Three-State Markov-Switching Model Overview

• The three-state Markov-switching model is a framework where an unobserved three-regime latent process drives state-dependent observation laws using a first-order Markov chain.
• It employs filtering, smoothing, and iterative estimation techniques (EM and Bayesian) to jointly infer regime probabilities and state-specific parameters for forecasting and ex post regime dating.
• The model’s flexibility extends to nonlinear covariate effects and high-dimensional data, while challenges include overfitting, weak identification, and label switching.

A three-state Markov-switching model is the $K=3$ specialization of a finite-state regime-switching framework in which an observed process evolves under regime-specific parameterizations while the governing regime is unobserved and follows a discrete first-order Markov chain. In the standard formulation, the model combines a latent state process, a $3\times 3$ transition matrix, and a state-dependent observation law. Its central attractions are the joint estimation of regime-specific parameters and of regime probabilities through filtering and smoothing, which support ex post regime dating, real-time monitoring, and forecast combinations that exploit regime persistence (Song et al., 2020).

1. Formal definition and state dynamics

Let the latent regime indicator be

$S_t\in\{1,2,3\}.$

The defining first-order Markov property is

$$\Pr(S_t=j\mid S_{t-1}=i,S_{t-2},\ldots,S_1)=\Pr(S_t=j\mid S_{t-1}=i)=p_{ij},$$

with transition matrix

$$P=[p_{ij}]_{3\times 3} = \begin{bmatrix} p_{11} & p_{12} & p_{13}\ p_{21} & p_{22} & p_{23}\ p_{31} & p_{32} & p_{33} \end{bmatrix}, \qquad p_{ij}\ge 0,\qquad \sum_{j=1}^3 p_{ij}=1.$$

In the benchmark exogenous specification, these transition probabilities are time-homogeneous. The same framework also accommodates non-homogeneous transition laws driven by covariates $v_t$, typically through a multinomial-logit parameterization,

$$p_{ij,t}= \frac{\exp(v_t\gamma_{ij})} {\exp(v_t\gamma_{i1})+\exp(v_t\gamma_{i2})+\exp(v_t\gamma_{i3})}, \qquad i,j\in\{1,2,3\},$$

with one $\gamma_{ij}$ normalized to zero in each row for identification (Song et al., 2020).

This formal structure is sufficiently broad to cover unrestricted recurrent switching, restricted transition architectures, and composite designs in which different parameter blocks are governed by separate Markov chains. A three-state model can therefore mean a single latent chain governing all switching, or one three-state component within a larger regime system. The same source also gives a three-state change-point example with monotonic transitions and an absorbing third state,

$$\begin{bmatrix} p_{11}&p_{12}&0\ 0&p_{22}&p_{23}\ 0&0&1 \end{bmatrix},$$

which is structurally distinct from an unrestricted recurrent Markov-switching specification (Song et al., 2020).

2. Observation equations and regime-dependent parameterization

The general measurement equation is

$y_t \sim F(x_t,S_t),$

so the conditional distribution of $3\times 3$0 depends on observed covariates $3\times 3$1 and on the latent regime. There is no restriction on $3\times 3$2: it may be discrete, continuous, or a mixture. A canonical example is the Gaussian regime-switching autoregression

$3\times 3$3

under which each regime $3\times 3$4 has its own parameter triple $3\times 3$5. More parsimonious designs are also standard, including switching means only,

$3\times 3$6

switching variances only,

$3\times 3$7

and fully switching autoregressive parameters. Parameters not indexed by $3\times 3$8 are common across regimes (Song et al., 2020).

The same three-state logic extends beyond linear Gaussian models. In Markov-switching generalized additive models, the latent process is an $3\times 3$9-state Markov chain; setting $S_t\in\{1,2,3\}.$0 yields a three-state specification with state-specific additive predictors

$S_t\in\{1,2,3\}.$1

where each smooth component can be represented by penalized B-splines. The standard parametric Gaussian switching regression is nested as the special case in which the state-specific functions are linear. This places the ordinary three-state Markov-switching regression inside a broader class of models that permit regime-specific nonlinear covariate effects and exponential-family observation laws (Langrock et al., 2014).

