Panel A of Table I reports the parameter values used in the prior constructed from the entire cross-section of stocks. Note that in the CAPM the prior correlation between a and (3 is positive. This occurs in spite of a negative cross-sectional correlation between the sample estimates a and /?, as has been observed in previous studies (e.g., Black, Jensen, and Scholes (1972)). That is, the off-diagonal element of the first matrix on the right-hand side of (18) is negative.
The positive correlation in the prior results from the fact that the average sampling covariance between a and /3, appearing in the second term on the right-hand side of (18), is also negative, and the difference results in a positive estimate of the cross-sectional covariance between a and f3. For the other two models, the prior correlations between a and the elements of j3 are generally negative. In particular, the prior correlation between a and the HML sensitivity (the last element of b in the FF model) is -0.55. This value, obtained here with individual stocks, is consistent with a similarly large negative correlation between a and HML sensitivities for industry portfolios observed by Fama and French (1997).
As noted earlier, the prior based on the entire cross-section can be viewed as economically non-informative compared to a prior that makes use of a cross-section selected according to one or more of the firm’s characteristics. For example, if a public utility’s cost of equity is to be estimated, the prior parameters can be obtained from a cross-section of utilities rather than the cross-section of all stocks.
Our second prior uses the cross-section of 186 utility firms (SIC codes between 4900 and 4999) with at least 48 months of data in the period from July 1963 through December 1995. The same approach described earlier for the entire cross-section is applied here, except that the off-diagonal elements of Vb are set to zero. The latter simplification and the 48-month data requirement are imposed in order to obtain a positive-definite prior covariance matrix for b with this smaller cross-section. Panel В of Table I reports the parameter values in this utility-specific prior.