I am planning to use grid-research to find optimal policy parameters according to welfare analysis. However, some parameters will change the number of eigenvalues larger than 1 in modulus. I am confused about this: why policy parameters will change the modulus? Since in my knowledge, only deep parameters could affect the rank condition.

No, the policy parametes define the policy rule that closes the model. Therefore, they are deep parameters. Think about a Taylor rule where you optimize the inflation feedback. As soon as you do not satisfy the Taylor principle, the BK conditions will not be satisfied anymore.

Thanks for your answer! That is exactly what I am confused. Actually, I know the coefficient of inflation in Taylor rule should be larger than one to meet BK condition. But I don’t know details about this principle. Thus when I put more targets into a augmented Taylor rule (for instance, a Taylor rule included financial factor), I can never know the reasonable value of policy coefficient and handle the results. I wonder where could I find the answer. Thanks in advance!

Well, I think I could describe my confusion more clearly. The paper attached is the one I referred. In its section 6, the authors do a grid-research to the policy parameters, for instance, [0,1] for the coefficient of financial factor. However, after having built a similar model, I could only run it within a much smaller interval (and neither 0 nor 1 can meet the rank condition). So my question is actually what should be concerned when building such a model and how can I get an interval as beautiful as shown in the attached paper. Is there any underlying tricks when designing policy rules?

That is hard to tell. The size of the stability/determinacy region can vary widly between models. What you need to make sure is that the non-solution of the model for some parameters is really driven by a violation of the BK conditions and not e.g. due numerically not being able to find the steady state. You can check this by trying some parameter combinations manually and checking the error code. A sign of problems is often if the determinacy region is not continuous.

Thanks so much for your reply. So there is no general method to control the stability region and the results of paper attached may just be kind of occasional. Furthermore, it is not even necessary for me to purchase a better-looking interval like that. Am I right?