Markov-Switching Rational Expectations CRE Pricing

Definition

A class of structural asset-pricing models in which a small New Keynesian macro block (output gap, inflation, short rate) is closed under rational expectations conditional on a finite-state Markov regime, and the resulting Markov-switching VAR (MS-VAR) is used as the state process for no-arbitrage pricing of long-lived cash-flow claims, in particular commercial real estate (CRE) cap rates. The defining feature is that investors anticipate possible regime changes, so current prices are forward-looking functions of the entire regime distribution rather than of the current regime alone.

Intuition

A pure regression of CRE cap rates on macro variables cannot decide whether a correlation reflects a discount-rate effect or a cash-flow effect, and cannot capture the anticipation of structural breaks. A structural model that requires investors to solve a forward-looking optimization conditional on knowing the Markov-chain transition matrix, and to price a perpetuity of growth-rate cash flows under that anticipation, mechanically separates the two channels and generates a measurable departure between model-implied and observed prices during periods of regime ambiguity. The model is “Markov-switching rational expectations” because the policy-rule and short-rate-volatility coefficients switch on a hidden Markov chain, and the chain itself is in the rational expectations agent’s information set.

Formal notation

State x_t = (g_t, π_t, r_t) (output gap, inflation, short rate). Compound regime S_t = (s_t^m, s_t^d) with two independent binary chains. Macro block under RE collapses (via Cho 2016) to a regime-conditional VAR

x_t = m(S_t) + Φ(S_t) x_{t−1} + Σ(S_t) ε_{t+1}

with nonlinear restrictions on (m, Φ, Σ) from the IS, Phillips curve, and Taylor-rule structural equations. Risk-neutral version

Φ^Q(S) = Φ(S) − Σ(S) Σ(S)' Π_x, m^Q(S) = m(S) − Σ(S) Σ(S)' Π_0

uses an affine market-price-of-risk specification. CRE income growth ν_{j,t} = a_j + γ_{j,π} π_t + γ_{j,g} g_t + ρ_j ν_{j,t−1} + u_{j,t} is contemporaneous in (π, g) so that conditional on a regime path, ln ξ_s ≡ ν_s − r_s is Normal — a property required for the Riccati pricing recursion.

Variants

• Two independent binary chains (Leather–Sagi): monetary-policy active/passive and discretion flexible/rigid, 4 compound regimes.
• Three independent chains (Bikbov–Chernov 2013): adds a third chain for output-gap and inflation volatility, used for samples that cross the Great Moderation.
• Single-regime affine reduction (Bekaert–Cho–Moreno 2010, Rudebusch–Wu 2008): degenerate single-regime case used for affine MS term-structure models; pricing factors are exponential-affine, no Riccati recursion.

Comparison

• vs. affine Markov-switching term-structure models (Bikbov–Chernov 2013; Dai–Singleton–Yang 2007): the latter price only fixed-coupon claims with exponential-affine pricing factors. Adding CRE forces the cash-flow exponent ν_s − r_s to be a non-trivial linear function of the macro state, which pushes the perpetuity pricing into the exponential-quadratic regime (see exponential-quadratic-asset-pricing-factors).
• vs. Gordon–Williams cap rate models: GW is static, single-regime, and cannot capture anticipation; this model is dynamic, multi-regime, and REE-consistent.
• vs. single-equation regression: a regression of cap rates on macro variables is misspecified (Sims 2001) under structural breaks; this model is the rational-expectations alternative.

When to use

When the asset of interest is a long-lived perpetuity-style claim (CRE, infrastructure, equity-like cash flows) whose income loadings on macro fundamentals interact non-trivially with monetary-policy regimes, and the goal is either (i) to identify the latent regime via the joint information in prices and macro variables, (ii) to decompose cap-rate movements into discount- rate vs. cash-flow channels, or (iii) to run regime-change counterfactuals on asset values and CRE-collateralized debt.

Known limitations

• Likelihood is profoundly sloppy (project measurements: condition number ~10²⁵–10²⁶ on the Hamilton surrogate at θ_truth and at ep06c_polished), with most directions essentially flat. Heuristic global search is not honest about this.
• The 39-quarter truncation plus constant-tail continuation value is an approximation whose error is not analytically bounded for the exponential-quadratic case.
• The CRE income loadings do not switch with the regime — a natural extension but expensive.
• The Cho 2016 determinacy gate (a no-bubble / transversality condition for the MS-RE forward solution) is the dominant feasibility constraint and does not have a simple closed form; the published parameter values in the source paper do not satisfy it without re-solving (project gotcha).
• No zero-lower-bound on the short rate.

Open problems

• A formal long-run convergence result for the multivariate exponential- quadratic Riccati operator under regime switching (Hansen–Scheinkman 2009 gives the right framework but the application is non-standard).
• A scalable global-optimization stack that does not lie about the basin geometry — currently the project’s planned next milestone.
• Out-of-sample CRE forecasting on alternative price series (Green Street, RCA).

Key papers

• leather-sagi-markov-switching-cre-asset-pricing (introduces the model with two independent binary chains and the joint TSM+CAP estimation)

My understanding

This is the core modeling object of the project. Everything else in the wiki — the Riccati derivation, the Cho–Moreno forward solution, the RBPF filter, the no-bubble condition, the basin-finder cascade, the global optimization pipeline — is either supporting machinery for this model or a methodological response to a difficulty it raises. The empirical surprise that justifies the entire construction is the LR test that adding CRE prices to the joint estimation significantly tightens monetary-policy regime identification (p < 10^{−10}): prices contain regime information that yields alone do not.