(Mis)Understanding Benign Overfitting in Equity Return Prediction

Hui Guo; Jiawei Huang; Runze Li; Yan Yu

Source-linked AI summary

(Mis)Understanding Benign Overfitting in Equity Return Prediction

Hui Guo, Jiawei Huang, Runze Li, Yan Yu

arXiv:2608.23761v1Aug 24, 2026stat.MLcs.LGstat.APstat.ME

TL;DR

The paper asks whether benign overfitting extends to equity return prediction despite limited evidence that standard predictors forecast returns. It combines theory and empirical analysis of highly flexible ridge and RFF models, finding that optimal large models converge to the historical average when feature slopes are zero. The conclusions are qualified by assumptions that differ from actual financial time-series conditions.

Problem
The paper examines whether benign overfitting in highly flexible nonlinear models provides genuine equity-return forecasting power when standard predictors have weak evidence of predictability.
Method
The paper develops asymptotic results under zero feature slopes and evaluates RFF ridge models across regularization levels, parameter-to-observation ratios, and training settings.
Results
The asymptotically best large model converges to the historical average, and the RFF model closely tracks that benchmark with negligible incremental forecasting value.
Takeaways & Limitations
In the absence of a true signal, highly flexible machine-learning architectures asymptotically collapse to the historical-average benchmark rather than improving equity-return forecasts.
Takeaways & Limitations
The theory relies on assumptions such as i.i.d. errors, while market returns are time-series data with persistent predictors, heteroskedasticity, and limited rolling windows.

Abstract

from arXiv · show

Highly overparameterized models often predict well despite interpolating training data in complex domains, challenging the classical bias--variance tradeoff. We investigate whether this ``benign overfitting'' phenomenon extends to equity return prediction. Consistent with recent statistical theory, we document two key phenomena: first, a double descent pattern in the ridgeless model's prediction risk; and second, that while the optimal ridge model consistently outperforms its ridgeless counterpart, this performance gap becomes negligible at large parameter-to-observation ratios. Ultimately, however, both models fail to outperform a simple historical average. This empirical evidence aligns with our asymptotic results under the null hypothesis of zero slope coefficients, suggesting that standard equity predictors lack true forecasting power---even within highly flexible, nonlinear machine learning architectures. These findings reconcile modern and classical machine learning in asset pricing: in the absence of a true signal, they asymptotically collapse to the historical average benchmark.

1 Introduction

The paper tests whether benign overfitting extends to equity return prediction, where standard predictors have weak out-of-sample evidence but flexible nonlinear models may recover signals. It finds double descent in ridgeless risk, yet complex models ultimately do not improve on the historical average.

Motivation: Benign overfitting describes models that nearly interpolate training data while maintaining or improving out-of-sample performance in highly overparameterized regimes.The relevant complexity ratio is γ = P/T; for ridge regression, effective degrees of freedom also incorporate shrinkage.
Motivation: Equity return predictability remains difficult to establish, while recent nonlinear machine learning results suggest flexible models may recover economically meaningful signals.The setting combines weak signals, persistent correlated predictors, potential nonlinearities, and severe overparameterization.
Empirical findings: The ridgeless RFF estimator exhibits double descent: test risk spikes near γ = P/T = 1, then declines as γ increases.The interpolation threshold is γ = 1.
Empirical findings: A heavily regularized ridge model has lower risk across γ and tracks the historical-average forecast, while optimal regularization bypasses the double-descent curve.Under zero true slope coefficients, the optimal ridge penalty diverges, z* = ∞, collapsing the model toward the historical average.
Implications: The historical average outperforms the complex ridgeless estimator, challenging the interpretation that greater complexity inherently extracts true equity-return signals.When features lack predictive power, optimally regularized architectures collapse back to the historical-average benchmark.
Contributions: The paper proves that zero slope coefficients imply a diverging optimal ridge penalty and establishes the historical average as the theoretically justified benchmark.Cross-validation achieves the same asymptotic prediction risk as the oracle-optimal penalty, and bootstrap results find no significant advantage for actual predictors over randomly resampled covariates.
Implications: The evidence indicates that RFF forecasting power can vanish after mean-centering predictors and that apparent performance may reflect correlation with the historical average.The same reliance on the historical average extends beyond RFFs to momentum strategies.

2 Model, Estimation, and Theoretical Properties

The paper models equity returns with high-dimensional random Fourier features and ridge regression, using effective degrees of freedom to characterize complexity. Under zero slope coefficients, theory links optimal regularization and prediction risk to the historical-average forecast, while cross-validation is shown to be asymptotically optimal.

2.1 Model: The model transforms low-dimensional predictors G_t into a high-dimensional random Fourier feature vector S_t and uses a linear model to predict excess returns.The feature dimension P can exceed the number of observations T.
2.2 Estimation: Ridge regression penalizes slope coefficients while leaving the intercept unpenalized, equivalently centering the data before fitting and recovering the intercept afterward.The ridgeless estimator is the z→0+ limit, corresponding to minimum-norm interpolation.
2.3 Effective Degrees of Freedom: A Measure of Model Complexity: Effective degrees of freedom combine predictor count and shrinkage into a scalar complexity measure, decreasing with stronger regularization.As z→∞, EDF_ridge(z) approaches 1, corresponding to a pure intercept or historical mean.
2.4 Theoretical Results under the Null of Zero Slope: Under the null hypothesis of zero slopes, the asymptotically optimal ridge penalty is z*=∞ and prediction error converges to σ^2.The historical average is therefore the optimal next-period forecast under the null.
2.5 Ridge Regression Penalty Parameter and Cross-Validation: Because true prediction error is unknown empirically, cross-validation selects the shrinkage penalty and achieves the same asymptotic prediction error as the optimally tuned estimator.The paper provides stronger uniform consistency results than the cited earlier theory.

