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The Long-Run Taylor Principle Revisited

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Problem

Two competing determinacy conditions exist for MSRE models: the long-run Taylor principle (LRTP) of Davig and Leeper (2007), which is a relatively simple eigenvalue condition on (P^{-1} kron I_n) diag(Gamma_1, Gamma_2), and the Farmer-Waggoner-Zha (FWZ) condition which is necessary and sufficient but requires a more complex matrix formulation. Cho and Moreno (2016) noted that the LRTP is only necessary for determinacy, not sufficient — FWZ (2010) provided an explicit counterexample. The paper asks: what additional restriction on the solution makes the LRTP sufficient as well as necessary?

Key idea

The LRTP is necessary and sufficient for determinacy if and only if the solution is restricted to not depend on the previous regime — i.e., y_t depends on s_t (current regime) and u_t (current shocks) but not on s_{t-1}. Under this restriction, the MSRE model reduces to the Davig-Leeper (DL) representation, which is a standard linear RE model in an augmented state vector, and the classical eigenvalue condition (all eigenvalues of Gamma_bar outside the unit circle) directly applies. The restriction is equivalent to requiring E(y_{t+1} | s_{t+1}, s_t, ..., u_t, ...) does not depend on s_t.

Method

  1. DL representation derivation: Starting from the MSRE model Gamma_{s_t} y_t = E(y_{t+1} | Omega_t) + Psi_{s_t} u_t, define regime-conditional variables y_{jt} (value y_t would take if s_t = j). Write out the system for each regime realization.
  2. Key step (Eq. 3.4): The conditional expectations E(y_{j,t+1} | s_t = i, Omega_t^{-s}) can in general depend on i. Only if they do NOT depend on i (the current regime) can the system be stacked into the DL representation diag(Gamma_1, Gamma_2) y_tilde_t = (P kron I_n) E(y_tilde_{t+1} | Omega_t^{-s}) + ... which is a standard linear RE model.
  3. Condition identification: The DL representation is valid iff y_{t+1} does not depend on s_t, i.e., the solution does not depend on the lagged regime.
  4. Proposition: Under the LRTP eigenvalue condition, the only non-explosive solution that does not depend on the previous regime is the regime-by-regime MSV solution y_t = Gamma_{s_t}^{-1} Psi_{s_t} u_t.

Results

  • Proposition: The long-run Taylor principle (generalized eigenvalue condition on (diag(Gamma_1, Gamma_2), P kron I_n)) is necessary and sufficient for determinacy within the class of solutions that do not depend on the previous regime.
  • The DL representation is valid only under this regime-independence restriction. Without it, the expectations E(y_{j,t+1} | s_t = i, ...) can depend on i, and the augmented linear system does not follow from the MSRE.
  • Lagged regimes are “sunspots” in the DL representation: they appear in the information set but the augmented system’s coefficients do not depend on them, so under the eigenvalue condition they are irrelevant for the solution.
  • If the economic theory underlying the MSRE does not imply regime-independence, the researcher must use the FWZ condition (or the MOD method of Cho 2020).

Limitations

  • The paper assumes 2 regimes for expositional clarity; the extension to h > 2 is straightforward but not written out.
  • The restriction “solution does not depend on previous regime” is a strong assumption that may not be justified by the underlying economic theory in most applications.
  • The paper does not provide new computational tools; it is a conceptual clarification.

Open questions

  • In which economic models does the regime-independence restriction hold naturally (i.e., is implied by the economic theory)?
  • How large is the gap between the LRTP region and the FWZ determinacy region in empirically calibrated models?

My take

A concise, elegant note that resolves a question left open in the Davig-Leeper vs. FWZ debate. The key insight — that the DL representation requires solutions to be independent of the previous regime, and this is precisely the restriction under which the LRTP becomes sufficient — is simple once stated but was missed in earlier work. For the CRE asset pricing project, this clarifies that the LRTP is not an adequate determinacy check: the project’s forward-method solver and absorbing-NBC proxy implement the FWZ/Cho conditions, which are strictly stronger than the LRTP.

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