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.

翻译：本文研究了高维近似因子模型中弱因子情形下主成分（PC）估计量的性质，其中因子载荷（$\boldsymbol{\Lambda}^0$）随横截面单元数量$N$呈次线性缩放，即对于某$\alpha \in (0,1)$，$\boldsymbol{\Lambda}^{0\top} \boldsymbol{\Lambda}^0 / N^\alpha$在极限下正定。尽管在强因子（即$\alpha=1$）情形下，这些估计量的一致性和渐近正态性已得到充分研究，但弱因子的统计性质仍鲜有探索。本文证明，在噪声依赖结构满足适当条件的前提下，PC估计量对任意$\alpha\in(0,1)$均保持一致性和渐近正态性。这一结果补充了Onatski（2012）关于$\alpha=0$时PC估计量不一致的早期结论，以及Bai与Ng（2023）建立$\alpha \in (1/2,1)$时PC估计量渐近正态性的最新工作。我们的证明策略融合了因子模型传统的基于特征值分解的方法，以及类似于矩阵补全等场景中使用的留一法分析思路。这种结合使我们能够处理比前者更弱的因子，同时放松后者通常要求的非相干性和独立性假设。