Estimation of Weak Factor Models

This paper proposes a novel estimation method for the weak factor models, a slightly stronger version of the approximate factor models of Chamberlain and Rothschild (1983), with large cross-sectional and time-series dimensions (N and T, respectively). It assumes that the kth largest eigenvalue of data covariance matrix grows proportionally to N^＜k＞ with unknown exponents 0 < _k 1 for k = 1,..., r. This is much weaker than the typical assumption on the recent factor models, in which all the r largest eigenvalues diverge proportionally to N. We apply the SOFAR method of Uematsu et al. (2019) to estimate the weak factor models and derive the estimation error bound. Importantly, our method yields consistent estimation of _k's as well. A finite sample experiment shows that the performance of the new estimator uniformly dominates that of the principal component (PC) estimator. We apply our method to analyze S&P500 firm security returns and find that the first factor is consistently near strong while the others are indeed weak. Another application demonstrates that forecasting bond yields based on our method outperforms that based on the PC.April 2019. Revised March 2020