DSGE Model Solutions as a Vector Time-Series process. Examples?

After much reading, here's the conclusion I got.

VAR models can be seen as State-Space models/representations.

Usually DSGE linearised solutions have a State-Space representation, with more restrictions than a regular VAR model would have. In fact, there are papers(I remember one from spanos testing the Smets and Wouters model) showing DSGE solutions don't pass the specification tests we usually do to the data.

For example, instead of assuming that the residuals follow a normal multivariate distribution, they show we would be better to assume a student-t multivariate distribution. Of course, the problem is that most textbooks dealing with multivariate time series - if I remember correctly lutkephöl is one of them - also assume normal distributions. But at least, the time series VAR analysis can be made robust, while I've never seen DSGE model estimations using student-t distributions instead of normal ones.

P.S.: this is more like a note to self, from my self studying the subject. If anyone knows more, be free to share.