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

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Forecasting asset returns is hard for reasons that are, at bottom, statistical. This paper gives a self-contained and mathematically rigorous account of why, and of the estimation theory that the difficulty demands. It formalizes the forecasting problem for a stationary, ergodic return process, decomposes predictive risk, and shows that the predictable component of returns is small relative to irreducible noise, so that estimation error governs out-of-sample performance and the finite-sample bias of predictive regression is of first order. It then develops the two technical obstacles that distinguish financial data from textbook settings. The first is serial dependence. Valid inference about forecasts requires limit theory for dependent sequences, variance estimators that are consistent under heteroskedasticity and autocorrelation, and the modern apparatus of predictive-ability testing. The second, and the central theme of the paper, is heavy tails. Returns are regularly varying with a finite tail index, so high moments may not exist, classical central-limit arguments and sample-moment estimators become fragile, and one must turn to extreme-value theory and to tail-index estimation. Particular attention is given to empirical confidence and prediction intervals. When asymptotic normality fails, validity must be recovered by resampling that respects both dependence and heavy tails, and by distribution-free conformal prediction with finite-sample coverage, extended to the non-exchangeable and non-stationary regime that finance inhabits. The paper closes with a workflow and a set of open problems. Its unifying message is that predictive inference on returns is a problem of estimation under dependence and near-infinite-variance tails, and that its solution is a disciplined combination of robust estimation, resampling and distribution-free coverage.

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Shehadi Candela, A. (2026). The Statistics of Return Forecasting: Heavy Tails, Estimation, and the Limits of Predictive Inference. QUAFI Working Paper No. 2026-03. QUAFI Research.

@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}
}

Historial de versiones

1. v1.113 sep 2026exposition revised throughout; structure and register brought in line with journal practice; no change to results or references
2. v1.01 may 2026first working-paper draft