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This paper investigates the properties of Quasi Maximum Likelihood estimationof an approximate factor model for an $n$-dimensional vector of stationary timeseries. We prove that the factor loadings estimated by Quasi Maximum Likelihoodare asymptotically equivalent, as $n\to\infty$, to those estimated viaPrincipal Components. Both estimators are, in turn, also asymptoticallyequivalent, as $n\to\infty$, to the unfeasible Ordinary Least Squares estimatorwe would have if the factors were observed. We also show that the usualsandwich form of the asymptotic covariance matrix of the Quasi MaximumLikelihood estimator is asymptotically equivalent to the simpler asymptoticcovariance matrix of the unfeasible Ordinary Least Squares. These results holdin the general case in which the idiosyncratic components are cross-sectionallyheteroskedastic, as well as serially and cross-sectionally weakly correlated.This paper provides a simple solution to computing the Quasi Maximum Likelihoodestimator and its asymptotic confidence intervals without the need of runningany iterated algorithm, whose convergence properties are unclear, andestimating the Hessian and Fisher information matrices, whose expressions arevery complex.

Asymptotic equivalence of Principal Components and Quasi Maximum Likelihood estimators in Large Approximate Factor Models