I am estimating a DSGE model with financial factors and a banking sector.
I want to introduce one observed variable more to my model, specifically, delinquency rate as proxy of probability of default, but when I do this the Log - posterior fall down from -3000 to -50000. I saw Log - posterior of another models and normally they are near to -500 so i think -50000 is very large and very poor. Do you recommend me to introduce this variable? Or should I descart it? What should I do if I want to introduce this variable?

It only makes sense to compare the log-posterior for different models with the SAME observed variables. Furthermore, by itself, the size of the log-posterior of a single model doesn’t mean anything. You can only use it for comparing models.
Usually, you decide about the list of the observed variables considering the fact that you want your model to explain.
If during model validation you remark that some unobserved variable has very counter intuitive behavior, you may consider adding an observed variable to remedy the problem.
If you had as many shocks as observed variables in the first model, you must then add also another shock to the model, and the fit of the original variables in the model may improve or deteriorate.
If you had more shocks that observed variables in the first mode, you don’t necessarily need to add another shock, but if you don’t, the fit of the other variables will deteriorate (think of it as an optimization problem to which you add an additional constraint).

I agree with @MichelJuillard on the selection of observables. You don’t simply discard observables, because the model does not fit them. What your experience suggests is that your model is not well able to explain the behavior of the observable you added. If that observable is an integral feature of your model, discarding it is no solution. The fit is as poor as before, you just don’t see it anymore.

@MichelJuillard’s point about comparing models is less clear to me. You compare models using the marginal data density, not the posterior density.