cointegration-analysis

Name: cointegration-analysis
Author: agiprolabs

235stars53forks

How to install cointegration-analysis?

agiprolabs/claude-trading-skills/cointegration-analysis

$npx -y skills add agiprolabs/claude-trading-skills --skill cointegration-analysis

Installs into the current project.

Summary

Cointegration testing for pairs trading using Engle-Granger, Johansen, and rolling stability analysis

Files · 1

View on GitHub

SKILL.md

1	# Cointegration Analysis
2
3	Cointegration testing identifies pairs of assets that share a long-run equilibrium
4	relationship, enabling statistical arbitrage and pairs trading strategies.
5
6	## What Is Cointegration?
7
8	Two price series are cointegrated when they are individually non-stationary
9	(random walks) but a linear combination of them is stationary (mean-reverting).
10	Intuitively, the prices may wander apart temporarily but are pulled back to an
11	equilibrium spread over time.
12
13	### Cointegration vs Correlation
14
15	\| Property \| Correlation \| Cointegration \|
16	\|---\|---\|---\|
17	\| Measures \| Short-term co-movement \| Long-run equilibrium \|
18	\| Stationarity \| Requires stationary returns \| Works with non-stationary prices \|
19	\| Time horizon \| Can change rapidly \| Stable over months/years \|
20	\| Trading use \| Momentum/trend signals \| Mean-reversion pairs trades \|
21	\| Failure mode \| Breaks in regime changes \| Breaks on structural shifts \|
22
23	Two assets can be highly correlated but not cointegrated (e.g., two unrelated
24	uptrends). Conversely, cointegrated assets may have low short-term correlation
25	during temporary divergences — which is exactly when pairs trades are entered.
26
27	### Why It Matters
28
29	- Pairs trading: Long the underperformer, short the outperformer, profit on convergence
30	- Statistical arbitrage: Systematic mean-reversion on spread z-scores
31	- Spread trading: Trade the spread directly as a synthetic instrument
32	- Risk hedging: Cointegrated hedge ratios minimize tracking error over time
33
34	## Methods
35
36	### 1. Engle-Granger Two-Step
37
38	The most common approach for two series.
39
40	Step 1 — Regress Y on X using OLS:
41
42	```
43	Y_t = α + β * X_t + ε_t
44	```
45
46	Step 2 — Test the residuals ε_t for stationarity using the ADF test.
47
48	- If residuals are stationary (p < 0.05) → Y and X are cointegrated
49	- β is the hedge ratio for the pairs trade
50	- α is the long-run mean of the spread
51
52	Important: Engle-Granger critical values differ from standard ADF critical
53	values. For n=2 series: 1% = -3.90, 5% = -3.34, 10% = -3.04.
54
55	Asymmetry warning: Testing Y~X can give a different result than X~Y. Always
56	test both directions and use the stronger result.
57
58	```python
59	from scipy import stats
60	import numpy as np
61	from statsmodels.tsa.stattools import adfuller
