Hidden Markov Model (HMM)

An HMM consists of two coupled stochastic layers: a hidden Markov chain governing regime transitions and an observable emission process generating returns conditional on the current regime. The hidden state sequence (S_t) evolves according to a first-order Markov property—the probability of being in a regime depends only on the immediately preceding state. Observed returns (R_t) are drawn from a state-dependent distribution, typically Gaussian with regime-specific mean (\mu_k) and variance (\sigma^2_k). This separation allows the model to capture distinct return-generating mechanisms for bull, bear, and sideways markets without requiring explicit regime labels.

The transition matrix (A = [a_{ij}]) encodes the stochastic dynamics of regime persistence and switching, where (a_{ij} = P(S_t = j \mid S_{t-1} = i)). Diagonal elements represent regime persistence—high values indicate stable, long-duration regimes typical of secular bull markets. Off-diagonal elements capture switching probabilities, which spike during market inflection points. Key properties:

• Each row sums to 1, forming a valid probability distribution
• The ergodic probability vector (\pi) satisfies (\pi A = \pi), giving the long-run fraction of time spent in each regime
• Expected regime duration for state (i) equals (1/(1 - a_{ii})), enabling quantification of regime stability

HMMs are defined by three fundamental computational tasks, each solved by a specific algorithm:

1. Likelihood Evaluation (Forward Algorithm) Given model parameters (\lambda = (A, \mu, \sigma)) and an observation sequence, compute (P(O \mid \lambda))—the probability the model generated the data. Used for model selection and comparing competing regime specifications.

2. State Decoding (Viterbi Algorithm) Find the single most probable sequence of hidden regimes (\hat{S} = \arg\max P(S \mid O, \lambda)). This produces a hard assignment of each time point to a specific market regime, enabling regime-conditional strategy backtesting.

3. Parameter Learning (Baum-Welch Algorithm) Given only observations, estimate (\lambda) that maximizes (P(O \mid \lambda)). This Expectation-Maximization variant iteratively refines parameter estimates without requiring labeled training data.

Each hidden state (k) is associated with an emission distribution (b_k(o_t) = P(O_t = o_t \mid S_t = k)) that characterizes the statistical behavior of returns within that regime. Common specifications include:

• Gaussian emissions: (R_t \mid S_t=k \sim \mathcal{N}(\mu_k, \sigma^2_k)), capturing regime-specific mean returns and volatility
• Student-t emissions: Heavy-tailed distributions for regimes with frequent outlier returns, such as crisis states
• Mixture emissions: Multi-modal distributions within a single regime for complex intra-regime dynamics

This structure directly quantifies the risk-return profile of each market environment, enabling dynamic position sizing based on the inferred regime.

While the Viterbi algorithm provides a single best path, the Forward-Backward algorithm computes the posterior probability (P(S_t = k \mid O_{1:T}, \lambda)) that the market was in regime (k) at time (t) given the entire observation sequence. This produces smoothed regime probabilities—a continuous signal between 0 and 1 for each state at every time point. Key advantages:

• Incorporates future information, not just past observations
• Provides uncertainty quantification around regime assignments
• Enables soft regime weighting in portfolio allocation rather than binary switching
• Reveals transition periods where the model is uncertain between two regimes

Selecting the optimal number of hidden states requires balancing fit against complexity. Standard approaches include:

• Akaike Information Criterion (AIC): Penalizes log-likelihood by number of parameters, (AIC = -2\log L + 2k)
• Bayesian Information Criterion (BIC): Applies a stronger penalty, (BIC = -2\log L + k\log N), favoring parsimony
• Cross-validated log-likelihood: Evaluates out-of-sample predictive performance on held-out periods
• Regime interpretability: States must map to economically meaningful market conditions (e.g., low-volatility bull, high-volatility bear, mean-reverting sideways)

Overfitting manifests as spurious short-duration regimes with near-zero persistence probabilities.