  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.
