Cointegration Analysis

What Is Cointegration?

Cointegration vs Correlation

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

Y_t = α + β * X_t + ε_t

• 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

from scipy import stats
import numpy as np
from statsmodels.tsa.stattools import adfuller

# Step 1: OLS regression
slope, intercept, _, _, _ = stats.linregress(x_prices, y_prices)
hedge_ratio = slope

# Step 2: Test residuals
residuals = y_prices - hedge_ratio * x_prices - intercept
adf_stat, p_value, _, _, crit_values, _ = adfuller(residuals, maxlag=None, autolag="AIC")

cointegrated = p_value < 0.05

2. Johansen Test

• 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

from statsmodels.tsa.vector_ar.vecm import coint_johansen

# data: T×N array of price series
result = coint_johansen(data, det_order=0, k_ar_diff=1)

# Trace statistic vs critical values (90%, 95%, 99%)
trace_stats = result.lr1          # Trace statistics
trace_crit = result.cvt           # Critical values
max_eigen_stats = result.lr2      # Max eigenvalue statistics
max_eigen_crit = result.cvm       # Critical values

# Cointegrating vectors
coint_vectors = result.evec

3. Phillips-Ouliaris

statsmodels.tsa.stattools.coint

from statsmodels.tsa.stattools import coint

# Returns: test statistic, p-value, critical values
t_stat, p_value, crit_values = coint(y_prices, x_prices)
cointegrated = p_value < 0.05

Practical Workflow

Step 1: Screen Pairs by Correlation

Step 2: Test Cointegration

Step 3: Estimate Hedge Ratio

references/methodology.md

Step 4: Compute Spread

spread = y_prices - hedge_ratio * x_prices - intercept
z_score = (spread - spread.mean()) / spread.std()

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

references/pairs_trading.md

Rolling Cointegration

Rolling Window Approach

window = 60
rolling_pvalues = []
rolling_hedges = []

for i in range(window, len(prices)):
    y_win = y_prices[i - window:i]
    x_win = x_prices[i - window:i]
    _, p_val, _ = coint(y_win, x_win)
    slope, intercept, _, _, _ = stats.linregress(x_win, y_win)
    rolling_pvalues.append(p_val)
    rolling_hedges.append(slope)

Monitoring Signals

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