Quasi Maximum Likelihood Estimation and Inference of Large Approximate Dynamic Factor Models via the EM algorithm

TL;DR

This paper develops a method for estimating large dynamic factor models using the EM algorithm, demonstrating consistency, asymptotic normality, and efficiency of the estimators as both sample size and cross-sectional dimension grow.

Contribution

It introduces a novel approach combining EM and Kalman smoothing for large dynamic factor models, proving asymptotic properties and efficiency of the estimators.

Findings

Estimated loadings are $ ext{√}T$-consistent and asymptotically normal.

Estimated factors are $ ext{√}n$-consistent and asymptotically normal.

Loadings are as efficient as PCA estimates; factors can be more efficient with sparse covariance.

Abstract

We study estimation of large Dynamic Factor models implemented through the Expectation Maximization (EM) algorithm, jointly with the Kalman smoother. We prove that as both the cross-sectional dimension, $n$, and the sample size, $T$, diverge to infinity: (i) the estimated loadings are $\sqrt T$-consistent, asymptotically normal and equivalent to their Quasi Maximum Likelihood estimates; (ii) the estimated factors are $\sqrt n$-consistent, asymptotically normal and equivalent to their Weighted Least Squares estimates. Moreover, the estimated loadings are asymptotically as efficient as those obtained by Principal Components analysis, while the estimated factors are more efficient if the idiosyncratic covariance is sparse enough.