This section extends results in Stambaugh (1997) and derives the posterior mean and variance-covariance matrix of Л in (31) and (32) when the likelihood function is given by (8) and the prior is given by (17). Recall that A contains the first К elements of в. Let Ф denote the data set consisting of F^ and Y^L\ the sample information about the moments of /“. Define the population counterparts to the quantities in (29) and (30) more.
If the prior parameter density for one or more of the models is improper, then obtaining nq can be problematic, since undefined constants appear in the numerator and/or denominator of (A.28). In our setting, even though the prior density for в and G in (17) is improper, model probabilities can still be defined, because that improper prior density can be made identical across the three models. In our setting, the parameter vector for model q can be partitioned as Sq = [<5q(i) <5(2)], where contains the elements of b and a and J(2) contains the elements of в and G. The elements of 59(i) differ in number and identity across models, but 6(2) can be made identical across models (as discussed below).
Therefore, even though p(6(2)) is defined only up to an undetermined constant, that constant appears in both the numerator and denominator of (A.28), and thus the ratio can be defined. As discussed by Kass and Raftery (1995), this treatment of ratios of improper priors for parameters that are common across models is due to Jeffreys (1961) and has been widely adopted.
The above statement that 5(2) can be made identical across models requires some clarification. As в and G are defined in Section I, they contain different elements across models.
Recall that they are the mean and covariance matrix of the vector of augmented factors, ft = [ft V’t I ’ where ft is an observation of the К factors for the given model and yt is an observation of the three long-history series. Although yt is the same across all three models, the factors differ across models. If, however, ff is instead defined as a larger vcctor containing yt and the union of all six factors across the three models, and в and G is defined as the mean and covariance matrix of that larger vector, all of the posterior moments reported are essentially unchanged. Specifically, h\ and H2 in (31) and (32) are redefined as submatrices of larger arrays, but their values are unchanged.
As a result, equation (31), which gives the posterior mean of A, is unaffected. In equation (32), which gives the posterior covariance matrix of A, the value of К changes to 6 (from either 1 or 3), but both S and L are large enough (390 and 840) such that any resulting changes in the reported standard deviations are trivial. Initially defining в and G in the above fashion would complicate the presentation of the methodology, so, given that the choice of definitions is essentially irrelevant to the empirical results, we adopt the simpler definition in Section I.