  This paper considers the maximum likelihood estimation of factor models of
high dimension, where the number of variables (N) is comparable with or even
greater than the number of observations (T). An inferential theory is
developed. We establish not only consistency but also the rate of convergence
and the limiting distributions. Five different sets of identification
conditions are considered. We show that the distributions of the MLE estimators
depend on the identification restrictions. Unlike the principal components
approach, the maximum likelihood estimator explicitly allows
heteroskedasticities, which are jointly estimated with other parameters.
Efficiency of MLE relative to the principal components method is also
considered.
