The estimate of a stock’s expected excess return is given by the posterior mean, Е(д|Ф), where ц is a function of the unknown parameters a, 3. and A (equation (2)) and Ф is the historical sample information available to the decision maker. The imprecision in the estimate of the expected excess return is characterized by the posterior variance, Уаг(,и|Ф). The posterior mean and variance are obtained by combining the sample information about the unknown parameters with the decision maker’s prior beliefs about those parameters. A key feature of the prior beliefs is the mispricing uncertainty about a, represented by the prior standard deviation, oa. We let aQ take different values on the interval (0, oc) in order to explore the role of mispricing uncertainty in estimating the cost of equity.
Prior beliefs about the elements of /? and their correlations with a are constructed by viewing the firm as a random draw from a cross-section of firms. The prior mean of (3, for example, is set equal to the average of the ordinary-least-squares (OLS) estimates of (3 for the firms in the cross-section. This cross-section can be selected either as a broad universe or as a subset of firms that share one or more characteristics with the firm whose cost of equity is to be estimated. That firm’s posterior mean of f3 is then “shrunk” away from its own OLS estimate and toward the cross-sectional mean, in a manner similar to that discussed by Vasicek (1973). If, as in an example presented later, the firm is a public utility and the cross-section consists of other utilities, then the given firm’s (3 is shrunk toward the average (3 for utilities. In estimating costs of equity for various industries, Fama and French (1997) follow a similar approach and shrink each industry’s (3 towrard the market-wide average j3.