Cointegration 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

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:

Y_t = α + β * X_t + ε_t

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 YX can give a different result than XY. Always test both directions and use the stronger result.

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

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

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

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.

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

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

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

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.