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Macro Fundamentals & Commercial Real Estate Price Dynamics

11 min read

Problem

Commercial real estate (CRE) is a $4T+ asset class to which regional banks have outsized exposure, but its empirical relationship to macro fundamentals — output growth, inflation, short rates — is poorly identified. CRE cap rates are smoothed appraisal-based series; inflation simultaneously raises discount rates and rental income (the two effects are of opposite sign and roughly the same magnitude); and the joint dynamics of macro variables and prices appear to undergo regime breaks (e.g. the 2000Q4 break in the regression of the ten-year yield on inflation, and the post-2000 reversal in the cap-rate / yield co-movement). Single-equation Gordon-Williams regressions and reduced-form linear regressions cannot disentangle the discount-rate channel from the cash-flow channel, and they cannot capture the anticipation effect: if investors price the possibility of a regime change, that anticipation feeds back into current prices in a way that no static model can represent. The paper asks: what is a structural, no-arbitrage-consistent pricing model for CRE that explicitly anticipates regime changes in macro fundamentals, and what does it imply for cap rates, risk premia, and CRE-collateralized mortgages?

Key idea

Build a Markov-switching rational-expectations (MS-RE) model that combines (i) a small New Keynesian macro block — IS, Phillips curve, monetary policy rule — in which the Taylor-rule coefficients and short-rate volatility are governed by two independent binary Markov chains (4 compound regimes interpretable as monetary-policy active/passive × discretion flexible/rigid), and (ii) a no-arbitrage asset-pricing block in which CRE income growth depends contemporaneously on inflation, the output gap, and lagged income, with risk-neutral dynamics that preserve conditional Normality given the regime path. The pricing operator for CRE is then exponential-quadratic in the macro state (rather than the affine form sufficient for bond pricing) because the long-run asset is a perpetuity of growth-rate-exposed cash flows, requiring a Riccati-style recursion in the latent quadratic loadings. The model is closed by Cho’s (2016) forward-solution conditions that pin down the Markov-switching VAR coefficients consistent with rational expectations, and an infinite-horizon continuation-value fixed point that anchors the 10-year truncated pricing recursion to the unconditional present-value identity. Joint estimation on bond and CRE prices is the source of the sharper monetary-policy regime identification — adding cap rates breaks degeneracies that yield-only Markov-switching term-structure models cannot resolve.

Method

Macro block. Three macro states x_t = (g_t, π_t, r_t): output gap, inflation, short rate. The structural equations are

IS: g_t = m_g + (1−μ_g) g_{t−1} + μ_g E_t g_{t+1} − φ (r_t − E_t π_{t+1}) + σ_g ε_t^g PC: π_t = (1−μ_π) π_{t−1} + μ_π E_t π_{t+1} + δ g_t + σ_π ε_t^π MP: r_t = m_r(s^m) + ρ(s^m) r_{t−1} + α̂(s^m) E_t π_{t+1} + β̂(s^m) g_t + σ_r(s^d) ε_t^r

with s^m, s^d ∈ {0,1} two independent binary Markov chains (4 compound regimes S_t = (s^m, s^d)). Cho (2016) gives the conditions under which this MS-RE system collapses to a regime-dependent VAR x_t = m(S_t) + Φ(S_t) x_{t−1} + Σ(S_t) ε_{t+1}, with m(S), Φ(S), Σ(S) satisfying nonlinear restrictions induced by the forward-looking expectation terms. Risk-neutral dynamics are constructed under a Duffee-style affine market price of risk: Φ^Q(S) = Φ(S) − Σ(S) Σ(S)' Π_x, m^Q(S) = m(S) − Σ(S) Σ(S)' Π_0, with constant Π_0, Π_x.

Pricing block. A claim to a perpetuity of growth-rate cash flows E_t[D_{t+1}/D_t] = e^{ν_t} has price-earnings ratio Q_t = Σ_{τ=t+1}^N E_t^Q[exp(Σ ν_s − Σ r_s)]. CRE income growth is modeled as ν_{j,t} = a_j + γ_{j,π} π_t + γ_{j,g} g_t + ρ_j ν_{j,t−1} + u_{j,t} for j ∈ {Apartment, Industrial, Office}, with u_{j,t} = w_{j,t} + σ_{j,Z} Z_t (idiosyncratic + single common factor Z_t) and a risk-neutral counterpart that shifts the constant by −ℓ_j = −λ_Z σ_{j,W}. Conditional on a Markov path (S_t, S_{t+1}, ...), both ln(ξ_s) = ν_s − r_s and its risk-neutral version are Normal, so E^Q[exp(Σ ln ξ_s)] is exponential-quadratic in the macro state through the moment-generating function of a multivariate Normal, evaluated via a Riccati recursion in the quadratic loadings. The infinite-horizon perpetuity is approximated by truncating at K = 39 quarters and replacing the tail by a constant continuation value Q_j^* solving the unconditional fixed point Q_j^* = Σ_{τ=t+1}^K E^Q[exp(Σ ν − Σ r)] + E^Q[exp(Σ ν − Σ r)] · Q_j^*.

