In the empirical results presented in the next section we compute the posterior variance of ц and its components, a and (3rX. For the latter quantity we report the unconditional variance as well as variances that condition on either j3 or A set equal to their posterior means. The conditional variances provide additional insight into the sources of uncertainty about the cost of equity. These variances of j3’\ are computed as
Cost-of-Equity Estimates: Posterior Moments A. Results for an Individual Stock
We first compute the moments of the posterior distribution for the expected excess return and its various components for the stock of a specific firm, Bay State Gas Company. One reason for this choice is that, across the three models, the OLS estimates of b for this company generally differ substantially from the cross-sectional averages, so the shrinkage effects discussed earlier can be illustrated. If we were instead to select a typical stock, b would be close to the cross-sectional average used to specify the prior mean, so any shrinkage effects would be minimal. In addition, selecting a utility allowrs us to compare results based on the first all-stock prior to those based on the second utility-specific prior.
As explained in the previous section, given the form of the likelihood and the assumed prior independence between the regression parameters (6 and cr) and the factor means (A), the posterior moments of the regression parameters depend only on the data used in the regression model. The monthly history of Bay State Gas begins in December 1974, so in this case, the regression-model data consist of monthly returns on the stock and the factors for the 253 months in the period from December 1974 through December 1995.
For Bay State Gas, the Metropolis-Hastings algorithm is used to compute the posterior means and standard deviations of the regression parameters, as described in the previous section. Panel A of Table II reports the posterior means and standard deviations of the CAPM a and (3. These posterior moments are reported for seven values of crQ, the prior standard deviation of a that characterizes a decision maker’s mispricing uncertainty about the given model.