Black Pine Lab · Methodology Vault
Wall Street gambles on averages. We trade on Stationarity. Black Pine Lab doesn't predict the future — we mathematically prove the tether.
Pair
TQQQ / XLK
Z-Score
-1.70σ
Signal
HOLD
Dual Confirm
PASS
OU Half-Life
6.24d
01 / Foundation
The Drunken Sailor Model
A drunk sailor staggers out of a bar. He wanders randomly — left, right, forward, back — with no predictable path. That's a non-stationary series. Attach him to the bar with a leash. He still staggers, but he always returns. That's stationarity: a series with a home.
We never trade the sailor (the price). We trade the leash (the spread). The leash is stationary. Mean-reversion is mathematically guaranteed.
ADF Equation
ΔYₜ = α + βt + γYₜ₋₁ + ΣδᵢΔYₜ₋ᵢ + εₜH₀: γ = 0 (unit root) | p < 0.05 → reject → stationary
KPSS opposing null: H₀ = stationary | p > 0.05 → fail to reject → stationary
TQQQ/XLK: ADF p=5.5e-05 | KPSS p=0.10 | DUAL_CONFIRMED
Dual Confirmation Gate
ADF Test
H₀: unit root present
p = 5.5e-05
REJECT → stationary
KPSS Test
H₀: series is stationary
p = 0.10
FAIL TO REJECT → stationary
Gate
Both tests agree
—
DUAL_CONFIRMED
02 / Pair Selection
The Man and the Dog
A man and his dog leave a bar. Both wander randomly — the man staggers, the dog sniffs every direction. Individually, neither is predictable. But they're connected by a leash. They may drift apart, but the leash always snaps them back together.
Correlation is a fluke. Cointegration is a contract. TQQQ and XLK are the man and the dog. Their prices diverge daily, but the mathematical bond holds over months. We exploit the divergence.
Engle-Granger Two-Step
yₜ = β·xₜ + α + εₜStep 2: test residuals ε for stationarity → if I(0), cointegrated
TQQQ/XLK: β=0.6255 | R²=0.9828 | Spread stationary: YES
Live Pair Stats
03 / Diagnosis
The Broken Compass Model
A compass with a unit root is broken. It spins freely with no magnetic north — every reading is as valid as the last, and the needle drifts wherever momentum takes it. A price series with a unit root behaves the same way: it has no equilibrium to return to.
GLD and GDX both failed this test in February 2026. Their AR(1) coefficients exceeded 1.0, meaning each day's price actively amplifies the previous day's deviation. That's not a random walk — that's an explosive process. We discarded them immediately.
AR(1) Unit Root Test
yₜ = φ·yₜ₋₁ + εₜφ = 1.0 → random walk (unit root)
φ > 1.0 → explosive process (discard immediately)
φ < 1.0 → mean-reverting (tradeable)
Explosive Process Audit — Feb 2026
GLD + GDX discarded as standalone instruments. Pair cointegration test pending before any re-inclusion.
04 / Signal Engine
The Rubber Band Velocity
Stretch a rubber band. The further you pull it, the harder it snaps back. The ECM γ coefficient is the thickness of that rubber band — it measures exactly how fast the spread corrects toward equilibrium each day.
A γ of −0.0085 means 0.85% of the deviation is corrected each day. The sign must be negative (confirming reversion, not divergence) and statistically significant (p < 0.05). Those two conditions are the entire signal gate.
ECM Equation
Δyₜ = γ·εₜ₋₁ + δ·Δxₜ + ηₜγ < 0 → mean-reverting | γ > 0 → diverging (discard)
Days to reversion ≈ −1 / γ | OU half-life = ln(2) / κ
γ = −0.0085, p = 6.5e-05 | OU κ=0.111, half-life=6.24d
ECM + OU Parameters
05 / Causality
The Shadow Model
A shadow always follows its object — it can't lead. But in markets, one asset can cast a shadow on another: its past movements contain information about the other's future. XLK is the object. TQQQ is the shadow.
Granger causality is not true causality — it is predictive precedence. If adding XLK's history to a model of TQQQ significantly improves the forecast, then XLK “Granger-causes” TQQQ. This validates the hedge direction: XLK is the anchor leg, TQQQ is the mean-reverting leg.
Granger F-Test
H₀: X does not Granger-cause YF-test on lagged X coefficients in VAR model
p < 0.05 → reject H₀ → X leads Y → hedge direction confirmed
XLK → TQQQ confirmed | Lags 1–5 tested | Long TQQQ / Short XLK
Causality Test Results
Shadow confirmed: XLK moves first. TQQQ follows. We enter when the shadow stretches too far from the object.
Pair Integrity Audit
Every candidate pair runs the full four-gate gauntlet. This is the live audit trail. Reviewed by Silas Blackpine (CIO) — 2026-02-21.
| Pair | ADF p-val | OU Half-Life | Sharpe | Verdict | Reason |
|---|---|---|---|---|---|
| TQQQ / XLK | 5.5e-05 | 6.24d | 1.688 | KEEP | Dual Confirmed — ADF+KPSS pass, OU κ=0.111, Sharpe=1.688 |
| GLD (solo) | 0.9624 | — | — | DROP | φ=1.0002 — Explosive process, no mean-reversion |
| GDX (solo) | 0.9682 | — | — | DROP | φ=1.0004 — Explosive process, no mean-reversion |
| IVV / SPY | <0.001 | — | — | WARN | Trivial pair — same index, zero statistical edge |
Source: Sprint_Review_Quant.md | Silas Blackpine | 2026-02-21 | Black Pine Lab
The Full Stack
A signal only reaches execution when all five layers confirm. This is the 20% of methodology that eliminates 80% of bad trades.
DUAL_CONFIRMED
Spread has a stable mean. Drunken sailor has a leash.
R² = 0.9828
Pair shares long-run equilibrium. Man and dog tethered.
φ < 1.0
No explosive process. Compass points home.
γ = −0.0085
Active pull to equilibrium. Rubber band confirmed.
XLK → TQQQ
Hedge direction confirmed. Shadow follows object.
Current System State
All five gates active — TQQQ/XLK under live monitoring — Z = −1.70σ
Methodology Disclosure
All statistical results shown are derived from historical data on the TQQQ/XLK pair using a 252-day OLS window and 30-day z-score normalization. Past cointegration does not guarantee future cointegration. The dual-confirmation gate, ECM gamma, and OU parameters are recomputed daily. Signals are informational only and do not constitute financial advice. Leveraged ETFs carry additional decay risk not captured by cointegration models alone. Backtest results (Sharpe 1.688, N=3 trades) are insufficient sample size for statistical inference — caveat documented per Sprint_Review_Quant.md, Silas Blackpine, 2026-02-21.