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The forward method as a solution refinement in rational expectations models

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Problem

Linear rational expectations (RE) models with predetermined variables generically admit multiple solutions: an infinity of non-fundamental (bubble / sunspot) solutions and a finite (but possibly multiple) set of fundamental solutions that depend only on state variables. The literature offers competing selection criteria — the minimum state variable (MSV) criterion of McCallum (1983), McCallum’s MOD solution (smallest generalized eigenvalues, 2004), and the E-stability criterion of Evans and Honkapohja (2001) — but no consensus on which fundamental REE is “the right one.” Even under determinacy, the unique stable fundamental solution is not automatically the economically sensible one (Cho and McCallum 2009). The paper asks: out of the potentially multiple fundamental REEs, which is the most economically sensible, and in what sense?

Key idea

Generalize the textbook forward method of recursive substitution — well-understood in univariate models without lagged variables (Cagan 1956, Samuelson 1958) — to a fully general multivariate linear RE system with predetermined variables. Define two model-implied properties:

  • the Forward Convergence Condition (FCC) — the coefficient sequences (M_k, Ω_k, Γ_k) produced by recursive forward substitution converge as k → ∞;
  • the No-Bubble Condition (NBC) — the expectational term lim_{k→∞} M_k E_t x_{t+k} = 0 when expectations are formed with the candidate solution.

The forward solution is then defined as the limit Ω* x_{t-1} + Γ* z_t of the forward representation under FCC, with no reference to the set of all REEs. Its existence and uniqueness are governed entirely by FCC, and it is automatically a member of the set of fundamental solutions S. The central theorem (Proposition 1) states: the forward solution is the unique fundamental REE satisfying the NBC. Consequently, all other fundamental REEs that may exist (including those returned by MSV, MOD, or E-stability) must violate the NBC and are therefore economically hard to interpret.

Method

Starting from the canonical multivariate model

x_t = A E_t x_{t+1} + B x_{t-1} + C z_t,        z_t = R z_{t-1} + ε_t

(where A, B are n × n, R has spectral radius < 1, and B_1 from the structural form has been inverted), the paper constructs a unique sequence of real-valued matrices {M_k, Ω_k, Γ_k} by recursive forward substitution under a regularity condition |I_n - A Ω_{k-1}| ≠ 0:

M_1 = A,    Ω_1 = B,    Γ_1 = C
M_k = (I_n - A Ω_{k-1})^{-1} A M_{k-1}
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)

so that the model can be rewritten as the forward representation

x_t = M_k E_t x_{t+k} + Ω_k x_{t-1} + Γ_k z_t      for all k = 1, 2, 3, ...

The forward convergence condition is satisfied iff (Ω_k, Γ_k) → (Ω*, Γ*) as k → ∞. Under FCC, the forward solution is x_t = Ω* x_{t-1} + Γ* z_t and (Ω*, Γ*) automatically belongs to the fundamental solution set S (defined by Ω = (I - AΩ)^{-1} B and Γ = (I - AΩ)^{-1}(C + AΓR)).

The no-bubble condition is lim_{k→∞} M_k E_t x_{t+k} = 0_{n×1}. Proposition 1 (Appendix B) proves the forward solution is the unique real-valued fundamental REE that satisfies the NBC. Proposition 2 (Appendix C) makes the diagnostic precise: if any other fundamental solution (Ω^(j), Γ^(j)) is used to evaluate the expectational term, then under FCC

lim_{k→∞} M_k E_t x_{t+k} = L^x(j) x_t + L^z(j) z_t ≠ 0

with L^x(j) = lim M_k Ω^(j)^k and L^z(j) = lim M_k Σ Ω^(j)^{k-i} Γ^(j) R^i. If FCC fails, the term explodes or oscillates. Either way, the offending REE is not a “true” bubble-free solution.

The paper also gives a graphical analysis (Section 3.3, Figs. 1–2) of the univariate case via the iteration v_{k+1} = 1 - θ/v_k with θ = ab, showing convergence/non-convergence regimes for the regularity condition and the FCC, and explaining why convergence is faster when the two roots of the matrix quadratic are well separated.

