Universal Factor Models

We propose a new factor analysis framework and estimators of the factors and loadings that are robust to certain weak factors in a large $N$ and large $T$ setting. Our framework, by simultaneously considering all quantile levels of the outcome variable, induces standard mean and quantile factor models, but the factors can have an arbitrarily weak influence on the outcome's mean or quantile at most quantile levels. Our method estimates the factor space at the $\sqrt{N}$-rate as long as each factor is strong at some unknown quantile level, and achieves $\sqrt{N}$- and $\sqrt{T}$-asymptotic normality for the factors and loadings based on a novel sample splitting approach that handles incidental nuisance parameters. We also develop a weak-factor-robust estimator of the number of factors and consistent selectors of factors of any tolerated level of influence on the outcome's mean or quantiles. Monte Carlo simulations demonstrate the effectiveness of our method.