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

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 $(n, T)\to \infty$.