Results

Section 5 applies the forward method to four economic examples — the first three based on a New-Keynesian (Phillips curve + IS + Taylor rule) model in (π_t, y_t) with natural rate AR(1) shock, the fourth a Dornbusch open-economy model from Evans–Honkapohja:

  • Example 1 (determinate, β = 1.5): Taylor principle holds. The model is determinate, FCC holds, the forward solution coincides with the unique stationary fundamental solution and with all three of MSV / MOD / E-stable. Ω_k, Γ_k converge by k=25.
  • Example 2 (indeterminate, β = 0.95, FCC holds): Three stable generalized eigenvalues, two distinct stationary fundamental solutions. The forward method picks (Ω(g_1, g_2), Γ(g_1, g_2)). The other fundamental solution (g_1, g_3) violates the NBC: L^x(g_1, g_3) ≠ 0 and L^z(g_1, g_3) ≠ 0 (concrete numerical values reported in the paper). MSV, MOD, and E-stability all happen to coincide with the forward solution here.
  • Example 3 (indeterminate, β = 0.9, FCC fails): A small change in β from 0.95 to 0.9 destroys forward convergence — Γ_k explodes (Table 2 shows γ_{k,11} grows from 0.27 to 14 332 by k=100) even though Ω_k still converges. The MSV criterion (verified by sweeping a constant α on B from 1 to 0, Table 3) selects a solution that violates the NBC. The forward method correctly refuses to certify any fundamental solution as economically sensible. E-stability also rejects both candidates here.
  • Example 4 (Dornbusch / Evans–Honkapohja): Three stationary fundamental solutions (ω = 0.7160, 0.7721, 0.9897). Two of them (0.7160 and 0.9897) pass E-stability. The forward solution is 0.7160; the other E-stable solution 0.9897 violates the NBC. MSV and MOD also pick 0.7160.

The numerical examples concretely demonstrate that the FCC is a brittle, model-specific property (small parameter perturbations can flip it), and that the popular selection criteria can silently return solutions failing the NBC even in simple, well-formulated economic models.

Limitations

  • The proof is restricted to the class of fundamental solutions; the paper takes no stand on whether non-fundamental (bubble / sunspot) REEs are ever the economically relevant equilibrium when FCC fails.
  • FCC is only a sufficient existence condition for the forward solution — it does not by itself guarantee dynamic stability of Ω*. The eigenvalues of Ω* must still be inside the unit circle for the forward solution to be stationary.
  • The regularity condition |I_n - A Ω_k| ≠ 0 is required at every k, not only in the limit; this is strictly stronger than the King–Watson (1998) condition that involves only Ω*. The paper notes Binder–Pesaran (1997) verify this informally on every economic example they examine.
  • The paper assumes the exogenous process z_t follows a known VAR(1) and that all structural parameters and steady states are known to the agents (so the model is in mean-deviation form without constants).
  • The relationship between the forward method and determinacy is left as a conjecture: an additional FCC for the constant term may be closely related to the determinacy condition.

Open questions

  • What is the formal relation between the FCC and standard determinacy conditions (Blanchard–Kahn, Klein, Sims)? The authors conjecture an additional FCC for constants is closely tied to determinacy.
  • When fundamental and non-fundamental REEs coexist, which class is more relevant to a given model?
  • What are the general theoretical relations (necessity, sufficiency) between the NBC refinement and the MSV / MOD / E-stability criteria?
  • Can E-stability be reformulated within the forward method framework as a refinement scheme?
  • Under what extensions (Markov-switching dynamics, regime-switching coefficient matrices) can the forward method and the FCC be generalized? (Not addressed by the paper but central to the CRE asset pricing project’s use of this method.)

My take

This paper is the canonical methodological reference for solving the multivariate linear RE block in the CRE asset pricing model. The forward method is exactly what is implemented in SimMdlPrices/get_forward_solution_msre and the FCC check is the determinacy gate inside check_determinancy_fmsre. The fact that the paper’s published Π_X parameters fail to satisfy the Cho–Moreno determinacy / FCC condition on this project’s parameter grid (logged in CLAUDE.md “Gotchas” as “Published θ fails RE determinacy — paper’s Π_X values don’t satisfy Cho-Moreno”) is a load-bearing observation: it shows that FCC is brittle in a way that matters for empirical work, exactly as Examples 2 → 3 in the paper foreshadow. The 70 350× gradient conditioning improvement from imposing the m_g IS-stationarity constraint, and the sensitivity of basin discovery to which fundamental REE the optimizer happens to land on, are downstream consequences of the same multiplicity that this paper formalizes.

The interesting open theoretical question for the project is the Markov-switching extension: regime-conditional (A, B, C, R) matrices break the simple (I - AΩ_{k-1})^{-1} recursion and require a higher-dimensional fixed-point iteration on a vector of regime-conditional Ω matrices. The CRE project’s compute_quadratic_pricing_factors_msvar Riccati recursion is the asset-pricing analogue; whether an MS-FCC and MS-NBC framework with the same uniqueness theorem holds is not settled in this paper.

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