Setting up the chain
The regimes come from two measurable properties: a rolling Hurst exponent (trending vs mean-reverting vs random) and a trend filter (price vs its 50-day average). That yields four states: Bullish Trend, Bearish Trend, Random/Neutral, and Mean-Reverting. Because regimes persist for days or weeks, the day-to-day sequence is dominated by self-transitions, which tell you nothing about structure. The interesting object is the jump chain: collapse each run of identical labels into one event and ask only, when a regime ends, what comes next?
The first-order matrix — and its trap
Counting run-boundary transitions over ten years of SPY gives a familiar first-order Markov matrix: when Bull ends it usually goes Neutral (~78%); Mean-Reverting essentially always exits through Neutral; Neutral splits toward Bull about half the time. Useful — but the first-order framing smuggles in an assumption: that the transition odds depend only on the current state, not on how you got there.
Testing the assumption
That assumption is testable. Condition on the current regime and ask whether the previous regime predicts the next one — a likelihood-ratio (G²) test of conditional independence, with a permutation test to keep the small run-count honest. On ten years of SPY jump-chain data the answer is unambiguous: first-order Markov is rejected (permutation p ≈ 0.0002), and nearly all of the signal concentrates in one place — the Neutral state.
Neutral is a pause, not a hub
Decompose what follows Neutral by what preceded it:
- Neutral entered from Bull → resumes Bull ≈ 81% of the time
- Neutral entered from Bear → resumes Bear ≈ 72%
- Neutral entered from Mean-Reverting → resumes MR ≈ 66%
The blended first-order row (“Neutral → Bull 52%”) is therefore wrong for every actual path — it averages three very different conditional distributions into one misleading number. The structural read: Neutral is a breather inside a persistent regime, not a memoryless junction that re-rolls the dice. When the market drifts into the random band, the base case is that the prior regime resumes.
Caveats
- Regime labels inherit the estimator's noise — different Hurst windows or thresholds shift the counts (though not the qualitative finding).
- A decade of daily data yields hundreds of runs, but the per-path slices are dozens, not thousands; the probabilities carry real sampling error.
- This describes label dynamics, not returns. “Bear tends to resume” is a statement about classification persistence — position sizing off it still requires a P&L-level test.