Forward Convergence Condition (FCC)

Definition

A linear rational expectations model is said to satisfy the Forward Convergence Condition (FCC) if the coefficient sequences $({\Omega }_k,{\Gamma }_k)$ of the state variables in the model’s forward representation $x_t=M_k{E}_t{x}_{t+k}+{\Omega }_k{x}_{t-1}+{\Gamma }_k{z}_t$ converge as the forward recursion goes to infinity, where the sequences are produced by the deterministic recursion described under forward-method-rational-expectations starting from $(M_1,{\Omega }_1,{\Gamma }_1)=(A,B,C)$.

Intuition

FCC is the multivariate generalization of “the present value of all future state-variable contributions is finite.” If the state-variable coefficients in the iterated forward representation stabilize, the model has a stable, model-implied relation between today’s endogenous variables and the state variables. If they explode or oscillate, then no such stable forward-looking relation exists, and the forward-method-rational-expectations declines to certify any candidate solution as economically sensible. Flood and Garber (1980) called the univariate version “process consistency” — an essential characteristic of any process pretending to serve as money — and Cho–Moreno’s FCC generalizes this.

Formal notation

Given the model $x_t=A{E}_t{x}_{t+1}+B{x}_{t-1}+C{z}_t,z_t=R{z}_{t-1}+{\epsilon }_t$, define $({\Omega }_k,{\Gamma }_k)$ recursively by

Ω_1 = B,    Γ_1 = C
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)

under the regularity condition $|I_n-A{\Omega }_{k-1}|\ne 0$ for all k. The model satisfies FCC iff there exist matrices $(\Omega \ast ,\Gamma \ast )$ such that ${\Omega }_k\to \Omega \ast$ and ${\Gamma }_k\to \Gamma \ast$. Cho–Moreno show that if FCC holds, then $(\Omega \ast ,\Gamma \ast )$ automatically satisfies the fundamental-solution conditions $\Omega \ast =(I-A\Omega \ast {)}^{-1}B$ and $\Gamma \ast =(I-A\Omega \ast {)}^{-1}(C+A\Gamma \ast R)$, so it is a member of the fundamental solution set S.

Variants

• Univariate without lags: FCC reduces to $|a\rho |<1$ for the model $x_t=a{E}_t{x}_{t+1}+z_t$ with $z_t=\rho z_{t-1}+{\epsilon }_t$.
• Univariate with one lag: FCC requires both convergence of ${\omega }_k$ and ${\gamma }_k$; the graphical iteration $v_{k+1}=1-\theta /v_k$ (with $\theta =ab$) explains the geometry.
• Multivariate: convergence of two coupled matrix sequences plus the per-step regularity condition $|I_n-A{\Omega }_{k-1}|\ne 0$.

Comparison

Versus standard determinacy (Blanchard–Kahn 1980, Klein 2000, Sims 2002): determinacy is a count of stable generalized eigenvalues vs. forward-looking variables. FCC is a stronger, dynamics-of-the-recursion property: it can fail even when the model is technically determinate, and conversely small parameter changes can flip FCC without changing the eigenvalue count. Cho–Moreno conjecture that an FCC for the constant term is closely related to determinacy.

Versus King–Watson (1998) existence condition: King–Watson require $|I-A\Omega \ast |\ne 0$ only at the limit. FCC’s regularity condition requires $|I-A{\Omega }_k|\ne 0$ at every step, which is strictly stronger.

When to use

• As a constructive existence check for the forward solution: if FCC holds, the forward solution exists and is unique.
• As a feasibility gate in estimation pipelines: nudging parameters into a non-FCC region produces no fundamental REE that is economically defensible.
• To explain why a determinacy-passing parameter set may still produce nonsensical economics (the determinacy condition counts roots; FCC checks the full recursion).

Known limitations

• FCC is sufficient but not necessary for the existence of fundamental solutions in S; mathematical fundamental solutions can exist when FCC fails (Cho–Moreno’s Example 3 with $\beta =0.9$). They simply violate the no-bubble-condition.
• FCC does not imply dynamic stability of $\Omega \ast$; $\Omega \ast$ may have unstable eigenvalues even when the recursion converges.
• Convergence rate is parameter-dependent and can be slow when the matrix-quadratic roots are near each other.
• The exogenous process $R$ is assumed known; the FCC is defined relative to this fixed $R$.

Open problems

• Necessary-and-sufficient conditions linking FCC to standard determinacy.
• Generalization to Markov-switching coefficient matrices where the recursion becomes a fixed-point on a vector of regime-conditional ${\Omega }^{(}s)$. (Used implicitly in the CRE asset pricing project but not formally proved.)
• A practical numerical tolerance for declaring $({\Omega }_k,{\Gamma }_k)$ “converged” — Cho–Moreno’s Example 1 reaches machine precision by k=25 but Example 4 (near-repeated roots) can take >1000 iterations.

Key papers

• cho-moreno-2010-forward-method-rational-expectations — formal definition and uniqueness theorem.

My understanding

In the CRE asset pricing pipeline, FCC is checked inside check_determinancy_fmsre after get_forward_solution_msre has run the recursion. The “Published θ fails RE determinacy” gotcha is exactly an FCC failure on the published parameter grid — and importantly, it is not flagged by a standard Blanchard–Kahn-style eigenvalue count, which is precisely the point Cho–Moreno make in their Example 3. The regularity condition $|I-A{\Omega }_k|\ne 0$ per step is also load-bearing: a single bad step kills the recursion. For the Markov-switching case the FCC needs to be reformulated as a contraction property of a regime-vector fixed-point map; this is a specific open theoretical question for the project.