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Solved. Thanks!

1. You don't get the plots because of the

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nograph

2. The model_comparison fails because the estimation does not properly work. You did not find a correct mode and the data is wrong. obs_y_h for example has a massive seasonal pattern

Solved. Thanks!

You should be working with different priors that do not imply asymptotes (e.g. at the unit root). In the unstable version I get

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Prior distribution for parameter theta_star has unbounded density!
Prior distribution for parameter rho_a_star has unbounded density!
Prior distribution for parameter phi_e_h has unbounded density!
Prior distribution for parameter phi_e_star has unbounded density!

That should explain the problems with rho_a_star.

Regarding the DSGE-VAR, the implied covariance matrix of the VAR's innovations, based on the artificial sample, is not positive definite. This suggests that your model is stochastically singular, i.e. the shocks you have left are not sufficient to get observables used in the VAR that are not exact linear combinations.