MOD Solution for MSRE Determinacy

Definition

The MOD (Minimum Of Modulus) solution of a Markov-switching rational expectations model is the MSV (minimum state variable) solution $x_t=Omeg{a}_1(s_t)x_{t-1}+Gamm{a}_1(s_t)z_t$ whose autoregressive coefficient matrices $Omeg{a}_1(s_t)$ have the smallest mean-square stability spectral radius among all MSV solutions: $rho(Ps{i}_{b}a{r}_{Omeg{a}_{1}kronOmeg{a}_1})=minrho(Ps{i}_{b}a{r}_{OmegakronOmega})$ over the full solution set S. The MOD solution plays the same role for MSRE models as generalized eigenvalues play for linear RE models: it provides both (a) the condition for non-existence of stable sunspot solutions and (b) the condition for uniqueness of the stable MSV solution.

Intuition

In a linear RE model, determinacy is checked by counting how many generalized eigenvalues are inside vs. outside the unit circle. In an MSRE model, there is no eigenvalue decomposition because the coefficient matrices switch with the regime. The MOD solution is the analog: it is the “least explosive” MSV solution, and its associated forward operator $F_1(s_t,s_{t+1})$ determines whether sunspot solutions can be mean-square stable. If the MOD solution itself is MSS and its forward operator rules out stable sunspots, the model is determinate.

Formal notation

For the MSRE model $x_t=E_t[A(s_t,s_{t+1})x_{t+1}]+B(s_t)x_{t-1}+C(s_t)z_t$, define the MSV solution set S as all $Omega(s_t)$ satisfying the fixed-point equation:

$Omega(s_t)={I_n-E_t[A(s_t,s_{t+1})Omega(s_{t+1})]}^{-1}B(s_t)$

Order the solutions by their mean-square stability spectral radius: $rho(Ps{i}_{b}a{r}_{Omeg{a}_{1}kronOmeg{a}_1})<=rho(Ps{i}_{b}a{r}_{Omeg{a}_{2}kronOmeg{a}_2})<=...$. The MOD solution is $Omeg{a}_1$. Its associated forward operator is $F_1(s_t,s_{t+1})={I_n-E_t[A(s_t,s_{t+1})Omeg{a}_1(s_{t+1})]}^{-1}A(s_t,s_{t+1})$.

The complete classification (Cho 2020, Proposition 3):

• Determinacy: $rho(Ps{i}_{b}a{r}_{Omeg{a}_{1}kronOmeg{a}_1})<1$ AND $rho(Ps{i}_{F_{1}kron{F}_1})<1$
• Indeterminacy: $rho(Ps{i}_{b}a{r}_{Omeg{a}_{1}kronOmeg{a}_1})<1$ AND $rho(Ps{i}_{F_{1}kron{F}_1})>=1$
• No stable solution: $rho(Ps{i}_{b}a{r}_{Omeg{a}_{1}kronOmeg{a}_1})>=1$

Variants

• LRE special case: When there is only one regime (P = 1), $Ps{i}_{b}a{r}_{OmegakronOmega}=OmegakronOmega$, so $rho(Ps{i}_{b}ar)<1$ iff $rho(Omega)<1$. The MOD solution is the solution with the smallest eigenvalues, and the classification reduces to the standard generalized eigenvalue count.
• Forward-looking models (B = 0): The MOD solution is trivially $Omega=0$ and $F=A$. Determinacy reduces to $rho(Ps{i}_{AkronA})<1$, recovering FWZ 2009 Theorem 2.
• Forward method approximation: The iterative recursion $Omeg{a}_{(k)}={I-E_t[AOmeg{a}_{(k-1)}]}^{-1}B$ starting from $Omeg{a}_{(1)}=B$ converges to the MOD solution in all tested cases, though no formal proof of equivalence exists for MSRE (it is proved for LRE).

Comparison

Versus the forward method (forward-method-rational-expectations): the forward method produces at most one solution by recursive substitution and selects the NBC-satisfying fundamental REE. In practice, it coincides with the MOD solution, but the MOD method has the theoretical advantage of providing necessary AND sufficient conditions, while the forward method’s conditions are only sufficient (Cho 2016).

Versus Groebner basis technique (Foerster et al. 2016): Groebner bases compute ALL MSV solutions, which is theoretically complete but computationally infeasible for n >= 3 (the cost grows as 2^N where N is the number of solutions). The MOD method needs only ONE solution.

Versus long-run Taylor principle (long-run-taylor-principle): the LRTP is only a necessary condition for determinacy (it checks first-moment stability, not second-moment); the MOD method is necessary and sufficient.

When to use

• When checking determinacy of an MSRE model for estimation or policy analysis.
• When the model has predetermined variables (lagged endogenous), where FWZ (2009) does not apply.
• When computational efficiency matters: the forward method + MOD identification takes ~10^{-3} to 10^{-2} seconds.

Known limitations

• The forward method is not proven to always converge to the MOD solution in MSRE (only in LRE). In rare “determinacy-inadmissible” cases (complex-valued MOD), the Groebner basis technique is needed.
• The MOD method classifies under mean-square stability; comparison with bounded stability remains open.
• The identification condition $rho(Ps{i}_{b}ar)\ast rho(Ps{i}_F)<1$ (“determinacy-admissible”) must hold for efficient computation; otherwise all MSV solutions must be enumerated.

Open problems

• Formal proof that the forward method converges to the MOD solution for MSRE models.
• Comparison of MOD-based MSS classification with bounded-stability classification (Barthelemy-Marx 2019).
• Extension to non-linear MSRE models.
• Empirical prevalence of the “unique stable MSV but indeterminate” phenomenon in estimated DSGE models.

Key papers

• determinacy-classification-markov-switching-rational-expectations — Cho (2020): introduces the MOD method and complete classification.
• understanding-markov-switching-rational-expectations-models — Farmer et al. (2009): foundational predecessor (forward-looking case only).

My understanding

The MOD method is the state of the art for MSRE determinacy checking. For the CRE asset pricing project, the forward-method solver in get_forward_solution_msre computes a solution that should coincide with the MOD solution (though this is not proven). The project’s check_determinancy_fmsre gate is effectively checking the MOD classification conditions. The Cho 2020 result that unique stable MSV does not imply determinacy under regime switching is a subtle but important caveat for the project’s 4-regime model.