The forward solution is the unique fundamental REE satisfying the no-bubble condition

Statement

In the class of fundamental rational expectations equilibria of a linear multivariate RE model with predetermined variables, the forward solution — the limit of the deterministic recursion (Ω_k, Γ_k) produced by recursive forward substitution starting from the structural matrices (B, C) — is the unique solution that satisfies the no-bubble condition lim_{k→∞} M_k E_t x_{t+k} = 0. Equivalently, every other real-valued fundamental REE that may exist in the model’s solution set S leaves a non-vanishing “bubble term” of the form L^x x_t + L^z z_t in the forward representation, even though it depends only on the state variables.

Evidence summary

Cho and Moreno (2010) prove this as Proposition 1 (Appendix B) by direct construction: the recursion Ω_k = (I_n - A Ω_{k-1})^{-1} B, Γ_k = (I_n - A Ω_{k-1})^{-1}(C + A Γ_{k-1} R), M_k = (I_n - A Ω_{k-1})^{-1} A M_{k-1} is uniquely determined by the structural matrices and is independent of any candidate solution. If it converges (FCC holds), the limit (Ω*, Γ*) automatically satisfies Ω* = (I - AΩ*)^{-1} B and is therefore in the fundamental solution set S. By construction M_k → 0 along this trajectory, so the NBC holds. Proposition 2 (Appendix C) computes the bubble residual along any other element of S explicitly: L^x(j) = lim_{k→∞} M_k Ω^(j)^k and L^z(j) = lim_{k→∞} M_k Σ Ω^(j)^{k-i} Γ^(j) R^i, both of which are non-zero whenever (Ω^(j), Γ^(j)) ≠ (Ω*, Γ*).

The paper provides four illustrative examples (Section 5): a determinate New-Keynesian model where the forward solution coincides with MSV/MOD/E-stable; an indeterminate variant where one of two stationary fundamental solutions explicitly violates the NBC with concrete numerical L^x, L^z; an indeterminate variant where FCC fails entirely and MSV/MOD return a solution that violates the NBC; and a Dornbusch open-economy example with three stationary solutions and two E-stable ones, only one of which satisfies the NBC.

Conditions and scope

• The model must be of the form x_t = A E_t x_{t+1} + B x_{t-1} + C z_t with z_t = R z_{t-1} + ε_t and R having spectral radius < 1.
• B_1 (the contemporaneous structural matrix) must be invertible.
• The regularity condition |I_n - A Ω_{k-1}| ≠ 0 must hold for every k of the recursion (strictly stronger than the King–Watson condition that requires it only at the limit).
• FCC must hold for the forward solution to exist; under FCC the uniqueness statement is automatic. If FCC fails, the claim is vacuous (no NBC-satisfying fundamental REE exists at all).
• The class is restricted to fundamental solutions. Non-fundamental (sunspot, bubble) REEs are by definition outside the claim’s scope.

Counter-evidence

None identified in the source paper. The proof is constructive and the four examples consistently illustrate the proposition. The only “counter-example-like” case (Example 3, β = 0.9) is one where FCC fails — there the claim is vacuously true because there is no NBC-satisfying REE for the proposition to identify.

Linked ideas

(none yet)

Open questions

• Does an analogous uniqueness theorem hold for the Markov-switching extension where structural matrices (A^(s), B^(s), C^(s)) depend on a regime process? This is the load-bearing question for the CRE asset pricing project’s _msre solver and is not addressed by Cho–Moreno (2010).
• Can the NBC uniqueness be generalized to models with constants where the steady state is not known to agents?
• What is the relation between the FCC + NBC framework and the determinacy condition (Blanchard–Kahn root count)?