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.

翻译：我们证明了在平稳时间序列的n维向量的近似因子模型中，当n→∞时，通过主成分分析估计的因子载荷与拟极大似然估计渐近等价。这两种估计量在n→∞时均与假设因子可观测时的不可行普通最小二乘估计量渐近等价。我们还证明拟极大似然估计的渐近协方差矩阵通常采用的夹心形式与不可行普通最小二乘的更简单的渐近协方差矩阵渐近等价。这为拟极大似然估计的渐近置信区间提供了一种简便的估计方法，无需估计Hessian矩阵和Fisher信息矩阵（其表达式非常复杂）。所有结果均适用于异质成分具有截面异方差性以及序列和截面弱相关性的普遍情形。