62
63	# Step 1: OLS regression
64	slope, intercept, _, _, _ = stats.linregress(x_prices, y_prices)
65	hedge_ratio = slope
66
67	# Step 2: Test residuals
68	residuals = y_prices - hedge_ratio * x_prices - intercept
69	adf_stat, p_value, _, _, crit_values, _ = adfuller(residuals, maxlag=None, autolag="AIC")
70
71	cointegrated = p_value < 0.05
72	```
73
74	### 2. Johansen Test
75
76	Tests multiple series simultaneously and returns the number of cointegrating
77	relationships. More powerful than Engle-Granger for >2 series.
78
79	- Based on a VAR model: ΔY_t = Π·Y_{t-1} + Σ Γ_i·ΔY_{t-i} + ε_t
80	- Tests the rank of the Π matrix
81	- Uses trace test and maximum eigenvalue test
82	- Returns: number of cointegrating vectors and the vectors themselves
83
84	```python
85	from statsmodels.tsa.vector_ar.vecm import coint_johansen
86
87	# data: T×N array of price series
88	result = coint_johansen(data, det_order=0, k_ar_diff=1)
89
90	# Trace statistic vs critical values (90%, 95%, 99%)
91	trace_stats = result.lr1 # Trace statistics
92	trace_crit = result.cvt # Critical values
93	max_eigen_stats = result.lr2 # Max eigenvalue statistics
94	max_eigen_crit = result.cvm # Critical values
95
96	# Cointegrating vectors
97	coint_vectors = result.evec
98	```
99
100	### 3. Phillips-Ouliaris
101
102	Similar to Engle-Granger but uses Phillips-Perron style test statistics
103	instead of ADF. More robust to heteroskedasticity and serial correlation in
104	the residuals. Available via `statsmodels.tsa.stattools.coint`.
105
106	```python
107	from statsmodels.tsa.stattools import coint
108
109	# Returns: test statistic, p-value, critical values
110	t_stat, p_value, crit_values = coint(y_prices, x_prices)
111	cointegrated = p_value < 0.05
112	```
113
114	## Practical Workflow
115
116	### Step 1: Screen Pairs by Correlation
117
118	Pre-filter using Pearson correlation > 0.7 to reduce the number of
119	cointegration tests (which are more expensive).
120
121	### Step 2: Test Cointegration
122
123	Run Engle-Granger in both directions. Use p < 0.05 threshold.
124
125	### Step 3: Estimate Hedge Ratio
126
127	Use OLS for simplicity. For production, consider Total Least Squares or
128	Dynamic OLS (see `references/methodology.md`).
129
130	### Step 4: Compute Spread
131
132	```python
133	spread = y_prices - hedge_ratio * x_prices - intercept
134	z_score = (spread - spread.mean()) / spread.std()
135	```
136
137	### Step 5: Test Spread for Mean Reversion
138
139	- ADF test: p < 0.05 confirms stationarity
140	- Hurst exponent: H < 0.5 indicates mean reversion (H ≈ 0.5 = random walk)
141	- Half-life: λ from AR(1) on spread; half-life = -ln(2)/ln(λ)
142	- Viable pairs: half-life between 5 and 60 days
143
144	### Step 6: Trade the Spread
145
146	If the spread is mean-reverting, it is a viable pairs trade can