3. Filtering, smoothing, likelihood, and forecasting

Likelihood construction follows hidden-Markov logic, with the latent regimes integrated out by the Hamilton filter. For $S_t\in\{1,2,3\}.$2, the predictive density recursion is

$S_t\in\{1,2,3\}.$3

Predicted state probabilities evolve according to

$S_t\in\{1,2,3\}.$4

or componentwise,

$S_t\in\{1,2,3\}.$5

After observing $S_t\in\{1,2,3\}.$6, Bayes’ rule yields filtered probabilities,

$S_t\in\{1,2,3\}.$7

and the sample likelihood is

$S_t\in\{1,2,3\}.$8

The paper distinguishes forecasted, filtered, and smoothed probabilities: $S_t\in\{1,2,3\}.$9, $$\Pr(S_t=j\mid S_{t-1}=i,S_{t-2},\ldots,S_1)=\Pr(S_t=j\mid S_{t-1}=i)=p_{ij},$$0, and $$\Pr(S_t=j\mid S_{t-1}=i,S_{t-2},\ldots,S_1)=\Pr(S_t=j\mid S_{t-1}=i)=p_{ij},$$1, respectively. Smoothed probabilities use the full sample and are therefore best for ex post regime dating; filtered probabilities are more relevant for real-time monitoring and forecasting. In Bayesian settings, smoothing is associated with FFBS (Song et al., 2020).

Forecasts are probability-weighted mixtures of state-specific forecasts. In the Gaussian AR case, a one-step-ahead conditional mean forecast has the schematic form

$$\Pr(S_t=j\mid S_{t-1}=i,S_{t-2},\ldots,S_1)=\Pr(S_t=j\mid S_{t-1}=i)=p_{ij},$$2

State persistence is encoded in the diagonal elements $$\Pr(S_t=j\mid S_{t-1}=i,S_{t-2},\ldots,S_1)=\Pr(S_t=j\mid S_{t-1}=i)=p_{ij},$$3; high diagonal probabilities imply long spells in a regime. The standard duration formula is directly implied by the Markov structure,

$$\Pr(S_t=j\mid S_{t-1}=i,S_{t-2},\ldots,S_1)=\Pr(S_t=j\mid S_{t-1}=i)=p_{ij},$$4

A three-state model can improve forecasts relative to a two-state model when the data exhibit an intermediate regime that two states would force into one of two extremes, but the gain depends on whether the richer regime structure is empirically meaningful (Song et al., 2020).

4. Estimation, model selection, and inferential issues

Maximum-likelihood estimation is described through an iterative EM algorithm. In the Gaussian three-state AR example, the parameter set is

$$\Pr(S_t=j\mid S_{t-1}=i,S_{t-2},\ldots,S_1)=\Pr(S_t=j\mid S_{t-1}=i)=p_{ij},$$5

together with the transition matrix $$\Pr(S_t=j\mid S_{t-1}=i,S_{t-2},\ldots,S_1)=\Pr(S_t=j\mid S_{t-1}=i)=p_{ij},$$6. The E-step uses filtering and smoothing to compute expected state occupancies and transitions; the M-step updates regime-specific parameters and transition probabilities subject to nonnegativity and row-sum constraints. Initialization requires an initial regime distribution $$\Pr(S_t=j\mid S_{t-1}=i,S_{t-2},\ldots,S_1)=\Pr(S_t=j\mid S_{t-1}=i)=p_{ij},$$7; for an ergodic three-state chain, a common choice is the stationary distribution $$\Pr(S_t=j\mid S_{t-1}=i,S_{t-2},\ldots,S_1)=\Pr(S_t=j\mid S_{t-1}=i)=p_{ij},$$8 defined by

$$\Pr(S_t=j\mid S_{t-1}=i,S_{t-2},\ldots,S_1)=\Pr(S_t=j\mid S_{t-1}=i)=p_{ij},$$9

