In the model uncertainty calculations presented in Tables VIII and IX we simply set the posterior model probabilities to be equal across models. Computing those probabilities using (A.28) is beyond the scope of this study. Moreover, if the set of prior model probabilities (7^’s) is the same for each stock, then the posterior probabilities would differ across stocks.
Rather than take that course, we instead specify equal posterior probabilities in order to simplify the analysis and, as discussed previously, obtain what is likely to be a generous assessment of model uncertainty. (Of course, assuming the same posterior model probabilities across stocks implies that the prior probabilities would differ across stocks.) Explicit posterior model probabilities would probably be more interesting in a multi-asset setting, and such an extension is a possible direction for future research.
Parameters Used in the Priors
In Panel A, for each stock with at least 24 months of data in the period 7/1963-12/1995, b is the ordinary-least-squares estimate of b defined by the regression
rt — [1 /t]b + et,
where rt is the excess return on the stock and ft is a vector of factors. The sample variance of the residuals from that regression is a2, an estimate of cr2, the variance of £t. In Panel B, b and a2 are obtained for every utility stock with at least 48 months of data in the above period. The prior mean of b, b, is computed as the cross-sectional average of the b’s, except that its first element, the mean of a, is set to zero. The prior standard deviations and correlations are obtained from Vb, which is computed as the cross-sectional covariance matrix of the b’s minus the cross-sectional average of the time-series sampling variances of the b’s.
The prior covariance matrix of b, Vb, is computed from Vb by varying the prior standard deviation of a (aa) between zero and infinity while preserving the correlation structure of Vb- In Panel B, the off-diagonal elements of Vb are set equal to zero in order to have that matrix be positive definite. The prior mean of a2 is computed as the cross-sectional average of the <x2’s, and the prior variance of a2 is computed as the difference between the cross-sectional variance of the <j2’s and the cross-sectional average of the time-series sampling variances of the <r2’s. Given the prior mean and variance of cr2, properties of the inverted gamma density imply the values of v and Sq, which are the two parameters used to define the prior density of a. Electronic Payday Loans Online