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 28 years of SPY (1998–2026, about 1,700 runs) 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 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 ≈ 77% of the time
- Neutral entered from Bear → resumes Bear ≈ 69%
- Neutral entered from Mean-Reverting → resumes MR ≈ 73%
Those are the 28-year pooled rates. The structure was first found on 2016–2026 alone (where Bull resumed 81% — that decade was the most persistently bullish of the sample) and then tested on the 18 earlier years it had never seen, 1998–2016, and on each half of that span separately. It held everywhere: Bull resumed 73–75%, Bear 67–69%, Mean-Reverting 71–78%, and the first-order model was rejected at the same significance in every window.
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).
- 28 years of daily data yields about 1,700 runs, but the per-path slices behind the resume rates are 170–320 events each; the probabilities still carry a few points of sampling error.
- The mix of regimes is not stable across eras even though the transition structure is: 1998–2007 was 24% Bull and 17% Mean-Reverting, 2016–2026 is 41% Bull and 9% Mean-Reverting. A pooled first-order matrix or occupancy share describes no single decade; the second-order conditional rates are the part that generalises.
- 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.