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

We provide an alternative derivation of the asymptotic results for the Principal Components estimator of a large approximate factor model. Results are derived under a minimal set of assumptions and, in particular, we require only the existence of 4th order moments. A special focus is given to the time series setting, a case considered in almost all recent econometric applications of factor models. Hence, estimation is based on the classical $n\times n$ sample covariance matrix and not on a $T\times T$ covariance matrix often considered in the literature. Indeed, despite the two approaches being asymptotically equivalent, the former is more coherent with a time series setting and it immediately allows us to write more intuitive asymptotic expansions for the Principal Component estimators showing that they are equivalent to OLS as long as $\sqrt n/T\to 0$ and $\sqrt T/n\to 0$, that is the loadings are estimated in a time series regression as if the factors were known, while the factors are estimated in a cross-sectional regression as if the loadings were known. Finally, we give some alternative sets of primitive sufficient conditions for mean-squared consistency of the sample covariance matrix of the factors, of the idiosyncratic components, and of the observed time series, which is the starting point for Principal Component Analysis.