QuanTAlib/lib/statistics/cointegration/Cointegration.md

Cointegration: Engle-Granger Two-Step Cointegration Test

Correlation tells you they move together. Cointegration tells you they're bound together. Two stocks can be uncorrelated yet cointegrated, or perfectly correlated yet destined to drift apart forever. The difference between 'similar direction' and 'shared destiny' is the difference between a tourist attraction and a gravitational orbit.

Property	Value
Category	Statistic
Inputs	Two series (A, B)
Parameters	period (default 20)
Outputs	Single series (ADF statistic)
Output range	Varies (see docs)
Warmup	period + 1 bars
PineScript	cointegration.pine

• The Cointegration indicator measures the long-run equilibrium relationship between two price series using the Engle-Granger two-step method with an...
• Similar: Correlation, Granger | Trading note: Tests if two series share a long-run equilibrium. Foundation of statistical arbitrage (pairs trading).
• Validated against TradingView PineScript reference and statistical property tests.

The Cointegration indicator measures the long-run equilibrium relationship between two price series using the Engle-Granger two-step method with an Augmented Dickey-Fuller (ADF) test. Unlike correlation, which measures short-term co-movement, cointegration tests whether two non-stationary series share a common stochastic trend—meaning they may diverge temporarily but are statistically bound to revert to their equilibrium relationship.

Historical Context

Cointegration was developed by Nobel laureates Clive Granger and Robert Engle in the 1980s, fundamentally changing how economists and traders think about relationships between time series. Their work addressed a critical problem: traditional regression on non-stationary data (like stock prices) produces spurious results—apparent relationships that are statistically meaningless.

The Engle-Granger (1987) two-step method remains the most widely used approach:

1. Estimate the cointegrating regression
2. Test the residuals for stationarity using the ADF test

This implementation follows the PineScript reference implementation, adapting the algorithm for O(1) streaming updates using running sums and ring buffers.

Architecture & Physics

The Mean-Reversion Mechanism

Cointegrated series exhibit an error-correction mechanism: when they diverge from equilibrium, market forces conspire to pull them back. This differs fundamentally from correlation:

Property	Correlation	Cointegration
Measures	Direction similarity	Long-run equilibrium
Horizon	Short-term	Long-term
Stability	Can vary over time	Structural relationship
Trading implication	Momentum	Mean-reversion

1. Linear Regression Component

The first step estimates the equilibrium relationship:

A_t = \alpha + \beta \cdot B_t + \epsilon_t

Where:

• \alpha = intercept (hedge ratio offset)
• \beta = slope coefficient (hedge ratio)
• \epsilon_t = residual (spread)

The regression coefficients are derived from correlation and standard deviations:

\beta = \rho_{AB} \cdot \frac{\sigma_A}{\sigma_B} \alpha = \bar{A} - \beta \cdot \bar{B}

2. Residual Calculation

The spread (residual) represents the deviation from equilibrium:

\epsilon_t = A_t - (\alpha + \beta \cdot B_t)

For cointegrated series, this spread should be stationary (mean-reverting).

3. Augmented Dickey-Fuller Test

The ADF test checks if residuals are stationary by testing for a unit root:

\Delta\epsilon_t = \gamma \cdot \epsilon_{t-1} + u_t

Where:

• \Delta\epsilon_t = \epsilon_t - \epsilon_{t-1} (first difference)
• \gamma = coefficient indicating mean-reversion speed
• u_t = regression error

The ADF statistic is:

\text{ADF} = \frac{\gamma}{\text{SE}(\gamma)}

Where \text{SE}(\gamma) = \sqrt{\frac{\text{Var}(u)}{\text{Var}(\epsilon_{t-1})}}

4. Interpretation

ADF Statistic	Interpretation
< -3.43	Strong cointegration (1% significance)
< -2.86	Cointegration (5% significance)
< -2.57	Weak cointegration (10% significance)
> -2.57	No evidence of cointegration

More negative values indicate stronger evidence that the series share a long-run equilibrium.

Mathematical Foundation

Running Statistics for O(1) Updates

This implementation maintains running sums for efficient streaming computation:

Means:

\bar{A} = \frac{\sum A_i}{n}, \quad \bar{B} = \frac{\sum B_i}{n}

Variances:

\sigma_A^2 = \frac{\sum A_i^2}{n} - \bar{A}^2, \quad \sigma_B^2 = \frac{\sum B_i^2}{n} - \bar{B}^2

Covariance:

\text{Cov}(A, B) = \frac{\sum A_i B_i}{n} - \bar{A} \cdot \bar{B}

Correlation:

\rho_{AB} = \frac{\text{Cov}(A, B)}{\sigma_A \cdot \sigma_B}

ADF Regression Statistics

The gamma coefficient is computed using running sums over period-1 observations:

\gamma = \frac{\text{Cov}(\Delta\epsilon, \epsilon_{t-1})}{\text{Var}(\epsilon_{t-1})}

Standard Error:

\text{SE}(\gamma)^2 = \frac{\sum(u_t)^2 / n}{\text{Var}(\epsilon_{t-1})}

where u_t = \Delta\epsilon_t - \gamma \cdot \epsilon_{t-1}

Performance Profile

Operation Count (Streaming Mode, Scalar)

Operation	Count	Cost (cycles)	Subtotal
ADD/SUB	25	1	25
MUL	12	3	36
DIV	8	15	120
SQRT	3	15	45
Buffer Access	8	3	24
FMA	8	4	32
Total	64	—	~282 cycles

Division and square root operations dominate the cost profile.

