We review Quasi Maximum Likelihood estimation of factor models for high-dimensional panels of time series. We consider two cases: (1) estimation when no dynamic model for the factors is specified (Bai and Li, 2016); (2) estimation based on the Kalman smoother and the Expectation Maximization algorithm thus allowing to model explicitly the factor dynamics (Doz et al., 2012). Our interest is in approximate factor models, i.e., when we allow for the idiosyncratic components to be mildly cross-sectionally, as well as serially, correlated. Although such setting apparently makes estimation harder, we show, in fact, that factor models do not suffer of the curse of dimensionality problem, but instead they enjoy a blessing of dimensionality property. In particular, we show that if the cross-sectional dimension of the data, $N$, grows to infinity, then: (i) identification of the model is still possible, (ii) the mis-specification error due to the use of an exact factor model log-likelihood vanishes. Moreover, if we let also the sample size, $T$, grow to infinity, we can also consistently estimate all parameters of the model and make inference. The same is true for estimation of the latent factors which can be carried out by weighted least-squares, linear projection, or Kalman filtering/smoothing. We also compare the approaches presented with: Principal Component analysis and the classical, fixed $N$, exact Maximum Likelihood approach. We conclude with a discussion on efficiency of the considered estimators.

翻译：我们回顾了面向高维时间序列面板数据因子模型的拟极大似然估计方法。我们考虑两种情况：(1) 未指定因子动态模型时的估计（Bai and Li, 2016）；(2) 基于卡尔曼平滑器和期望最大化算法的估计，从而允许显式建模因子动态（Doz et al., 2012）。我们关注的是近似因子模型，即允许特质成分存在轻度截面相关和序列相关。尽管此类设定表面上看使估计更困难，但我们证明因子模型实际上并未遭受维数灾难问题，反而具有维数祝福性质。具体而言，我们证明：若数据的截面维度 $N$ 趋于无穷，则 (i) 模型仍可识别，(ii) 使用精确因子模型对数似然导致的误设误差将消失。此外，若同时令样本量 $T$ 趋于无穷，我们还能一致估计模型所有参数并进行推断。对潜在因子的估计同样如此，可采用加权最小二乘法、线性投影或卡尔曼滤波/平滑法实现。我们还将所提方法与主成分分析及经典固定 $N$ 精确极大似然法进行了比较。最后讨论了所考虑估计量的有效性。