Markov-Switching Model

Definition

A Markov-switching (or regime-switching) model is a state-space model in which the parameters governing the observation density depend on a discrete latent state $S_t$ that evolves as a first-order Markov chain. Conditional on the regime, the data are described by an ordinary linear-Gaussian model; marginally, the model captures abrupt structural shifts — recessions, exchange-rate regimes, high- vs. low-volatility episodes — that a single-regime model cannot. Introduced to econometrics by Hamilton (1989), the framework replaces the continuous latent state of a linear state-space model with a finite collection of regimes, so the conditional likelihood is computed by a discrete analogue of the Kalman filter known as Hamilton's filter.

Key Ideas

Discrete latent state: The unobserved regime $S_t \in \{1,\ldots,m\}$ indexes which parameter vector $\theta_{S_t}$ generates $y_t$ . The observation density is $f(y_t \mid Y_{t-1}, S_t = j, \theta)$ — a single-regime model conditional on $S_t$ .
First-order Markov transition: Regimes persist according to a transition matrix $P = [p_{ij}]$ with $p_{ij} = \Pr(S_t = j \mid S_{t-1} = i)$ and $\sum_j p_{ij} = 1$ . Expected regime duration is $1/(1 - p_{ii})$ .
State-space grounding: A Markov-switching model is a non-linear, non-Gaussian state-space model whose state is categorical rather than continuous. When a continuous latent component (e.g. an unobserved-components trend) is also present, the two states are filtered jointly and the exact mixture filter is approximated by the Kim (1994) filter — the regime-switching counterpart of the Kalman filter.
Model taxonomy: Regimes may shift the conditional mean (switching-mean models, Hamilton 1989), the innovation variance (switching-variance / Markov heteroskedasticity), the autoregressive (AR) dynamics, or any block of these independently.
Identification and label switching: Regime labels are exchangeable, so an ordering or sign restriction (e.g. $\mu_1 < \mu_2$ , or "state 1 = high variance") is needed to pin down a unique parameterisation — exactly as in the Mixture Model.
The unidentified-nuisance-parameter problem: Under the null of a single regime, the transition probabilities are unidentified ("nuisance parameters present only under the alternative"), so the likelihood-ratio statistic does not have a standard $\chi^2$ distribution (Davies 1977; Hansen 1992).

How It Works

State-Space Form

The canonical Hamilton (1989) switching-mean autoregression writes a series as deviations from a regime-dependent mean:

$y_t - \mu_{S_t} = \sum_{k=1}^{r} \phi_k\,(y_{t-k} - \mu_{S_{t-k}}) + \varepsilon_t, \qquad \varepsilon_t \sim \mathcal{N}(0, \sigma^2)$

with $\mu_{S_t} = \mu_0 + \mu_1 S_t$ and $S_t \in \{0,1\}$ . The pair (observation equation, Markov state equation) is a state-space system; the only departure from the linear-Gaussian case is that the "state" $S_t$ is discrete.

Hamilton's Filter (Conditional Likelihood)

The filter recursively tracks the probability of each regime given data through $t$ . Starting from the chain's stationary distribution:

Prediction: $\Pr(S_t = j \mid Y_{t-1}) = \sum_{i} p_{ij}\,\Pr(S_{t-1} = i \mid Y_{t-1})$
Update after observing $y_t$ : $\Pr(S_t = j \mid Y_t) = \frac{f(y_t \mid S_t = j, Y_{t-1})\,\Pr(S_t = j \mid Y_{t-1})}{\sum_{k} f(y_t \mid S_t = k, Y_{t-1})\,\Pr(S_t = k \mid Y_{t-1})}$

The denominator is the one-step predictive density of $y_t$ , so the sample log-likelihood accumulates as

$\ell(\theta) = \sum_{t=1}^{T} \log\!\left[\sum_{j} f(y_t \mid S_t = j, Y_{t-1})\,\Pr(S_t = j \mid Y_{t-1})\right],$

which is maximised numerically to obtain the maximum likelihood estimator (MLE). This parallels the Kalman filter's prediction-error decomposition, with summation over discrete regimes replacing the Gaussian integral.

Smoothing

Full-sample regime inference $\Pr(S_t = j \mid Y_T)$ comes from a backward pass (Kim 1994 smoother), the discrete analogue of the Rauch-Tung-Striebel Kalman smoother.

Estimation Strategies

Maximum likelihood via Hamilton's filter for pure regime-switching models; the Kim filter adds a collapsing step when continuous and discrete states coexist (e.g. unobserved-components models), keeping the Gaussian-mixture count bounded.
Bayesian via Data Augmentation: treat the regime path $S_{1:T}$ as missing data and sample it inside a Gibbs Sampler. Albert-Chib (1993) draw each $S_t$ from its full conditional; Chib (1996) draws the whole path jointly via a forward-filter/backward-sample (FFBS) recursion — the discrete-state analogue of the Carter-Kohn (1994) simulation smoother — sharply reducing autocorrelation. Conditional on $S_{1:T}$ , the regime-specific parameters and the transition matrix have conjugate (Normal, Inverse-Gamma, Dirichlet/Beta) posteriors.

Why It Matters

Provides a parsimonious, interpretable way to model abrupt change — turning points, policy regimes, volatility states — without the over-parameterisation of fully time-varying-parameter models.
The filter delivers, as a by-product, smoothed regime probabilities that date events directly (e.g. recession dates matching the National Bureau of Economic Research (NBER) chronology).
Markov-switching heteroskedasticity is a known confound for classical inference: it inflates the size of unit-root tests and mimics the near-unit-root persistence of generalized autoregressive conditional heteroskedasticity (GARCH), so failing to model it leads to spurious conclusions about persistence and integration.
It is the shared engine behind a family of applied models in this wiki — the Markov-Switching VAR, the Plucking Model, the Great Moderation break analysis, and Markov-switching treatments of Purchasing Power Parity.

Open Questions

Choosing the number of regimes $m$ : testing $m$ vs. $m+1$ regimes is non-standard because of the unidentified-nuisance-parameter problem; marginal-likelihood / Bayes-factor comparison is the principled but costly alternative.
Specifying transition dynamics: fixed transition probabilities can be generalised to duration-dependent or covariate-driven (time-varying) transitions, at the cost of identification and computation.
Scaling to large systems: the number of regime-specific parameters grows quickly with the cross-sectional dimension, limiting multivariate applications.