The structure of the covariance matrix for b, Ф(сг) in (11), produces a prior that is essentially a hybrid of two more standard alternative priors for the regression model. In one alternative, the normal density for b and the inverted-gamma density for cr2 are independent, so that no part of the covariance matrix for b involves a2 (e.g., Chib and Greenberg (1996)). As explained above, this prior would make a independent of cr2 and hence assign greater probability to high Sharpe ratios. In the other alternative, the well-known natural-conjugate prior, the marginal prior for a2 is still inverted gamma, but the entire covariance matrix of b is proportional to a2 (e.g., Zellner (1971), Chapter 3). In the formula for the posterior mean of (3 for that case, the relative weights on the sample estimate and the prior mean do not depend on sample information about a. That is, j3 is given no more weight when the sample residual variance is small than when it is large, and that property is unappealing. Vasicek (1973) argues that the natural-conjugate prior is inappropriate when the prior parameters are estimated from a cross-section of stocks.
We assume that the regression parameters are independent of the moments of /“, the augmented set of factors:
which is the standard diffuse prior used to represent “non-informative” beliefs about the parameters of a multivariate normal distribution (e.g., Box and Tiao (1973)).
In order to construct the prior distribution for the regression parameters in (9) and (10), we specify the elements in b and V& and the scalar quantities s0 and v. (Note from (11) through (15) that Vb, Sq and и determine the conditional covariance matrix Ф(ег).) The prior values are chosen with the objective that the prior mean of b for any given stock be the mean of b in a given cross-section of stocks and that the prior unconditional covariance matrix of b for that stock, Vb, be the covariance matrix of b in the cross-section. Similarly, the prior mean and variance of a2 for the stock, determined by So and is, correspond to moments of a2 in the cross-section.