Endogenous transition probability

[husangkim]

Dear Junior Maih.

I wanted to ask some questions about endogenous switching/transition probabilities (ESPs) in RISE. I have read the thread #217 (#217), and this is a follow-up:

In my experiments, when I specify ESPs in .rs file, such as:

• coef_tp_1_2 = 1/(1 + exp((c - pi))
• coef_tp_2_1 = 1/(1 + exp((pi - c))
  where c is the steady state value of inflation pi (e.g., c=0).

The first-order solution I obtain is identical to the solution I get when I instead fix the probabilities at their steady-state values, for example:

• coef_tp_1_2 = 0.5
• coef_tp_2_1 = 0.5

My understanding is that, at first order, the perturbation solution only uses the ESP evaluated at the expansion point (steady state). The solutions become different when I calibrate c a different value from the steady state. My questions are

1. If the first-order solution under ESPs coincides with the solution under constant probabilities equal to the steady-state ESP values, is it valid to:

• Solve the model using constant probabilities set equal to those steady-state values.
• During simulation, compute the exact ESP each period using the nonlinear formula above,
• Draw the regime using these time-varying probabilities (ESP),
• Apply the corresponding regime-specific linear policy functions from the constant-probability solution?
  In other words, at first order, is anything lost by solving with fixed probabilities but simulating with the fully nonlinear ESP?

2. Relatedly, this may be model-dependent, but if ESPs only enter the policy functions at second order and above, does using a first-order approximation significantly understate their impact?

Thank you very much for the clarification.

[jmaih]

Dear Husang, At first order, your intuition is correct: the policy functions of an endogenous-switching model coincide with those of an exogenous-switching model *as long as the transition probabilities match at the steady state*. This happens because the policy functions are approximated, and their first-order expansion only uses the transition matrix evaluated at the expansion point. However, this is only half of the solution. A regime-switching DSGE model has *two components*: 1. *Policy functions* (approximated) 2. *Transition matrix* (not approximated) Even if the policy functions are identical, the *transition matrix is not*. Under endogenous switching, the transition probabilities are time-varying and depend on the state of the economy. This means the *simulated dynamics differ* because the regime sequence is generated from a transition matrix that evolves over time. RISE already handles this automatically. You do *not* need to compute probabilities by hand or adjust anything during simulation. So the answer to your main question is: - Yes, the first-order policy functions will be identical if the steady-state probabilities coincide. - No, you do not lose anything by solving with steady-state probabilities: RISE will still use the time-varying ESPs when simulating. - And yes, the *impact of endogenous switching appears in the simulations through the time-varying transition matrix*, even at first order. - At higher orders, the policy functions themselves also differ. Best regards, Junior

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On Thu, Nov 27, 2025 at 9:10 PM husangkim ***@***.***> wrote: *husangkim* created an issue (jmaih/RISE_toolbox#228) <#228> Dear Junior Maih. I wanted to ask some questions about endogenous switching/transition probabilities (ESPs) in RISE. I have read the thread #217 <#217> (#217 <#217>), and this is a follow-up: In my experiments, when I specify ESPs in .rs file, such as: - coef_tp_1_2 = 1/(1 + exp((c - pi)) - coef_tp_2_1 = 1/(1 + exp((pi - c)) where c is the steady state value of inflation pi (e.g., c=0). The first-order solution I obtain is identical to the solution I get when I instead fix the probabilities at their steady-state values, for example: - coef_tp_1_2 = 0.5 - coef_tp_2_1 = 0.5 My understanding is that, at first order, the perturbation solution only uses the ESP evaluated at the expansion point (steady state). The solutions become different when I calibrate c a different value from the steady state. My questions are 1. If the first-order solution under ESPs coincides with the solution under constant probabilities equal to the steady-state ESP values, is it valid to: - Solve the model using constant probabilities set equal to those steady-state values. - During simulation, compute the exact ESP each period using the nonlinear formula above, - Draw the regime using these time-varying probabilities (ESP), - Apply the corresponding regime-specific linear policy functions from the constant-probability solution? In other words, at first order, is anything lost by solving with fixed probabilities but simulating with the fully nonlinear ESP? 2. Relatedly, this may be model-dependent, but if ESPs only enter the policy functions at second order and above, does using a first-order approximation significantly understate their impact? Thank you very much for the clarification. — Reply to this email directly, view it on GitHub <#228>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/AATKBT6GZWEESBZT3X6YVMD365LD7AVCNFSM6AAAAACNM4PZMSVHI2DSMVQWIX3LMV43ASLTON2WKOZTGY3TEOBYHEYTANI> . You are receiving this because you are subscribed to this thread.Message ID: ***@***.***>

[husangkim]

Thank you so much for the detailed response!