Fixed-order PCA: Theory for Overestimated Factor Models

Yuan Liao; Xin Tong; Wanjie Wang; Dacheng Xiu

doi:10.48550/arxiv.2605.18448

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Fixed-order PCA: Theory for Overestimated Factor Models

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Fixed-order PCA: Theory for Overestimated Factor Models

Yuan Liao, Xin Tong, Wanjie Wang and Dacheng Xiu

ArXiv.org

Cornell University

05/18/2026

DOI: 10.48550/arxiv.2605.18448

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Preprint (Author's original) This preprint has not been evaluated by subject experts through peer review. Preprints may undergo extensive changes and/or become peer-reviewed journal articles. Open Access

Abstract

We develop asymptotic theory for principal component analysis (PCA) of a high-dimensional factor model in which the working dimensionRis fixed and only required to satisfyR ≥ r , whereris the true number of factors. Building on anisotropic local laws from random matrix theory, we show that the ``extra'' empirical eigencomponents beyond ther -th are asymptotically noise-governed, incoherent, and nearly orthogonal to the factor loadings. We introduce two rotations, an expandedr× RmapH'and a compressedR× rmapH⁺ , and establish consistency of the estimated factors under both. As an application, we analyze a factor-augmented regression for treatment-effect inference and prove√T̅ -asymptotic normality for every fixedR ≥ 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 boundingrfrom above.

Mathematics - Statistics Theory

Statistics - Theory

Details

Title: Subtitle: Fixed-order PCA: Theory for Overestimated Factor Models
Creators: Yuan Liao - University of Iowa
Xin Tong - National University of Singapore
Wanjie Wang - National University of Singapore
Dacheng Xiu - University of Chicago
Resource Type: Preprint
Publication Details: ArXiv.org
DOI: 10.48550/arxiv.2605.18448
ISSN: 2331-8422
Publisher: Cornell University; Ithaca, New York
Language: English
Date posted: 05/18/2026
Academic Unit: Economics
Record Identifier: 9985164631002771

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