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Selecting and Testing Asset Pricing Models: A Stepwise Approach

Guanhao Feng, Wei Lan, Hansheng Wang, Jun Zhang

TL;DR

This paper tackles the problem of selecting asset pricing models that price cross-sectional returns while considering unselected candidate factors. It introduces a forward stepwise evaluation (FSE) to add factors and a backward stepwise evaluation (BSE) to prune redundancies, guided by SR^2 gains and GRS improvements, and halted by a high-dimensional alpha (HDA) test. The framework identifies an efficient, minimal model that jointly prices test assets and candidate factors, with theoretical guarantees on screening consistency and practical validation on U.S. equity data (1973–2021) across 97 factors and 285 test assets. Empirically, an 8-factor model (MKT, REG, PEAD, HMLM, STR, ILR, SMB, EPRD) emerges as the central solution, pricing unselected factors well and delivering strong out-of-sample investment performance (annualized Sharpe ~2.73 in-sample, ~1.53 OOS), while maintaining robustness to transaction costs. The approach provides a rigorous, economically grounded path to model selection in asset pricing, with broad applicability to factor testing, portfolio optimization, and future integration with machine-learning baselines.

Abstract

The asset pricing literature emphasizes factor models that minimize pricing errors but overlooks unselected candidate factors that could enhance the performance of test assets. This paper proposes a framework for factor model selection and testing by (i) selecting the optimal model that spans the joint efficient frontier of test assets and all candidate factors, and (ii) testing pricing performance on both test assets and unselected candidate factors. Our framework updates a baseline model (e.g., CAPM) sequentially by adding or removing factors based on asset pricing tests. Ensuring model selection consistency, our framework utilizes the asset pricing duality: minimizing cross-sectionally unexplained pricing errors aligns with maximizing the Sharpe ratio of the selected factor model. Empirical evidence shows that workhorse factor models fail asset pricing tests, whereas our proposed 8-factor model is not rejected and exhibits robust out-of-sample performance.

Selecting and Testing Asset Pricing Models: A Stepwise Approach

TL;DR

This paper tackles the problem of selecting asset pricing models that price cross-sectional returns while considering unselected candidate factors. It introduces a forward stepwise evaluation (FSE) to add factors and a backward stepwise evaluation (BSE) to prune redundancies, guided by SR^2 gains and GRS improvements, and halted by a high-dimensional alpha (HDA) test. The framework identifies an efficient, minimal model that jointly prices test assets and candidate factors, with theoretical guarantees on screening consistency and practical validation on U.S. equity data (1973–2021) across 97 factors and 285 test assets. Empirically, an 8-factor model (MKT, REG, PEAD, HMLM, STR, ILR, SMB, EPRD) emerges as the central solution, pricing unselected factors well and delivering strong out-of-sample investment performance (annualized Sharpe ~2.73 in-sample, ~1.53 OOS), while maintaining robustness to transaction costs. The approach provides a rigorous, economically grounded path to model selection in asset pricing, with broad applicability to factor testing, portfolio optimization, and future integration with machine-learning baselines.

Abstract

The asset pricing literature emphasizes factor models that minimize pricing errors but overlooks unselected candidate factors that could enhance the performance of test assets. This paper proposes a framework for factor model selection and testing by (i) selecting the optimal model that spans the joint efficient frontier of test assets and all candidate factors, and (ii) testing pricing performance on both test assets and unselected candidate factors. Our framework updates a baseline model (e.g., CAPM) sequentially by adding or removing factors based on asset pricing tests. Ensuring model selection consistency, our framework utilizes the asset pricing duality: minimizing cross-sectionally unexplained pricing errors aligns with maximizing the Sharpe ratio of the selected factor model. Empirical evidence shows that workhorse factor models fail asset pricing tests, whereas our proposed 8-factor model is not rejected and exhibits robust out-of-sample performance.
Paper Structure (52 sections, 4 theorems, 47 equations, 6 figures, 14 tables, 2 algorithms)

This paper contains 52 sections, 4 theorems, 47 equations, 6 figures, 14 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Methodology
  3. Contribution and comparison with existing methods.
  4. Empirical Findings
  5. Other Related Literature
  6. Methodology: Stepwise Evaluation
  7. Efficient Factor Models and the Optimal Model
  8. Stepwise Evaluation Framework
  9. Factor Selection Criterion: Maximal $\text{SR}^2$
  10. Two non-nested model comparisons.
  11. Multiple non-nested model comparisons.
  12. Model Stop Criterion: High-dimensional Alpha Test
  13. Baseline Model and Stepwise Evaluation
  14. Toy Model for Illustrations
  15. Data
  16. ...and 37 more sections

Key Result

Theorem 1

Under model (eq: true_model) and Assumptions assump1-assump4 in Appendix sec:assumption, suppose the threshold value of the HDA test satisfies $z_{\lambda}^1 \lesssim T N^{-1/2- n_2 }$, we have as $(N, T) \to \infty$, and $|\mathcal{M}_b^F| =O(N^{n_1+2n_2})$ with $n_1 + 2n_2 < 1/3$, where $n_1$ and $n_2$ are defined in Assumption assump1.

Figures (6)

  • Figure 1: Expanded Factor Models (FSE Process)
  • Figure 2: Expanded Factor Models (FF3 Example)
  • Figure 3: Reduced Factor Models (BSE Process)
  • Figure 4: Reduced Factor Models (FF3$^F$ Example)
  • Figure A.1: Illustration Example of Stepwise Evaluation
  • ...and 1 more figures

Theorems & Definitions (4)

  • Theorem 1: Screening Consistency
  • Theorem 2: Selection Consistency
  • Lemma 1
  • Lemma 2