High-dimensional probabilistic PCA with strong and weak factors: Estimation of noise variance

Factor models are widely used in economics for modeling panel data. In this paper, we consider a consistent estimator of the homoscedastic noise variance in high-dimensional factor models, also known as probabilistic principal component models. Given that the maximum likelihood estimator of noise variance has a downward bias, and to our best knowledge, most of the literature requires that the variances of principal components are bounded, which motivates us to study the case when the variances of principal components can grow to infinity as the dimension p increases. This also corresponds to a factor model, which includes both strong and weak factors—a very common situation in practice. The superiority of the proposed estimator is checked by several Monte Carlo experiments. Moreover, we also consider a goodness-of-fit testing problem for probabilistic principal component models in a high-dimensional setting. The proposed result is valid without the Gaussian assumption, and the variances of the principal components are allowed to diverge to infinity. Its good performance is confirmed by a simulation study and real data analysis.