  Standard high-dimensional factor models assume that the comovements in a
large set of variables could be modeled using a small number of latent factors
that affect all variables. In many relevant applications in economics and
finance, heterogenous comovements specific to some known groups of variables
naturally arise, and reflect distinct cyclical movements within those groups.
This paper develops two new statistical tests that can be used to investigate
whether there is evidence supporting group-specific heterogeneity in the data.
The first test statistic is designed for the alternative hypothesis of
group-specific heterogeneity appearing in at least one pair of groups; the
second is for the alternative of group-specific heterogeneity appearing in all
pairs of groups. We show that the second moment of factor loadings changes
across groups when heterogeneity is present, and use this feature to establish
the theoretical validity of the tests. We also propose and prove the validity
of a permutation approach for approximating the asymptotic distributions of the
two test statistics. The simulations and the empirical financial application
indicate that the proposed tests are useful for detecting group-specific
heterogeneity.
