Kapetanios Unit Root Test

Python implementation of the Kapetanios (2005) unit root test with up to m structural breaks.

Overview

This package provides a comprehensive implementation of the unit root test proposed by George Kapetanios (2005), which tests for a unit root against the alternative hypothesis of stationarity with up to m structural breaks. The test is an extension of the Zivot and Andrews (1992) test and uses a sequential procedure to endogenously determine break dates.

Key Features

• Multiple Structural Breaks: Test for up to 5 structural breaks
• Flexible Break Types:
  • Model A: Breaks in intercept only
  • Model B: Breaks in trend only
  • Model C: Breaks in both intercept and trend
• Sequential Search: Computationally efficient sequential procedure (Bai & Perron, 1998)
• Automatic Lag Selection: AIC, BIC, or t-statistic methods
• Windows Compatible: Fully tested on Windows, Linux, and macOS

Installation

From PyPI (recommended)

pip install kapetanios-test

From source

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

Quick Start

import numpy as np
from kapetanios_test import kapetanios_test

# Generate sample data (random walk with structural break)
np.random.seed(42)
T = 200
y = np.cumsum(np.random.randn(T))
y[100:] += 5  # Add level break at t=100

# Run the test
result = kapetanios_test(
    y,
    max_breaks=3,      # Test for up to 3 breaks
    model='C',         # Allow breaks in both intercept and trend
    trimming=0.15,     # Trimming parameter
    lag_selection='aic'  # Use AIC for lag selection
)

# Display results
print(result)

Usage

Basic Usage

The simplest way to use the package is through the kapetanios_test() function:

from kapetanios_test import kapetanios_test

result = kapetanios_test(y, max_breaks=2, model='A')

Advanced Usage

For more control, use the KapetaniosTest class:

from kapetanios_test import KapetaniosTest

# Initialize test
test = KapetaniosTest(
    max_breaks=3,
    model='C',
    trimming=0.15,
    max_lags=None,  # Automatically determined
    lag_selection='aic'
)

# Run test
result = test.fit(y)

# Access results
print(f"Test statistic: {result.statistic}")
print(f"Break dates: {result.break_dates}")
print(f"Reject null: {result.reject_null}")

Model Types

• Model A: Breaks in intercept only

  result = kapetanios_test(y, model='A')
  

• Model B: Breaks in trend only

  result = kapetanios_test(y, model='B')
  

• Model C: Breaks in both intercept and trend (recommended)

  result = kapetanios_test(y, model='C')
  

Lag Selection Methods

# Akaike Information Criterion (default)
result = kapetanios_test(y, lag_selection='aic')

# Bayesian Information Criterion
result = kapetanios_test(y, lag_selection='bic')

# Sequential t-statistic (Perron's method)
result = kapetanios_test(y, lag_selection='t-stat')

Interpretation

Hypotheses

• Null Hypothesis (H₀): The series contains a unit root with drift (non-stationary)
• Alternative Hypothesis (H₁): The series is trend-stationary with up to m structural breaks

Decision Rule

Reject H₀ if the test statistic is less than (more negative than) the critical value at your chosen significance level.

result = kapetanios_test(y)

if result.reject_null:
    print("Reject H0: Evidence of stationarity with structural breaks")
else:
    print("Fail to reject H0: Evidence consistent with unit root")

Output

The KapetaniosResult object contains:

• statistic: The test statistic (minimum t-statistic)
• pvalue: Approximate p-value
• critical_values: Dictionary with 1%, 5%, and 10% critical values
• break_dates: List of estimated break dates (indices)
• n_breaks: Number of breaks detected
• model_type: Type of model ('A', 'B', or 'C')
• lags: Number of lags used
• reject_null: Boolean indicating rejection at 5% level

Methodology

The test follows the sequential procedure outlined in Kapetanios (2005):

1. Step 1: Search for a single break across all possible dates
2. Step 2: Select break with minimum SSR
3. Step 3: Conditional on first break, search for second break
4. Step 4: Repeat until m breaks are found
5. Step 5: Compute minimum t-statistic across all searches
6. Step 6: Compare with critical values

