arXiv 8 Feb 2024 · Econometrics · publishedJournal of Econometrics (2025) · 3 citations (OpenAlex)
arXiv:2402.05789 · PDF · DOI · OpenAlex · Extracted main text
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.
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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. and Ng, S (2023) Approximate factor models with weaker loadings | 0.950 | 14 | 6 | 86% |
| 2 | Onatski, A (2012) Asymptotics of the principal components estimator of large factor models with weakly influential factors | 0.928 | 4 | 3 | 100% |
| 3 | Bai, J (2003) Inferential theory for factor models of large dimensions | 0.822 | 9 | 3 | 56% |
| 4 | Ma, C., Wang, K., Chi, Y., and Chen, Y (2020) Implicit regularization in nonconvex statistical estimation: Gradient descent converges linearly for phase retrieval, matrix com… | 0.585 | 3 | 3 | 33% |
| 5 | Chen, J., Liu, D., and Li, X (2020) Nonconvex rectangular matrix completion via gradient descent without $l_2,$ regularization | 0.523 | 7 | 3 | 14% |
| 6 | Chen, Y., Fan, J., Ma, C., and Yan, Y (2019) Inference and uncertainty quantification for noisy matrix completion | 0.511 | 3 | 2 | 33% |
| 7 | Chen, Y., Chi, Y., Fan, J., Ma, C., and Yan, Y (2020) Noisy matrix completion: Understanding statistical guarantees for convex relaxation via nonconvex optimization | 0.511 | 2 | 2 | 50% |
| 8 | Bai, J. and Ng, S (2002) Determining the number of factors in approximate factor models | 0.511 | 2 | 1 | 100% |
| 9 | Bai, J., Ng, S., et al (2008) Large dimensional factor analysis | 0.511 | 2 | 1 | 100% |
| 10 | Abbe, E., Fan, J., Wang, K., and Zhong, Y (2020) Entrywise eigenvector analysis of random matrices with low expected rank | 0.405 | 1 | 1 | 100% |
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