Spectral Radius Test for Markov Pricing Operator (Asset-Pricing No-Bubble Condition)

Definition

A finite-dimensional, computationally cheap test for whether the infinite sum of dividend strips Q_{j,t} = ∑_{n=1}^∞ D_n(z_t, S_t) converges in a regime-switching exponential-affine asset pricing model. The exact statement is that the positive pricing operator

(T f)_i(z) = e^{h' z} ∑_j π_{ij} E^Q[f_j(z') | z, S_{t+1} = j]

has spectral radius r(T) < 1. Because T acts on functions of the continuous state and is infinite-dimensional, this is approximated by a S × S transfer matrix M whose entry is

M_{ij} = π_{ij} · exp(c(j))

where c(S) is the per-regime asymptotic growth rate of the path-conditional log strip price under a steady-state Riccati loading. Convergence is then declared iff ρ(M) < 1. By the Perron-Frobenius theorem this is exactly the convergence condition for e_i' M^n 1 to decay exponentially.

Intuition

After the Riccati loading β_n has converged (which happens exponentially fast under ρ(K) < 1), each step in regime S multiplies the path-conditional strip price by approximately exp(c(S)). The unconditional strip price is therefore a Markov-modulated geometric process whose growth rate per period is the Lyapunov exponent of the random matrix sequence {π_{ij} exp(c(j))}. By Perron-Frobenius, that Lyapunov exponent is the log of the spectral radius of the deterministic matrix M_{ij} = π_{ij} exp(c(j)). The strip sum converges iff that radius is below 1.

The conceptual move is “first kill the continuous-state degree of freedom by passing to the steady-state Riccati loading, then handle the residual regime-switching combinatorics with a finite Perron-Frobenius argument.”

Formal notation

Let Φ_z^Q(S), Σ_z(S), m_z^Q(S) be the augmented-state risk-neutral parameters in regime S, and h = (0, 0, −1, 1)' the net cash-flow-minus-discount selector (Section 4 of the source note). Choose a steady-state loading B^*(S) (two natural choices below). The per-regime asymptotic log growth rate is

c(S) = m_z^Q(S)' B^*(S) + ½ B^*(S)' Σ_z(S) Σ_z(S)' B^*(S),

and the transfer matrix is

M_{ij} = π_{ij} · exp(c(j)).

The asset-pricing no-bubble condition (NBC) is

ρ(M) < 1.

The cumulant approximation λ* = ∑_S π_S^U c(S) + ½ σ²_{LR,regime} < 0 (Section 7.7) and the Lyapunov form λ* = h' z̄^Q + ½ σ²_LR < 0 (Section 7.3) are alternate but weaker / second-order forms of the same condition.

Variants

• Absorbing loading: B^{abs}(S) = (I − Φ_z^Q(S)')^{−1} h. Treats each regime as if it persists forever; conservative bias (overestimates ρ(M), fewer false positives, but excludes some valid parameter regions).
• Stacked loading: B̃(S) = the S-th block of (I − K)^{−1} Δ, where K is the stacked persistence matrix from Section 6.4. Averages across regime transitions at the first-moment level; better median accuracy near the boundary, slight risk of false positives.
• Cumulant approximation: λ* < 0, second-order Lyapunov exponent. Cheap, but agrees with ρ(M) < 1 only when no individual regime has c(S) > 0. Diverges when one regime is “expansionary” because the exponential of accumulated positive growth can dominate even when the average is negative.
• Strong sufficient form: max_S c(S) < 0. If every regime contracts in expectation, no transition pattern can produce divergence.
• Necessary form: ∑_S π_S^U c(S) < 0. The ergodic-average growth rate must be negative.
• Smooth NBC barrier: −log(1 − ρ(M(θ))), added to the negative log-likelihood as a log-barrier (Section 7.8), provides a differentiable feasibility constraint for MLE/MAP optimization.

