(Mis)Understanding Benign Overfitting in Equity Return Prediction

Hui Guo; Jiawei Huang; Runze Li; Yan Yu

stat.MLAug 24, 2026

(Mis)Understanding Benign Overfitting in Equity Return Prediction

Authors: Hui Guo, Jiawei Huang, Runze Li, Yan Yu

Organizations: Carl H. Lindner College of Business, University of Cincinnati · Católica Lisbon School of Business and Economics, Universidade Católica Portuguesa · Department of Statistics, Pennsylvania State University

Abstract

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.

Figures & tables

Figure 1: Out-of-sample test risk (mean squared error) against

\gamma=P/T

. Each color corresponds to a different training window length (

T=12

60

, and

120

months). Solid lines show the ridgeless estimator (

z\to 0^{+}

); dashed lines show the heavily regularized benchmark (

z=10^{9}

), which virtually overlay their historical averages respectively. The vertical dashed line at

\gamma=1

marks the interpolation threshold.

Panel A: 1931–1974
	Portfolio	$k$	$\text{OOS }R^{2}$	SR	Corr	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$
		$(z=10^{k})$
1	Market			0.46
2	History			0.52		3.00
3	RFF	9	0.00	0.52	1.00	3.00	1.05
4	Linear	9	0.00	0.52	1.00	3.00	-0.24

Table 1: Market-Timing Performance: 12-Month Training Windows

Figure 2: Newey–West $t$ -Statistic of $\alpha_{2}$ for the RFF Model across regularization values: 12-Month Training Windows

Panel A: 1931–1974
	Portfolio	SR	Corr	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$
1	History	0.52		2.99
2	TsMom	0.54	0.95	2.93	-0.21
3	VolMom	0.52	0.64	2.49	0.64
4	DW	0.55	0.93	3.38	1.65
5	PV	0.54	0.41	2.32	0.96

Table 2: Historical Average Benchmark and Momentum Strategies: 12-Month Training Windows

	SR	OOS $R^{2}$	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$
Original Data	0.43	$-$ 0.01	2.82	0.67
Percentile	33.00	0.00	8.00	3.00
Mean	0.43	0.00	2.88	1.33
Median	0.43	0.00	2.88	1.36
Std	0.01	0.00	0.05	0.36
5th Pct	0.42	0.00	2.81	0.77

Table 3: Bootstrap Null Distribution of Out-of-Sample Performance: 12-Month Window

Panel A: 1931–1974
			Non-centered Predictors (KMZ)				Centered Predictors
	Portfolio	$k$	SR	Corr	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$	SR	Corr	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$
		$(z=10^{k})$
1	Market		0.46				0.46
2	History		0.52		3.00		0.52		3.00
3	RFF	9	0.27	0.37	0.78	-0.18	0.18	0.04	1.13	1.02

Table 4: Market-Timing Performance: 12-Month Training Windows with KMZ No Intercept with Non-centered Market Return

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Figure A1: Newey–West $t$ -Statistic of $\alpha_{2}$ for the RFF Model across regularization values: 60-Month Training Windows

Figure A2: Newey–West $t$ -Statistic of $\alpha_{2}$ for the RFF Model across regularization values: 120-Month Training Windows

Panel A: 1935–1974
	Portfolio	$k$	$\text{OOS }R^{2}$	SR	Corr	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$
		$(z=10^{k})$
1	Market			0.52
2	History			0.58		1.44
3	RFF	9	0.00	0.58	1.00	1.44	0.73
4	Linear	9	0.00	0.58	1.00	1.44	1.88

Appendix

Table A1: Market-Timing Performance: 60-Month Training Windows

Panel A: 1940–1974
	Portfolio	$k$	$\text{OOS }R^{2}$	SR	Corr	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$
		$(z=10^{k})$
1	Market			0.53
2	History			0.70		2.12
3	RFF	9	0.00	0.70	1.00	2.12	0.53
4	Linear	9	0.00	0.70	1.00	2.12	1.94

Appendix

Table A2: Market-Timing Performance: 120-Month Training Windows

Panel A: 1935–1974
	Portfolio	SR	Corr	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$
1	History	0.58		1.48
2	TsMom	0.57	1.00	1.38	-1.44
3	VolMom	0.50	0.89	1.21	-0.22
4	DW	0.55	0.94	1.54	0.25
5	PV	0.49	0.77	0.37	-0.25

Appendix

Table A3: Historical Average Benchmark and Momentum Strategies: 60-Month Training Windows