Bond block. ZCB price B_{t,τ} = E_t^Q[exp(−Σ_{s=t}^{τ−1} r_s)] is exponential-affine in x_t conditional on regime path; treasury strip yields enter the likelihood with i.i.d. Normal measurement error.

Estimation. The data are quarterly 1992Q1–2014Q2 (90 obs in the paper version): short rate, 2/5/10-year treasury strip yields, output gap (quadratically detrended log real GDP per capita), PCE inflation, and per-quarter aggregated NCREIF property values and NOI growth for Apartment, Industrial, and Office portfolios. Cap rates are computed as end-of-quarter property value divided by trailing four-quarter NOI. Maximum likelihood: a Sobol grid of 1,000,000 points in parameter space is evaluated, the top 1,000 are passed to a local optimizer, and the global maximum across these is reported. The likelihood is constrained to parameterizations under which the Cho forward solution exists and the regime-conditional VARs are mean-square stable under both physical and risk-neutral measures, with δ, φ ≥ 0. Two estimation variants are reported: a TSM-only model (treasury strips only, comparable to Bikbov–Chernov 2013) and a CAP model (treasuries + CRE).

Results

  • Adding CRE prices sharpens regime identification. The CAP model’s regime posteriors are visibly more persistent than the TSM-only model’s, and the likelihood ratio test rejects equality of the macro parameters across the two estimations at p < 10^{−10} (Chi-Squared, 34 df). A second LR test rejects the null that the CRE-block parameters are pinned down by macro variables alone at p < 10^{−10}.
  • Four interpretable regimes. Two policy regimes (Active / Passive: high vs. low Taylor-rule responsiveness to inflation expectations and the output gap) crossed with two volatility regimes (Flexible / Rigid: high vs. low short-rate volatility). The estimated post-2011 period sits in Passive-Rigid — lowest cap rates, highest prices, lowest real estate risk premium.
  • Regime “impulse responses”. A sudden change from Active-Flexible into Passive- Rigid implies a 100–150 bp drop in expected cap rates (a 15–20% increase in property values). The mirror move is somewhat more muted.
  • Bond pricing fit. Treasury strip pricing errors in the CAP model are roughly 40 bp — comparable to but slightly worse than Bikbov–Chernov 2013 (~26 bp without CRE), the standard erosion observed when adding more assets to the joint estimation.
  • CRE pricing fit. Real-estate cap-rate pricing errors are 5–7%. Income growth is closely tracked. The most pronounced model–data divergence is 2005–2010 when CRE prices across all categories sit persistently above their model-implied values — the authors explicitly interpret this as a measurable departure of prices from fundamentals, peaking in mid-2007.
  • Inflation channel reverses sign in multivariate setting. Univariate regressions give CRE income growth a positive loading on inflation; in the joint estimation (after controlling for the output gap, with which inflation is positively correlated) the inflation loading is much smaller and can become negative. This is the central message about why a structural model is necessary.
  • Real estate risk premium. Quarterly systematic real-estate volatility σ_{i,Z} ≈ 30 bp, with λ_Z = 0.42 implying a quarterly real-estate-specific risk premium of ~15 bp (~60 bp annualized). The risk premium is highest in the Active-Flexible regime and lowest in the Passive-Rigid regime, consistent with the cap-rate ordering.
  • CRE mortgage spreads. Ten-year IO mortgages on diversified CRE portfolios show

    100 bp spread differences across extreme policy scenarios at high LTVs. Apartment mortgages have the smallest spreads (consistent with the GSE-supported segment). A regime change from Passive-Rigid into Active-Flexible can imply ~30% loss on existing CRE loan portfolios at high LTV — a quantitative monetary-policy / bank-fragility channel.

