Statement
A forward-looking Markov-switching rational expectations model has a unique mean-square stable equilibrium (determinacy) if and only if no non-MSV solution component w_t = Lambda_{s_{t-1},s_t} w_{t-1} + V_{s_t} V'_{s_t} gamma_t satisfying the structural constraint Gamma_i V_i = sum_j p_{i,j} V_j Phi_{i,j} has a mean-square stable spectral radius r_sigma(M_1(Phi_{i,j})) < 1. Equivalently (Cho 2020), determinacy requires rho(Psi_{F_1 kron F_1}) < 1 where F_1 is the forward operator associated with the MOD solution.
Evidence summary
Farmer et al. (2009) prove this for the purely forward-looking case (Theorem 2) by decomposing all solutions as MSV + Markov-switching MA component and applying MJLS mean-square stability theory. Cho (2020) generalizes to models with lagged variables by showing that rho(Psi_{F kron F}) <= 1 is both necessary and sufficient for sunspot non-existence (Proposition 1), with an analytically constructed maximally-stable sunspot achieving equality in (14). Foerster (2016) provides extensive numerical confirmation in a calibrated NK model.
Conditions and scope
- Forward-looking MSRE (FWZ 2009) or general MSRE with predetermined variables (Cho 2020).
- Mean-square stability as the stability concept.
- Under bounded stability, different (and less tractable) conditions apply.
- The condition is on the spectral radius of an Nn^2 x Nn^2 operator, which is computationally feasible for small-to-moderate n and N.
Counter-evidence
None. The conditions are proved as necessary and sufficient.
Linked ideas
(none yet)
Open questions
- Whether the same characterization extends to non-linear MSRE models.
- Efficient algorithms for the general constrained optimization when k_i > 1 (beyond the scalar case of FWZ Proposition 1).