L2-Convergence of the Population Principal Components in the Approximate Factor Model

We prove that under the condition that the eigenvalues are asymptotically well separated and stable, the normalised principal components of a r-static factor sequence converge in mean square. Consequently, we have a generic interpretation of the principal components estimator as the normalised principal components of the statically common space. We illustrate why this can be useful for the interpretation of the PC-estimated factors, performing an asymptotic theory without rotation matrices and avoiding singularity issues in factor augmented regressions.