Question about RISE solvers and multiple solutions

[husangkim]

Dear Junior Maih.

Thank you very much for your work on RISE and for making it such a valuable tool. I had a question regarding how the RISE solvers handle cases with multiple solutions.

In the example codes, it seems that there are four available solvers:
solvers = {'mnk', 'fwz', 'mn', 'mfi'};

1. How does each solver deal with multiple solutions? Specifically, do they store all available solutions, or only one? If I obtain only a single solution after running solve(), is there a way to verify that the solution is unique?

2. Among the solvers listed above, which would you recommend as the most robust baseline option?

Thank you very much for your time and guidance.

Best regards,

Husang Kim

[jmaih]

Dear Husang, Regarding your question on multiple solutions: all the solvers in RISE are iterative solvers designed for nonlinear problems in which *multiple* solutions may exist. None of them attempts to enumerate or store all possible solutions. Each solver returns *one* solution — the particular fixed point it converges to — and this depends on several factors: the internal mechanics of the algorithm, the structure of the model, the initial conditions, and the convergence criteria. This contrasts sharply with constant-parameter linear rational-expectations models, where we can characterize uniqueness and stability ex-ante using analytical criteria. In Markov-switching linear rational-expectations systems, stability cannot be assessed *before* solving the model; we cannot know beforehand whether the model admits a unique stable solution. RISE does not provide a built-in method to verify uniqueness once you leave the constant-parameter case. A common practical approach is to run the solver from different initial conditions and check whether it converges to the same fixed point. Consistent convergence raises confidence in uniqueness, although it can never guarantee it. There is one exception: the *groebner* solver. It borrows techniques from computational algebraic geometry and can, in small models, compute *all* algebraic solutions. However, it is extremely costly and scales poorly. If you use it to compute all solutions, you can then test ex post whether only one of them is stable (if uniqueness is a priority for you). Personally, I care more about *stability* than uniqueness. There is no evidence that real-world data are generated in environments where agents always face a unique choice. Saying that a particular parameterization yields a unique stable solution is one thing; insisting that a model must have a unique stable solution to be meaningful is quite another. As for the solvers: - *mnk* — Newton algorithm using Kronecker products (efficient for small systems, but storage and computation explode as the system grows) by Maih (2015). - *fwz* — Newton algorithm following Farmer, Waggoner and Zha (2011). - *mn* — Newton algorithm without Kronecker products (similar to mnk but scalable to larger systems) by Maih (2015). - *mfi *— Functional iteration by Maih (2015) with very good properties; this is the default and generally the most robust choice. Best, Junior

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On Wed, Nov 5, 2025 at 5:45 AM husangkim ***@***.***> wrote: *husangkim* created an issue (jmaih/RISE_toolbox#223) <#223> Dear Junior Maih. Thank you very much for your work on RISE and for making it such a valuable tool. I had a question regarding how the RISE solvers handle cases with multiple solutions. In the example codes, it seems that there are four available solvers: solvers = {'mnk', 'fwz', 'mn', 'mfi'}; 1. How does each solver deal with multiple solutions? Specifically, do they store all available solutions, or only one? If I obtain only a single solution after running solve(), is there a way to verify that the solution is unique? 2. Among the solvers listed above, which would you recommend as the most robust baseline option? Thank you very much for your time and guidance. Best regards, Husang Kim — Reply to this email directly, view it on GitHub <#223>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AATKBTZLFD2I6X4QF5EHORL33F6FLAVCNFSM6AAAAACLFPOW22VHI2DSMVQWIX3LMV43ASLTON2WKOZTGU4DSMRWHEYDQMI> . You are receiving this because you are subscribed to this thread.Message ID: ***@***.***>

[husangkim]

Thank you for the deep and insightful response.

Please permit me some more follow-up questions. Is there a way to do "different initial conditions and check whether it converges to the same fixed point" test? Also, other than those four, the Grobner basis solution is not available I guess?

The toolbox has been very helpful for my research. I really appreciate your contribution.

Best,

Husang Kim.

[jmaih]

Hi Husang, Regarding your follow-up questions: There is *no general guarantee* that different initial conditions will converge to the same fixed point. Whether they do depends on the model's structure, the presence of multiple equilibria, and the algorithm's internal mechanics. Because of this, running the solver from multiple starting points is only a *practical heuristic*, not a formal uniqueness test. In RISE, you can experiment with different initializations using the option *solve_initialization*. The default choice is: - *'backward'* — the most robust option in general, assuming the model is purely backward-looking for initialization. Other available settings are: - *'zeros'* — initializes all state variables at zero - *'random'* — initializes at randomized values Switching among these can give you some insight into whether the solver consistently finds the same solution, although divergence or convergence to different fixed points is entirely possible depending on the model. Concerning the Groebner-basis approach: RISE does include an algebraic solver, *dsge_groebner*, which attempts to compute *all* algebraic solutions of the system. However, it is only useful for *very small* models and is *computationally very expensive*. For this reason, it is not part of the default solver list and is rarely practical for empirical work. But if you truly need to enumerate all possible solutions and the model is simple enough, dsge_groebner is available. Best, Junior

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On Fri, Nov 7, 2025 at 2:23 AM husangkim ***@***.***> wrote: *husangkim* left a comment (jmaih/RISE_toolbox#223) <#223 (comment)> Thank you for the deep and insightful response. Please permit me some more follow-up questions. Is there a way to do "different initial conditions and check whether it converges to the same fixed point" test? Also, other than those four, the Grobner basis solution is not available I guess? The toolbox has been very helpful for my research. I really appreciate your contribution. Best, Husang Kim. — Reply to this email directly, view it on GitHub <#223 (comment)>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AATKBT7CW34XP6YJ5BSBPP333PX7VAVCNFSM6AAAAACLFPOW22VHI2DSMVQWIX3LMV43OSLTON2WKQ3PNVWWK3TUHMZTKMBQGA3DGMRRGY> . You are receiving this because you commented.Message ID: ***@***.***>

[husangkim]

Thank you so much for your kind and sincere response. I really appreciate that you tried to make this toolbox so complete!