Equations (25) and (26) essentially imply that the shrinkage weights applied to (3 and /3 in determining the posterior mean of (3 depend on the prior precision about (3 as compared to the sample precision of the regression estimates, and the latter increases in the sample length T. For Bay State Gas, with T = 253, the posterior mean of (3 is much closer to the least-squares estimate of 0.42 than the prior mean of 1.12, and the posterior mean moves only slightly, from 0.47 to 0.45, as aa goes from zero to infinity. For smaller values of T, the posterior mean of j3 is shrunk more toward the prior mean. The slight dependence on aa arises from correlation in the posterior between a and f3, both through pap in the prior and through the off-diagonal elements in the first row of X’X.
The expected excess return has ct as one of its components. Panel В of Table II reports posterior moments for the other component, /3’A, and the overall expected excess return, ц. Recall that information about A is contained not only in the available histories of returns on the factors but also in the longer histories of other series that are correlated with the factors. The first part of Panel В reports posterior moments based on the longer period from January 1926 through December 1995, whereas the second part reports moments based on the shorter period beginning in July 1963. The posterior mean of A, A, is 8.05% based on the longer period but only 5.52% based on the shorter period.
This difference reflects the fact that the average return on the NYSE portfolios is higher over the 1926-95 period than during the shorter 1963-95 period. Given the high positive correlations betwreen the NYSE indexes and the Fama-French NYSE-AMEX-Nasdaq index, the posterior mean of the latter index is adjusted upwrard (see Stambaugh (1997)). This adjustment produces a cost-of-equity estimate for Bay State Gas that is above the shorter-period estimate by about 1.2%.
For the overall period, the posterior mean of the expected excess return p is about 3.8% based on a strict CAPM (aa = 0) and, given the behavior of the posterior mean of a discussed above, the posterior mean of p. remains between 3% and 4% for values of aQ smaller than 5%. That is, prior uncertainty about Bay State Gas’s CAPM mispricing (a) that seems substantial in economic terms still results in a posterior mean fairly close to the CAPM value. As will be demonstrated below, this observation generalizes across stocks and across the three pricing models considered. best online payday loans