Kolari, Liu, and Zhang (KLH) (2021) conducted out-of-sample (or one month-ahead), cross-sectional Fama and MacBeth (1973) tests

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Kolari, Liu, and Zhang (KLH) (2021) conducted out-of-sample (or one month-ahead), cross-sectional Fama and MacBeth (1973) tests of a number of different test asset portfolios. The final form of the empirical ZCAPM was estimated using one year of daily returns. The U.S. Treasury bill rate proxied the riskless rate. To begin their analyses, KLH reported evidence on the relation between beta risk and zeta risk with respect to one-month-ahead stock returns. Using 97 stock portfolios, the empirical ZCAPM was run using daily returns for one year to obtain βi ,a and Zi ,a estimates for each portfolio. The actual returns Rit+1 in the next one month were retained. The process was repeated by rolling forward one month to get new estimates of βi ,a and Zi ,a as well as the next one-month returns Ri t+1 for portfolios. Repeating this process from 1964 to the end of 2018, they generated a series of 648 monthly values for βi ,a, Zi ,a, and Rit+1. Next, the average values of these values were computed for each portfolio. What did they find?

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