Journal of Econometrics 99 (2000) 347}348
Reply to Nobile
We thank Nobile for his comments. We feel that they add substantially to the usefulness of the paper. In our reply, we will follow Nobile's comment num-bering system.
The"rst comment augments the material for choosing the ID prior presented in Section 5.2.1 of the paper. Nobile's idea of imposing symmetry on the variances as well as the means of the elements ofRseems quite sensible. With the additional constraint we need only choose the single quantityi, to specify
the prior onR. This makes the choice of prior quite easy which is an important consideration in a Bayesian analysis. As noted in Section 5.2.1, the relative variability ofUandc, plays an important role is determining the prior
distribu-tion of the correladistribu-tions. Nobile's approach seems automatically to strike an appropriate balance. We simulated from the prior using o"3, i"6, and
Nobile's suggested vale ofq, and got a very reasonable prior distribution for the
correlation.
The second comment points out that while the paper presents a new approach to specifying the prior and computing the posterior for the MNP model, valuable contributions are also made to our understanding of the original NID approach. The paper does not argue that the NID approach be abandoned, which approach proves to be most valuable in practice remains to be seen.
The third and "nal comment of the introduction warns against the use of improper priors. For these models, the posterior even with improper priors is proper.
Section 2 of the comment provides and example where the ID Gibb's sampler proposed in the paper results in draws that are highly autocorrelated. In Section 6.2 of the paper we present a much more dramatic example of failure. The sampler actually gets stuck Nobile incorrectly states that `Although similar di$culties were encountered by MPR in their examples, they were not of the present gravitya. Section 7 of the paper gives some insight into how the ID Gibb's sample runs into trouble. The weight that the prior puts on the smallest eigenvalue is critical in determining the performance of the ID sampler. It is important to note that if there are convergence problems for a speci"c dataset, this will be immediately obvious to the investigator. We have presented a num-ber of examples of problems for which the ID sampler performs well.
Section 3 of the comment presents an alternative method for specifying a prior and drawing the corresponding poster with a constrained diagonal. Although the suggested alternative does not seem to have any clear advantage over the ID approach of the paper, we think it is a very nice contribution. It has the basic advantages of the ID approach: it is easy to implement by drawing from standard distributions and the prior may be easier to assess due to analytical tractability.
Robert E. McCulloch, Peter E. Rossi
Graduate School of Business University of Chicago, 1101 East 58thStreet, Chicago, IL 60637, USA E-mail address:peter.rossi@gsb.uchicago.edu (P.E. Rossi).
