Multi-regime Markov-switching models with time-varying transition probabilities: An application to U.S. Treasury yields

Abstract: This paper studies Markov-switching (MS) models with time-varying transition probabilities (TVTP) under various specifications of the transition probability matrix. Especially, we extend the two-regime common-variance setting of the Generalized Autoregressive Score (GAS) model from (Bazzi et al., 2017) to the general $K$-regime case with regime-specific means and variances. Our study contains comprehensive Monte Carlo simulations and we developed an open-source R package, \texttt{multiregimeTVTP}, for data simulation and parameter estimation. We find that the regime means, variances, and transition probabilities are reliably recovered, whereas the TVTP driving coefficients are harder to identify. Another finding from our paper is that the GAS score coefficient appears to be statistically non-identifiable, due to a ridge in the joint likelihood surface $(σ^2,A)$. In addition, we find that one-step point forecasts are remarkably robust to TVTP misspecification, but filtered regime probabilities are not, so correct specification matters most for characterizing regime dynamics rather than short-horizon forecasting. An empirical application to U.S. Treasury zero-coupon yield changes at four maturities (1961-2024) shows that an exogenous specification driven by the lagged yield level dominates the constant and lagged-change models in fit, while the GAS specification fails to converge, with $\hat{A}$ collapsing to zero, reflecting the same identifiability issue observed in simulation.

• The paper extends the Markov-switching framework by incorporating time-varying transition probabilities in a K-regime setting to enhance regime inference in U.S. Treasury yields.
• It systematically compares three TVTP mechanisms—lagged observables, exogenous covariates, and GAS dynamics—through extensive Monte Carlo simulations demonstrating robust parameter recovery.
• Empirical results favor the exogenous covariate model for yield curves, highlighting limitations in GAS-driven dynamics and offering insights for risk management and forecasting.

Multi-Regime Markov-Switching Models with Time-Varying Transition Probabilities: Application and Methodological Advances

Introduction

This paper provides a systematic extension and empirical evaluation of Markov-switching (MS) models with time-varying transition probabilities (TVTP) in the context of U.S. Treasury yield dynamics. The methodological contribution extends the Generalized Autoregressive Score (GAS) framework of Bazzi et al. [2017] to accommodate a general $K$-regime setting with regime-specific means and variances. Three distinct mechanisms for the evolution of the transition probability matrix—covariate driven, exogenous covariate driven, and score-driven (GAS)—are assessed through comprehensive Monte Carlo analysis and empirical application.

Model Extensions and Specification

The classical MS framework assumes regime switching with constant transition probabilities. While tractable, constant transition probabilities are insufficient for capturing the dependence of underlying regime changes on evolving economic conditions or exogenous variables. Diebold et al. [1994] and Filardo [1994] introduced TVTP models where transition probabilities are directly influenced by lagged observations or exogenous covariates. Creal et al. [2013] and Bazzi et al. [2017] further developed the GAS framework, using the score of the observation density's likelihood to adaptively update transition probabilities.

This study formalizes and compares three variants for $f_t$, the K(K-1) vector that parametrizes transition dynamics:

• Model I: $f_t = a + b \, y_{t-1}$, a function of lagged observables.
• Model II: $f_t = \beta + \gamma X_{t-1}$, a function of exogenous covariates.
• Model III: $f_t = w + A s_{t-1} + B f_{t-1}$, a score-driven process, with $s_t$ as the scaled score of the conditional likelihood, i.e., generalized autoregressive score (GAS) dynamics.

All models employ a multinomial logistic link for mapping unconstrained parameters to transition probabilities.

Monte Carlo Simulation Study

A robust Monte Carlo design evaluates statistical properties and estimation performance, including parameter recovery, interval coverage, forecast accuracy, and robustness to model misspecification. Scenarios cover $K=2$ and $K=3$ regimes, both with common and regime-specific variances, and all transition mechanism types. The simulation framework is implemented in an open-source R package, multiregimeTVTP, providing reproducible tools for simulation, filtering, and estimation under all model specifications.

