The reduced-rank beta in linear stochastic discount factor models

In a linear stochastic discount factor model, failure of the full-rank conditions affects the standard statistical inference of coefficients. We propose a novel risk measurement, the reduced-rank beta, which is the risk sensitivity to the effective part of factors for the full-rank covariance matrix. Our reduced-rank beta is a generalisation of the standard beta when the full-rank condition is not satisfied. By considering the Fama–French five-factor (FF5) model for the US equity market, the failure of the full-rank condition is found to affect beta estimates. We demonstrate the reduced-rank beta has important empirical implications for model reductions and anomaly explanations.