Estimating the beta-return relationship by considering the sign and the magnitude of daily returns

We test the beta-return relationship under the working assumption that beta is realized under constrained but innovative trading environments. Specifically, we estimate a residual beta risk as the difference between a probability-weighted realized beta and an ordinary least squares (OLS) beta, and test the beta-return relationship using daily returns on the U.S. stock market factor and 30 U.S. industries. Our estimates of the market risk premium using the cross-sectional regression (CSR) of Fama and MacBeth (1973) over a period spanning from 1926/07 to 2014/12 are in line with the central prediction of the capital asset pricing model (CAPM) that the realized return is linearly related to beta.