Test Assets and Weak Factors

We show that two important issues in empirical asset pricing—the presence of weak factors and the selection of test assets—are deeply connected. Since weak factors are those to which test assets have limited exposure, an appropriate selection of test assets can improve the strength of factors. Building on this insight, we introduce supervised principal component analysis (SPCA), a methodology that iterates supervised selection, principal-component estimation, and factor projection. It enables risk premia estimation and factor model diagnosis even when weak factors are present and not all factors are observed. We establish SPCA's asymptotic properties and showcase its empirical applications.