Statement
Each of the three standard solution selection criteria for linear rational expectations models — the minimum state variable (MSV) criterion of McCallum (1983), the MOD criterion (smallest generalized eigenvalues) of McCallum (2004), and the E-stability criterion of Evans and Honkapohja (2001) — can return a fundamental REE that violates the no-bubble condition lim_{k→∞} M_k E_t x_{t+k} = 0. Such failures occur in well-formulated, economically motivated models, including standard New-Keynesian models with non-Taylor-principle parameter values and the Dornbusch open-economy model.
Evidence summary
Cho and Moreno (2010), Section 5, provide explicit numerical counter-examples:
- Example 3 (New-Keynesian, β = 0.9): MSV (verified by sweeping a constant α multiplying B from 1 to 0; Table 3) and MOD select the fundamental solution
(Ω(g_1, g_2), Γ(g_1, g_2)). But FCC fails —Γ_kexplodes from 0.27 at k=1 to 14 332 at k=100 (Table 2) — and consequently both fundamental solutions in S violate the NBC. The MSV/MOD criteria silently certify a bubble-carrying REE. (E-stability happens to also reject both candidates here, so it is not implicated in this specific example, but Example 4 implicates it elsewhere.) - Example 4 (Dornbusch / Evans–Honkapohja open economy): Three stationary fundamental solutions (ω = 0.7160, 0.7721, 0.9897). Two of them (ω = 0.7160 and ω = 0.9897) pass the E-stability criterion. Only the forward solution
ω = 0.7160satisfies the NBC;ω = 0.9897is E-stable but violates the NBC. Hence E-stability is not a sufficient condition for the NBC.
The general mechanism is identified in Proposition 2: any fundamental REE that is not the forward solution leaves a non-vanishing bubble residual L^x x_t + L^z z_t in the forward representation, even though it depends only on state variables. The MSV / MOD / E-stability criteria are not designed to detect this residual, so they can return offending solutions whenever the model is in a region where a non-forward fundamental REE happens to win their respective ranking.
The paper also footnotes (footnote 20) that McCallum (2004) himself shows an example where the MSV solution differs from the unique stationary fundamental (MOD) solution, and Cho–Moreno verify that in that example the MSV solution violates the NBC.
Conditions and scope
- The claim is existential: at least one counter-example exists for each criterion in standard economic settings. The paper does not characterize how prevalent such failures are in estimated DSGE models.
- The counter-examples in Section 5 use small (2-equation, univariate-with-lag) models. Whether similar failures occur in higher-dimensional empirical DSGE applications is not directly tested.
- For Examples 1 and 2, all three criteria coincide with the forward solution and the NBC holds; the claim is not that these criteria always fail, only that they can fail.
Counter-evidence
Cho–Moreno’s own footnote 20 acknowledges that in well-formulated economic models with FCC holding, the MSV criterion is “highly likely” to satisfy the NBC. So the failure mode is typically tied to either FCC-failure regions or to specific model classes where multiple stationary fundamentals coexist. No counter-evidence in the strict sense is presented in the paper.
Linked ideas
(none yet)
Open questions
- What is the empirical prevalence of NBC-violating MSV/MOD/E-stable selections across estimated DSGE models in the literature? (Cho-Moreno do not characterize this.)
- Are there sufficient conditions on the model structure that guarantee MSV / MOD / E-stability coincide with the forward solution?
- Does the same failure mode occur in Markov-switching RE models (relevant to the CRE asset pricing project)? Cho (2020) shows that in MSRE, a unique stable MSV solution does NOT imply determinacy — stable sunspots can coexist — which is a related but distinct phenomenon from the NBC violation in LRE. See determinacy-classification-markov-switching-rational-expectations.