COSTS OF EQUITY CAPITAL: Prior Parameters 2

The statistics b and a2, computed for each stock, are used to construct the prior parameters 6, V&, Sq, and v. The prior mean of b, 6, is set equal to the cross-sectional average of the b’s, except that the first element, a, is set to zero. The prior covariance matrix of 6, Vb,
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where E(b) is the sample cross-sectional covariance matrix of the b’s. The second term in (18) is the average across stocks of the usual estimate for the sampling variance of b, where of and (X’X)i are based on the observations available for stock i. (The bar denotes an average across stocks.) As noted by Fama and French (1997), under standard assumptions, Vb is an estimate of the cross-sectional covariance of the 6’s. For all three models, it happens that V*, is positive definite (not guaranteed in general). To construct the matrix Vb, as represented in (12), Vp is set equal to the corresponding submatrix of Vb, and pap is taken from the correlation matrix associated with VRather than set cr2 equal to the (1,1) element of Vj,, however, we instead let it take a wide range of values, ranging from zero to infinity.10 Each value of oa is then combined with the fixed values of Vp and pap, using (12), to form the matrix Vb used in the prior.

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Equation (20) is the analog to equation (18). The first term on the right-hand side is the sample cross-sectional variance of the cr2,s, while the second term is the cross-sectional average of the estimates of the sampling variance of a2. That is, of denotes the estimated residual variance for stock i, based on I\ observations for that stock, and the estimated sampling variance of