  We study the estimation of a high dimensional approximate factor model in the
presence of both cross sectional dependence and heteroskedasticity. The
classical method of principal components analysis (PCA) does not efficiently
estimate the factor loadings or common factors because it essentially treats
the idiosyncratic error to be homoskedastic and cross sectionally uncorrelated.
For efficient estimation it is essential to estimate a large error covariance
matrix. We assume the model to be conditionally sparse, and propose two
approaches to estimating the common factors and factor loadings; both are based
on maximizing a Gaussian quasi-likelihood and involve regularizing a large
covariance sparse matrix. In the first approach the factor loadings and the
error covariance are estimated separately while in the second approach they are
estimated jointly. Extensive asymptotic analysis has been carried out. In
particular, we develop the inferential theory for the two-step estimation.
Because the proposed approaches take into account the large error covariance
matrix, they produce more efficient estimators than the classical PCA methods
or methods based on a strict factor model.
