Identification theory for compound (product-of-independent) Markov chains in regime-switching asset pricing

Motivation

The Leather-Sagi MS-RE model uses two independent binary Markov chains — one for monetary policy stance (active/passive) and one for wage rigidity (flexible/rigid) — producing 4 compound regimes. This product structure imposes strong restrictions on the 4x4 transition matrix: instead of 12 free probabilities (unrestricted 4-state chain), there are only 4 free probabilities (2 persistence parameters per binary chain). The question is: when is this product structure identified from asset price data?

No formal results exist in the literature. The question matters for three reasons:

1. If the product structure is not identified, the model may be misspecified without detection — the data cannot tell whether the 4 regimes arise from 2 independent sources or from 1 correlated source.
2. If the product structure IS identified, the independence restriction provides a powerful falsifiable prediction: regime transitions in the two chains should be conditionally independent given the state.
3. Existing literature (DSY 2007, ABW 2008, Bikbov-Chernov) uses single Markov chains exclusively. The compound-chain construction is genuinely novel and deserves formal treatment.

Hypothesis

The product structure of 2 independent binary Markov chains is generically identified from a sufficiently rich observation menu (macro variables + Treasury yields + CRE cap rates), but NOT identified from macro variables alone or from macro + short rate alone. The identifying power comes from the cross-sectional dimension of the asset price panel: different-maturity yields and different- property-type cap rates load differently on the two chains, breaking the equivalence with a single 4-state chain.

Problem Statement for Formal Treatment

Setup

Let $S_t=(S_t^{(1)},S_t^{(2)})$ be a compound regime process where $S_t^{(1)}\in \{1,2\}$ and $S_t^{(2)}\in \{1,2\}$ are independent binary Markov chains with transition matrices ${\Pi }^{(1)}$ and ${\Pi }^{(2)}$. The compound state $s_t\in \{1,2,3,4\}$ indexes the product $(S_t^{(1)},S_t^{(2)})$, and the compound transition matrix is:

${\Pi }_{ij}^C={\Pi }_{i_1{j}_1}^{(1)}\cdot {\Pi }_{i_2{j}_2}^{(2)}$

where $i=(i_1,i_2)$, $j=(j_1,j_2)$ under the natural bijection.

The continuous state $X_t\in {\mathbb{R}}^n$ follows a regime-switching VAR:

$X_{t+1}={\mu }_{s_{t+1}}+{\Phi }_{s_{t+1}}{X}_t+{\Sigma }_{s_{t+1}}{\epsilon }_{t+1},{\epsilon }_{t+1}\sim N(0,I)$

The observation vector $Y_t$ includes:

• Macro variables: linear in $X_t$ (no regime dependence in observation equation)
• Treasury yields at maturities ${\tau }_1,\dots ,{\tau }_J$: affine in $X_t$ with regime-dependent loadings $a_{s_t}(\tau )+b_{s_t}(\tau {)}^{\prime }{X}_t$
• CRE cap rates for property types $k=1,\dots ,K$: exponential-quadratic in $X_t$ with regime-dependent loadings

Question 1 (Local Identification)

Under what conditions on $({\mu }_s,{\Phi }_s,{\Sigma }_s,a_s(\cdot ),b_s(\cdot ))$ and the observation menu $Y_t$ is the compound structure locally identified at a generic parameter point ${\theta }_0$?

Formally: let ${\mathcal{M}}_C$ be the compound model (4 free transition parameters) and ${\mathcal{M}}_U$ be the unrestricted 4-state model (12 free transition parameters). Both generate the same family of observed likelihoods $p(Y_{1:T}\mid \theta )$. Is there a ${\theta }_0\in {\mathcal{M}}_C$ such that the Fisher information matrix $I({\theta }_0)$ under ${\mathcal{M}}_C$ has full rank, AND no $\tilde{\theta }\in {\mathcal{M}}_U\setminus {\mathcal{M}}_C$ generates the same observed distribution?

Conjecture: Local identification holds generically when: (a) The regime-conditional VAR parameters $({\mu }_s,{\Phi }_s,{\Sigma }_s)$ are distinct across at least 3 of the 4 compound states (not just across the 2 states of each binary chain), AND (b) The observation menu includes at least 2 asset prices with different regime loadings (e.g., Treasury yields at 2 maturities, or yields + CRE cap rates).

