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Determinacy and Classification of Markov-switching Rational Expectations Models

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Problem

Prior MSRE determinacy results were incomplete. Farmer et al. (2009) provided necessary and sufficient conditions only for forward-looking models without predetermined variables. Cho (2016) gave sufficient (but not necessary) conditions using the forward solution. Foerster et al. (2016) computed all MSV solutions via Groebner bases but this was computationally prohibitive for models with n >= 3. No complete, tractable classification of general MSRE models (including models with lagged endogenous variables) into the three exhaustive cases — determinacy, indeterminacy, no stable solution — existed.

Key idea

The MOD (minimum of modulus) solution — the MSV solution Omega_1(s_t) with the smallest mean-square stability spectral radius rho(Psi_bar_{Omega_1 kron Omega_1}) — completely classifies all MSRE models via two spectral conditions: (1) rho(Psi_bar_{Omega_1 kron Omega_1}) < 1 (MOD solution is MSS), and (2) rho(Psi_{F_1 kron F_1}) < 1 (no stable sunspots exist). The MOD solution plays the same role for MSRE models as generalized eigenvalues do for linear RE models. Crucially, the full solution set need not be computed — only the MOD solution is needed.

Method

  1. Solution decomposition: Any solution x_t = Omega(s_t) x_{t-1} + Gamma(s_t) z_t + w_t where w_t satisfies a purely forward-looking restriction w_t = E_t[F(s_t, s_{t+1}) w_{t+1}] (Eqs. 3-6).
  2. MOD solution definition (Definition 1): The MSV solution minimizing rho(Psi_bar_{Omega kron Omega}) over all solutions to the matrix fixed-point Omega(s_t) = {I - E_t[A(s_t,s_{t+1}) Omega(s_{t+1})]}^{-1} B(s_t).
  3. Proposition 1 (sunspot non-existence): No MSS sunspot w_t exists iff rho(Psi_{F kron F}) <= 1. This is proved by constructing the maximally stable sunspot Lambda_m analytically.
  4. Proposition 2 (MSV uniqueness): If rho(Psi_bar_{Omega_1 kron Omega_1}) * rho(Psi_{F_1 kron F_1}) < 1, then Omega_1 is the unique real-valued MOD solution and no other MSV solution is MSS.
  5. Proposition 3 (complete classification): Determinacy iff both conditions hold; indeterminacy iff MOD is MSS but rho(Psi_{F_1 kron F_1}) >= 1; no stable solution iff MOD is not MSS.
  6. Efficient computation: The forward method (iterating Omega_{(k)} = {I - E_t[A Omega_{(k-1)}]}^{-1} B from B) finds the MOD solution in ~10^{-3} to 10^{-2} seconds; the Groebner basis technique is needed only for rare determinacy-inadmissible cases.

Results

  • Complete classification (Table 1): All MSRE models are partitioned into determinacy, indeterminacy, and no stable solution by two spectral conditions on the MOD solution alone.
  • Unique MSV does NOT imply determinacy under regime switching: A model can have a unique stable MSV solution but admit a continuum of stable sunspot solutions (unlike LRE models where uniqueness of MSV implies determinacy). This is a general phenomenon arising from the strict inequality in rho(Psi_bar kron) * rho(Psi_F kron) >= rho(Psi_{Omega' kron F}).
  • Long-run Taylor principle is only sufficient for first-moment stability, which is strictly weaker than MSS (second-moment stability). The LRTP region strictly contains the determinacy region.
  • Fiscal-monetary policy switching example: Full classification of the Leeper (1991) model with switching AM/PF and PM/AF policy mixes, demonstrating all three cases including indeterminacy with a unique stable MSV.
  • Equivalence to standard methods for LRE: When there is only one regime, the MOD method reduces to standard generalized eigenvalue conditions (Table 2 vs. gensys).

Limitations

  • The forward method and MOD solution may not coincide in theory (no proof of equivalence for MSRE, though they agree in all experiments).
  • Determinacy-inadmissible models (complex-valued or repeated MOD solutions) require the computationally expensive Groebner basis technique.
  • The paper uses mean-square stability; comparison with boundedness-based classification (Barthelemy-Marx 2019) is left for future work.

Open questions

  • Formal proof that the forward method always converges to the MOD solution in MSRE models.
  • Extension to non-linear MSRE models beyond first-order perturbation approximation.
  • Empirical prevalence of the “unique stable MSV + indeterminate” phenomenon in estimated DSGE models.

My take

This paper completes the MSRE determinacy theory started by Farmer et al. (2009). The MOD method is the state of the art: it is necessary and sufficient, computationally efficient (one solution needed, not all), and handles models with predetermined variables. For the CRE asset pricing project, this validates the forward-method solver as the correct tool — the forward solution nearly always coincides with the MOD solution, and the project’s check_determinancy_fmsre is checking exactly the conditions of Proposition 3. The subtle result that unique stable MSV does not imply determinacy under regime switching is directly relevant to the project’s 4-regime model.

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