cointegration-analysis

Files (5)

• SKILL.md
• methodology.md · references/
• pairs_trading.md · references/
• pairs_backtest.py · scripts/
• test_cointegration.py · scripts/

SKILL.md (first lines)

# 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

| Property | Correlation | Cointegration |
|---|---|---|
| Measures | Short-term co-movement | Long-run equilibrium |
| Stationarity | Requires stationary returns | Works with non-stationary prices |
| Time horizon | Can change rapidly | Stable over months/years |
| Trading use | Momentum/trend signals | Mean-reversion pairs trades |
| Failure mode | Breaks in regime changes | Breaks on structural shifts |

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 Y~X can give a different result than X~Y. Always
test both directions and use the stronger result.

```python
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

```python
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`.

```python
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).

From Serennity007/awesome-stock-quant-skills, MIT.

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