the log marginal data density in mode_computer=6

[sfking]

Dear all:

I am now estimating an model using mode_compute=6, and the results in oo_.Marginal Density is
LaplaceApproximation= 588
ModifiedHarmonicMean=592

the question is which one is the log marginal data density, and could I choose the mode with higher results if I estimate alternative mode specification with the results

LaplaceApproximation= 588
ModifiedHarmonicMean=592

Thanks

[jpfeifer]

Both the Laplace approximation and the modified harmonic mean estimator are ways to compute the marginal data density. As computing the marginal data density involves solving a complicated integral, these two methods for tackling the issue have been proposed. In theory, they should yield identical results as they measure the same thing. In practice, both involve approximations and may yield differing results. It is hard to tell which one to prefer. Footnote 11 of Smets/Wouters (2007) states:

As discussed in John Geweke (1998), the Metropolis-
Hastings-based sample of the posterior distribution can be
used to evaluate the marginal likelihood of the model.
Following Geweke (1998), we calculate the modified harmonic
mean to evaluate the integral over the posterior
sample. An alternative approximation is the Laplace approximation
around the posterior mode, which is based on a
normal distribution. In our experience, the results of both
approximations are very close in the case of our estimated
DSGE model. This is not too surprising, given the generally
close correspondence between the histograms of the posterior
sample and the normal distribution around the estimated
mode for the individual parameters. Given the large advantage
of the Laplace approximation in terms of computational
costs, we will use this approximation for comparing
alternative model specifications in the next section.