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

On Estimation and Inference of Large Approximate Dynamic Factor Models via the Principal Component Analysis