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.

翻译：我们为主成分估计在大规模近似因子模型中的渐近结果提供了一种替代推导。这些结果是在最少的假设条件下推导得出的，特别地，我们仅要求四阶矩的存在性。我们重点关注时间序列背景，这是近年来几乎所有因子模型计量经济学应用中考虑的情况。因此，估计基于经典的$n\times n$样本协方差矩阵，而非文献中常用的$T\times T$协方差矩阵。实际上，尽管这两种方法渐近等价，但前者更符合时间序列背景，且能立即写出主成分估计更直观的渐近展开式，表明它们与最小二乘估计等价，只要$\sqrt n/T\to 0$且$\sqrt T/n\to 0$，即载荷矩阵的估计如同因子已知时的时间序列回归，而因子的估计如同载荷已知时的截面回归。最后，我们给出因子、特质成分及观测时间序列样本协方差矩阵均方一致性的若干替代性充分条件，这是主成分分析的出发点。