Panel A: 1940–1974
	Portfolio	SR	Corr	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$
1	History	0.70		2.15
2	TsMom	0.69	1.00	2.08	-1.25
3	VolMom	0.60	0.95	1.37	-1.28
4	DW	0.66	0.97	1.83	-0.50
5	PV	0.45	0.83	-0.60	-1.17

Appendix

Table A4: Historical Average Benchmark and Momentum Strategies: 120-Month Training Windows

	SR	OOS $R^{2}$	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$
Original Data	0.46	$-$ 0.00	1.54	1.41
Percentile	19.00	16.00	87.00	0.00
Mean	0.47	0.00	1.44	2.13
Median	0.47	$-$ 0.00	1.44	2.15
Std	0.01	0.00	0.09	0.26
5th Pct	0.45	$-$ 0.00	1.29	1.70

Appendix

Table A5: Bootstrap Null Distribution of Out-of-Sample Performance: 60-Month Window

	SR	OOS $R^{2}$	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$
Original Data	0.56	0.00	2.46	1.90
Percentile	40.00	100.00	100.00	74.00
Mean	0.56	$-$ 0.00	2.04	1.72
Median	0.56	$-$ 0.00	2.03	1.67
Std	0.01	0.00	0.11	0.27
5th Pct	0.55	$-$ 0.00	1.89	1.35

Appendix

Table A6: Bootstrap Null Distribution of Out-of-Sample Performance: 120-Month Window

Panel A: 1935–1974
		k	Non-centered Predictors (KMZ)				Centered Predictors
	Portfolio	$(z=10^{k})$	SR	Corr	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$	SR	Corr	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$
1	Market		0.52				0.52
2	History		0.58		1.44		0.58		1.44
3	RFF	9	0.36	0.40	1.04	0.70	0.09	-0.04	0.83	0.71
4	Linear	9	0.39	0.82	0.30	-0.92	0.25	-0.03	2.02	1.86

Appendix

Table A7: Market-Timing Performance: 60-Month Training Windows with KMZ No Intercept with Non-centered Market Return

Panel A: 1940–1974
		k	Non-centered Predictors (KMZ)				Centered Predictors
	Portfolio	$(z=10^{k})$	SR	Corr	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$	SR	Corr	$t_{\alpha_{1}}$	$t_{\alpha_{2}}$
1	Market		0.53				0.53
2	History		0.70		2.12		0.70		2.12
3	RFF	9	0.38	0.40	1.06	0.63	0.08	-0.05	1.09	0.51
4	Linear	9	0.53	0.86	0.74	-0.56	0.21	-0.19	2.56	1.93

Appendix

Table A8: Market-Timing Performance: 120-Month Training Windows with KMZ No Intercept with Non-centered Market Return

Benign Overfitting

Benign overfitting studies how overparameterized models can fit noisy training data yet still predict accurately on new examples.

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Sep 6, 2021J/K move · Enter open · S save

A Farewell to the Bias-Variance Tradeoff? An Overview of the Theory of Overparameterized Machine Learning

Yehuda Dar, Vidya Muthukumar, Richard G. Baraniuk

Faculty of Computer and Information Science, Ben-Gurion University · Schools of Electrical and Computer Engineering, and Industrial and Systems Engineering, Georgia Institute of Technology · Electrical and Computer Engineering Department, Rice University

Double Descent Neural Network Generalization Benign Overfitting Statistical Learning Theory Overparameterization

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The last decade of progress in machine learning (ML), especially the deep learning era, has raised a number of scientific questions that challenge the longstanding dogma of the field. One of the most important riddles was the good empirical generalization of overparameterized models. Overparameterized models are highly complex with respect to the size of the training dataset, which enables them to perfectly fit (i.e., interpolate) even noisy training data. Such interpolation of noisy data is traditionally associated with detrimental overfitting, and yet a wide range of interpolating models -- from simple linear models to deep neural networks -- have been observed to generalize remarkably well on fresh test data. Indeed, the discovery of the double descent phenomenon has revealed that highly overparameterized models can improve over the best underparameterized model in test performance. Understanding learning in this overparameterized regime required new theory and foundational empirical studies, even for the simplest case of the linear model. The underpinnings of this understanding have been laid in foundational analyses of overparameterized linear regression and related statistical learning tasks, mostly published between 2018 and 2022, which resulted in precise analytic characterizations of double descent. This paper provides an overview of the theory of overparameterized ML (henceforth abbreviated as TOPML) by focusing on explaining the most foundational findings through a statistical signal processing perspective. We emphasize the unique aspects that define the TOPML research area as a subfield of modern ML theory and outline interesting open frontiers that remain.

Abstract

Benign Overfitting

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