In all three models, the posterior means of A are affected substantially by augmenting the factor histories, which begin in July 1963, with the longer histories of additional series that begin in 1926. These effects on posterior means indicate an important reliance on the information in the longer histories of the additional variables, but the posterior standard deviations of (3’X for the longer period are generally of about the same magnitude, or even slightly larger, than the posterior standard deviations for the shorter period. This outcome might seem puzzling, but the comparison of posterior standard deviations does not really provide a sensible measure of the additional information provided by the longer histories. The reason is that the longer histories can also provide additional information about uncertainty.
In particular, since the sample volatility of the long-history series is higher prior to 1963 than after, the posterior beliefs about the factors’ variances center on higher values when based on the overall period. This increase in posterior variance of the factors, ceteris paribus, raises the posterior variance of A, the vector of factor means. In effect, more information can reveal greater volatility, and thus greater uncertainty, than otherwise perceived. That effect then works in opposition to the more obvious one (also present): longer histories provide more information about factor means and, ceteris paribus, lower their posterior variances.
When aa is very large, the posterior standard deviation of a is fairly close to the usual frequentist standard error for the estimated regression intercept. In that case, not surprisingly, the posterior uncertainty about a dominates the posterior uncertainty about the expected excess return. At lower values of aa, the posterior standard deviation of a is typically about 1/2 to 3/4 of aa.
For example, when aa = 5%, the posterior standard deviation of a is just over 3% in all three models. The difference between the posterior standard deviation of fj, and the posterior standard deviation of (3’X arises due to uncertainty about a. In general, for values of