Question 2 (Testable Restriction)

Given identification, can the compound structure be tested against the unrestricted alternative via a standard LR or Wald test?

The restricted model ${\mathcal{M}}_C$ nests inside ${\mathcal{M}}_U$ as a nonlinear restriction on $\Pi$: the compound ${\Pi }^C$ has rank structure ${\Pi }^C={\Pi }^{(1)}\otimes {\Pi }^{(2)}$. This is a smooth 8-dimensional manifold (4 persistence parameters + 4 label-switching parameters, the latter nuisance) inside the 12-dimensional simplex of unrestricted 4-state transition matrices.

Test statistic: $LR=2[\ell ({\hat{\theta }}_U)-\ell ({\hat{\theta }}_C)]$ with 8 degrees of freedom (12 - 4 free probabilities, but label-switching reduces effective df — exact df depends on the invariance structure).

Question 3 (Observation Menu Sufficiency)

What is the minimal observation menu that identifies the product structure?

Sub-questions:

• (3a) Do macro variables + short rate alone identify the product structure? (Conjecture: NO — the short rate is a single linear combination of $X_t$, and with only 3 macro observables + 1 yield, the 4 regime-conditional observation densities may not distinguish compound from unrestricted.)
• (3b) Do macro variables + yield curve (multiple maturities) suffice? (Conjecture: YES, if the yield loadings $b_s(\tau )$ differ across the 4 compound states at different rates for different $\tau$.)
• (3c) Does adding CRE cap rates provide additional identifying power beyond the yield curve? (Conjecture: YES, because the exponential-quadratic CRE loadings encode the product structure differently than the affine yield loadings — the quadratic term creates cross-moments that the affine term cannot replicate.)

Suggested Proof Approaches

Approach A (Algebraic): Work with the second-order properties of $Y_t$ under the two models. The autocovariance sequence ${\Gamma }_Y(h)=E[Y_t{Y}_{t+h}^{\prime }]$ under ${\mathcal{M}}_C$ has the product structure baked into the regime-mixing weights. Show that the mapping from $({\Pi }^{(1)},{\Pi }^{(2)})$ to $\{{\Gamma }_Y(h){\}}_{h=0}^H$ is injective for sufficiently large $H$ and rich $Y_t$.

Approach B (Numerical Rank): Compute the Fisher information matrix $I(\theta )$ at a grid of parameter points in ${\mathcal{M}}_C$, verifying full rank numerically. This is the BC (2013) approach — constructive but point-specific. The project’s Hamilton filter + RBPF infrastructure can evaluate the likelihood and its numerical derivatives.

Approach C (Spectral): Represent the observation process as a hidden Markov model with emission kernel $f(Y_t\mid X_t,s_t)$. Use the spectral identifiability theory for HMMs (Anandkumar et al. 2012, Hsu-Kakade-Zhang 2012) to characterize when the emission distributions across the 4 compound states are distinguishable, and when the Kronecker structure of ${\Pi }^C$ is recoverable from the observable moments.

Relevant Literature

• identification-under-regime-switching — BC-style numerical local identification
• bikbov-chernov-monetary-policy-regimes-term-structure — 8-regime compound model (closest published)
• Anandkumar, Hsu, Kakade (2012) — spectral HMM identification
• Allman, Matias, Rhodes (2009) — generic identifiability of finite mixture models
• understanding-markov-switching-rational-expectations-models — FWZ 2009, compound chain construction

Expected outcome

A theorem (or clean conjecture with numerical verification) characterizing when the product-of-independent-chains structure is identified from asset price data. At minimum, numerical verification at the project’s DGP parameter point.

Risks

• The algebraic approach may require regularity conditions that are hard to verify for the exponential-quadratic pricing factors.
• The spectral HMM approach may not apply directly because the emission kernel is state-space-valued (continuous $X_t$), not finite.
• Even if the theory works, the practical identification quality (finite-sample Fisher conditioning) may be poor — similar to the existing curvature findings (condition number ~10^25).

Pilot results

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Lessons learned

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