3 Empirical Results

Across regularization choices, subsamples, and training windows, RFF forecasts do not improve on the historical-average benchmark. Apparent timing performance is largely absorbed by that benchmark, while heavy regularization mechanically shrinks the model toward zero slopes and the historical average.

3.2 RFF Model versus Intercept-Only Model: At z = 10^9, the RFF forecast has correlation 1.00 with the historical-average benchmark and an essentially zero R^2.The heavily regularized model converges toward the historical average.
3.2 RFF Model versus Intercept-Only Model: The RFF and intercept-only models yield nearly identical Sharpe ratios and CAPM alpha statistics, indicating no marginal predictive content beyond the unconditional mean.This comparison is reported for the 12-month training window.
3.2 RFF Model versus Intercept-Only Model: Cross-validation selects the maximum regularization value of 10^9 in 47.68% of months, pushing slope coefficients toward zero and collapsing the model onto the historical average.The cross-validated portfolio has out-of-sample R^2 of −0.01 across both sub-periods and the full sample.
3.2 RFF Model versus Intercept-Only Model: Across the three periods, all t_alpha2 statistics remain below the 10% threshold, so neither ridgeless nor regularized RFF models enhance performance beyond the historical-average managed benchmark.This robustness holds across regularization parameters and sub-sample periods.
3.3 RFF Model and Momentum Strategies: Momentum strategies likewise produce no significant abnormal returns after controlling for the historical-average benchmark, and their out-of-sample performance is largely explained by benchmark variation.In the post-1975 sample, none delivers a statistically significant CAPM alpha.

4 Conclusion

The paper finds that overparameterization can produce double descent without granting equity-return forecasting power. Under weak or absent signal, flexible models converge toward the historical-average benchmark rather than delivering a virtue of complexity.

Highly parameterized machine-learning models motivate testing whether benign overfitting extends to market-return prediction.
The ridgeless random Fourier features estimator exhibits double descent, but this pattern does not imply that large models outperform small ones.
When feature slope coefficients are zero, the optimal ridge penalty is infinite, so the best large model converges to the historical average.
The RFF model closely tracks the historical average and exhibits negligible incremental market-timing ability.
The results do not reject the broader promise of machine learning because equity-return relationships are time-varying and may require architectures tailored to those dynamics.

A.1 Assumption and Proof of Theorem 1: High-dimensional 𝑃Asymptotics

Under the null of market-return unpredictability, the high-dimensional ridge theory identifies infinite optimal shrinkage and shows that model error vanishes while prediction error approaches the noise variance.

The proof studies high-dimensional ridge asymptotics under isotropic and general covariance features, using population-centered notation.
The limiting ridge-risk minimizer is z*=γσ2/r2, where r2=∥β∥2; under β=0, r2=0 and the optimal penalty is infinite.
For fixed z>0, variance remains nonvanishing even though the squared-bias term vanishes under the null hypothesis of return unpredictability.
As z approaches infinity, the asymptotic bias and variance terms both vanish, including under general covariance matrices with finite feature-spectrum moments.
At the optimal z*=∞, model error tends to zero and prediction error tends to σ2 as P,T→∞ with P/T→γ∈(0,∞).

A.2 Assumption, Sketch and Remark of Proof of Theorem 2: CV Uniform Consistency

The second theorem establishes that generalized cross-validation consistently selects ridge shrinkage asymptotically, matching the prediction error of the optimally tuned ridge estimator under stated high-dimensional assumptions.

The theorem assumes a linear signal-plus-independent-error model with covariance-transformed predictor rows and finite higher moments.
The covariance matrix is deterministic positive definite with eigenvalues uniformly bounded between rmin>0 and rmax<∞.
GCV replaces each hat-matrix diagonal element with tr(Hc(z))/T, improving numerical stability while using effective degrees of freedom as a complexity measure.
The GCV-selected parameter is ẑGCV=arg minz∈Z GCV(z).
The GCV-tuned ridge estimator achieves the same asymptotic prediction error as the optimally tuned ridge estimator.

B Predictor Variables

The predictor set combines the fourteen Goyal–Welch variables with the term spread and related bond, inflation, variance, equity-issuance, valuation, and yield measures.

The study uses the fourteen predictor variables from Goyal and Welch (2008) alongside the term spread.
Valuation measures include dividend-price, dividend-yield, earnings-price, dividend-payout, and book-to-market ratios.
The Treasury bill rate and long-term yield are included as interest-rate predictors.
Default yield spread is the BAA–AAA corporate-bond yield difference, while default return spread compares long-term corporate and government bond returns.
The predictors include long-term government-bond return, monthly inflation, stock-market variance, and net equity expansion.

C Additional Tables and Figures

This section provides additional market-timing tables, bootstrap distributions, and figures across 60- and 120-month training windows. It documents model definitions, benchmark conventions, and inference procedures used in these analyses.

Additional tables and figures examine training sample lengths of 60 and 120 months.
The market-timing tables compare the historical average, momentum strategies, linear models, and nonlinear RFF models with 12,000 features.RFF denotes the nonlinear model, while Linear denotes the model using the 15 original predictors.
Bootstrap tables report original-data performance percentiles and pooled percentiles across training windows, alongside distributional summaries from 100 replications.Each replication uses 100 independently drawn sets of random Fourier features.
Reported outcomes include annualized Sharpe ratios, out-of-sample R^2, and Newey–West t-statistics for intercepts with and without the historical average.The intercept statistics use four Newey–West lags and correspond to regressions on the market or on the market plus the historical average.
The appendix distinguishes non-centered, centered, and fully demeaned predictor specifications, with regularization indexed by z = 10^k and a ridgeless case at z = 0.Cross-validation selects z for the CV specification.