Model Selection for General Multi-Level Group Factor Models with Global, Regional and Local Factors

This paper considers a general multi-level group factor model in which there are R different cross-sectional groups of entities, and latent factors may influence all or only a subset of the R groups. There are r^G global factors that affect all R groups; r^{H(k)} intermediate-level regional factors that affect some k out of the R groups (for k = R − 1, R − 2, ..., 3, and 2, respectively); and r^F group-specific local factors that only affect a single group. We propose a novel projection-based approach to estimating the global factors and corresponding factor loadings. Two new methods called the “projection-based modified eigenvalue ratio” (ProjectionMER) and the “projection-based eigenvalue thresholding” (ProjectionET) are proposed for consistent estimation of the value of r^G. Then, we provide sequential algorithms for estimations of the regional structure, the number of regional/local factors in each region/group, the regional and local factors, and corresponding factor loadings. The proposed methods are easy to implement, and they allow for serial and cross-sectional correlations in the error terms and permit correlations in regional and local factors. Moreover, they do not require prior knowledge of the group affiliation of regional and local factors. Our Monte Carlo simulation results provide promising evidence for the applicability of the methods. We apply the methods to the study of international stock comovements