COSTS OF EQUITY CAPITAL: Stochastic Setting

Let r denote the T x 1 vector of returns on the stock of the firm whose cost of equity is to be estimated. In many cases, the stock’s return history, or at least the portion of that history used in the estimation, may be shorter than the history of the factors. It is assumed that there are S observations of the factors, with S > T. Let F^ denote the T x К matrix containing the T observations of the factors corresponding to the same periods as the returns in r. The regression disturbance et in (1) is assumed to be, in each period t, an independent realization from a normal distribution with zero mean and variance a2, so the most recent T observations of the returns and the factors obey the regression relation
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where b’ = \a /?’], X = [l? e contains the T regression disturbances, lt is a T-vector of l’s, It is a T x T identity matrix, and the notation is read “is distributed as.”

In addition to the S observations of the К factors, there exist L observations of Kl variables that are correlated with the factors. If L > S, then, as shown by Stambaugh (1997), the longer histories of these additional variables contain information about A, the A’ x 1 vector of factor means, beyond that contained in the factor histories alone. Let yt denote the Kl x 1 vector containing the observations of the additional variables in period t, and let Y^ denote the L x Kl matrix containing all L observations of yt. For each of the S periods over which both ft and yt are observed, define the “augmented” set of factors ff = [f[ y^], and assume that
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w’here the realizations are independent across t, в’ — [А’ вг2], and G denotes the covariance matrix of /“. For the L — S periods in which only yt is observed, it is also assumed that
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again with independent realizations across t, where G22 is the corresponding submatrix of G. That is, the marginal distribution of yt is given by (5) for all L periods. Finally, it is assumed that /“ is independent of e for all t.
Given the above assumptions, it follows that the likelihood function for the parameters (b, а, в, G) can be factored as
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