This paper investigates the properties of Quasi Maximum Likelihood estimation of an approximate factor model for an $n$-dimensional vector of stationary time series. We prove that the factor loadings estimated by Quasi Maximum Likelihood are asymptotically equivalent, as $n\to\infty$, to those estimated via Principal Components. 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. All these results hold in the general case in which the idiosyncratic components are cross-sectionally heteroskedastic, as well as serially and cross-sectionally weakly correlated. The intuition behind these results is that as $n\to\infty$ the factors can be considered as observed, thus showing that factor models enjoy a blessing of dimensionality.

翻译：本文研究了$n$维平稳时间序列向量近似因子模型的拟最大似然估计性质。我们证明，当$n\to\infty$时，拟最大似然估计的因子载荷与主成分估计的因子载荷渐近等价。这两种估计量在$n\to\infty$时又均与不可行的普通最小二乘估计量（假设因子可观测时可得）渐近等价。我们还证明，拟最大似然估计量渐近协方差矩阵的常见三明治形式与不可行普通最小二乘估计量的更简单渐近协方差矩阵渐近等价。这些结果在特质成分存在横截面异方差性、序列弱相关及横截面弱相关的一般情形下均成立。这些结论背后的直观解释是：当$n\to\infty$时，因子可被视为可观测变量，这表明因子模型具有维度优势特性。