Bayesian estimation starts from the complete-data likelihood

$$P=[p_{ij}]_{3\times 3} = \begin{bmatrix} p_{11} & p_{12} & p_{13}\ p_{21} & p_{22} & p_{23}\ p_{31} & p_{32} & p_{33} \end{bmatrix}, \qquad p_{ij}\ge 0,\qquad \sum_{j=1}^3 p_{ij}=1.$$0

and alternates sampling from $$P=[p_{ij}]_{3\times 3} = \begin{bmatrix} p_{11} & p_{12} & p_{13}\ p_{21} & p_{22} & p_{23}\ p_{31} & p_{32} & p_{33} \end{bmatrix}, \qquad p_{ij}\ge 0,\qquad \sum_{j=1}^3 p_{ij}=1.$$1 and $$P=[p_{ij}]_{3\times 3} = \begin{bmatrix} p_{11} & p_{12} & p_{13}\ p_{21} & p_{22} & p_{23}\ p_{31} & p_{32} & p_{33} \end{bmatrix}, \qquad p_{ij}\ge 0,\qquad \sum_{j=1}^3 p_{ij}=1.$$2 using MCMC and FFBS (Song et al., 2020).

Several inferential difficulties are intrinsic. Identification can be delicate, especially when testing whether switching is present at all; the likelihood can be multimodal and, in larger models, even unbounded; and Bayesian inference faces label switching because permutations of labels $$P=[p_{ij}]_{3\times 3} = \begin{bmatrix} p_{11} & p_{12} & p_{13}\ p_{21} & p_{22} & p_{23}\ p_{31} & p_{32} & p_{33} \end{bmatrix}, \qquad p_{ij}\ge 0,\qquad \sum_{j=1}^3 p_{ij}=1.$$3 leave the likelihood invariant. These problems are especially consequential when the regimes are to be given substantive labels such as recession, normal growth, and expansion, or low, medium, and high volatility. The same source therefore treats the number of states $$P=[p_{ij}]_{3\times 3} = \begin{bmatrix} p_{11} & p_{12} & p_{13}\ p_{21} & p_{22} & p_{23}\ p_{31} & p_{32} & p_{33} \end{bmatrix}, \qquad p_{ij}\ge 0,\qquad \sum_{j=1}^3 p_{ij}=1.$$4 as a model-selection problem rather than a convention: frequentist work uses information criteria such as AIC, and Bayesian work uses marginal data densities. A three-state specification offers more flexibility than a two-state model, but it should be justified by the application and the data rather than imposed mechanically (Song et al., 2020).

A complementary line of work addresses the same issue nonparametrically. Fuzzy clustering has been proposed as a pre-estimation diagnostic for the number of Markov-switching states, on the argument that fuzzy memberships can reproduce Hamilton-filter state inference. The procedure compares candidate values of $$P=[p_{ij}]_{3\times 3} = \begin{bmatrix} p_{11} & p_{12} & p_{13}\ p_{21} & p_{22} & p_{23}\ p_{31} & p_{32} & p_{33} \end{bmatrix}, \qquad p_{ij}\ge 0,\qquad \sum_{j=1}^3 p_{ij}=1.$$5 through cluster-validity indices such as PC, PE, MPC, ASW, ASWF, and XB. In a quarterly U.S. GDP growth application, all six indices selected $$P=[p_{ij}]_{3\times 3} = \begin{bmatrix} p_{11} & p_{12} & p_{13}\ p_{21} & p_{22} & p_{23}\ p_{31} & p_{32} & p_{33} \end{bmatrix}, \qquad p_{ij}\ge 0,\qquad \sum_{j=1}^3 p_{ij}=1.$$6; the corresponding MS(3) model was interpreted as boom, expansion, and contraction, and AIC and BIC both favored MS(3) over MS(2) (Otranto et al., 2023).

