When can weak latent factors be statistically inferred?

The factor model, a pivotal tool for analyzing large panel data, has become a significant topic in finance and economics research (e.g., Chamberlain and Rothschild, 1983; Fama and French, 1993; Stock and Watson, 2002; Bai and Ng, 2002; Giglio and Xiu, 2021; Fan et al., 2021b). The estimation and inference for factor models are crucial in economic studies, particularly in areas like asset pricing and return forecasting. In the era of big data, the factor model has gained increased prominence in capturing the latent common structure for large panel data, where both the cross-sectional and temporal dimensions are ultra-high (see, e.g., recent surveys Bai and Wang, 2016; Fan et al., 2021a). Principal component analysis (PCA), known for its simplicity and effectiveness, is closely connected with the factor model and has long been a key research topic of interest in the econometric community (e.g., Stock and Watson, 2002; Bai and Ng, 2002; Bai, 2003; Onatski, 2012; Fan et al., 2013; Bai and Ng, 2013, 2023). As pointed out by Giglio et al. (2023), most theoretical guarantees for the PCA approach to factor analysis rely on the pervasiveness assumption (e.g., Bai and Ng, 2002; Bai, 2003). This assumption requires the signalto-noise ratio (SNR), which measures the factor strength relative to the noise level, to grow with the rate of N - the square root of the cross-sectional dimension. However, many real datasets in economics do not exhibit sufficiently strong factors to meet this pervasiveness assumption. When the SNR grows slower than N, the resulting model is often called the weak factor model (e.g., Onatski, 2009, 2010). Extensive research has been dedicated to the weak factor model (e.g., Onatski, 2012; Bailey et al., 2021; Freyaldenhoven, 2022; Uematsu and Yamagata, 2022a,b; Bai and Ng, 2023; Jiang et al., 2023; Choi and Yuan, 2024), among which