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cointegration-analysis

235stars53forks

View on GitHub

How to install cointegration-analysis?

agiprolabs/claude-trading-skills/cointegration-analysis

$npx -y skills add agiprolabs/claude-trading-skills --skill cointegration-analysis

Installs into the current project.

Summary

Cointegration testing for pairs trading using Engle-Granger, Johansen, and rolling stability analysis

Files · 1

View on GitHub

SKILL.md

1	# Cointegration Analysis
2
3	Cointegration testing identifies pairs of assets that share a long-run equilibrium
4	relationship, enabling statistical arbitrage and pairs trading strategies.
5
6	## What Is Cointegration?
7
8	Two price series are cointegrated when they are individually non-stationary
9	(random walks) but a linear combination of them is stationary (mean-reverting).
10	Intuitively, the prices may wander apart temporarily but are pulled back to an
11	equilibrium spread over time.
12
13	### Cointegration vs Correlation
14
15	\| Property \| Correlation \| Cointegration \|
16	\|---\|---\|---\|
17	\| Measures \| Short-term co-movement \| Long-run equilibrium \|
18	\| Stationarity \| Requires stationary returns \| Works with non-stationary prices \|
19	\| Time horizon \| Can change rapidly \| Stable over months/years \|
20	\| Trading use \| Momentum/trend signals \| Mean-reversion pairs trades \|
21	\| Failure mode \| Breaks in regime changes \| Breaks on structural shifts \|
22
23	Two assets can be highly correlated but not cointegrated (e.g., two unrelated
24	uptrends). Conversely, cointegrated assets may have low short-term correlation
25	during temporary divergences — which is exactly when pairs trades are entered.
26
27	### Why It Matters
28
29	- Pairs trading: Long the underperformer, short the outperformer, profit on convergence
30	- Statistical arbitrage: Systematic mean-reversion on spread z-scores
31	- Spread trading: Trade the spread directly as a synthetic instrument
32	- Risk hedging: Cointegrated hedge ratios minimize tracking error over time
33
34	## Methods
35
36	### 1. Engle-Granger Two-Step
37
38	The most common approach for two series.
39
40	Step 1 — Regress Y on X using OLS:
41
42	```
43	Y_t = α + β * X_t + ε_t
44	```
45
46	Step 2 — Test the residuals ε_t for stationarity using the ADF test.
47
48	- If residuals are stationary (p < 0.05) → Y and X are cointegrated
49	- β is the hedge ratio for the pairs trade
50	- α is the long-run mean of the spread
51
52	Important: Engle-Granger critical values differ from standard ADF critical
53	values. For n=2 series: 1% = -3.90, 5% = -3.34, 10% = -3.04.
54
55	Asymmetry warning: Testing Y~X can give a different result than X~Y. Always
56	test both directions and use the stronger result.
57
58	```python
59	from scipy import stats
60	import numpy as np
61	from statsmodels.tsa.stattools import adfuller
62
63	# Step 1: OLS regression
64	slope, intercept, _, _, _ = stats.linregress(x_prices, y_prices)
65	hedge_ratio = slope
66
67	# Step 2: Test residuals
68	residuals = y_prices - hedge_ratio * x_prices - intercept
69	adf_stat, p_value, _, _, crit_values, _ = adfuller(residuals, maxlag=None, autolag="AIC")
70
71	cointegrated = p_value < 0.05
72	```
73
74	### 2. Johansen Test
75
76	Tests multiple series simultaneously and returns the number of cointegrating
77	relationships. More powerful than Engle-Granger for >2 series.
78
79	- Based on a VAR model: ΔY_t = Π·Y_{t-1} + Σ Γ_i·ΔY_{t-i} + ε_t
80	- Tests the rank of the Π matrix
81	- Uses trace test and maximum eigenvalue test
82	- Returns: number of cointegrating vectors and the vectors themselves
83
84	```python
85	from statsmodels.tsa.vector_ar.vecm import coint_johansen
86
87	# data: T×N array of price series
88	result = coint_johansen(data, det_order=0, k_ar_diff=1)
89
90	# Trace statistic vs critical values (90%, 95%, 99%)
91	trace_stats = result.lr1 # Trace statistics
92	trace_crit = result.cvt # Critical values
93	max_eigen_stats = result.lr2 # Max eigenvalue statistics
94	max_eigen_crit = result.cvm # Critical values
95
96	# Cointegrating vectors
97	coint_vectors = result.evec
98	```
99
100	### 3. Phillips-Ouliaris
101
102	Similar to Engle-Granger but uses Phillips-Perron style test statistics
103	instead of ADF. More robust to heteroskedasticity and serial correlation in
104	the residuals. Available via `statsmodels.tsa.stattools.coint`.
105
106	```python
107	from statsmodels.tsa.stattools import coint
108
109	# Returns: test statistic, p-value, critical values
110	t_stat, p_value, crit_values = coint(y_prices, x_prices)
111	cointegrated = p_value < 0.05
112	```
113
114	## Practical Workflow
115
116	### Step 1: Screen Pairs by Correlation
117
118	Pre-filter using Pearson correlation > 0.7 to reduce the number of
119	cointegration tests (which are more expensive).
120
121	### Step 2: Test Cointegration
122
123	Run Engle-Granger in both directions. Use p < 0.05 threshold.
124
125	### Step 3: Estimate Hedge Ratio
126
127	Use OLS for simplicity. For production, consider Total Least Squares or
128	Dynamic OLS (see `references/methodology.md`).
129
130	### Step 4: Compute Spread
131
132	```python
133	spread = y_prices - hedge_ratio * x_prices - intercept
134	z_score = (spread - spread.mean()) / spread.std()
135	```
136
137	### Step 5: Test Spread for Mean Reversion
138
139	- ADF test: p < 0.05 confirms stationarity
140	- Hurst exponent: H < 0.5 indicates mean reversion (H ≈ 0.5 = random walk)
141	- Half-life: λ from AR(1) on spread; half-life = -ln(2)/ln(λ)
142	- Viable pairs: half-life between 5 and 60 days
143
144	### Step 6: Trade the Spread
145
146	If the spread is mean-reverting, it is a viable pairs trade can

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