Let denote the posterior variance of ц obtained under model q. When estimating the expected excess return, the decision maker is left with overall uncertainty given by the unconditional posterior variance across models:
The first term on the right-hand side of (42), the expected value across models of the posterior variance of /i, is essentially the average within-model uncertainty about the expected excess return. This component of the overall uncertainty is analyzed in the previous section. Link The second term on the right-hand side of (42), the variance across models of the posterior mean of /л, can be termed “model” uncertainty, or the component of the overall posterior variance of ii attributable to uncertainty about which model to use.
Calculation of posterior model probabilities (7rg’s) is beyond the intended scope of this study. As noted at the outset, we focus more on issues related to using various factor-based models for cost-of-equity estimation rather than on issues related to testing such models or evaluating their relative merits. In order to illustrate the calculation of model uncertainty, we consider various sets of candidate models and, for each set, the 7t9’s are made equal across models.
When only two models are entertained, model uncertainty is bounded above by the value we report with equal 7iy s. With three models, the greatest of those bounds for the three possible two-model combinations is the upper bound on model uncertainty for the three-model combination. Although assigning equal probabilities across the three models generally results in a value somewhat less than that upper bound, that simple specification still provides a fairly generous assessment of model uncertainty that is useful in revealing its potential importance relative to the within-model parameter uncertainty discussed previously.