  This paper studies the principal components (PC) estimator for high
dimensional approximate factor models with weak factors in that the factor
loading ($\boldsymbol{\Lambda}^0$) scales sublinearly in the number $N$ of
cross-section units, i.e., $\boldsymbol{\Lambda}^{0\top} \boldsymbol{\Lambda}^0
/ N^\alpha$ is positive definite in the limit for some $\alpha \in (0,1)$.
While the consistency and asymptotic normality of these estimates are by now
well known when the factors are strong, i.e., $\alpha=1$, the statistical
properties for weak factors remain less explored. Here, we show that the PC
estimator maintains consistency and asymptotical normality for any
$\alpha\in(0,1)$, provided suitable conditions regarding the dependence
structure in the noise are met. This complements earlier result by Onatski
(2012) that the PC estimator is inconsistent when $\alpha=0$, and the more
recent work by Bai and Ng (2023) who established the asymptotic normality of
the PC estimator when $\alpha \in (1/2,1)$. Our proof strategy integrates the
traditional eigendecomposition-based approach for factor models with
leave-one-out analysis similar in spirit to those used in matrix completion and
other settings. This combination allows us to deal with factors weaker than the
former and at the same time relax the incoherence and independence assumptions
often associated with the later.
