Forward method (rational expectations solver)

Definition

The forward method is a constructive solution technique for linear rational expectations (RE) models. It rewrites the structural equation

$$x_t=A{\mathbb{E}}_t{x}_{t+1}+B{x}_{t-1}+C{z}_t\text{\hspace{1em}(1)}$$

as a forward representation

$$x_t=M_k{\mathbb{E}}_t{x}_{t+k}+{\Omega }_k{x}_{t-1}+{\Gamma }_k{z}_t\text{\hspace{1em}(2)}$$

for arbitrary horizon $k$, by iteratively substituting ${\mathbb{E}}_t{x}_{t+i+1}=M_1{\mathbb{E}}_t{x}_{t+i+2}+{\Omega }_1{\mathbb{E}}_t{x}_{t+i}+{\Gamma }_1{R}^i{z}_t$ into itself. The coefficient triple $(M_k,{\Omega }_k,{\Gamma }_k)$ is generated by a deterministic recursion in the structural matrices alone:

$$\begin{array}{rl}{M}_1& =A,{\Omega }_1=B,{\Gamma }_1=C\\ M_k& =(I-A{\Omega }_{k-1}{)}^{-1}A{M}_{k-1}\\ {\Omega }_k& =(I-A{\Omega }_{k-1}{)}^{-1}B\\ {\Gamma }_k& =(I-A{\Omega }_{k-1}{)}^{-1}(C+A{\Gamma }_{k-1}R)\end{array}\text{\hspace{1em}(3)}$$

provided the regularity condition $|I-A{\Omega }_{k-1}|\ne 0$ holds at every step. The forward solution is the limit $({\Omega }^{\ast },{\Gamma }^{\ast })$ if the forward-convergence-condition holds, and is by construction a member of the fundamental REE set; it is also the unique fundamental REE that satisfies the no-bubble-condition.

Intuition

Rational forward-looking agents recursively eliminate the next-period expectation in favor of (a) farther-future expectations and (b) state variables. If iterating this elimination produces a stable model-implied relation between today’s endogenous variables and the state variables, that relation is “what the agents would actually believe.” It does not require knowing the full set of mathematically possible solutions in advance; the recursion is uniquely determined by the structural parameters and is executed without committing to any particular $\Omega$ first.

Formal notation

For the canonical model (Eq. 1) with $z_t=R{z}_{t-1}+{\epsilon }_t$, define recursively $(M_k,{\Omega }_k,{\Gamma }_k)$ as in Eq. 3. The forward representation is given by Eq. 2. If $({\Omega }_k,{\Gamma }_k)\to ({\Omega }^{\ast },{\Gamma }^{\ast })$, the forward solution is

$$x_t={\Omega }^{\ast }{x}_{t-1}+{\Gamma }^{\ast }{z}_t\text{\hspace{1em}(4)}$$

and $({\Omega }^{\ast },{\Gamma }^{\ast })\in \mathcal{S}$ (the fundamental solution set defined by $\Omega =(I-A\Omega {)}^{-1}B$ and $\Gamma =(I-A\Omega {)}^{-1}(C+A\Gamma R)$).

Variants

• Univariate without predetermined variables (Cagan 1956, Samuelson 1958): the textbook case $x_t=a{\mathbb{E}}_t{x}_{t+1}+z_t$ with closed-form forward solution ${\gamma }^{\ast }=1/(1-a\rho )$.
• Univariate with one lag: the simplest case where multiple fundamental solutions can exist; admits a graphical analysis via the iteration $v_{k+1}=1-\theta /v_k$ with $\theta =ab$.
• Multivariate with predetermined variables (Cho and Moreno 2010): the full generalization treated in cho-moreno-2010-forward-method-rational-expectations.
• Markov-switching extension (CRE asset pricing project, not in the original paper): regime-conditional structural matrices $(A^{(s)},B^{(s)},C^{(s)})$ require a vector-of-matrices fixed-point iteration on regime-conditional ${\Omega }^{(s)}$. Used in the project’s get_forward_solution_msre.

Comparison

Versus eigenvalue / generalized Schur (QZ) methods (Blanchard–Kahn 1980, Uhlig 1997, King–Watson 1998, Klein 2000, Sims 2002): the eigenvalue methods first characterize all elements of S via generalized eigenvalue decomposition and then need a separate selection criterion (MSV, MOD, E-stability) to pick one. The forward method avoids the selection problem entirely — it produces at most one solution by construction, and that solution is the unique NBC-satisfying fundamental REE.

Versus backward induction with terminal conditions (Binder–Pesaran 1997, Driskill 2006): both impose some form of zero terminal condition on future expectations, but the forward method does not need an explicit terminal condition and generalizes to multivariate models with predetermined variables.

When to use

• When you want a single, economically defensible fundamental REE without manually enumerating the full solution set.
• When the model is multivariate with both lagged predetermined variables and forward-looking expectations.
• When the alternative selection criteria (MSV, MOD, E-stable) disagree, or when you suspect the model may be in a regime where the criteria silently violate the NBC.
• As a determinacy / feasibility gate (FCC convergence check) for parameter grids in estimation pipelines.

Known limitations

• FCC is sufficient for existence but not for dynamic stability of the forward solution: ${\Omega }^{\ast }$ may have eigenvalues outside the unit circle even when the recursion converges.
• The regularity condition $|I-A{\Omega }_{k-1}|\ne 0$ must hold for every $k$, not only the limit; this is strictly stronger than the King–Watson condition on ${\Omega }^{\ast }$ alone.
• The recursion does not handle the constant term separately; an additional FCC for the constant is conjectured to be related to determinacy.
• When FCC fails, the method produces no solution at all — even if other fundamental solutions exist mathematically. (This is interpreted as a feature: those solutions all violate the NBC and are economically suspect.)
• Convergence speed depends on how separated the roots of the matrix quadratic are; near-repeated roots can take >1000 recursions for three-decimal precision.

Open problems

• Markov-switching generalization (regime-dependent structural matrices) is needed for the CRE asset pricing project but is not covered in the 2010 paper. The Riccati-recursion analogue lives in compute_quadratic_pricing_factors_msvar.
• Necessary-and-sufficient conditions linking FCC to standard determinacy conditions (Blanchard–Kahn, Klein, Sims).
• Extensions to non-linear or models with constants where steady states are not known.

Key papers

• cho-moreno-2010-forward-method-rational-expectations — the canonical multivariate generalization, FCC + NBC + uniqueness theorem.
• determinacy-classification-markov-switching-rational-expectations — Cho (2020) uses the forward method as the primary computational tool for the MOD solution; shows the forward solution coincides with the MOD in all experiments (formal proof open).
• understanding-markov-switching-rational-expectations-models — FWZ (2009) derives the MSV solution that is the forward method’s target in MSRE models.

My understanding

In the CRE project’s pipeline (construct_structural_matrices_macro → transform_struct_to_rf → get_forward_solution_msre → check_determinancy_fmsre → augment_macro_fsmsre_nu → asset pricing), the forward method is the load-bearing solver. The “Published $\theta$ fails RE determinacy” gotcha — paper’s ${\Pi }_X$ values don’t satisfy the Cho–Moreno determinacy condition on the project’s parameter grid — is a direct consequence of how brittle FCC is across nearby parameter values, exactly as the Cho–Moreno Examples 2 → 3 ($\beta =0.95\to 0.9$) demonstrate. The Markov-switching extension _msre is the project’s own contribution; whether the same uniqueness-of-NBC-satisfying-solution holds in the MS case is an open theoretical question.