Actually, There is No Rotational Indeterminacy in the Approximate Factor Model

TL;DR

This paper proves that in the approximate factor model, principal components converge in mean square without the need for rotation, providing a new asymptotic theory that simplifies factor estimation.

Contribution

It establishes convergence of principal components in the approximate factor model without rotation matrices, refining the understanding of factor identification.

Findings

Principal components converge in mean square (up to sign) as n→∞.

Factors space is consistently estimated with finite T as n→∞.

Consistency of individual factors requires both n and T to go to infinity.

Abstract

We show that in the approximate factor model the population normalised principal components converge in mean square (up to sign) under the standard assumptions for $n\to \infty$. Consequently, we have a generic interpretation of what the principal components estimator is actually identifying and existing results on factor identification are reinforced and refined. Based on this result, we provide a new asymptotic theory for the approximate factor model entirely without rotation matrices. We show that the factors space is consistently estimated with finite $T$ for $n\to \infty$ while consistency of the factors a.k.a the $L^2$ limit of the normalised principal components requires that both $(n, T)\to \infty$.