Asymptotic equivalence of Principal Components and Quasi Maximum Likelihood estimators in Large Approximate Factor Models

TL;DR

This paper demonstrates that in large approximate factor models, Quasi Maximum Likelihood and Principal Components estimators become asymptotically equivalent, simplifying inference even with complex idiosyncratic components.

Contribution

It establishes the asymptotic equivalence of Quasi Maximum Likelihood and Principal Components estimators in large approximate factor models with heteroskedastic and weakly correlated errors.

Findings

Quasi Maximum Likelihood estimators are asymptotically equivalent to Principal Components.

Both estimators are asymptotically equivalent to the unfeasible OLS estimator with observed factors.

The asymptotic covariance matrices of the estimators are also equivalent.

Abstract

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. All 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. The intuition behind these results is that as $n\to\infty$ the factors can be considered as observed, thus showing that factor models enjoy a blessing of dimensionality.