Cointegration

The core property of cointegration is that while individual asset price series are non-stationary (containing unit roots), a specific linear combination of them is stationary. This means the spread between the assets has a constant mean and variance over time, reverting to equilibrium rather than wandering randomly. Formally, if two I(1) series X and Y are cointegrated, there exists a coefficient β such that X - βY is I(0). This stationary spread is the tradable signal in pairs trading.

Cointegration implies a long-run equilibrium relationship between the assets. Economic forces or shared risk factors bind the series together, preventing them from diverging indefinitely. When the spread deviates from this equilibrium, it is expected to revert, creating a trading opportunity. This contrasts with correlation, which measures short-term directional co-movement but says nothing about long-run convergence. Two assets can be highly correlated yet not cointegrated if their prices drift apart over time.

The Engle-Granger test is a classic method for detecting cointegration:

• Step 1: Regress one asset price on the other using OLS to estimate the cointegrating vector: Y = α + βX + ε
• Step 2: Test the residuals ε for a unit root using the Augmented Dickey-Fuller (ADF) test. If the residuals are stationary, the series are cointegrated. This method is simple but assumes one asset is the dependent variable, which can produce different results depending on the ordering.

The Johansen test extends cointegration analysis to multiple time series simultaneously, overcoming the limitations of the Engle-Granger approach. It uses a Vector Error Correction Model (VECM) framework to identify the number of cointegrating relationships (the cointegrating rank) in a system of n variables. The test provides two statistics:

• Trace statistic: Tests the null of r or fewer cointegrating vectors
• Maximum eigenvalue statistic: Tests the null of exactly r cointegrating vectors against r+1 This method is symmetric and does not require choosing a dependent variable.

Cointegrated systems exhibit an error correction mechanism: short-term deviations from the long-run equilibrium are corrected over time. The ECM model captures this dynamic by including the lagged residual (the error correction term) as a regressor:

• ΔY_t = α(Y_{t-1} - βX_{t-1}) + γΔX_t + ε_t The coefficient α measures the speed of adjustment back to equilibrium. A negative and significant α confirms that the dependent variable adjusts to correct disequilibrium. This framework separates long-run equilibrium dynamics from short-run fluctuations.

The half-life of mean reversion quantifies how quickly a cointegrated spread reverts to its mean after a deviation. It is calculated from the error correction coefficient α as:

• Half-Life = -ln(2) / ln(1 + α) A shorter half-life indicates faster reversion and is generally more desirable for trading, as it implies more frequent round-trip trades. However, very short half-lives may indicate noise rather than a genuine tradable signal. Typical half-lives for viable pairs range from a few days to several weeks.

The estimated time it takes for a cointegrated spread to revert halfway back to its long-term equilibrium mean after a deviation. This metric directly dictates the optimal trading horizon.

• Calculation: Derived from an AR(1) model fitted to the spread: Half-Life = -ln(2) / ln(θ), where θ is the autoregressive coefficient.
• Trading Implication: A half-life of 3 days suggests a short-term mean-reversion strategy; a half-life of 30 days suggests a slower, longer-horizon approach.
• Practical Use: If the half-life is too long (e.g., >100 days), the strategy ties up capital inefficiently. If too short (e.g., <1 hour), transaction costs may erode profits.

A time series is stationary if its statistical properties—mean, variance, and autocorrelation—are constant over time. Cointegration requires the spread to be stationary, even if the individual assets are not.

• Augmented Dickey-Fuller (ADF) Test: The primary statistical test for stationarity. A p-value < 0.05 rejects the null hypothesis of a unit root (non-stationarity).
• Why It Matters: Trading a non-stationary spread is dangerous; it has no equilibrium to revert to and can wander arbitrarily far, leading to unbounded losses.
• Common Pitfall: Structural breaks in the relationship can cause a previously stationary spread to become non-stationary, invalidating the strategy.

A multivariate extension of cointegration testing that can identify multiple cointegrating relationships within a basket of more than two assets, unlike the two-step Engle-Granger method.

• Trace Statistic: Tests the null hypothesis of r or fewer cointegrating vectors against the alternative of more than r.
• Maximum Eigenvalue Statistic: Tests the null of r cointegrating vectors against the specific alternative of r+1.
• Portfolio Application: Essential for constructing statistical arbitrage baskets of 3+ stocks, where multiple equilibrium relationships may exist simultaneously.

The foundational methodology for testing cointegration between two time series. Simple to implement but limited to identifying a single cointegrating relationship.

• Step 1: Regress one asset's price on the other using OLS to estimate the long-run equilibrium relationship: Y_t = α + βX_t + ε_t.
• Step 2: Test the residuals (ε_t) for stationarity using the ADF test. If stationary, the series are cointegrated.
• Limitation: The test result can depend on which asset is chosen as the dependent variable, and it cannot handle more than one cointegrating vector.

The most direct application of cointegration: a market-neutral strategy that simultaneously buys the underperforming asset and sells short the overperforming asset in a cointegrated pair.

• Entry Signal: When the spread widens beyond a threshold (e.g., 2 standard deviations from the mean), short the winner and go long the loser.
• Exit Signal: When the spread reverts back to its mean or crosses to the opposite threshold.
• Risk: The relationship can permanently break down due to a merger, regulatory change, or fundamental business shift, causing the spread to diverge without reverting.

The Z-Score normalizes the cointegration spread to a standard scale, measuring how many standard deviations the current spread is from its historical mean. This is the core signal generator.

• Formula: Z = (Spread_t - μ_spread) / σ_spread
• Bollinger Bands: A visualization tool that plots a moving average of the spread with upper and lower bands at ±k standard deviations. A breach of the bands triggers a trade.
• Calibration: A Z-score threshold of ±2.0 is common, but optimal thresholds should be determined via walk-forward optimization to avoid overfitting.