Consistency between loglikelihoods and business cycle statistics for DSGE model comparison

Dear Johannes,
Thank you very much for your helpful guidance, I am grateful.
To compare between two DSGE models, we can use Bayesian model comparison based on loglikelihods and comparison of business cycle statistics based on moments of observable variables. Should the results between these two exercises be similar and completely consistent? or slight inconsistent results for some observable variables are allowed due to different nature of these two model comparison tools?
Fir example, baseline DSGE model has the larger loglikelihoods than its variant DSGE model 1, for consistency, do we also expect that variances of all observable variables in baseline model to be the most similar to those statistics of data? and that variances of all observables in variant DSGE models 1 to be less similar to those of statistics of data?
Thank you very much and look forward to hearing from you.
Best regards,

  1. It’s not about the log-likelihood, but the marginal data density.
  2. The topic has been discussed e.g. at