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Understanding Markov-Switching Rational Expectations Models

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Problem

Reduced-form Markov-switching VARs (Hamilton 1989) are widely used to model structural shifts, but no known set of necessary and sufficient conditions existed for determinacy in forward-looking Markov-switching rational expectations (MSRE) models. Davig and Leeper (2007) provided conditions for a linear representation of the MSRE but Farmer, Waggoner, and Zha (2008a) showed these conditions did not apply to the original MSRE model. The paper asks: when does an MSRE model with purely forward-looking variables have a unique mean-square stable equilibrium?

Key idea

All solutions to the forward-looking MSRE model Gamma_{s_t} y_t = E_t y_{t+1} + Psi_{s_t} u_t can be decomposed as the sum of (a) a regime-conditional MSV (minimum state variable) solution G_{s_t} u_t and (b) a Markov-switching first-order process w_t = Lambda_{s_{t-1},s_t} w_{t-1} + V_{s_t} V'_{s_t} gamma_t (Theorem 1). The determinacy question then reduces to asking whether this Markov-switching component w_t has any mean-square stable realization. The answer is given by the spectral radius of the operator M_1(Phi_{i,j}) = [p_{i,j} Phi_{i,j} kron Phi_{i,j}]: determinacy holds iff r_sigma(M_1(Phi_{i,j})) >= 1 for all valid (V_i, Phi_{i,j}) satisfying the structural constraint Gamma_i V_i = sum_j p_{i,j} V_j Phi_{i,j} (Theorem 2, Corollary 1).

Method

  1. Characterization Theorem (Theorem 1): Prove that every solution to the MSRE is the MSV solution plus a Markov-switching moving-average component.
  2. Spectral radius criterion (Theorem 2): Show the component is MSS iff r_sigma(M_1(Phi_{i,j})) < 1, directly applying MJLS mean-square stability theory (Costa-Fragoso-Marques 2004).
  3. Constrained optimization reformulation (Corollary 1): Recast the determinacy check as: for each admissible dimension {k_1,...,k_h}, minimize r_sigma(M_1(Phi_{i,j})) subject to the structural constraints; if the optimum is always >= 1, the model is determinate.
  4. Special case k_i = 1 (Proposition 1): For scalar sunspot dimensions, the structural constraint reduces to a nonlinear equation det(diag(Gamma_i) - (diag(c_i) P kron I_n)) = 0 and the MSS condition becomes r_sigma(diag(c_i^2) P) < 1. Graphical analysis via the correspondence c_1 = psi(c_2).
  5. Application to new-Keynesian model (Section IX): Three examples showing that (a) a sufficiently active monetary policy ensures determinacy (alpha_2 = 2.5), (b) a weakly active policy causes indeterminacy (alpha_2 = 1.05, p_11 = 0.75), and (c) more persistent passive regimes expand the indeterminacy region (alpha_2 = 1.05, p_11 = 0.90).

Results

  • Necessary and sufficient conditions for determinacy in forward-looking MSRE models with no predetermined variables (Theorem 2 + Corollary 1).
  • Mean-square stability is the correct stability concept for MSRE, not bounded stability: MSS allows temporarily explosive regimes that are reined in by sufficiently frequent switching to stable regimes. Bounded stability has no tractable characterization (CFM Example).
  • Sunspot characterization: All non-MSV solutions have the Markov-switching MA structure of Eqs. (15)-(16), and are MSS iff the spectral radius condition (19) holds.
  • Indeterminacy can spillover: In the NK model, even an active monetary policy regime can exhibit a wide range of sunspot dynamics (persistence parameter c_2 from -1 to >1) when the passive regime is sufficiently persistent.
  • The paper provides the first algorithm for partitioning the parameter space of MSRE models into determinate and indeterminate regions.

Limitations

  • Purely forward-looking models only: The paper does not cover MSRE models with predetermined (lagged endogenous) variables. This is noted as the main open question.
  • Computational cost of the general case: For k_i > 1, the constrained optimization is non-trivial. The special case k_i = 1 is amenable to graphical analysis but is not general.
  • MSS vs. bounded stability debate: The paper argues for MSS but acknowledges that some economists prefer bounded stability; the two concepts disagree in important cases.

Open questions

  • Extension to models with predetermined variables (addressed later by Farmer et al. 2011, Cho 2014/2020, Foerster et al. 2016).
  • Efficient algorithms for the general constrained optimization (Corollary 1) beyond the k_i = 1 special case.
  • Whether indeterminate MSRE models can be estimated by partitioning the parameter space into determinate/indeterminate regions with separate likelihood constructions.

My take

This is the foundational paper for MSRE determinacy theory. The key insight — reducing the determinacy question to mean-square stability of a Markov-switching sunspot process — is elegant and connects the economics RE literature to the MJLS engineering literature. The paper’s limitation to forward-looking models was addressed by Cho (2020) with the MOD method. For the CRE asset pricing project, this paper provides the theoretical underpinning for the check_determinancy_fmsre gate: the spectral radius condition is what the project’s absorbing-NBC proxy operationalizes.

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