Yuan Liao, Xin Tong, Wanjie Wang, Dacheng Xiu
arXiv 18 May 2026 · Mathematics — Statistics Theory
arXiv:2605.18448 · PDF · DOI · OpenAlex · Extracted main text
We develop asymptotic theory for principal component analysis (PCA) of a high-dimensional factor model in which the working dimension $R$ is fixed and only required to satisfy $R \ge r$, where $r$ is the true number of factors. Building on anisotropic local laws from random matrix theory, we show that the “extra” empirical eigencomponents beyond the $r$-th are asymptotically noise-governed, incoherent, and nearly orthogonal to the factor loadings. We introduce two rotations, an expanded $r\times R$ map $H'$ and a compressed $R\times r$ map $H^{+}$, and establish consistency of the estimated factors under both. As an application, we analyze a factor-augmented regression for treatment-effect inference and prove $\sqrt{T}$-asymptotic normality for every fixed $R \ge r$. These results provide a theoretical underpinning for the common empirical practice of adopting a conservative upper bound on the number of factors, and shift the analytical burden from consistent dimension selection to the milder requirement of bounding $r$ from above.
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The works this paper leans on most, across its whole bibliography — not restricted to papers in our corpus. Ranked by composite intensity, which combines how often a work is mentioned, how many sections mention it, and how much of that falls in the main text rather than the appendix.
| Reference | Intensity | Mentions | Sections | Main text | |
|---|---|---|---|---|---|
| 1 | Bai, J (2003) Inferential theory for factor models of large dimensions | 1.000 | 6 | 3 | 100% |
| 2 | Knowles, Antti and Yin, Jun (2017) Anisotropic local laws for random matrices | 0.763 | 18 | 5 | 44% |
| 3 | Moon, R. and Weidner, M (2015) Linear regression for panel with unknown number of factors as interactive fixed effects | 0.737 | 3 | 2 | 100% |
| 4 | Abbe, Emmanuel and Fan, Jianqing and Wang, Kaizheng and Zhong, Yiqiao (2020) Entrywise eigenvector analysis of random matrices with low expected rank | 0.644 | 2 | 2 | 100% |
| 5 | Belloni, Alexandre and Chernozhukov, Victor and Hansen, Christian (2014) Inference on treatment effects after selection among high-dimensional controls | 0.644 | 2 | 2 | 100% |
| 6 | Chernozhukov, Victor and Chetverikov, Denis and Demirer, Mert and Du… (2016) Double machine learning for treatment and causal parameters | 0.644 | 2 | 2 | 100% |
| 7 | Fan, Jianqing and Wang, Weichen and Zhong, Yiqiao (2018) An $ _infty$ eigenvector perturbation bound and its application to robust covariance estimation self | 0.644 | 2 | 2 | 100% |
| 8 | Fan, Jianqing and Li, Kunpeng and Liao, Yuan (2021) Recent developments in factor models and applications in econometric learning self | 0.511 | 2 | 2 | 50% |
| 9 | Bai, J. and Ng, S (2002) Determining the number of factors in approximate factor models | 0.511 | 2 | 1 | 100% |
| 10 | French, Eric and Jones, John Bailey (2011) The effects of health insurance and self-insurance on retirement behavior | 0.511 | 2 | 1 | 100% |
Showing the top 10 of 37 scored citations.