In comparing the model in (5b), allowing for time variation in the factor sensitivities, and the static specification in (5a), for each economic sector, the model with time variation is strongly preferred over the model with static unconditional factor sensitivities. For each case, the Bayes factor exceeds 4, and for 9 of the 10 sectors, the Bayes factor exceeds 5, indicating very strong evidence in favor of model (5b) over model (5a). Based on the marginal likelihood values, it is also clear that allowing for heteroskedastic errors in (5c) results in a much more probabilistic model for each sector than that of the unconditional variance model in (5a). In this case, the Bayes factor favoring model (5c) over model (5a) exceeds 28 for each of the economic sectors.
(Insert Table 4 here)
Finally, it is also shown in table 4 that the model in (5), incorporating both time varying factor sensitivities and Markov switching heteroskedastic errors, is preferred to all other models by a Bayes factor of at least 3 for eight of the ten economic sectors. For Consumer Staples and Health Care, model (5c) is preferred to model (5).
Empirical Results for the Fully Specified Model
Table 5 summarizes results from the Bayesian Gibbs sampling estimation of the parameters in model (5). The sample period for the simulation is January 12, 1990 to June 10, 2002. For each model for each parameter, the posterior mean, median and standard deviation are reported. For each sector there is strong evidence for persistent volatility regimes as indicated by the values for
p and q greater than 0.9 for each of the tested portfolios. The standard error terms, vk , are
generally of small size, indicating a slow evolution process for the factor loadings.
(Insert Table 5 here)
Given the estimated parameters, inference and prediction of the state vector, T , can proceed.
Because we are ultimately interested in the predictability of portfolio returns, we would like to focus on the behavior of out of sample factor loadings as opposed to the backward looking path