Key numerical findings:

• Regime means, variances, and transition probabilities are reliably estimated across all data-generating processes (DGPs), with diminishing bias and RMSE as sample size increases.
• TVTP driving coefficients ($\alpha, \gamma$, and $A$) are substantially harder to identify, especially in regime-rich ($f_t$0) settings; at least $f_t$1 is needed for acceptable RMSE.
• In the GAS model, the score coefficient matrix $f_t$2 is empirically non-identifiable; profile likelihood analyses reveal a pronounced ridge in the $f_t$3 surface, with MLEs collapsing to $f_t$4 independently of the data-generating parameter. This is a structural, not computational, identifiability problem.
• Short-horizon point forecast accuracy (e.g., one-step-ahead) is robust to misspecification of TVTP dynamics—accuracy differences are typically below 1% across models. This reflects the dominance of regime means in conditional expectation for short horizons.
• Filtered regime probability accuracy is highly sensitive to TVTP specification. Especially in $f_t$5, misspecification can nearly double MSE in regime probability estimates. Correct specification is therefore critical for state-based inference or long-horizon forecasting.

Coverage rates for Wald confidence intervals on TVTP parameters are systematically below nominal levels, further reflecting challenges in estimating dynamic parameters under moderate sample sizes and regime complexity.

Empirical Application to U.S. Treasury Yields

The methodology is applied to reconstructed U.S. Treasury zero-coupon yield changes at maturities of 1, 12, 36, and 72 months (monthly, 1961–2024). The focus is on yield curve changes, rendered stationary via differencing.

Four specifications are compared: constant transition probabilities (baseline), TVTP driven by lagged yield changes (Model I), TVTP driven by lagged yield level (Model II: Exogenous), and GAS (Model III).

Empirical findings:

• Across all maturities, the exogenous covariate model (lagged yield level as driver) uniformly dominates in-sample fit by AIC and BIC, superseding both constant and TVP (lagged change) specifications.
• GAS specification is empirically infeasible: for all maturities and 100 random initializations per fit, either no convergence occurs or $f_t$6 collapses to zero, identically reducing to the constant-probability model. Results mirror the simulation findings and confirm the lack of empirical identification of score-driven transition dynamics in yield curve changes.
• Filtered regime classification reveals robust segmentation into low-, moderate-, and high-volatility states. In all cases, variance ordering aligns with economic expectations (e.g., rare, high-volatility regime during stress periods).
• Regime-specific means and variances under the best (exogenous) specification are stable and interpretable, with minimal overlap of regime distributions, indicating robust univariate discrimination.

Practical and Theoretical Implications

The strong identifiability and fit of exogenous-driven TVTP is supportive of regime-switching dynamics in fixed income being responsive to lagged yield levels (not simply lagged changes), aligning with economic expectations of mean-reversion and the influence of prevailing rate environments on volatility regimes.

The statistical fragility of GAS-driven transition probabilities, both in simulation and in real data, signals intrinsic identifiability limitations in jointly learning transition dynamics and regime conditional variances when the score process is used as the sole information carrier. This limitation has broader implications for dynamic score-based switching models in low signal-to-noise settings or when regime persistence is high.

Forecasts at short horizons are, by construction, insensitive to transition mechanism misspecification, but model choice critically affects regime inference and long-horizon forecasting, making TVTP specification especially relevant for applications such as risk management, stress testing, and financial stability assessment.

Future Directions

Several open questions and avenues are highlighted:

• Investigating alternative or hybrid functional forms for TVTP (cyclicality, regime duration effects, or mean reversion), particularly in regime-rich settings or multi-variate contexts.
• Exploring bootstrapped or robustified standard error estimation methods for TVTP parameters, given the frequent undercoverage of conventional confidence intervals.
• Application to high-frequency financial processes and cross-asset regime dynamics, where forward-looking or market-based covariates may better inform TVTP than lagged observables.
• Extending the framework to state-dependent or regime-dependent risk premia in term structure modeling.

Conclusion

This paper provides a formal and transparent extension of Markov-switching models with time-varying transition probabilities to the general K-regime, regime-specific variance case. Through both simulation and application to nearly six decades of fixed income data, it demonstrates robust parameter recovery for means and variances and substantiates the superiority of exogenous (lagged-level-driven) TVTP in empirical fit for Treasury yields. The identification problem of the GAS mechanism is highlighted both theoretically and practically, serving as a cautionary note for unrestricted score-driven regime modeling in economic time series. A comprehensive and extensible implementation is provided via the multiregimeTVTP R package, enabling wider adoption and further study.