Inferential Theory for Pricing Errors with Latent Factors and Firm Characteristics

TL;DR

This paper introduces a new framework for analyzing pricing errors in factor models that incorporate latent factors and firm characteristics, providing improved estimation, inference, and empirical insights into mispricing components.

Contribution

It develops a generalized model and estimators that address orthogonality and basis issues, with a rigorous inferential theory accommodating many characteristics.

Findings

Strong evidence of inside and outside alphas in US stocks

Inside alpha shows industry-level co-movements

Outside alpha reflects idiosyncratic shocks beyond fundamentals

Abstract

We study factor models that combine latent factors with firm characteristics and propose a new framework for modeling, estimating, and inferring pricing errors. Following Zhang (2024), our approach decomposes mispricing into two distinct components: inside alpha, explained by firm characteristics but orthogonal to factor exposures, and outside alpha, orthogonal to both factors and characteristics. Our model generalizes those developed recently such as Kelly et al. (2019) and Zhang (2024), resolving issues of orthogonality, basis dependence, and unit sensitivity. Methodologically, we develop estimators grounded in low-rank methods with explicit debiasing, providing closed-form solutions and a rigorous inferential theory that accommodates a growing number of characteristics and relaxes standard assumptions on sample dimensions. Empirically, using U.S. stock returns from 2000-2019, we document strong evidence of both inside and outside alphas, with the former showing industry-level co-movements and the latter reflecting idiosyncratic shocks beyond firm fundamentals. Our framework thus unifies statistical and characteristic-based approaches to factor modeling, offering both theoretical advances and new insights into the structure of pricing errors.