Recall that E(cr2) is the prior mean of cr2, which in this case is equal to 0.016 on a monthly basis, using (14) and the CAPM values in Panel A of Table I. If crQ = 5% on an annualized basis, then the corresponding monthly value of aa used in the calculations is 0.0042 (= .05/12). For T = 253, as with Bay State Gas, equation (40) implies wa = 0.78. That is, even with mispricing uncertainty of 5% per annum, and a value of T that is fairly large compared to those often used in practice, the prior mean of a is still given heavy weight in computing the posterior mean of a.
Of course, as cra becomes large, we see from (40) that the sample estimate a is given increasingly greater weight, as illustrated by the results in Table II. Alternatively, a would also be given greater weight if the prior mean E(cr2) were lower. Such might be the case, for example, if one were to estimate the expected excess return on an asset known a priori to possess lower residual variance, such as a diversified portfolio of stocks. For the typical individual stock, however, as will be further demonstrated in the next subsection, a is given heavy weight only when the prior mispricing uncertainty is very high.
The prior mean a is set to zero in this study, but, as noted earlier, one could instead set a to a non-zero value, possibly the average sample a for a cross-section of stocks that share similar characteristics with the given stock. The degree of shrinkage toward that prior mean would, for a given <ra, be otherwise similar to that demonstrated here.