Comparison

• Vs. macro NBC of Cho-Moreno / Bikbov-Chernov (lim_n E_t[A_n x_{t+n}] = 0): different object. The macro NBC constrains the physical-measure dynamics to rule out explosive sunspot solutions, and is verified via Cho’s forward convergence conditions on the regime-switching rational expectations system. The asset-pricing NBC constrains the risk-neutral dynamics together with risk premia to produce finite asset valuations. The two are logically independent: a parameter set can satisfy one and fail the other. Both must hold.
• Vs. per-regime eigenvalue restrictions (max_S ρ(Φ^Q(S)) < 1): per-regime stability is necessary but not sufficient for stacked stationarity (Section 7.9). Two individually stable matrices can produce an unstable Markov-modulated product. The correct stationarity check is ρ(K) < 1 on the stacked persistence matrix; the asset-pricing NBC ρ(M) < 1 is then an additional restriction enforcing finite valuations.
• Vs. Hansen-Scheinkman operator approach (long-term risk operator): the operator T here is exactly an instance of Hansen-Scheinkman’s positive multiplicative pricing operator, and r(T) < 1 is their long-run zero-coupon yield being positive. The Leather-Sagi note’s contribution is the explicit, computable finite-dimensional approximation ρ(M).
• Vs. directly iterating the Riccati and watching for blow-up: much smoother. The Riccati blow-up is a cliff in the likelihood and provides no usable gradient; 1 − ρ(M(θ)) is smooth (away from eigenvalue crossings) and gives the optimizer a clean repulsive force.

When to use

• As the production no-bubble gate in the inner loop of MLE/MAP estimation for any regime-switching exponential-affine asset pricing model.
• As a smooth log-barrier in penalized log-likelihoods, replacing try { iterate Riccati } catch { reject } patterns.
• As a diagnostic that identifies which regime is responsible for a violation, via the per-regime c(S) values. In CRE-NK applications, the typical culprit is the passive / wage-rigid regime where short rates are low and the convexity term dominates discounting.
• As a feasibility classifier label for Sobol-archive generation (e.g., the project’s GBT classifier achieves AUC 0.997 using the absorbing spectral test as the ground-truth label).

Known limitations

• Approximation bias. Neither the absorbing nor the stacked loading is the true eigenfunction of T. Both are exponentially accurate (O(ρ(K)^n) error in the strip tail) but neither is exact. The stacked loading has roughly half the median error of absorbing across 100 validation draws (median absolute error in ρ(M): stacked 0.004 vs. absorbing 0.007).
• Conservative bias of the absorbing loading: it tightens the feasible region beyond what a perfectly accurate operator test would allow, excluding some economically valid parameters. In the project, this is treated as an acceptable cost for tractability.
• Eigenvalue crossings. ρ(M(θ)) is smooth in θ only away from points where two eigenvalues collide. At a crossing the gradient is undefined, which can hurt L-BFGS but is benign for derivative-free optimizers (BOBYQA, NelderMead).
• Cumulant truncation (λ* < 0) can disagree with ρ(M) < 1 when one regime is expansionary; the spectral test is the right one.
• No analytic certificate of correctness near the boundary — close to ρ(M) = 1, the recommended belt-and-suspenders is to also iterate the actual Riccati and check Q_j^* numerically.

Open problems

• A computable closed-form expression for the exact eigenfunction of T, beyond the absorbing / stacked approximations.
• A principled hybrid of stacked and absorbing loadings that is uniformly more accurate than either.
• Sharp analytic bounds on the gap r(T) − ρ(M) as a function of regime persistence and Σ_z.
• Higher-cumulant corrections analogous to Bikbov-Chernov’s two-cumulant expansion of λ*, to characterize when the spectral test itself is loose.

Key papers

• riccati-equations-leather-sagi — derives the spectral radius condition ρ(M) < 1 (Section 7.6), the layered diagnostic hierarchy (Section 7.7), the smooth NBC barrier (Section 7.8), and the stacked-stationarity result ρ(K) < 1 (Section 7.9). Validates the absorbing and stacked spectral proxies on 100 parameter draws.

My understanding

This is the test that turned the project’s CRE estimation problem from “fragile, with cliffs” into “smoothly constrained.” The 99.1%-accurate absorbing spectral proxy that is now the production no-bubble gate (vs. 63.3% for the deprecated get_lim_η heuristic) is exactly Section 7.6 of the source note with the absorbing loading, and 2.87× more Sobol points pass it than passed the old heuristic. The right mental model is: ρ(K) < 1 is the structural feasibility constraint on the dynamics; ρ(M) < 1 is the valuation feasibility constraint on top of that. Both are needed; neither implies the other.