Regime-Switching Vector Autoregression (MS-VAR)

Unlike a standard VAR where the autoregressive coefficient matrices are fixed, MS-VAR allows these matrices to vary across regimes. Each regime has its own set of lagged interaction parameters.

• Mechanism: The model estimates separate coefficient matrices for each hidden state, capturing how the dynamic relationships between variables change when the economy shifts from expansion to contraction.
• Example: In a low-volatility regime, the feedback from interest rates to equity returns may be weak; in a crisis regime, the same coefficient may become strongly negative and highly significant.
• Benefit: Prevents the averaging of distinct dynamic structures, avoiding misleading impulse response functions that would result from fitting a single linear model to regime-mixed data.

The unobservable regime variable follows a first-order Markov process, meaning the probability of being in a given regime at time t depends only on the regime at time t-1.

• Transition Probability Matrix: A square matrix where each entry p_ij represents the probability of moving from regime i to regime j. High diagonal values imply regime persistence.
• Expected Duration: Calculated as 1/(1-p_ii), this tells you how long the system is expected to remain in a given state. A duration of 20 quarters for an expansion regime indicates strong persistence.
• Filtered vs. Smoothed Probabilities: Filtered probabilities use only information up to time t; smoothed probabilities use the entire sample, providing the best inference of the regime at each point in time.

The variance-covariance matrix of the innovations is allowed to differ across regimes, directly modeling heteroskedasticity and volatility clustering in a multivariate setting.

• Crisis Identification: A regime with a significantly inflated covariance matrix identifies periods of market turmoil where all series exhibit elevated volatility and increased cross-correlation.
• Contemporaneous Relationships: Changes in the covariance structure reveal how the instantaneous co-movement between variables intensifies during stress—a phenomenon known as correlation breakdown.
• Structural Identification: Regime-switching in the error structure can be exploited to identify structural shocks without relying on controversial short-run or long-run restrictions, using heteroskedasticity for identification.

MS-VAR models can specify either regime-dependent intercepts or regime-dependent means, which have different implications for the dynamic adjustment path after a regime switch.

• MSI Specification: Regime-dependent intercepts cause an immediate one-time jump in the level of the series when a regime change occurs.
• MSM Specification: Regime-dependent means imply a smooth adjustment toward the new long-run mean, as the autoregressive dynamics govern the transition path.
• Practical Relevance: Choosing between MSI and MSM affects forecasts. An MSM model predicts a gradual recovery after a recession, while an MSI model implies an abrupt rebound, making the distinction critical for scenario analysis in macro stress testing.

The EM algorithm is the workhorse for estimating MS-VAR models, iteratively maximizing the likelihood in the presence of unobserved regime states.

• E-Step: Computes the smoothed probabilities of being in each regime at each time point, given the current parameter estimates. This uses the Hamilton filter, a forward-backward recursive algorithm.
• M-Step: Updates the parameter estimates—coefficient matrices, covariance matrices, and transition probabilities—using weighted least squares where the weights are the smoothed regime probabilities.
• Convergence: The algorithm alternates until the log-likelihood stabilizes. Multiple starting values are essential to avoid local maxima, a common pitfall in high-dimensional MS-VAR models with many parameters.

Standard impulse response functions trace the effect of a shock through a linear system. In MS-VAR, regime-dependent IRFs show how the propagation of a shock depends critically on the prevailing regime.

• Asymmetric Responses: A monetary policy shock may have a muted effect on output in a high-growth regime but a magnified effect in a recessionary regime, capturing the well-documented asymmetry in policy transmission.
• State-Dependent Propagation: The IRF is calculated conditional on remaining in a given regime, or generalized to account for possible regime switches during the response horizon.
• Application: Central banks use regime-dependent IRFs to assess whether interest rate changes will have the intended effect given the current phase of the business cycle, informing state-contingent forward guidance.

The stochastic engine driving regime dynamics. This matrix defines the probabilities of moving from one regime to another, quantifying the expected persistence and switching frequency of market states.

• Constant Transition Probabilities: Standard MS-VAR assumes fixed probabilities.
• Ergodic Probability: The long-run proportion of time spent in each regime, derived from the matrix.
• Expected Duration: Calculated as 1/(1-p_ii), indicating how long a regime typically lasts.
• Time-Varying Transition Probability (TVTP): An extension where probabilities depend on observable exogenous variables like the VIX or yield spreads.

The iterative optimization workhorse for calibrating MS-VAR models when the regime sequence is unobservable. It finds maximum likelihood estimates of parameters in models with latent variables.

• E-Step (Expectation): Computes the smoothed probability of being in each regime at each time point given current parameter estimates.
• M-Step (Maximization): Updates the VAR coefficients, covariance matrices, and transition probabilities to maximize the expected log-likelihood.
• Hamilton Filter: A recursive algorithm used within the E-step to calculate filtered and smoothed state probabilities.
• Convergence: Iterates until the log-likelihood improvement falls below a tolerance threshold.

Analyzes how a shock to one variable propagates through the system, conditional on the prevailing regime. This reveals fundamentally different economic dynamics in expansion versus recession.

• Asymmetric Propagation: A monetary policy shock may have a stronger effect on output in a recession than in an expansion.
• Generalized Impulse Response Functions (GIRF): Accounts for the possibility that a shock itself can trigger a regime switch.
• State-Dependent Multipliers: Quantifies how the impact of a fiscal stimulus varies across credit cycle regimes.
• Model Comparison: Contrast regime-dependent responses with a linear VAR to demonstrate the value of the switching framework.

An extension that extracts a small number of latent common factors driving a large panel of time series, where the factor loadings or dynamics shift across regimes.

• Dimensionality Reduction: Summarizes information from hundreds of economic indicators into a single 'business cycle' factor.
• Regime-Switching Loadings: The sensitivity of individual series to the common factor changes with the regime.
• Nowcasting: Uses high-frequency data to estimate the current state of the economy, accounting for regime shifts.
• Synchronization Analysis: Measures the degree of co-movement across international business cycles, identifying decoupling regimes.

Generates density forecasts that are a weighted average of regime-specific predictions, providing a full probabilistic view of future outcomes rather than a single point estimate.

• Predictive Density: A mixture distribution combining forecasts from each VAR regime weighted by predicted regime probabilities.
• Fan Charts: Visual representation of forecast uncertainty that widens as the horizon extends, often showing multi-modal distributions.
• Scenario Analysis: Construct forecasts conditional on remaining in a specific regime to stress-test portfolio outcomes.
• Turning Point Prediction: Uses the filtered probability of entering a recession regime as a leading indicator.

A probabilistic framework that incorporates prior beliefs about regime parameters and uses Markov Chain Monte Carlo (MCMC) methods to estimate the posterior distribution of states and coefficients.

• Gibbs Sampling: Iteratively draws from conditional posterior distributions for regimes, VAR parameters, and transition probabilities.
• Prior Specification: Allows the analyst to impose shrinkage or economic constraints on regime-specific dynamics.
• Model Uncertainty: Naturally handles uncertainty about the number of regimes through reversible-jump MCMC or marginal likelihood comparison.
• Multi-Move Sampling: Efficiently samples the entire regime sequence jointly rather than one period at a time.