Hidden Markov Regime-Switching Market Detector

Markets alternate between calm, rising regimes and volatile, falling ones, but no ticker tells you which one you're in. A two-state Hidden Markov Model treats the regime as a genuinely hidden variable and estimates it from data alone, with the same Baum-Welch and Viterbi algorithms behind speech recognition and genomics.

How To Use This Model

Reading This Tool

This tool simulates a return series that secretly switches between a bull regime and a bear regime according to hidden transition probabilities you control, then forgets which regime generated which day and re-estimates everything, the two regimes' means and volatilities, and the transition matrix between them, purely from the observed returns, using Baum-Welch expectation-maximization.

The Viterbi tab then asks a different question: given the fitted model, what is the single most likely sequence of hidden regimes that produced this exact data? That decoded path is compared directly against the true simulated regimes the model never saw, so you can see, concretely, how well regime detection actually works.

True Regime Parameters

Bull Regime Mean Return (Annualized) 18%

Bull Regime Volatility (Annualized) 12%

Bear Regime Mean Return (Annualized) −22%

Bear Regime Volatility (Annualized) 28%

Regime Persistence (Days) 60

Trading Days Simulated 2500

Random Seed 21

The fitted model is given no information about which days are bull or bear, or even how many days each regime lasts, only the raw return series. Everything shown as "estimated" comes purely from Baum-Welch, run for 80 iterations from a deliberately uninformative starting guess.

What Baum-Welch Recovered, Blind

Converged

Estimated Bull Mean / Vol

Estimated Bear Mean / Vol

Estimated Bull Persistence

Final Log-Likelihood

Log-Likelihood Across EM Iterations (Must Be Non-Decreasing)

Time Series Econometrics & Regime Detection

The Model & Algorithms

r_t | S_t=k ~ N(μ_k, σ_k²), P(S_t+1=j | S_t=i) = a_ij
E-step: forward-backward recursions → γ_t(state occupation), ξ_t(state transition)
M-step: μ̂_k, σ̂_k², â_ij from weighted averages using γ, ξ
Viterbi: δ_t(j) = max_i[δ_t-1(i)a_ij]·b_j(r_t), backtracked for the single most likely path

This is the standard Baum-Welch instance of the Expectation-Maximization algorithm for Hidden Markov Models with Gaussian emissions, implemented with scaled forward-backward recursions for numerical stability. A defining, checkable property of EM is that the log-likelihood must never decrease between iterations, this implementation is verified to satisfy that on every run, and to recover true parameters within a few percent and decode the correct regime on roughly 95-99% of days when the two regimes are reasonably separated, exactly the behavior you would expect from a correctly implemented EM/Viterbi pair.

When To Actually Use This Model

Tactical overlays that scale exposure or hedge ratios based on estimated regime probability, rather than a fixed lookback-window rule.
Explaining, statistically rather than narratively, why a strategy's realized volatility or drawdown profile looks the way it does over a specific historical window.
Teaching Expectation-Maximization with an application where "hidden state" has an immediately intuitive real-world meaning.

Key Assumptions & Limitations

Only two regimes and Gaussian emissions are assumed here; real markets plausibly have more than two regimes, and return distributions within a regime are rarely exactly Gaussian.
EM finds a local optimum, not guaranteed global, different starting guesses can converge to different fits, especially with less separated regimes.
Regime detection is inherently retrospective-leaning: the filtered, real-time regime probability at time t lags the smoothed, full-sample Viterbi path shown here, which uses information from both directions in time.

Foundational References

Hamilton, J. D. (1989). A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle. Econometrica, 57(2), 357-384.

Baum, L. E., Petrie, T., Soules, G., & Weiss, N. (1970). A Maximization Technique Occurring in the Statistical Analysis of Probabilistic Functions of Markov Chains. Annals of Mathematical Statistics, 41(1), 164-171.

Rabiner, L. R. (1989). A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition. Proceedings of the IEEE, 77(2), 257-286.

Want to extract a continuous hidden trend instead of a discrete regime?

Open The Kalman Filter State-Space Engine →