Limitations

  • Data window starts in 1992Q1. Excludes the pre-Great-Moderation period; the authors deliberately omit the third (output-gap volatility) regime that Bikbov–Chernov (2013) include for that reason. A White test on the 1992–2014 sample cannot reject homoskedasticity in g, π at the 10% level, so this is consistent, but it limits cross-period generalization.
  • No zero lower bound. Conditional on the regime path the short rate is Gaussian, so the model has no ZLB and no QE-era nonlinearity — explicitly flagged as a candidate explanation for the persistent 2015–2017 model–price gap.
  • Truncated continuation value. The 39-quarter truncation plus constant tail Q_j^* is an analytical approximation with bounded accuracy at long horizons. Bikbov–Chernov’s exponential-affine formula is accurate to ~40 periods; the exponential-quadratic case is not formally bounded in the paper.
  • Single common CRE factor. All cross-sectional CRE risk premia load on one systematic shock Z_t, which is a strong restriction; sector heterogeneity beyond that one factor is forced into idiosyncratic measurement error.
  • No regime switching in the CRE income block. The income loadings (a_j, γ_{j,π}, γ_{j,g}, ρ_j) are constant across the four compound regimes — arguably the most natural extension but expensive to estimate.
  • Likelihood is highly non-linear and the global-max search is heuristic (1M Sobol → top-1k local search). The downstream project’s experience confirms this is the binding bottleneck for honest global recovery on simulated data.

Open questions

  • Is the 2005–2010 model–price gap an economic bubble or a missing pricing factor? The paper takes the former interpretation but explicitly notes it cannot rule out the latter without a richer model.
  • Does regime-switching in the CRE income coefficients improve identification? Untested in the paper.
  • What is the spectral / no-bubble condition for the multivariate exponential- quadratic Riccati operator under regime switching? The paper checks mean-square stability of the regime-conditional VAR under both P and Q, but the long-run convergence of the quadratic Riccati factors under regime switching is not formally characterized — this is the project’s open theoretical question.
  • Out-of-sample CRE price forecasting under the regime-switching model has not been validated against alternative pricing series (Green Street, RCA).
  • GSE subsidy decomposition. The Apartment mortgage spread is suspiciously low; the authors attribute part of it to the implicit GSE subsidy but do not decompose it.

My take

This is the central paper of the project. It is the source of the model specification used throughout SimMdlPrices/: 55 structural parameters (54 free under the m_g IS-stationarity constraint), 3 macro states, 4 compound regimes, 3 income types, exponential-quadratic Riccati pricing factors. Several of the project’s hardest open problems descend directly from the paper:

  • The Cho–Moreno determinacy gate is the dominant feasibility constraint in the parameter space; the published Π_X values do not satisfy Cho–Moreno (project gotcha).
  • The infinite-horizon Riccati tail is the project’s most-debated approximation (the T_bar choice; the geometric tail operator η_H · r/(1−r) is the leading hypothesis for the operator-floor null bottleneck in Exp 12, 2026-04-12).
  • The likelihood is profoundly sloppy — condition number ~10²⁵–10²⁶ on the Hamilton surrogate at both θ_truth and the current incumbent ep06c_polished, with most directions effectively flat. The paper’s heuristic Sobol-then-local optimization is not honest about this; the project’s basin-finder cascade and the in-progress global optimization pipeline are the response.
  • The RBPF (Rao-Blackwellized particle filter on regime histories) is not in the paper itself — the paper uses a Hamilton-style filter on (x_t, S_t). The RBPF is the project’s most important methodological contribution beyond the paper, and the ~1572-nat gap between Hamilton-optimal and RBPF-optimal likelihoods is the evidence that a serious filter is needed once cap rates are in the panel.
  • The CAP-vs-TSM LR-test result (cap rates sharpen regime identification) is the empirical hook that justifies the entire project: real-estate prices do contain information that pure term-structure models cannot extract.

Things to flag for future ingestion as separate notes (not part of this paper page): the Riccati derivation appendix lives in riccati_equations_leather_sagi.md; the Cho 2016 forward-solution conditions in cho_moreno_2010_forward_method_rational_expectations.md; Hansen-Scheinkman 2009 for the long-run / spectral framework that should eventually formalize the no-bubble condition for the quadratic case.

Background concepts referenced but not yet in the wiki (intentionally not creating new pages to respect the seminal-paper concept hard limit; will be created when other papers in the queue introduce them more centrally): no-bubble condition / transversality, Cho 2016 forward solution, Markov-switching VAR, small New Keynesian model, Rao- Blackwellized particle filter on regime histories, capitalization rate as a financial quantity, and the Bikbov–Chernov 2013 affine MS term-structure model on which the paper builds.

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