Determining the Rank of the Beta Matrix in a Factor Model with Factor-Candidate Regressors

Abstract

Many variables have been proposed as common risk factors driving asset returns, which we refer to as “empirical factors” to distinguish them from true latent factors. We examine how many true latent factors are correlated with the empirical factors by estimating the rank of the beta matrix corresponding to the empirical factors. To obtain more comprehensive empirical results, we analyze data with large numbers of portfolios and individual stock returns. While many rank estimators are available in the literature, most of them are designed for data with a relatively small number of asset returns. Thus, we develop and use a new rank estimator that is consistent regardless of the number of response variables. Analyzing twenty-six empirical factors, we find that they capture at most five different latent risk factors. Additionally, most of the multifactor asset pricing models we consider are found to have very limited power to identify risk premiums.