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Regime Shifts in a Dynamic Term Structure Model of U.S. Treasury Bond Yields

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Problem

Standard Gaussian dynamic term structure models (DTSMs) impose constant parameters on the risk-neutral dynamics of yields, but US Treasury yield data exhibit clear breaks in volatility, persistence, and slope across monetary-policy regimes (e.g., the pre/post-Volcker period and the early-2000s “conundrum”). Earlier regime-switching term structure work either (a) used constant Markov transition probabilities — which cannot generate state-dependent expected duration of regimes or capture the link between the macro state and regime persistence — or (b) sacrificed no-arbitrage closed-form bond pricing in order to obtain state-dependent transitions. The open question this paper addresses is whether one can simultaneously have:

  1. No-arbitrage bond pricing in closed form,
  2. State-dependent Markov transition probabilities (so the macro/yield state itself drives how likely a regime change is), and
  3. Priced regime-shift risk (so the regime-switch event carries its own market price of risk separate from the diffusive factor risk).

Key idea

Build a discrete-time Gaussian DTSM in which a finite-state Markov chain governs the parameters of the short-rate process and the market prices of risk, where the Markov chain’s transition matrix is itself an explicit function of the latent factors. Show that — under the right exponential-affine parametrization of the market prices of both factor risk and regime-shift risk — bond prices remain exponential-affine in the factors conditional on the current regime, with regime-specific loadings that solve a coupled system of recursions. This pins down the structure under which closed-form pricing survives state-dependent regime transitions, and lets the authors decompose the term premium into a factor-risk component and a regime-shift-risk component.

Method

Setup. Let s_t ∈ {1,...,M} be a discrete latent regime, and X_t ∈ R^N be a vector of latent (or observable) Gaussian factors. Under the physical measure P,

  • Within regime s_t = i, X_{t+1} = μ_i^P + Φ_i^P X_t + Σ_i ε_{t+1}^P with ε_{t+1}^P ~ N(0, I).
  • Short rate is regime-affine: r_t = δ_{0, s_t} + δ_1' X_t.
  • The Markov transition matrix is state-dependent: Π_{ij}(X_t) = P(s_{t+1}=j | s_t=i, X_t), with each row parametrized so that probabilities depend smoothly on X_t (the paper uses a logistic / multinomial-logit form keeping closed-form tractability).

Pricing kernel. The stochastic discount factor jointly prices factor risk and regime-shift risk:

  • Factor-risk component: standard essentially-affine market price of risk λ_i(X_t) = λ_{0,i} + λ_{1,i} X_t, generating an essentially-affine change of measure within each regime.
  • Regime-shift-risk component: a separate Radon–Nikodym derivative twists the state-dependent transition matrix Π(X_t) into a risk-neutral transition matrix Π^Q(X_t), with regime-shift market prices of risk that are themselves exponential-affine in X_t. Crucially, the regime-shift market price of risk is allowed to be nonzero — earlier work usually set it to zero, which forced Π^Q = Π^P and zeroed out the regime-shift risk premium by assumption.

Bond pricing. Conditional on the current regime, the price of an n-period zero-coupon bond is exponential-affine:

P_t^{(n)}(s_t = i) = exp( A_n(i) + B_n(i)’ X_t )

with regime-specific coefficient functions A_n(·), B_n(·) solving a coupled recursion of the form

B_{n+1}(i) = -δ_1 + (Φ_i^Q)’ Σ_j Π_{ij}^Q(X_t) B_n(j)

A_{n+1}(i) = -δ_{0,i} + Σ_j Π_{ij}^Q(X_t) [A_n(j) + B_n(j)’ μ_i^Q + (1/2) B_n(j)’ Σ_i Σ_i’ B_n(j)] + (regime-jump correction terms)

Because Π^Q(X_t) depends on X_t, the recursion is not exactly affine in X_t unless Π^Q is approximated; the paper handles this by either (a) using a log-linear approximation of log Π^Q(X_t) around the unconditional mean, which preserves exact affinity in X_t, or (b) solving the recursion numerically. Either way, the resulting bond yields are linear in X_t within each regime, which lets the authors apply a standard regime-switching Kalman / Hamilton filter to estimate the model by maximum likelihood on observed Treasury yields.

Estimation. Joint estimation of the regime parameters, factor dynamics, market prices of factor and regime-shift risk on monthly US Treasury zero-coupon yields spanning multiple maturities and a sample period that brackets the major US monetary policy regime changes. Identification of the regime-shift risk premium comes from the spread between the time-series-average term premium and the cross-sectional shape of yields in each regime.

Decomposition. The fitted model decomposes the unconditional term premium into:

  • Pure expectations component (E_t[r] minus r_t),
  • Factor-risk premium (compensation for diffusive shocks within a regime),
  • Regime-shift risk premium (compensation for the possibility of moving to a high-rate / high-volatility regime).

