Regime-Switching Covariance

The statistical workhorse behind regime-switching covariance. An HMM assumes the market occupies one of N discrete latent states at any time, with transitions governed by a stochastic transition matrix. The model simultaneously infers the probability of being in each regime and the state-conditional covariance matrix. Key components:

• Emission probabilities: Asset returns given the current regime
• Transition probabilities: Likelihood of shifting from bull to bear or crisis
• Baum-Welch algorithm: Expectation-maximization for parameter estimation
• Viterbi decoding: Reconstructing the most likely historical regime path

An extension that combines regime-switching dynamics with time-varying volatility clustering. Unlike standard HMMs with constant state-conditional variance, MS-GARCH allows volatility within each regime to evolve via a GARCH(1,1) process. This captures both:

• Abrupt shifts: Sudden transitions to high-volatility crisis states
• Persistence effects: Volatility clustering within a given regime
• Leverage effects: Asymmetric responses to positive vs. negative shocks Critical for risk parity because it prevents underestimation of tail risk during turbulent periods.

A two-stage estimation framework for time-varying correlations that pairs naturally with regime-switching volatility models. Stage one fits univariate GARCH models to each asset. Stage two estimates a dynamic correlation matrix using standardized residuals. When combined with regime detection:

• Regime-conditional DCC: Correlations estimated separately per regime
• Smooth-transition DCC: Correlations evolve continuously with regime probabilities
• Asymmetric DCC: Captures the well-documented phenomenon that correlations spike during market crashes This addresses the core failure of static risk parity during crises.

A regularization technique that stabilizes regime-conditional covariance estimates, especially when regime samples are sparse. Shrinkage blends the sample covariance matrix with a structured target matrix (e.g., constant correlation, single-factor, or identity). The shrinkage intensity can be:

• Regime-specific: Higher shrinkage in rare crisis states with few observations
• Time-varying: Adjusted based on the effective sample size per regime
• Bayesian: Incorporating prior beliefs about covariance structure Without shrinkage, regime-switching models overfit noise in short-lived states, producing unstable risk parity weights.

The practical application layer where regime-switching covariance meets portfolio construction. Rather than using a single risk parity allocation, the portfolio's risk budget per asset shifts based on the inferred regime probability:

• Bull regime: Higher risk budget to equities and credit
• Bear regime: Shift risk budget toward government bonds and gold
• Crisis regime: Reduce overall leverage and allocate to safe-haven currencies Implementation requires solving a convex optimization problem with regime-weighted covariance: Σ_weighted = Σ p(regime_i) × Σ_regime_i

The observable market features used as inputs to infer the current regime. These go beyond simple return data to capture multidimensional market states:

• VIX futures term structure: Contango vs. backwardation signals stress
• Credit spreads: High-yield OAS widening indicates credit regime shift
• Cross-asset correlations: Correlation breakdown signals regime change
• Treasury yield curve slope: Inversion predicts recessionary regime
• Currency volatility: FX vol spikes flag global risk-off events
• Liquidity measures: Bid-ask spread widening indicates market stress These features serve as the emission variables in the HMM framework.