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. 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. This paper provides a simple solution to computing the Quasi Maximum Likelihood estimator and its asymptotic confidence intervals without the need of running any iterated algorithm, whose convergence properties are unclear, and estimating the Hessian and Fisher information matrices, whose expressions are very complex.

翻译：本文研究了适用于$n$维平稳时间序列向量的近似因子模型的拟极大似然估计性质。我们证明，当$n\to\infty$时，拟极大似然估计的因子载荷与主成分估计在渐近意义下等价。这两种估计量同时与当因子可观测时不可行普通最小二乘估计量在$n\to\infty$条件下渐近等价。进一步表明，拟极大似然估计量的渐近协方差矩阵通常使用的三明治形式，与不可行普通最小二乘估计量更简洁的渐近协方差矩阵渐近等价。上述结论在异质成分具有截面异方差性、序列相关与截面弱相关的一般情形下成立。本文为计算拟极大似然估计量及其渐近置信区间提供了简便方案，无需运行任何收敛性质不明确的迭代算法，也无需估计形式极其复杂的海森矩阵与费舍尔信息矩阵。