Cointegration Test: The Statistical Foundation of Pairs Trading

Cointegration identifies pairs of assets that share a stable long-run relationship — they may diverge in the short term, but revert to a stable spread over time. It is the mathematical foundation of pairs trading and statistical arbitrage, distinguishing temporary mispricing from permanent divergence.

Section 1: Core Mechanics

A cointegrated pair of assets has a property that correlation alone cannot confirm: their price spread is mean-reverting. You can subtract one from the other (with the right hedge ratio) and get a stationary series — one with a stable mean and variance over time. This is what makes pairs trading viable as a systematic strategy.

The critical distinction between correlation and cointegration: Two assets can be highly correlated (both rise in bull markets) but not cointegrated (their spread widens without reverting). Cointegrated assets share a common stochastic trend — they are mathematically linked in a way that prevents indefinite divergence.

Formula

The Johansen Test checks for cointegration among N time series. The two test variants are:

Trace Statistic:

Maximum Eigenvalue Statistic:

Where is the sample size, are the ordered eigenvalues from the reduced-rank regression, and is the number of cointegrating vectors being tested (null hypothesis: r = 0 means no cointegration).

If the test statistic exceeds the critical value at the 95% confidence level, reject the null hypothesis of no cointegration.

For a pair of two assets, the Engle-Granger two-step method is simpler: run OLS regression of Price_A on Price_B, then test the residuals for stationarity using the ADF unit root test. If residuals are stationary — cointegration exists.

Inputs

• Two (or more) price series: Must be non-stationary I(1) processes — confirm with ADF unit root test first. If the series are stationary (I(0)), cointegration is not applicable.
• Lookback window: Minimum 252 days; 2–5 years preferred for stable estimates
• Lag order for VECM: Selected using AIC/BIC criterion in the Vector Error Correction Model

Parameters

Output

• Test decision: Cointegrated (fail to reject null = no cointegration; reject null = cointegration exists)
• Hedge ratio: The slope coefficient from OLS regression of Price_A on Price_B — how many units of B to sell for each unit of A held long
• Spread series: Price_A minus hedge_ratio times Price_B — should be stationary if cointegrated
• Z-score: Standardized spread — (spread - mean) / standard deviation of spread

Visual Behavior

Plot the Z-score of the spread as a separate panel. A stationary Z-score oscillates around 0 within a stable band. Entry zones appear when the Z-score breaches ±2.0. Exit signals appear when the Z-score returns to 0.

Section 2: Interpretation & Signals

Correlation vs. Cointegration — Side by Side

Step-by-Step Pairs Trade Execution

Step 1 — Screen: Find candidate pairs with Pearson r above 0.7 over the past 252 days. This is a necessary but not sufficient screen.

Step 2 — Verify I(1): Run ADF unit root test on each price series. Both must be non-stationary (ADF fails to reject null = series has a unit root = I(1)). If either series is stationary, cointegration testing is invalid.

Step 3 — Test cointegration: Run Johansen test (or Engle-Granger) at 95% confidence. Confirm the test statistic exceeds the critical value.

Step 4 — Calculate hedge ratio: Use OLS regression: Price_A = alpha + beta times Price_B + residual. The slope beta is the hedge ratio.

Step 5 — Build the spread: Spread = Price_A minus beta times Price_B. Calculate rolling mean and rolling standard deviation of the spread.

Step 6 — Z-score: Z = (Spread minus rolling_mean) / rolling_std.

Step 7 — Trade signals:

• Z below -2.0: Long Asset A, short (beta units of) Asset B — spread expected to revert upward
• Z above +2.0: Short Asset A, long (beta units of) Asset B — spread expected to revert downward
• Z between -0.5 and +0.5: Exit position — spread has reverted to mean

Z-Score Signal Zones

Chart — Pairs Spread Z-Score (XOM vs CVX)

Section 3: Pass vs. Live — Real-Time Reliability

The Johansen test result is fixed once calculated on the historical window. The Z-score of the spread updates daily as new prices arrive. There is no repaint. The key real-time risk is relationship breakdown — not a calculation artifact.

Section 4: Practical Use Cases

Real example: XOM and CVX have been cointegrated over multiple years as the two largest US oil majors sharing the same commodity price input, capital structure, and regulatory environment. In April 2023, the Z-score of their spread dropped to -2.3, signaling XOM had lagged CVX unusually. Entering long XOM, short CVX in beta-adjusted equal dollar amounts, the spread reverted to 0 by July 2023 — a clean 3-month pairs trade with minimal market directional exposure.

Section 5: Pseudo Code

INPUT: price_a[], price_b[], confidence=0.95

PROCESS:
  Step 1: Verify both series are I(1) — non-stationary
            from statsmodels.tsa.stattools import adfuller
            adf_a = adfuller(price_a)
            adf_b = adfuller(price_b)
            # Both should FAIL to reject null (p > 0.05 = non-stationary = I(1))

  Step 2: Run Johansen cointegration test
            from statsmodels.tsa.vector_ar.vecm import coint_johansen
            result = coint_johansen(endog=[[a, b] for a, b in zip(price_a, price_b)], det_order=0, k_ar_diff=1)
            trace_stat = result.lr1[0]   # Trace statistic for r=0
            crit_value = result.cvt[0, 1]  # 95% critical value
            cointegrated = (trace_stat > crit_value)

  Step 3: Calculate hedge ratio via OLS (if cointegrated)
            import numpy as np
            hedge_ratio = np.polyfit(price_b, price_a, 1)[0]

  Step 4: Build spread and Z-score
            spread = price_a - hedge_ratio * price_b
            rolling_mean = rolling_average(spread, window=60)
            rolling_std = rolling_stdev(spread, window=60)
            z_score = (spread - rolling_mean) / rolling_std

  Step 5: Generate signals
            if z_score[-1] < -2.0: signal = "LONG SPREAD (long A, short B)"
            if z_score[-1] > +2.0: signal = "SHORT SPREAD (short A, long B)"
            if abs(z_score[-1]) < 0.5: signal = "EXIT"

OUTPUT: cointegrated (bool), hedge_ratio (float), spread[], z_score[], signal (str)
EDGE CASES:
  - If either series is I(0) (stationary): cointegration test is invalid — return error
  - If test does not reject null at 95%: try 90% or extend the historical window
  - Recalculate hedge ratio monthly — it drifts as the relationship evolves
  - Stop-loss if abs(z_score) exceeds 3.5 and has not reversed within 10 bars

Section 6: Parameters & Optimization

Johansen Test Configuration

Z-Score Window Choices

Section 7: Synergies & Conflicts

Section 8: Common Mistakes

Section 9: Cheat Sheet