Cointegration testing for pairs trading using Engle-Granger, Johansen, and rolling stability analysis

Cointegration Analysis Cointegration testing identifies pairs of assets that share a long run equilibrium relationship, enabling statistical arbitrage and pairs trading strategies. What Is Cointegration? Two price series are cointegrated when they are individually non stationary (random walks) but a linear combination of them is stationary (mean reverting). Intuitively, the prices may wander apart temporarily but are pulled back to an equilibrium spread over time. Cointegration vs Correlation Property Correlation Cointegration Measures Short term co movement Long run equilibrium Stationarity Requires stationary returns Works with non stationary prices Time horizon Can change rapidly Stable over months/years Trading use Momentum/trend signals Mean reversion pairs trades Failure mode Breaks in regime changes Breaks on structural shifts Two assets can be highly correlated but not cointegrated (e.g., two unrelated uptrends). Conversely, cointegrated assets may have low short term correlation during temporary divergences — which is exactly when pairs trades are entered. Why It Matters Pairs trading : Long the underperformer, short the outperformer, profit on convergence Statistical arbitrage : Systematic mean reversion on spread z scores Spread trading : Trade the spread directly as a synthetic instrument Risk hedging : Cointegrated hedge ratios minimize tracking error over time Methods 1. Engle Granger Two Step The most common approach for two series. Step 1 — Regress Y on X using OLS: Step 2 — Test the residuals ε t for stationarity using the ADF test. If residuals are stationary (p < 0.05) → Y and X are cointegrated β is the hedge ratio for the pairs trade α is the long run mean of the spread Important : Engle Granger critical values differ from standard ADF critical values. For n=2 series: 1% = 3.90, 5% = 3.34, 10% = 3.04. Asymmetry warning : Testing Y~X can give a different result than X~Y. Always test both directions and use the stronger result. 2. Johansen Test Tests multiple series simultaneously and returns the number of cointegrating relationships. More powerful than Engle Granger for 2 series. Based on a VAR model: ΔY t = Π·Y {t 1} + Σ Γ i·ΔY {t i} + ε t Tests the rank of the Π matrix Uses trace test and maximum eigenvalue test Returns: number of cointegrating vectors and the vectors themselves 3. Phillips Ouliaris Similar to Engle Granger but uses Phillips Perron style test statistics instead of ADF. More robust to heteroskedasticity and serial correlation in the residuals. Available via statsmodels.tsa.stattools.coint . Practical Workflow Step 1: Screen Pairs by Correlation Pre filter using Pearson correlation 0.7 to reduce the number of cointegration tests (which are more expensive). Step 2: Test Cointegration Run Engle Granger in both directions. Use p < 0.05 threshold. Step 3: Estimate Hedge Ratio Use OLS for simplicity. For production, consider Total Least Squares or Dynamic OLS (see references/methodology.md ). Step 4: Compute Spread Step 5: Test Spread for Mean Reversion ADF test : p < 0.05 confirms stationarity Hurst exponent : H < 0.5 indicates mean reversion (H ≈ 0.5 = random walk) Half life : λ from AR(1) on spread; half life = ln(2)/ln(λ) Viable pairs: half life between 5 and 60 days Step 6: Trade the Spread If the spread is mean reverting, it is a viable pairs trade candidate. See references/pairs trading.md for entry/exit rules and risk management. Rolling Cointegration Cointegration relationships can break down over time due to structural changes, regime shifts, or evolving market dynamics. Rolling Window Approach Test cointegration on rolling 60–90 day windows: Monitoring Signals Signal Healthy Warning Stop Trading Rolling p value < 0.05 0.05–0.10 0.10 Hedge ratio drift < 10% change 10–25% change 25% change Spread half life 5–60 days 60–120 days 120 days or < 5 Crypto Pairs Candidates Layer 1 Correlation SOL vs ETH — L1 sector beta, often cointegrated during trending markets SOL vs AVAX — alternative L1 correlation Stablecoins USDC vs USDT — should be perfectly cointegrated (peg arbitrage) Useful as a sanity check for your cointegration pipeline Liquid Staking Derivatives mSOL vs jitoSOL — both track SOL staking yield stSOL vs mSOL — Lido vs Marinade staking Same Sector Tokens DEX tokens: RAY vs ORCA Lending tokens: cross protocol comparison Meme tokens: rarely cointegrated, high risk Common Pitfalls 1. Spurious cointegration — Two trending series (both up in a bull market) may appear cointegrated. Always test on sufficient data ( 200 observations) and check out of sample stability. 2. Structural breaks — A fundamental change (protocol upgrade, tokenomics change) can permanently break cointegration. Monitor rolling p values. 3. Look ahead bias — Estimating the hedge ratio on the full sample and then backtesting on the same sample inflates results. Always use walk forward estimation. 4. Too short sample — Cointegration tests need 100 observations minimum, ideally 200, to have reasonable power. 5. Ignoring transaction costs — Pairs trades involve 4 transactions per round trip. At 0.3% per leg, that is 1.2% in costs that the spread must overcome. 6. Asymmetric cointegration — The relationship may only hold in one direction or one regime. Consider threshold cointegration models for production use. Integration with Other Skills correlation analysis — Pre screening pairs by correlation before cointegration testing mean reversion — Trading the cointegrated spread using mean reversion entry/exit rules vectorbt — Backtesting pairs strategies with walk forward validation regime detection — Identifying when cointegration regimes shift volatility modeling — Spread volatility forecasting for dynamic position sizing Files References references/methodology.md — Engle Granger details, Johansen derivation, hedge ratio estimation methods, spread construction references/pairs trading.md — Entry/exit rules, risk management, performance metrics, crypto specific considerations Scripts scripts/test cointegration.py — Full cointegration test pipeline with ADF, Hurst, half life, rolling stability, and demo mode scripts/pairs backtest.py — Walk forward pairs trading backtest with synthetic data and performance reporting

cointegration-analysis