MS-GARCH

MS-GARCH combines two unobserved stochastic processes: a Markov chain governing discrete regime transitions and a GARCH process modeling conditional variance within each regime. This dual structure allows volatility to exhibit both abrupt shifts and gradual decay.

• The Markov chain captures sudden market reclassifications (e.g., calm to turbulent)
• The GARCH component preserves volatility clustering and persistence within each state
• Unlike standard GARCH, the model does not assume a single, stable variance process

A critical implementation challenge: the conditional variance at time t depends on the entire history of regimes, not just the current state. This path-dependence arises because the GARCH recursion uses lagged variances that were themselves generated under previous regimes.

• The variance at any point is a function of all past regime-specific parameters
• Standard filtering requires integrating over all possible regime paths
• Gray's (1996) collapsing procedure aggregates path-dependent variances using conditional regime probabilities to maintain tractability

Each regime possesses its own set of GARCH parameters, allowing fundamentally different volatility behavior across states. A low-volatility regime may exhibit high persistence, while a crisis regime shows explosive short-term reactions.

• Regime 1 (Bull): Low baseline volatility, moderate persistence, symmetric news impact
• Regime 2 (Bear/Crisis): Elevated baseline volatility, high reactivity to shocks, potential leverage effects
• Parameters estimated simultaneously via maximum likelihood with the EM algorithm or Bayesian MCMC

MS-GARCH outputs filtered and smoothed probability sequences for each regime at every time point. Smoothed probabilities use the full sample information to retrospectively classify regimes, providing the most accurate state identification.

• Filtered probabilities: real-time inference using only past and current data
• Smoothed probabilities: refined estimates using future observations (via the Kim smoother)
• Traders use these probabilities to dynamically adjust position sizing and hedging ratios

Multi-step volatility forecasts integrate uncertainty over future regime paths. The model produces a probability-weighted average of forecasts from each possible regime sequence, rather than a single deterministic projection.

• k-step ahead forecast weights all 2^m possible regime paths by their transition probabilities
• Captures the risk of regime shifts occurring within the forecast horizon
• Produces fatter-tailed predictive distributions than single-regime GARCH, better matching observed market behavior during transitions

MS-GARCH can incorporate asymmetric volatility responses within each regime through specifications like the GJR-GARCH or EGARCH variants. This allows the model to capture the stylized fact that negative returns increase future volatility more than positive returns of equal magnitude.

• Asymmetry parameters can differ across regimes
• Bear markets often exhibit stronger leverage effects than bull markets
• Enables precise modeling of volatility feedback loops during market downturns

The foundational framework where time-series parameters switch between a finite number of regimes governed by an unobservable Markov chain. Unlike MS-GARCH, the basic Markov switching model typically assumes constant variance within regimes, whereas MS-GARCH extends this by allowing autoregressive conditional heteroskedasticity to persist within each state.

• Captures structural breaks like bull/bear transitions
• Regime changes follow a first-order Markov process
• Parameters estimated via maximum likelihood or Bayesian methods

A stochastic matrix that defines the probabilities of moving between volatility regimes in an MS-GARCH framework. Each element P(i,j) represents the probability of transitioning from regime i to regime j, directly controlling the expected duration and persistence of high and low volatility states.

• Diagonal elements determine regime stickiness
• Ergodic probabilities derived from the matrix give long-run state frequencies
• Time-varying variants allow transition probabilities to depend on exogenous variables like the VIX

The empirical phenomenon where large price changes tend to follow large changes and small changes follow small changes. This stylized fact motivates the GARCH component within MS-GARCH, as volatility exhibits persistence within a regime before a structural shift occurs.

• Measured by significant autocorrelation in squared returns
• GARCH(1,1) captures within-regime clustering
• Regime switching explains sudden shifts in volatility levels that simple GARCH cannot

An iterative optimization method essential for calibrating MS-GARCH models when the regime sequence is unobserved. The algorithm alternates between computing expected state probabilities given current parameter estimates and maximizing the likelihood to update parameters.

• E-step: Infer regime probabilities using forward-backward smoothing
• M-step: Update GARCH parameters for each regime
• Handles the latent variable problem inherent in regime-switching models

A tail-risk measure that calculates expected losses conditional on the current market regime identified by an MS-GARCH model. Unlike static VaR, this approach acknowledges that risk distributions differ fundamentally between low-volatility trending states and high-volatility turbulent states.

• Provides state-dependent capital reserves
• Prevents underestimation of risk during crisis regimes
• Integrates volatility forecasting with regime awareness for stress testing

An alternative to MS-GARCH where volatility follows a latent stochastic process rather than a deterministic GARCH recursion, with parameters switching across regimes. While MS-GARCH models volatility as conditionally deterministic given past shocks, regime-switching SV treats volatility as having its own random innovations.

• Captures both within-regime stochastic variation and across-regime shifts
• Estimated via MCMC or particle filters
• More flexible but computationally intensive than MS-GARCH