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

We prove that in an approximate factor model for an $n$-dimensional vector of stationary time series the factor loadings estimated via Principal Components are asymptotically equivalent, as $n\to\infty$, to those estimated by Quasi Maximum Likelihood. Both estimators are, in turn, also asymptotically equivalent, as $n\to\infty$, to the unfeasible Ordinary Least Squares estimator we would have if the factors were observed. We also show that the usual sandwich form of the asymptotic covariance matrix of the Quasi Maximum Likelihood estimator is asymptotically equivalent to the simpler asymptotic covariance matrix of the unfeasible Ordinary Least Squares. This provides a simple way to estimate asymptotic confidence intervals for the Quasi Maximum Likelihood estimator without the need of estimating the Hessian and Fisher information matrices whose expressions are very complex. All our results hold in the general case in which the idiosyncratic components are cross-sectionally heteroskedastic as well as serially and cross-sectionally weakly correlated.