Following standard practices, Kolari, Liu, and Zhang (KLH) (2021) implemented Fama–MacBeth cross-sectional regression tests. The analyses were rolled forward one month at a time to generate a series of estimated market prices of beta and zeta risk loadings (denoted λa and λRD, respectively).

To comparatively evaluate the performance of the ZCAPM, they ran tests for the CAPM, Carhart (1997) four-factor model, and Fama and French (1992, 1993, 2015, 2018) three-, five-, and six-factor models. For the often used 25 size and BM sorted portfolios from Kenneth French’s data website, they produced graphs comparing realized excess returns to fitted excess returns for the three-factor model of Fama and French (1992, 1993) versus the empirical ZCAPM. What did these graphs show? What about the results for 47 industry portfolios? What do these results mean?