General Specification
In the above,

and it is assumed that Vb is positive definite. In order to have Ф(<т) be positive definite, we also require
where the equality of the second and third expressions follows from the properties of the inverted gamma distribution for a,4

In specifying the parameters for the above priors, we use an empirical-Bayes procedure that relies on data for a cross-section of individual stocks. The effects of “mispricing” uncertainty are investigated by entertaining a wide range of values for aa. Details of that approach are provided in Subsection C.2.
Observe in (11) that the conditional prior variance of a is proportional to a2, the variance of et. This feature of our prior recognizes that a high value of |ct| accompanied by a low value of a2 implies a high Sharpe ratio for some combination of the asset, the factor-mimicking positions, and cash (earning the riskless rate).5 In particular, {a/a)2 is the difference between the maximum squared Sharpe ratio for such a combination and the maximum squared Sharpe ratio for combinations of only the factor-mimicking positions and cash.6 Following MacKinlay (1995), a prior positive association between a and a is imposed to reduce the probability of high Sharpe ratios as compared to priors that treat those parameters as independent. In contrast, we do assume independence between /5 and a in the absence of a compelling a priori argument to the contrary.