Results

  • Statistical fit: the state-dependent-transition regime-switching DTSM significantly improves fit (likelihood ratio + out-of-sample yield-prediction metrics) over both (i) single-regime Gaussian DTSMs and (ii) regime-switching DTSMs with constant transition probabilities. The state-dependence of Π is not a nuisance — it carries economically meaningful information about how the level/slope of the curve forecasts regime change.
  • Regime identification: estimated regimes line up cleanly with monetary policy episodes (in particular the 1979–82 Volcker disinflation period is a separate regime with sharply higher short-rate volatility and a different Taylor-rule-like feedback from factors to short rates).
  • Regime-shift risk premium is economically large: at long maturities the regime-shift component accounts for a substantial share of the term premium — comparable to or larger than the within-regime factor-risk premium — and is highly time-varying. Imposing a zero regime-shift risk premium (the standard earlier-literature assumption) materially understates compensation for regime uncertainty in long-maturity yields.
  • Time-varying expected regime durations: because Π(X_t) depends on the factors, expected regime durations move with the macro state — the model captures the fact that high yields and a steep curve forecast a higher probability of switching to a tightening regime, which a constant-Π model cannot reproduce.
  • Excess-return decomposition: expected excess holding-period returns on long Treasuries are decomposed into a factor-risk component and a regime-shift risk component; both vary with the state, and the regime-shift component is the primary driver of the level of the long-end term premium during transition windows.

Limitations

  • Discrete-time, Gaussian factors only: cannot directly accommodate stochastic-volatility factors within a regime (CIR-style square-root dynamics) without breaking the closed-form-affine bond pricing.
  • Logistic-form Π(X_t): the state-dependence is parametric and chosen for tractability; the closed-form-affine bond price requires a log-linear approximation of Π^Q(X_t) (or numerical recursion) to remain exactly affine in X_t.
  • Latent factors only: the paper does not explicitly link X_t to observed macro variables (output gap, inflation), so the regime interpretation is ex-post identification, not a structural macro-finance model.
  • M=2 regimes in the empirical work: the paper estimates a two-regime specification; extending to more regimes is conceptually straightforward but computationally heavier and identification of additional regime-shift risk prices becomes thin.
  • No-bubble / determinacy is not the binding issue here (it is a reduced-form pricing model, not a forward-looking RE solution), so the paper sidesteps the Cho–Moreno-style determinacy questions that arise in NK macro-finance models.

Open questions

  • How should the framework be extended when factor dynamics are themselves driven by a forward-looking RE solution (e.g., an NK output-gap / inflation / short-rate block) so that regime persistence and macro determinacy become coupled?
  • What is the right way to price regime-shift risk when the Markov chain is decomposed into multiple independent binary chains (e.g., monetary-policy regime × wage-rigidity regime), so that there are 4+ compound states but only 2 economically meaningful prices of regime risk?
  • Can the closed-form-affine pricing recursion be extended to quadratic pricing factors (Riccati-style) without losing tractability when Π(X_t) depends on X_t?
  • How robust is the regime-shift risk premium estimate to the choice of log-linear approximation of Π^Q(X_t) versus a fully numerical solution?

My take

DSY 2007 is the closest published predecessor to this project’s compound-regime NK macro-finance pricing model. Three things from this paper are load-bearing for the CRE asset pricing model:

  1. The regime-shift risk premium is not a footnote. Setting it to zero (the default in many earlier regime-switching DTSMs) materially distorts long-end pricing. The current project’s RBPF likelihood already reflects this through the priced compound-regime kernel; DSY 2007 is the canonical reference for why that price has to be there.

  2. State-dependent transitions vs constant Π is a real economic distinction, not a modeling preference. The current project uses two binary chains with constant transition probabilities (4 compound states); DSY 2007 says the state-dependent generalization improves fit on yields alone. Whether the same is true on cap rates / CRE prices is an open empirical question — and a natural follow-up experiment.

  3. The closed-form-affine recursion under regime switching is the analog of the project’s exponential-quadratic Riccati pricing. The DSY 2007 derivation is for affine pricing factors and 2 regimes; the project extends this to exponential-quadratic pricing factors and 4 compound states with the compute_quadratic_pricing_factors_msvar Riccati recursion. The mathematical machinery (regime-coupled forward recursion, regime-specific loadings) is the same — only the algebra is different.

The single most important caveat is that DSY 2007’s state-dependent Π(X_t) requires a log-linear approximation of log Π^Q(X_t) to keep the bond price exactly affine in X_t. The current project uses constant Π precisely because this approximation step is awkward inside an RBPF where the regime path has to be sampled in closed form. If this project later moves to state-dependent transitions, the approximation choice has to be revisited carefully — especially since the project’s no-bubble / NBC feasibility check is already sensitive to small perturbations in Π_X (see the published-θ NBC failure recorded in gotchas).

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