Large factor model estimation by nuclear norm plus ℓ1 norm penalization

This paper provides a comprehensive estimation framework via nuclear norm plus ℓ1 norm penalization for high-dimensional approximate factor models with a sparse residual covariance. The underlying assumptions allow for non-pervasive latent eigenvalues and a prominent residual covariance pattern. In that context, existing approaches based on principal components may lead to misestimate the latent rank. On the contrary, the proposed optimization strategy recovers with high probability both the covariance matrix components and the latent rank and the residual sparsity pattern. Conditioning on the recovered low rank and sparse matrix varieties, we derive the finite sample covariance matrix estimators with the tightest error bound in minimax sense and we prove that the ensuing estimators of factor loadings and scores via Bartlett’s and Thomson’s methods have the same property. The asymptotic rates for those estimators of factor loadings and scores are also provided.