How do central banks use IRFs of DSGE models and VARs

[jpfeifer]

1. You description above is wrong. It’s not about optimality. If you estimate a DSGE model on actual data, you typically still assume that the agents in it are behaving optimal.
2. The econometric difference between a VAR and a DSGE model is mostly the presence of cross-equation restriction. See e.g.

Bayesian Estimation in Dynare versus B-VAR

I am not sure I understand the question. First of all, the DSGE model generally is not a VAR. It is a VARMA. If you are lucky, you can represent it as a finite order VAR in your observables (see e.g. the ABCD-test of Fernandez-Villaverde et al. and the fundamentalness literature around Lippi-Reichlin). Second, the coefficients in your VAR are unrestricted. This differs from the DSGE model that will feature cross-equation restrictions (e.g. the discount factor entering for example coefficients …

3. From a central bank’s perspective, the biggest difference between the two is the fact that DSGE models are robust to the Lucas critique. That means you can do (policy) counterfactuals in them. That is not valid with VARS as they are not structural models.
4. A second advantage of DSGE models for central banks is that they allow you to tell narratives about how the world works in a consistent way. That eases communication to policy-makers and the public. With a VAR it’s mostly about which curve goes up or down, while the DSGE model tells you about the underlying economic decisions.
5. That of course comes with a downside as the structure of the model implies restriction on the data that may not be satisfied in practice (due to model misspecification). In that case, DSGE-VARs allow relaxing the rigid cross-equation restrictions of DSGE-models while still using informative priors in a VAR. But they are mostly useful for forecasting.