Estimation of Weak Factor Models

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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 (&#8810;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. (2019) to consider consistent estimation of the weak factors structure, where the k-th largest eigenvalue grows proportionally to N&#945;k with some unknown exponents 0<&#945;k&#8804;1 for k=1,…,r. Importantly, our method enables us to consistently estimate &#945;k as well. In our finite sample experiment, the performance of the new estimator uniformly dominates that of the principal component (PC) estimators in terms of mean absolute loss, and its superiority gets larger as the common components become weaker. We apply our method to analyze S&P500 firm security monthly returns from January 1984 to April 2018, and the results show that the first factor is consistently near strong, whilst the second to the fourth exponents vary over months between 0.90 and 0.65 and they cross. In another application, we consider out-of-sample performance of forecasting regressions for bond yield using extracted factors by our method and by the PC, and the forecasting performance test concludes that our method outperforms the PC method