This paper studies Quasi Maximum Likelihood estimation of Dynamic Factor Models for large panels of time series. Specifically, we consider the case in which the autocorrelation of the factors is explicitly accounted for, and therefore the model has a state-space form. Estimation of the factors and their loadings is implemented through the Expectation Maximization (EM) algorithm, jointly with the Kalman smoother.~We prove that as both the dimension of the panel $n$ and the sample size $T$ diverge to infinity, up to logarithmic terms: (i) the estimated loadings are $\sqrt T$-consistent and asymptotically normal if $\sqrt T/n\to 0$; (ii) the estimated factors are $\sqrt n$-consistent and asymptotically normal if $\sqrt n/T\to 0$; (iii) the estimated common component is $\min(\sqrt n,\sqrt T)$-consistent and asymptotically normal regardless of the relative rate of divergence of $n$ and $T$. Although the model is estimated as if the idiosyncratic terms were cross-sectionally and serially uncorrelated and normally distributed, we show that these mis-specifications do not affect consistency. Moreover, the estimated loadings are asymptotically as efficient as those obtained with the Principal Components estimator, while the estimated factors are more efficient if the idiosyncratic covariance is sparse enough.~We then propose robust estimators of the asymptotic covariances, which can be used to conduct inference on the loadings and to compute confidence intervals for the factors and common components. Finally, we study the performance of our estimators and we compare them with the traditional Principal Components approach through MonteCarlo simulations and analysis of US macroeconomic data.
翻译:本文研究 Qasi 最大隐隐性估计 动态因子模型对于大型时间序列板来说, Qasi 最大隐性估计 。 具体地说, 我们考虑的是, 这些因素的自动调整是明确的, 因此模型有状态空间表。 估计因素及其负荷是通过期待最大化算法, 与 Kalman 平滑器一起实施的 。 ~ 我们证明, 作为面板的尺寸和样本大小, 美元与不精确值相差, 直至对数术语 :(一) 估计的负荷是 $\ sqrt T$ 不变的, 因此, 如果 美元/ n\\ 到 0 美元; 估计的参数是 $ 最大化, 那么, 美元的内值和内值的内值是 。