Memory Footprint

Component	Size
Main buffers (2× period)	16 × period bytes
ADF buffers (2× period-1)	16 × (period-1) bytes
Running sums	80 bytes
State variables	64 bytes
Total per instance	~32 × period + 144 bytes

For period=20: ~784 bytes per indicator instance.

Quality Metrics

Metric	Score	Notes
Accuracy	9/10	Matches Engle-Granger methodology
Timeliness	6/10	Requires full period for stable estimates
Robustness	8/10	Handles edge cases (NaN, zero variance)
Interpretability	7/10	Requires understanding critical values

Validation

Library	Status	Notes
TA-Lib	N/A	No cointegration implementation
Skender	N/A	No cointegration implementation
Tulip	N/A	No cointegration implementation
Ooples	N/A	No cointegration implementation
TradingView	✅	Matches PineScript reference implementation
Statistical	✅	Validated against expected properties

Note: Cointegration is typically found in econometrics packages (statsmodels, R's urca) rather than TA libraries. This implementation focuses on streaming computation suitable for real-time trading.

Use Cases

1. Pairs Trading

Identify cointegrated pairs for mean-reversion strategies:

• Entry: When spread deviates significantly from mean
• Exit: When spread reverts to equilibrium
• Stop: When cointegration breaks down

2. Statistical Arbitrage

Build market-neutral portfolios using cointegrated baskets:

• Long undervalued leg, short overvalued leg
• Position sizing based on hedge ratio (β)

3. Risk Management

Monitor cointegration stability:

• Degrading ADF statistics signal relationship breakdown
• Adjust positions before pairs diverge permanently

4. Index Tracking

Construct synthetic indices from cointegrated components:

• Track expensive ETFs with cheaper alternatives
• Exploit tracking errors

API Usage

Streaming Mode (Bi-Input)

var coint = new Cointegration(period: 20);
foreach (var (priceA, priceB) in pricePairs)
{
    var result = coint.Update(priceA, priceB);
    if (coint.IsHot && result.Value < -2.86)
    {
        Console.WriteLine($"Cointegrated at 5% level: ADF = {result.Value:F2}");
    }
}

Batch Mode

var seriesA = new TSeries();
var seriesB = new TSeries();
// ... populate series ...
var results = Cointegration.Calculate(seriesA, seriesB, period: 20);

Span Mode (Zero Allocation)

double[] pricesA = new double[1000];
double[] pricesB = new double[1000];
double[] output = new double[1000];
// ... populate inputs ...
Cointegration.Batch(pricesA.AsSpan(), pricesB.AsSpan(), output.AsSpan(), period: 20);

Bar Correction Support

var coint = new Cointegration(20);

// New bar
coint.Update(100.0, 50.0, isNew: true);  // ADF = -2.5

// Same bar corrected (e.g., real-time tick update)
coint.Update(101.0, 51.0, isNew: false); // Recalculates without advancing state

Common Pitfalls

1. Confusing Correlation with Cointegration: High correlation does not imply cointegration. Two trending stocks can be 99% correlated but not cointegrated (spurious regression). Conversely, mean-reverting pairs may have low correlation but strong cointegration.

2. Warmup Period: The indicator requires period + 1 bars before producing valid results. During warmup, IsHot returns false and results may be NaN.

3. Critical Values: ADF critical values are approximate: -3.43 (1%), -2.86 (5%), -2.57 (10%). These differ from standard t-distribution values due to the unit root null hypothesis.

4. Zero-Variance Edge Cases: Perfectly linear relationships (A = β×B + α with no noise) produce zero-variance residuals, resulting in NaN. This is mathematically correct—perfect cointegration has no estimation uncertainty.

5. Non-Stationarity Requirement: Both input series should be integrated of order 1 (I(1))—non-stationary but with stationary first differences. Applying cointegration to already-stationary series is meaningless.

6. Period Selection: Short periods (10-20) respond faster but may produce unstable estimates. Longer periods (50-100) are more stable but slower to adapt. Consider the expected holding period for your trading strategy.

7. Structural Breaks: Cointegration can break down due to fundamental changes (mergers, regulatory shifts, market regime changes). Monitor ADF statistics over time and be prepared to exit when the relationship deteriorates.

8. Memory per Instance: Each indicator instance allocates ~32×period bytes for buffers. For scanning many pairs, consider batch processing or pooling.

When to Use Cointegration

Use it when:

• Building pairs trading or statistical arbitrage strategies
• Identifying mean-reversion opportunities across related instruments
• Validating hedge ratios for portfolio construction
• Monitoring relationship stability over time

Skip it when:

• Series are already stationary (use correlation instead)
• Looking for momentum/trend signals
• Short-term (intraday) trading where co-movement matters more than equilibrium
• One-off analysis where econometrics packages (statsmodels) are more appropriate

References

• Engle, R.F. and Granger, C.W.J. (1987). "Co-integration and Error Correction: Representation, Estimation, and Testing." Econometrica, 55(2), 251-276.
• Dickey, D.A. and Fuller, W.A. (1979). "Distribution of the Estimators for Autoregressive Time Series with a Unit Root." Journal of the American Statistical Association, 74(366), 427-431.
• TradingView. "Cointegration Indicator (PineScript)." TradingView Community Scripts.
• Vidyamurthy, G. (2004). "Pairs Trading: Quantitative Methods and Analysis." Wiley Finance.