Model Comparison Bayesian Estimation (again)

[Peter Zar]

Thanks!

[Peter Zar]

The problem typically is that the way the marginal data density is computed, you need the prior to integrate to 1. But often there is implicit prior truncation due to e.g. the Blanchard-Kahn conditions not being satisfied. Estimating different regimes is a case like that, because you a priori impose a 0 prior density to parameter draws falling into other regimes when you reject these draws (instead of using the non-zero prior density given by your explicitly specified prior). When you want to compute the marginal data densities for the particular regimes, you therefore need to keep track of the prior mass in each regime and adjust for it not being equal to 1.

Sorry to ask again, I thought it was clear yesterday: Isn't that always the case? For instance in every NKM we have some sort of Taylor principle and usually some indeterminacy cut-off, say above 1. So normal model comparison would also not work?

I think I still do not fully understand the specific nature of having different regimes. The following is from a paper that cite Sims (2003) "Probability models for monetary policy decisions" and the training sample method:
"This poses a challenge to our analysis because, as in Lubik and Schorfheide (2004), we determine which region is favoured by the data based on the posterior probability of each region. This statistic is influenced by the prior distributuion"

So again, it is probably something with the regions but I still cannot make up my mind. Any helpful suggestions are appreciated!

Thanks again

[jpfeifer]

Yes, that is always the case, but most of the time, we do not do model comparison. Also, for easy models we can use a prior distribution that excludes the indeterminacy region (like a uniform distribution from 1 to 5 for the inflation feedback coefficient, where indeterminacy happens for values below 1).

If you read the discussion to An/Schorfheide (2006), the discussants are worried about the implicit prior truncation and An/Schorfheide respond that their implicit prior only truncates 3% of the mass and argue that this does not affect their conclusions.

The problem is not having different regimes, but rather having a prior distribution that includes two regimes, but then estimating the marginal data density for the regimes separately. Because in that case, you try to compare to "models" that now have an implicit prior that does not integrate to 1 (because the mass is split across the two regimes).

[Peter Zar]

I think now I got it, thanks so much! Also for the reference!

Best,
Peter