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.