Regime Detection and Switching AI Prompt

Detect and model market regime switches in this financial time series.

Time series: {{time_series}}
Regime definition goal: {{goal}} (vol regimes, trend/mean-reversion, risk-on/risk-off, etc.)

1. Hidden Markov Model (HMM) regime detection:

   Specification for 2-state HMM:
   - State S_t ∈ {1, 2} (hidden, not directly observable)
   - Emission distribution: R_t | S_t = k ~ N(μ_k, σ²_k)
   - Transition matrix: P = [[p_{11}, p_{12}], [p_{21}, p_{22}]]
     p_{11} = P(stay in state 1 | currently in state 1)

   Estimation via EM algorithm (Baum-Welch):
   - Report: μ_1, σ_1, μ_2, σ_2, transition matrix P
   - Viterbi algorithm: most likely state sequence
   - Smoothed probabilities: P(S_t = k | all data) — softer than Viterbi

   Model selection:
   - 2 vs 3 states: compare BIC or AIC
   - 2-state typically: bull (high μ, low σ) and bear (low/negative μ, high σ)
   - 3-state may add: transition (moderate μ, rising σ)

2. Markov-Switching Regression:
   R_t = μ_{S_t} + φ_{S_t} R_{t-1} + ε_{S_t}
   - Different AR(1) coefficient in each regime
   - Captures mean-reversion in some regimes and momentum in others
   - Hamilton (1989) filter for real-time regime probability

3. Threshold and SETAR models:
   SETAR (Self-Exciting Threshold AR):
   - Regime is determined by whether a variable exceeds a threshold
   - R_t = α_1 + φ_1 R_{t-1} + ε_t if R_{t-d} ≤ threshold (regime 1)
   - R_t = α_2 + φ_2 R_{t-1} + ε_t if R_{t-d} > threshold (regime 2)
   - Self-exciting: the lagged return itself determines the regime
   - Suptest (Andrews) for the threshold location

4. Practical regime classification:
   Simple rule-based approach for transparency:
   - Regime = function of: trailing volatility, VIX level, credit spreads, or trend indicator
   - Pros: interpretable, auditable, does not require re-estimation
   - Cons: less statistically rigorous than HMM

5. Using regimes for portfolio management:
   - Regime-conditional strategy performance: does the alpha strategy perform differently across regimes?
   - Regime-conditional asset allocation: what is the historically optimal allocation in each regime?
   - Real-time regime probabilities: current P(S_t = bear) — use as a risk aversion dial
   - Transition probability: P(bear next month | bull this month) — forward-looking risk indicator

Return: HMM parameter estimates, smoothed regime probabilities, regime-conditional statistics, regime-conditional strategy performance, and current regime assessment.