co-eco

Quasi-likelihood ratio tests for cointegration, cobreaking, and cotrending

A Python implementation of the econometric tests proposed by Carrion-i-Silvestre and Kim (2019) for testing cointegration, cobreaking, and cotrending in time series data with structural breaks.

Overview

This package provides three types of tests:

1. Robust Cointegration Test (Q_r): Tests the null hypothesis of cointegration regardless of whether structural breaks are present (robust to the presence or absence of breaks).

2. Joint CI/CB Test (Q_cb): Tests the joint null hypothesis of cointegration AND cobreaking (breaks cancel out).

3. Joint CI/CT Test (Q_ct): Tests the joint null hypothesis of cointegration AND cotrending (both breaks and linear trends cancel out). Only applicable for Model II.

Additionally, Dmax tests are provided for determining the number of breaks when testing cobreaking or cotrending.

Installation

pip install co-eco

Or install from source:

git clone https://github.com/merwanroudane/coeco.git
cd coeco
pip install -e .

Quick Start

import numpy as np
from co_eco import CKTest

# Generate example data
np.random.seed(42)
T = 200
x = np.cumsum(np.random.randn(T, 1), axis=0)  # I(1) regressor
y = 0.5 * x.flatten() + np.cumsum(np.random.randn(T))  # Cointegrated

# Run tests
test = CKTest(y, x, model=2, klags=2, kleads=2)
results = test.run(num_breaks=1)

# Display results
print(results)

Detailed Usage

Model Specifications

• Model I: Mean shifts only (no linear trend)
• Model II: Linear trend with intercept shifts

Known Break Dates

from co_eco import ck_test_known_breaks

Q_r, Q_cb, Q_ct = ck_test_known_breaks(
    y, x, 
    model=2, 
    break_dates=[100],  # Known break at t=100
    klags=2, 
    kleads=2
)

Unknown Break Dates

from co_eco import ck_test_unknown_1break, ck_test_unknown_2breaks

# One unknown break
Q_r, Q_cb, Q_ct, Tbhat = ck_test_unknown_1break(y, x, model=2, klags=2, kleads=2)

# Two unknown breaks
Q_r, Q_cb, Q_ct, Tbhat = ck_test_unknown_2breaks(y, x, model=2, klags=2, kleads=2)

Using the CKTest Class

from co_eco import CKTest

test = CKTest(y, x, model=2, klags=2, kleads=2)

# Run for specific number of breaks
results_1break = test.run(num_breaks=1)
print(results_1break.is_cointegrated(0.05))  # True if CI holds at 5% level

# Run for all break specifications (0, 1, 2)
all_results = test.run_all(max_breaks=2)

# Run Dmax tests
dmax_cb, dmax_ct = test.run_dmax(max_breaks=2)
print(dmax_cb)

Output Formats

# Get summary
print(results.summary())

# Get LaTeX table
latex_table = results.to_latex()

# Get dictionary
results_dict = results.to_dict()

Example: US Budget Sustainability

from co_eco import CKTest
from co_eco.data import load_us_budget

# Load data
R, E, B = load_us_budget()

# Test for cointegration between expenditures and revenues
test = CKTest(E, R.reshape(-1, 1), model=2, klags=3, kleads=3)

# Run tests with 1 break
results = test.run(num_breaks=1)
print(results)

# Run tests with 2 breaks
results_2 = test.run(num_breaks=2)
print(results_2)

Critical Values

Critical values are provided for:

• Significance levels: 1%, 5%, 10%
• Number of breaks: 0, 1, 2
• Number of regressors: 1-5
• Both Model I and Model II

from co_eco import get_critical_values, get_lambda_bar

# Get critical value
cv = get_critical_values('robust_ci', m=1, model=2, px=1, significance_level=0.05)

# Get lambda bar parameter
lbar = get_lambda_bar(m=1, model=2, px=1)

API Reference

Main Functions

Utility Functions

Classes

Mathematical Background

The tests are based on quasi-likelihood ratio principles. Under the null hypothesis of cointegration:

$$y_t = \beta' x_t + \alpha' d_t + v_t$$

where:

• $y_t$ is the dependent variable
• $x_t$ are stochastic regressors (I(1))
• $d_t$ are deterministic components (intercepts, trends, step dummies)
• $v_t$ is I(0) under cointegration, I(1) under no cointegration

The test statistics are:

• Q_r: Robust CI test (invariant to break parameters)
• Q_cb: Joint test with $m \log(T)$ adjustment
• Q_ct: Joint test with additional adjustment for trend

Compatibility

This package is an exact translation of the original MATLAB code by Carrion-i-Silvestre and Kim (2019). Results should match the MATLAB implementation up to numerical precision.

Citation

If you use this package in your research, please cite:

@article{carrion2019quasi,
  title={Quasi-likelihood ratio tests for cointegration, cobreaking, and cotrending},
  author={Carrion-i-Silvestre, Josep Llu{\'\i}s and Kim, Dukpa},
  journal={Econometric Reviews},
  volume={38},
  number={6},
  pages={681--709},
  year={2019},
  publisher={Taylor \& Francis},
  doi={10.1080/07474938.2018.1528416}
}

And this Python implementation:

@software{coeco2024,
  author = {Roudane, Merwan},
  title = {co-eco: Python implementation of Carrion-i-Silvestre and Kim (2019) cointegration tests},
  year = {2024},
  url = {https://github.com/merwanroudane/coeco}
}

References

• Carrion-i-Silvestre, J.L. and Kim, D. (2019). Quasi-likelihood ratio tests for cointegration, cobreaking, and cotrending. Econometric Reviews, 38(6), 681-709.
• Andrews, D.W.K. (1991). Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation. Econometrica, 59(3), 817-858.
• Saikkonen, P. (1991). Asymptotically Efficient Estimation of Cointegration Regressions. Econometric Theory, 7(1), 1-21.

Author

Dr Merwan Roudane
Email: merwanroudane920@gmail.com
GitHub: https://github.com/merwanroudane/coeco

License

MIT License - see LICENSE for details.