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 (1,1) and (2,3) must be the same). If you want to account for MA components and cross-equation restrictions, you cannot use OLS and/or equation by equation estimation.