  In this paper, a statistical model for panel data with unobservable grouped
factor structures which are correlated with the regressors and the group
membership can be unknown. The factor loadings are assumed to be in different
subspaces and the subspace clustering for factor loadings are considered. A
method called least squares subspace clustering estimate (LSSC) is proposed to
estimate the model parameters by minimizing the least-square criterion and to
perform the subspace clustering simultaneously. The consistency of the proposed
subspace clustering is proved and the asymptotic properties of the estimation
procedure are studied under certain conditions. A Monte Carlo simulation study
is used to illustrate the advantages of the proposed method. Further
considerations for the situations that the number of subspaces for factors, the
dimension of factors and the dimension of subspaces are unknown are also
discussed. For illustrative purposes, the proposed method is applied to study
the linkage between income and democracy across countries while subspace
patterns of unobserved factors and factor loadings are allowed.
