Dake Zhang, Assistant Professor of Finance at Antai College of Economics and Management (ACEM), Shanghai Jiao Tong University, together with Stefano Giglio of Yale University and Dacheng Xiu of the University of Chicago, has published a new study that addresses one of the most persistent challenges in empirical asset pricing: the "weak-factor problem."
The paper, "Test Assets and Weak Factors," appeared in February 2025 in The Journal of Finance, the field's leading journal. It proposes Supervised Principal Component Analysis (SPCA), a method that selectively filters test assets before extracting factors — an approach analogous to curating high-quality training data before feeding it into a large language model.
Why test-asset choice matters
Estimating factor risk premiums is a core task in asset pricing, and researchers must select a set of test assets to do so. Conventional models assume that factors exert strong explanatory power across all assets. In reality, however, a factor may only be exposed in a small subset of a large asset pool, causing it to appear statistically weak — and rendering standard risk-premium estimates unreliable.
The study challenges a long-held assumption: that factor strength is an intrinsic property of the factor itself. Instead, the authors show that whether a factor appears strong or weak depends fundamentally on which test assets are used. A liquidity factor, for instance, looks weak when tested against market-cap and book-to-market sorted portfolios — because those portfolios have already diversified away liquidity risk. The same factor becomes strongly explanatory when liquidity-sorted portfolios are used as test assets.
How SPCA works
Traditional PCA is unsupervised: it draws on the full universe of test assets to identify components with the greatest overall variation, allowing weak-factor signals to be drowned out by idiosyncratic noise. SPCA overcomes this through an iterative three-step process: it first screens assets by their correlation with the target factor, retaining only highly informative ones; it then extracts principal components from this curated subset; finally, it projects out explained variation and repeats the process on residuals.
The result is a framework that not only surfaces weak-factor signals but also detects and controls for latent omitted factors. Theoretical derivations and finite-sample simulations confirm that SPCA delivers unbiased and consistent risk-premium estimates even when unobserved weak variables are present.
Two practical contributions
The method has two key applications. First, it can recover the true premiums of non-traded macroeconomic factors — such as consumption growth or market liquidity — by adaptively selecting the most sensitive stocks and constructing optimal mimicking portfolios that strip out measurement noise. Second, SPCA serves as a diagnostic tool: by comparing maximum Sharpe ratios from extracted latent components against those from observed factors, researchers can identify whether a pricing model is missing important drivers or whether its test-asset set lacks representativeness.
"There are no inherently weak factors," the authors conclude, "only inadequately chosen test perspectives."
About the author
Dake Zhang holds a Ph.D. in Econometrics and Statistics from the University of Chicago Booth School of Business, where he also earned an M.B.A. He received an M.A. in Statistics from the University of Chicago and a B.S. in Mathematics from Tsinghua University. His research focuses on financial machine learning, empirical asset pricing, factor models, and reinforcement learning.
Zhang shared thoughtful advice for students pursuing cross-disciplinary studies in the AI era. He believes the booming development of large language models greatly lowers interdisciplinary learning barriers and empowers curious young researchers to explore diverse fields.
“Explore broadly to identify topics that genuinely spark your curiosity, then dive deep,” Zhang suggested, encouraging students to discover their own research interests through continuous exploration and practice.
