Determining the Rank of the Beta Matrix in a Factor Model with

This paper proposes a new method to estimate the rank of the beta matrix in a factor model. We consider the case in which possible factor variables, which we call factor-candidate variables, are observed. The idiosyncratic error terms are allowed to be correlated both over different cross section units and over different time periods. For the factor model, estimating the rank of the beta matrix is equivalent to estimating the number of the relevant factors among the factor-candidate variables. The estimator we propose is easy to use because it is computed with the eigenvalues of the inner product of an estimated beta matrix. Simulation results show it works well even if small samples are used. We confirm that all of the Fama-French (1993) three factors have power to explain US stock returns. In contrast, only one or two among the five factors of Chen, Roll, and Ross (1986) have explanatory power. When we combine the three factors of Fama-French (FF) together with the five factors of Chen, Roll and Ross (CRR) we find that a factor not captured by FF is captured by CRR. Furthermore, we find that momentum factors capture a source of risk not captured by either FF or CRR. Finally, we find evidence for five factors in US stock returns among the eleven factor candidates used