5. Major extensions and related formulations

The standard three-state model implies geometric dwell times because, in a first-order homogeneous chain, remaining in regime $$P=[p_{ij}]_{3\times 3} = \begin{bmatrix} p_{11} & p_{12} & p_{13}\ p_{21} & p_{22} & p_{23}\ p_{31} & p_{32} & p_{33} \end{bmatrix}, \qquad p_{ij}\ge 0,\qquad \sum_{j=1}^3 p_{ij}=1.$$7 for $$P=[p_{ij}]_{3\times 3} = \begin{bmatrix} p_{11} & p_{12} & p_{13}\ p_{21} & p_{22} & p_{23}\ p_{31} & p_{32} & p_{33} \end{bmatrix}, \qquad p_{ij}\ge 0,\qquad \sum_{j=1}^3 p_{ij}=1.$$8 periods has probability proportional to $$P=[p_{ij}]_{3\times 3} = \begin{bmatrix} p_{11} & p_{12} & p_{13}\ p_{21} & p_{22} & p_{23}\ p_{31} & p_{32} & p_{33} \end{bmatrix}, \qquad p_{ij}\ge 0,\qquad \sum_{j=1}^3 p_{ij}=1.$$9. Explicit-duration Markov-switching models generalize this by augmenting the latent process with duration variables. The monograph on explicit-duration models organizes these augmentations into decreasing-count, increasing-count, and count-duration encodings, each designed to represent different information about segment boundaries and resets. In this formulation, a three-state model remains a three-regime process, but duration distributions need no longer be geometric, and changepoint or reset semantics can be built directly into the latent-state structure (Chiappa, 2019).

Markov-switching state-space models provide a second major extension. In the linear Gaussian case,

$v_t$0

a three-state specification is obtained by setting $v_t$1 and estimating regime-specific state-transition and observation matrices, innovation covariances, and transition probabilities. Because exact likelihood evaluation would require accounting for all $v_t$2 regime histories, the paper develops approximate Kim-filter-based maximum likelihood, EM, and parametric bootstrap procedures, together with initialization strategies and convergence acceleration suited to high-dimensional spatio-temporal data (Degras et al., 2021).

In large-dimensional factor models, regime switching can act on the loading structure rather than directly on a univariate mean or variance. The relevant framework writes the observed panel as a factor model with regime-specific loading matrices and exploits an equivalent linear representation: latent factors are first recovered by PCA, then loadings and transition probabilities are estimated through an EM algorithm based on a modified Baum-Lindgren-Hamilton-Kim filter and smoother. Although the formal development is two-state, the model and the algorithm extend naturally to three states, with regime-specific loading blocks and a $v_t$3 transition matrix (Barigozzi et al., 2022).

Other extensions alter the meaning of the latent state. Independent-regime models treat some regimes as separate latent AR(1) processes and augment the hidden chain with counters recording the time since each AR regime was last visited; this yields exact forward, backward, and EM algorithms for two- and three-regime electricity-price models (Bean et al., 2019). Coupled nonhomogeneous hidden Markov models for epidemic surveillance use three conceptual states—absence, endemic, outbreak—and allow transition probabilities to depend on covariates and neighboring outbreaks; clone states are added to enforce minimum endemic and outbreak durations, so the conceptual three-state process is represented internally by a seven-state hidden chain (Douwes-Schultz et al., 2023).

6. Interpretation, applications, and common misconceptions

Interpretation in a three-state model is never based on transition probabilities alone. The core empirical principle is joint reading of regime-specific parameters and time-varying regime probabilities. In macroeconomic applications, a two-state expansion–recession model can be refined to recession, moderate growth, and strong growth; in finance, low-, medium-, and high-volatility regimes are a common three-state interpretation. A recent order-flow study of the Korean stock market implements a three-state Markov-switching model with Bull, Normal, and Crisis regimes. The paper reports 528 Bull days with mean return $v_t$4 and volatility $v_t$5, 598 Normal days with mean return $v_t$6 and volatility $v_t$7, and 95 Crisis days with mean return $v_t$8 and volatility $v_t$9. In the associated regime-conditional predictive regression, foreign-investor order flow is most informative in Crisis, with

$$p_{ij,t}= \frac{\exp(v_t\gamma_{ij})} {\exp(v_t\gamma_{i1})+\exp(v_t\gamma_{i2})+\exp(v_t\gamma_{i3})}, \qquad i,j\in\{1,2,3\},$$0

so the crisis-state slope is approximately $$p_{ij,t}= \frac{\exp(v_t\gamma_{ij})} {\exp(v_t\gamma_{i1})+\exp(v_t\gamma_{i2})+\exp(v_t\gamma_{i3})}, \qquad i,j\in\{1,2,3\},$$1 times the bull-state slope (Kang, 9 Jan 2026).

