Jianqing Fan, Yuling Yan, Yuheng Zheng
arXiv 4 Jul 2024 · Statistics — Methodology · 5 citations (OpenAlex)
arXiv:2407.03616 · PDF · DOI · OpenAlex · Extracted main text
This article establishes a new and comprehensive estimation and inference theory for principal component analysis (PCA) under the weak factor model that allow for cross-sectional dependent idiosyncratic components under the nearly minimal factor strength relative to the noise level or signal-to-noise ratio. Our theory is applicable regardless of the relative growth rate between the cross-sectional dimension $N$ and temporal dimension $T$. This more realistic assumption and noticeable result require completely new technical device, as the commonly-used leave-one-out trick is no longer applicable to the case with cross-sectional dependence. Another notable advancement of our theory is on PCA inference $ - $ for example, under the regime where $N\asymp T$, we show that the asymptotic normality for the PCA-based estimator holds as long as the signal-to-noise ratio (SNR) grows faster than a polynomial rate of $\log N$. This finding significantly surpasses prior work that required a polynomial rate of $N$. Our theory is entirely non-asymptotic, offering finite-sample characterizations for both the estimation error and the uncertainty level of statistical inference. A notable technical innovation is our closed-form first-order approximation of PCA-based estimator, which paves the way for various statistical tests. Furthermore, we apply our theories to design easy-to-implement statistics for validating whether given factors fall in the linear spans of unknown latent factors, testing structural breaks in the factor loadings for an individual unit, checking whether two units have the same risk exposures, and constructing confidence intervals for systematic risks. Our empirical studies uncover insightful correlations between our test results and economic cycles.
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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 | 1.000 | 20 | 6 | 100% |
| 2 | Choi, J. and Yuan, M (2024) High dimensional factor analysis with weak factors | 1.000 | 15 | 5 | 100% |
| 3 | Jiang, P., Uematsu, Y., and Yamagata, T (2023) Revisiting asymptotic theory for principal component estimators of approximate factor models | 1.000 | 15 | 5 | 100% |
| 4 | Fan, J., Liao, Y., and Mincheva, M (2013) Large covariance estimation by thresholding principal orthogonal complements self | 1.000 | 6 | 4 | 100% |
| 5 | Onatski, A (2012) Asymptotics of the principal components estimator of large factor models with weakly influential factors | 1.000 | 6 | 3 | 100% |
| 6 | Bai, J (2003) Inferential theory for factor models of large dimensions | 1.000 | 5 | 3 | 100% |
| 7 | Yan, Y., Chen, Y., and Fan, J (2024) Inference for heteroskedastic pca with missing data self | 0.965 | 10 | 4 | 90% |
| 8 | Breitung, J. and Eickmeier, S (2011) Testing for structural breaks in dynamic factor models | 0.874 | 6 | 2 | 100% |
| 9 | Bai, J. and Ng, S (2002) Determining the number of factors in approximate factor models | 0.874 | 5 | 2 | 100% |
| 10 | Onatski, A (2010) Determining the number of factors from empirical distribution of eigenvalues | 0.843 | 3 | 3 | 100% |
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