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No-Bubble Condition (NBC)

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Definition

A rational expectations equilibrium (REE) is said to satisfy the No-Bubble Condition (NBC) if the term involving the expectation of future endogenous variables in the model’s forward representation vanishes as the forward recursion goes to infinity, when expectations are formed using the candidate solution. Formally, for the multivariate model x_t = A E_t x_{t+1} + B x_{t-1} + C z_t with forward representation x_t = M_k E_t x_{t+k} + Ω_k x_{t-1} + Γ_k z_t, the NBC is

lim_{k → ∞} M_k E_t x_{t+k} = 0_{n × 1}

evaluated at the candidate REE.

Intuition

The expectational term M_k E_t x_{t+k} is the present-value contribution of the endogenous variable arbitrarily far in the future. If the agents’ beliefs about the system place no weight on the very distant future of the endogenous variable, that term vanishes; if it does not vanish, then today’s x_t depends on a “bubble” — anything unrelated to the model’s fundamentals. Cho–Moreno show that this bubble term need not be exotic: it can survive even when the candidate solution is itself a fundamental REE depending only on state variables. In that case the bubble term takes the form L^x x_t + L^z z_t with L^x = lim M_k Ω^k, which is hard to interpret as anything other than a bubble.

Formal notation

Univariate without lags: lim_{k→∞} a^k E_t x_{t+k} = 0.

Univariate with one lag: lim_{k→∞} m_k E_t x_{t+k} = 0, where m_k = (1 - a ω_{k-1})^{-1} a m_{k-1}.

Multivariate: lim_{k→∞} M_k E_t x_{t+k} = 0_{n×1}, with M_k from the recursion in forward-method-rational-expectations.

For any non-forward fundamental solution (Ω^(j), Γ^(j)) under FCC,

lim_{k→∞} M_k E_t x_{t+k} = L^x(j) x_t + L^z(j) z_t,
L^x(j) = lim_{k→∞} M_k Ω^(j)^k,
L^z(j) = lim_{k→∞} M_k Σ_{i=1}^{k} Ω^(j)^{k-i} Γ^(j) R^i.

Cho–Moreno’s Proposition 2 shows these limits are non-zero when (Ω^(j), Γ^(j)) ≠ (Ω*, Γ*).

Variants

  • Asset pricing: the no-bubble condition on a stock price p_t = β E_t (p_{t+1} + d_{t+1}) is lim_{k→∞} β^k E_t p_{t+k} = 0 — the textbook statement that the stock price equals the discounted present value of dividends.
  • Fiscal: the no-Ponzi-game condition on government debt is the NBC applied to the government’s intertemporal budget constraint.
  • Real business cycle / Ramsey: the transversality condition on capital is the NBC applied to the household’s optimization problem.
  • Open economy: the zero boundary condition on net foreign assets is the NBC for an open-economy model.

These all share the same mathematical content: the present-value contribution of an endogenous variable arbitrarily far in the future is zero. Cho–Moreno’s contribution is to elevate the NBC from an assumption used to rule out non-fundamental sunspot solutions into a solution refinement criterion among fundamental REEs, and to prove the forward solution is the unique fundamental REE that satisfies it.

Comparison

Versus MSV (McCallum 1983), MOD (McCallum 2004), E-stability (Evans–Honkapohja 2001): these are alternative selection criteria over fundamental REEs. Cho–Moreno show by example that all three can return solutions that violate the NBC, even in well-formulated economic models. In their Example 3 (β = 0.9 New-Keynesian model), the MSV and MOD criteria pick a solution that violates the NBC because FCC fails. In Example 4 (Dornbusch / Evans–Honkapohja), two different solutions pass E-stability but only one satisfies the NBC.

Versus the rank condition (McCallum 1983): the rank condition rules out some fundamental solutions on technical grounds; the NBC is sharper because it directly identifies which solution is bubble-free.

When to use

  • As the definitive solution refinement when multiple fundamental REEs exist and selection criteria disagree.
  • As a sanity check on any candidate solution returned by an eigenvalue-decomposition method: directly compute M_k E_t x_{t+k} along the candidate solution and verify it vanishes.
  • As an interpretive lens: a non-zero lim M_k E_t x_{t+k} = L^x x_t + L^z z_t reveals that the candidate solution depends on a state-variable bubble, even if it looks like a “pure” fundamental REE.

Known limitations

  • The NBC is not a refinement scheme for non-fundamental (sunspot) REEs — those are defined to violate it. The NBC’s domain is the class of fundamental REEs only.
  • Cho–Moreno’s uniqueness theorem (the forward solution is the only NBC-satisfying fundamental REE) requires the regularity condition |I_n - A Ω_{k-1}| ≠ 0 at every step.
  • The NBC is sensitive to whether the model is in mean-deviation form; if constants are not handled properly (i.e., the agents do not know the steady state), the NBC may need an additional condition for the constant term.
  • Verifying the NBC numerically requires either iterating M_k for many k (expensive in high dimensions) or evaluating L^x = lim M_k Ω^k directly via eigendecomposition.

Open problems

  • General theoretical relations (necessity, sufficiency) between the NBC refinement and MSV / MOD / E-stability.
  • Markov-switching extension: how does the NBC generalize when structural matrices depend on a regime process?
  • Whether the NBC has a natural reformulation in terms of asset-pricing risk-neutral measures (relevant to the CRE project’s term-structure pricing pipeline).

Key papers

My understanding

The NBC is the central economic-meaning lemma behind the CRE asset pricing project’s choice of solver. The project uses the forward method (get_forward_solution_msre) precisely because Cho–Moreno’s Proposition 1 guarantees its output is the unique fundamental REE that does not depend on a bubble in the future endogenous variables — which is a non-negotiable property for a no-arbitrage asset pricing model where bubble terms in macro variables would propagate into the term-structure factor loadings via the Riccati recursion. The “Published θ fails RE determinacy” gotcha is, viewed through this lens, a symptom: the published parameter values land in a region where any candidate fundamental solution would violate the NBC, and the project’s RE solver correctly refuses to certify a solution there. The Markov-switching analogue of NBC is an open question and is not addressed by this paper.

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