Estimating Weak Factor Models

In this paper, we propose a novel consistent estimation method for the approximate factor model of Chamberlain and Rothschild (1983), with large cross-sectional and time-series dimensions ($N$ and $T$, respectively). Their model assumes that the $r$ ($\ll N$) largest eigenvalues of data covariance matrix grow as $N$ rises without specifying each diverging rate. This is weaker than the typical assumption on the recent factor models, in which all the $r$ largest eigenvalues diverge proportionally to $N$, and is frequently referred to as the weak factor models. We extend the sparse orthogonal factor regression (SOFAR) proposed by Uematsu et al.\ (2017) to consistently estimate the weak factors structure, where the $k$-th largest eigenvalue grows proportionally to $N^{a_k}$ with some unknown $0\leq>