Efficient estimation of approximate factor models via penalized maximum likelihood

We study an approximate factor model in the presence of both cross sectional dependence and heteroskedasticity. For efficient estimations it is essential to estimate a large error covariance matrix. We estimate the common factors and factor loadings based on maximizing a Gaussian quasi-likelihood, through penalizing a large covariance sparse matrix. The weighted ℓ1 penalization is employed. While the principal components (PC) based methods estimate the covariance matrices and individual factors and loadings separately, they require consistent estimation of residual terms. In contrast, the penalized maximum likelihood method (PML) estimates the factor loading parameters and the error covariance matrix jointly. In the numerical studies, we compare PML with the regular PC method, the generalized PC method (Choi 2012) combined with the thresholded covariance matrix estimator (Fan et al. 2013), as well as several related methods, on their estimation and forecast performances. Our numerical studies show that the proposed method performs well in the presence of cross-sectional dependence and heteroskedasticity.