive Inference

Agustín Shehadi Candela · QUAFI Research · 2026-05-01 · v1.0

Abstract

A self-contained, mathematically rigorous account of why forecasting asset returns is hard, and of the estimation theory the difficulty demands. The paper formalizes the forecasting problem for a stationary, ergodic return process and decomposes predictive risk to show that estimation error governs out-of-sample performance and the finite-sample bias of predictive regression is first-order. It then develops the two technical obstacles that distinguish financial data from textbook settings: serial dependence — requiring limit theory for dependent sequences, HAC variance estimation, and modern predictive-ability testing — and heavy tails, the central theme, where returns are regularly varying with a finite tail index so high moments may not exist and classical central-limit and sample-moment arguments become fragile, motivating extreme-value theory (Fisher–Tippett–Gnedenko, Pickands–Balkema–de Haan) and tail-index estimation. Particular attention is given to empirical confidence and prediction intervals: when asymptotic normality fails, validity is recovered by resampling that respects dependence and heavy tails (block and stationary bootstraps, subsampling) and by distribution-free conformal prediction with finite-sample coverage, extended to the non-exchangeable, non-stationary regime finance inhabits. It closes with a rigorous workflow and open problems. Classical theorems are stated with assumptions and attributed to their sources; elementary results are proved.

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JEL codes

Cite

@techreport{shehadi2026statistics,
  author      = {Shehadi Candela, Agust\'in},
  title       = {The Statistics of Return Forecasting: Heavy Tails, Estimation, and the Limits of Predictive Inference},
  institution = {QUAFI Research},
  type        = {QUAFI Working Paper},
  number      = {2026-03},
  year        = {2026}
}

Preliminary working paper; circulated for discussion. The views are the author's own. Not investment advice.