Three-state designs also appear in volatility modeling across scales. A recent EUR/USD study estimates three independent AR(1)-MS-GARCH models, each with Calm, Turbulent, and Crisis regimes, on daily, 4-hour, and hourly data. On the shorter horizons the transition probabilities are allowed to vary through Filardo-style TVTP, and the three filtered probability vectors are combined by outer product into a $$p_{ij,t}= \frac{\exp(v_t\gamma_{ij})} {\exp(v_t\gamma_{i1})+\exp(v_t\gamma_{i2})+\exp(v_t\gamma_{i3})}, \qquad i,j\in\{1,2,3\},$$2-state cross-scale tensor. The paper reports expected dwell times of 45.1 days, 13.9 days, and 7.6 days for the daily Calm, Turbulent, and Crisis states, respectively, and documents improved out-of-sample volatility forecasting relative to a single-regime GARCH benchmark, including a smoothed RMSE reduction from $$p_{ij,t}= \frac{\exp(v_t\gamma_{ij})} {\exp(v_t\gamma_{i1})+\exp(v_t\gamma_{i2})+\exp(v_t\gamma_{i3})}, \qquad i,j\in\{1,2,3\},$$3 to $$p_{ij,t}= \frac{\exp(v_t\gamma_{ij})} {\exp(v_t\gamma_{i1})+\exp(v_t\gamma_{i2})+\exp(v_t\gamma_{i3})}, \qquad i,j\in\{1,2,3\},$$4 and a Diebold–Mariano statistic of $$p_{ij,t}= \frac{\exp(v_t\gamma_{ij})} {\exp(v_t\gamma_{i1})+\exp(v_t\gamma_{i2})+\exp(v_t\gamma_{i3})}, \qquad i,j\in\{1,2,3\},$$5 with $$p_{ij,t}= \frac{\exp(v_t\gamma_{ij})} {\exp(v_t\gamma_{i1})+\exp(v_t\gamma_{i2})+\exp(v_t\gamma_{i3})}, \qquad i,j\in\{1,2,3\},$$6 (Chaudhary, 4 Jun 2026).

A common misconception is that every “three-state Markov-switching model” is a latent-regime hidden Markov model of the Hamilton type. The literature uses the phrase more broadly. In the “three-state herding model of the financial markets,” the three states are observable agent groups—fundamentalists, pessimistic chartists, and optimistic chartists—and the market state is the composition vector of these groups rather than a single latent regime label (Kononovicius et al., 2012). In the Method of Undetermined Markov States for linear DSGE models, the three-state chain is an exact representation of impulse-response dynamics with impact, medium-run, and absorbing long-run states and transition matrix

$$p_{ij,t}= \frac{\exp(v_t\gamma_{ij})} {\exp(v_t\gamma_{i1})+\exp(v_t\gamma_{i2})+\exp(v_t\gamma_{i3})}, \qquad i,j\in\{1,2,3\},$$7

so the Markov states are a representation of the solution path rather than exogenous structural regimes (Roulleau-Pasdeloup, 2022). These examples show that the defining commonality is the use of a three-state Markov mechanism, not a unique economic interpretation.

In its standard econometric sense, however, the three-state Markov-switching model remains the $$p_{ij,t}= \frac{\exp(v_t\gamma_{ij})} {\exp(v_t\gamma_{i1})+\exp(v_t\gamma_{i2})+\exp(v_t\gamma_{i3})}, \qquad i,j\in\{1,2,3\},$$8 member of a hidden Markov family in which the observation law is regime dependent, the regime process is first-order Markov, the likelihood is built by filtering the latent chain, and empirical interpretation relies on combining state-specific parameters with forecasted, filtered, or smoothed regime probabilities. Its utility lies in allowing a richer regime partition than a two-state model while preserving tractable probabilistic structure; its main risks are overfitting, weak identification, unstable estimation, and ambiguity in state labeling when the data do not support three genuinely distinct regimes (Song et al., 2020).