Quasi Maximum Likelihood Estimation of High-Dimensional Factor Models

Factor models are some of the most common dimension reduction techniques in time series econometrics. They are based on the idea that each element of a set of N time series is made of a common component driven by few latent factors capturing the main comovements among the series, plus idiosyncratic components oen representing just measurement error or at most being weakly cross-sectionally correlated with the other idiosyncratic components. When N is large the factors can be retrieved by cross-sectional aggregation of the observed time series. This is the so-called blessing of dimensionality, meaning that having N growing to infinity poses no estimation problem but in fact is a necessary condition for consistent estimation of the factors and for identification of the common and idiosyncratic components. There exist two main ways to estimate a factor model: principal component analysis and maximum likelihood estimation. The former method is more recent and more common in econometrics, but the latter, which is the classical approach, has many appealing features such as allowing one to impose constraints, deal with missing values, and explicitly model the dynamic of the factors. Maximum likelihood estimation of large factor models has been studied in two influential papers: Doz et al.ʼs “A Quasi Maximum Likelihood Approach for Large Approximate Dynamic Factor Models” and Bai and Liʼs “Maximum Likelihood Estimation and Inference for Approximate Factor Models of High Dimension.” The latter considers the static case, which is closer to the classical approach and no model for the factors is assumed, and the former is more general: it considers estimation combined with the use of Kalman filtering techniques, which has grown popular in macroeconomic policy analysis. Those two papers, together with other recent results, have brought new asymptotic results for which a synthesis is provided. Special attention is paid to the set of assumptions, which is taken to be the minimal set of assumptions required to get the results.