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Semiparametric Conditional Factor Models in Asset Pricing

Qihui Chen, Nikolai Roussanov, Xiaoliang Wang

TL;DR

This paper develops semiparametric conditional latent factor models for asset pricing by estimating nonlinear alpha and beta surfaces as functions of observed characteristics. It introduces regressed-PCA, a sieve-based, two-step procedure that first extracts characteristic-pure-play portfolios and then applies PCA to obtain latent factors, enabling nonzero alphas and time-varying loadings while accommodating unbalanced panels. The authors establish strong asymptotic results and a bootstrap inference framework, and demonstrate via US stock data that the model uncovers sizable pricing errors that are decreasing over time, with nonlinear specifications delivering substantial improvements in both in-sample and out-of-sample performance. Empirically, regressed-PCA factors consistently price a wide range of testing portfolios better than IPCA and traditional FF5/KNS factors, and trading strategies based on the estimated alphas and factors yield high Sharpe ratios, underscoring the method’s practical relevance for disentangling mispricing from risk and for robust factor construction.

Abstract

We introduce a simple and tractable methodology for estimating semiparametric conditional latent factor models. Our approach disentangles the roles of characteristics in capturing factor betas of asset returns from ``alpha.'' We construct factors by extracting principal components from Fama-MacBeth managed portfolios. Applying this methodology to the cross-section of U.S. individual stock returns, we find compelling evidence of substantial nonzero pricing errors, even though our factors demonstrate superior performance in standard asset pricing tests. Unexplained ``arbitrage'' portfolios earn high Sharpe ratios, which decline over time. Combining factors with these orthogonal portfolios produces out-of-sample Sharpe ratios exceeding 4.

Semiparametric Conditional Factor Models in Asset Pricing

TL;DR

This paper develops semiparametric conditional latent factor models for asset pricing by estimating nonlinear alpha and beta surfaces as functions of observed characteristics. It introduces regressed-PCA, a sieve-based, two-step procedure that first extracts characteristic-pure-play portfolios and then applies PCA to obtain latent factors, enabling nonzero alphas and time-varying loadings while accommodating unbalanced panels. The authors establish strong asymptotic results and a bootstrap inference framework, and demonstrate via US stock data that the model uncovers sizable pricing errors that are decreasing over time, with nonlinear specifications delivering substantial improvements in both in-sample and out-of-sample performance. Empirically, regressed-PCA factors consistently price a wide range of testing portfolios better than IPCA and traditional FF5/KNS factors, and trading strategies based on the estimated alphas and factors yield high Sharpe ratios, underscoring the method’s practical relevance for disentangling mispricing from risk and for robust factor construction.

Abstract

We introduce a simple and tractable methodology for estimating semiparametric conditional latent factor models. Our approach disentangles the roles of characteristics in capturing factor betas of asset returns from ``alpha.'' We construct factors by extracting principal components from Fama-MacBeth managed portfolios. Applying this methodology to the cross-section of U.S. individual stock returns, we find compelling evidence of substantial nonzero pricing errors, even though our factors demonstrate superior performance in standard asset pricing tests. Unexplained ``arbitrage'' portfolios earn high Sharpe ratios, which decline over time. Combining factors with these orthogonal portfolios produces out-of-sample Sharpe ratios exceeding 4.
Paper Structure (45 sections, 49 theorems, 244 equations, 11 figures, 34 tables)

This paper contains 45 sections, 49 theorems, 244 equations, 11 figures, 34 tables.

Table of Contents

  1. Introduction
  2. Estimation Method
  3. Regressed-PCA
  4. Key Properties
  5. Comparing Methods
  6. Asset Pricing Interpretation
  7. Comparison with Portfolio Sorting
  8. Econometric Analysis
  9. Asymptotic Properties
  10. Weighted Bootstrap
  11. Specification Test
  12. Empirical Analysis
  13. Data and Methodology
  14. Empirical Results
  15. Model Estimation
  16. ...and 30 more sections

Key Result

Theorem 4.1

Suppose Assumptions Ass: Basis-Ass: Improvedrates hold. Let $\hat{a}$, $\hat{B},\hat{F}$, $\hat{\alpha}(\cdot)$, and $\hat{\beta}(\cdot)$ be given in Eqn: Estimators. Assume (i) $N\to\infty$; (ii) $T\geq K+1$ ($T$ may stay fixed or grow simultaneously with $N$); (iii) $J\to\infty$ with $J^{2}\xi^{2} where $\kappa>1/2$ is a constant representing the smoothness of $\alpha(\cdot)$ and $\beta(\cdot)$.

Figures (11)

  • Figure 1: $95\%$ confidence intervals for coefficients in $\alpha(\cdot)$ under linear specifications of $\alpha(\cdot)$ and $\beta(\cdot)$ with 36 characteristics
  • Figure 2: Estimates of coefficients in $\beta(\cdot)$ under linear specifications of $\alpha(\cdot)$ and $\beta(\cdot)$ with 36 characteristics (blue: significant at the $5\%$ level; red: insignificant)
  • Figure 3: $95\%$ confidence intervals for $\|a\|^2$, $R^{2}_{f}$, $R^{2}_{f,T,N}$, and $R^{2}_{f,N,T}$ (\ref{['Eqn: R24']}-\ref{['Eqn: R26']}) with $K=10$: subsample analysis
  • Figure 4: Annualized realized excess returns and Sharpe ratios of the pure-alpha portfolio with $K=10$: subsample analysis
  • Figure E.1: Histograms of the 2nd entry in $\sqrt{NT}(\hat{a} - a)$ (blue) and $\sqrt{NT}(\hat{a}^{\ast} - \hat{a})$ (yellow, based on the first simulation replication) when $\theta = 1$, $\delta = 0.5$, and $\rho = 0.3$
  • ...and 6 more figures

Theorems & Definitions (49)

  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Theorem A.1
  • Theorem C.1
  • Lemma C.1
  • Lemma C.2
  • Lemma C.3
  • Lemma C.4
  • ...and 39 more