Common Factors and Specific Factors

In this paper we study factor models for security returns on financial markets, where some pervasive factors are common across all securities and other pervasive factors prevail only within some groups of securities but not in others. This kind of structured factors allow a more nuanced analysis of determinants of the security returns, in particular, they allow to study clustering structures in security returns as well as their determinants. The clustering structure provides a natural way to group the securities and to interpret common factors and group-specific factors. We give conditions under which the common factor space and the group-specific factor spaces can be identified, and propose an effective procedure to estimate the unobservable structure in the factor space. Concretely, the procedure will determine the unknown number of groups, endogenously classify securities into groups, determine the number of common factors across all groups as well as the number of group-specific factors in each group, and estimate the common factors and the group-specific factors. The estimated factor structure will provides a more meaningful interpretation of the estimated factors in practical applications.