Linear VARs treat every quarter the same. Recessions aren't just low-growth quarters on the same linear trajectory as expansions. Output drops faster in downturns than it rises in booms. Volatility spikes during crises. The Phillips curve may steepen. Hamilton (1989) built a model that takes this asymmetry seriously: the economy occupies one of \(M\) discrete regimes at each date, transitions between regimes follow a first-order Markov chain, and the autoregressive dynamics are regime-specific. His original application to US real GNP growth identified two regimes - expansion and recession - with transition probabilities that closely tracked NBER business-cycle dates. This model extends the reduced-form VAR, already documented in the VAR page of this platform.

The mechanism is a hidden Markov model (HMM) applied to a time-series regression. A latent state variable \($s_t \in$ \{1, 2, $\ldots, M$\}\) governs which set of parameters is active at time \(t\). The transition between states follows \($P(s_t = j | s_{t-1} = i) = p_{ij}$\), and the collection of transition probabilities forms the \(M $\times M$\) transition matrix \(P\). Conditional on being in regime \($s_t = m$\), the VAR has regime-specific intercepts \($c^{(m)}$\), lag coefficients \($A_j^{(m)}$\), and error covariance \($\Sigma^{(m)}$\). The econometrician doesn't observe \($s_t$\) directly - it must be inferred from the data.

Inference on the latent state uses the Hamilton filter (forward recursion) and the Kim smoother (backward recursion). The Hamilton filter computes the filtered probability \($P(s_t = m | y_{1:t})$\) - the probability of being in regime \(m\) given data up to time \(t\). The Kim smoother computes the smoothed probability \($P(s_t = m | y_{1:T})$\) - the probability given the full sample. Smoothed probabilities are the standard reporting output. When the two-regime model is applied to US GDP growth, the smoothed recession probability spikes above 0.8 during every NBER recession and stays below 0.2 during expansions.

Markov-switching models are standard at central banks for recession-probability monitoring. The Federal Reserve Bank of Atlanta publishes a GDP-based recession indicator derived from a Markov-switching model. The ECB uses MS-VARs to study asymmetric monetary policy transmission across business-cycle phases. Chauvet and Hamilton (2006) maintain a monthly coincident recession indicator for the US based on a multivariate MS model using employment, income, industrial production, and sales. Academic researchers use MS-VARs for exchange-rate dynamics (Engel and Hamilton 1990), stock-market volatility regimes (Turner et al. 1989), and fiscal policy analysis under regime uncertainty.

The model consists of two interlocking pieces: a regime-specific VAR and a Markov chain governing regime transitions. The regime-specific VAR at time \(t\) is \($y_t = c^{(s_t)} + A_1^{(s_t)} y_{t-1} + \cdots + A_p^{(s_t)} y_{t-p} + u_t^{(s_t)}$\) where \(u_t^{(s_t)} \sim N(0, \Sigma^{(s_t)})\). Intercepts, lag coefficients, and error covariances can all switch, though parsimony often restricts switching to a subset (e.g., intercepts and volatility switch, lag coefficients don't).

The Markov chain is characterized by the transition matrix \(P\) with \($p_{ij} = P(s_t = j | s_{t-1} = i)$\), where rows sum to one. For a two-regime model, \(P\) has two free parameters: the probability of staying in expansion \($p_{11}$\) and the probability of staying in recession \($p_{22}$\). Expected regime durations are \(1/(1 - $p_{mm})$\): if \($p_{22} = 0.8$\), the expected recession duration is 5 quarters. The ergodic probabilities - the unconditional fraction of time spent in each regime - are derived from the left eigenvector of \(P\) associated with eigenvalue 1.

Estimation by maximum likelihood uses the EM algorithm. The E-step runs the Hamilton filter and Kim smoother to compute smoothed probabilities and regime-conditional expectations of sufficient statistics. The M-step updates the VAR parameters in each regime by weighted OLS (weighted by smoothed probabilities) and updates the transition probabilities from the smoothed transition counts. Convergence is declared when the log-likelihood increment falls below a threshold. Bayesian estimation via MCMC is an alternative: the regime path \($s_{1:T}$\) is drawn from its full conditional using the forward-filtering, backward-sampling (FFBS) algorithm, and the regime-specific parameters are drawn from their conditional posteriors.

Hamilton's (1989) original application estimated a two-regime AR(4) for US real GNP growth from 1951 to 1984. The model identified a high-growth expansion regime (mean growth ≈ 1.2% per quarter, probability of staying ≈ 0.90) and a low-growth recession regime (mean growth ≈ -0.4% per quarter, probability of staying ≈ 0.75). The smoothed recession probabilities lined up with NBER recession dates with striking accuracy, providing the first formal statistical basis for what business-cycle analysts had been doing informally. The Federal Reserve Bank of Atlanta's GDP-based recession indicator and Chauvet-Hamilton (2006) monthly coincident indicator are direct descendants of this work.

MS-VARs are used to study asymmetric transmission. Ehrmann, Ellison, and Valla (2003) estimated a Markov-switching SVAR for the US to test whether monetary policy has different effects in recessions versus expansions. Hubrich and Tetlow (2015) at the ECB used MS-VARs to study financial stress and its nonlinear impact on the real economy. The Bank of Japan uses MS models to date the transitions in and out of the deflationary regime that characterized the 1990s-2010s.

The model breaks down when the number of regimes is uncertain, when the regimes are not recurrent (a one-time structural break doesn't fit the MS framework well), or when the number of variables is large. The parameter count scales as \(M $\times$\) (VAR parameters per regime) + \(M(M-1)\) transition probabilities, which becomes unwieldy for \(M > 3\) or large VARs. For gradual structural change rather than discrete regime switches, TVP-VARs are more appropriate.

Testing for the number of regimes is non-standard because under the null of \(M=1\) (no switching), the transition probabilities are not identified, violating regularity conditions for the likelihood ratio test. Hansen (1992) derived bounds for the test, but they are conservative. Carrasco, Hu, and Ploberger (2014) proposed an optimal test. In practice, researchers often compare 2-regime and 3-regime specifications using information criteria (AIC, BIC) or Bayesian marginal likelihoods, acknowledging that formal tests are imperfect.