The asset-pricing no-bubble condition for regime-switching CRE models reduces to a finite spectral-radius test

Statement

In a regime-switching exponential-affine asset pricing model, the asset-pricing no-bubble condition — that the infinite sum of dividend strips Q_{j,t} = ∑_{n=1}^∞ D_n is finite — is equivalent to the spectral radius r(T) < 1 of an infinite-dimensional positive pricing operator T. This operator condition admits a finite-dimensional, computationally cheap approximation ρ(M) < 1, where M is an S × S transfer matrix M_{ij} = π_{ij} exp(c(j)) built from the Markov transition matrix and the per-regime asymptotic log growth rates c(S) of the path-conditional strip price.

The reduction is exact in the operator-theoretic statement and approximate (with error O(ρ(K)^n), exponentially small in regime persistence) in the finite-dimensional form. Both the absorbing and the stacked steady-state Riccati loadings give a usable, empirically conservative spectral test; the stacked loading is more accurate near the boundary, and the absorbing loading is a safer backstop.

Evidence summary

• Theoretical (strong): Section 7.6 of riccati-equations-leather-sagi derives the operator T, identifies its spectral radius with the convergence condition for the strip sum, and constructs M via Perron-Frobenius after passing to the steady-state Riccati loading. Section 7.3 gives the cumulant approximation λ* < 0, which is shown in Section 7.7 to be strictly weaker than ρ(M) < 1 when one regime is expansionary.
• Empirical (strong, in-region): Section 7.11 reports a validation experiment across 100 parameter draws from 4 strata of the Leather-Sagi-relevant region. The stacked loading achieves median absolute error 0.004 in ρ(M) and 97% classification accuracy; the absorbing loading is slightly less accurate (0.007) but more conservative (no false positives in this experiment).
• Production deployment (strong): In the wider CRE asset pricing project, the absorbing spectral proxy is now the production no-bubble gate. It achieves 99.1% accuracy versus 63.3% for the deprecated get_lim_η heuristic, and 2.87× more Sobol parameter draws pass it. The deprecated heuristic is documented as such in the project’s main CLAUDE.md.
• Optimization deployment (moderate): Section 7.8 shows 1 − ρ(M(θ)) is smooth in θ (away from eigenvalue crossings) and gives a clean log-barrier for penalized MLE. This replaces the catastrophic Riccati-divergence cliff that the project’s earlier estimation pipeline had to tiptoe around.

Conditions and scope

See conditions in frontmatter. Important scope notes:

• The test is distinct from the macro NBC of Cho-Moreno and Bikbov-Chernov, which constrains the physical-measure dynamics of the rational expectations VAR to rule out explosive sunspot solutions. The two NBCs are logically independent — a parameter set can satisfy one and fail the other. Both must hold for the model to be sensible. (Source note Section 7.10.)
• The test requires stacked stationarity ρ(K) < 1 as a precondition. Per-regime stability max_S ρ(Φ^Q(S)) < 1 is necessary but not sufficient; this is a well-known MJLS subtlety (source note Section 7.9). Without stacked stationarity, the steady-state Riccati loading does not exist and the spectral radius ρ(M) is undefined.
• Validation accuracy is region-dependent. The 97% figure is for the Leather-Sagi-relevant region; outside that region (e.g., very low regime persistence, or income autocorrelations close to 1) the spectral approximation may degrade.
• The test is not exact — it is an O(ρ(K)^n) approximation to the true operator spectral radius r(T). It is empirically conservative, but a region of technically feasible parameters is excluded.

Counter-evidence

• Cumulant expansion can disagree with the spectral test when one regime has c(S) > 0. In that case the cumulant condition λ* < 0 is too lenient; the spectral test correctly rejects.
• Loading-choice disagreement near the boundary. Stacked and absorbing loadings can give different verdicts on a small set of borderline parameters. The recommended hybrid (stacked primary, absorbing safety backstop) papers over this but does not resolve the underlying fact that neither is the true eigenfunction.
• No tight analytic gap bound. The error r(T) − ρ(M) is bounded by O(ρ(K)^n) qualitatively but no sharp constant is known.

Linked ideas

(Tracker: ideas that propose to compute the exact operator spectrum, build a higher-order spectral expansion, or find a parameter-free hybrid loading would link here.)

Open questions

• Is there a tractable closed form for the exact eigenfunction of T, beyond the absorbing / stacked approximations?
• Can the validation accuracy of the spectral proxy be characterized analytically as a function of regime persistence Π and shock volatilities Σ?
• How does the smooth NBC barrier interact with other smooth penalties in the project’s MLE pipeline (m_g IS-stationarity constraint, eigenvalue penalties on Φ(S))?
• What is the right generalization to multi-asset payoffs with cross-asset regime structure beyond the simple block-triangular augmentation of (x, ν_j)?