The model estimated is:

Δyₜ = μ₀ + μ₁t + αyₜ₋₁ + Σⱼ cⱼΔyₜ₋ⱼ + Σᵢ φᵢDUᵢ,ₜ + Σᵢ ψᵢDTᵢ,ₜ + εₜ

Where:

• DUᵢ,ₜ: Intercept break dummies
• DTᵢ,ₜ: Trend break dummies
• α: Coefficient of interest (H₀: α = 0)

Examples

Example 1: Testing Economic Time Series

import pandas as pd
from kapetanios_test import kapetanios_test

# Load GDP data
df = pd.read_csv('gdp_data.csv')
gdp = df['real_gdp'].values

# Test for unit root with up to 2 breaks
result = kapetanios_test(
    gdp,
    max_breaks=2,
    model='C',
    trimming=0.15
)

print(result)

Example 2: Monte Carlo Simulation

import numpy as np
from kapetanios_test import kapetanios_test

# Simulate stationary process with break
np.random.seed(123)
T = 150
t = np.arange(T)
epsilon = np.random.randn(T)

# AR(1) with structural break
y = np.zeros(T)
y[0] = epsilon[0]
for i in range(1, T):
    if i < 75:
        y[i] = 0.5 + 0.1 * t[i] + 0.7 * y[i-1] + epsilon[i]
    else:
        y[i] = 2.0 + 0.1 * t[i] + 0.7 * y[i-1] + epsilon[i]  # Break at t=75

result = kapetanios_test(y, max_breaks=3, model='C')
print(f"Detected breaks at: {result.break_dates}")

Example 3: Comparing Different Models

from kapetanios_test import kapetanios_test

# Test all three models
models = ['A', 'B', 'C']
for model in models:
    result = kapetanios_test(y, max_breaks=2, model=model)
    print(f"\nModel {model}:")
    print(f"  Statistic: {result.statistic:.4f}")
    print(f"  Reject H0: {result.reject_null}")

Critical Values

Critical values are from Table I in Kapetanios (2005) and are automatically used based on:

• Model type (A, B, or C)
• Maximum number of breaks
• Significance level (1%, 5%, 10%)

Limitations

1. Sample Size: Requires sufficient observations relative to number of breaks and trimming parameter
2. Trimming: The trimming parameter must ensure adequate observations between breaks
3. Power: Power decreases as the maximum number of breaks increases
4. Null Hypothesis: Breaks are only considered under the alternative hypothesis, not under the null

Requirements

• Python >= 3.8
• NumPy >= 1.20.0
• Pandas >= 1.3.0
• SciPy >= 1.7.0
• statsmodels >= 0.13.0

References

Primary Reference

Kapetanios, G. (2005). Unit-root testing against the alternative hypothesis of up to m structural breaks. Journal of Time Series Analysis, 26(1), 123-133.

Related Literature

• Bai, J., & Perron, P. (1998). Estimating and testing linear models with multiple structural changes. Econometrica, 66(1), 47-78.
• Zivot, E., & Andrews, D. W. K. (1992). Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis. Journal of Business & Economic Statistics, 10(3), 251-270.
• Perron, P. (1989). The great crash, the oil price shock, and the unit root hypothesis. Econometrica, 57(6), 1361-1401.

Contributing

Contributions are welcome! Please feel free to submit a Pull Request.

1. Fork the repository
2. Create your feature branch (git checkout -b feature/AmazingFeature)
3. Commit your changes (git commit -m 'Add some AmazingFeature')
4. Push to the branch (git push origin feature/AmazingFeature)
5. Open a Pull Request

Testing

Run the test suite:

pytest tests/

With coverage:

pytest --cov=kapetanios_test tests/

License

This project is licensed under the MIT License - see the LICENSE file for details.

Author

Dr. Merwan Roudane

• Email: merwanroudane920@gmail.com
• GitHub: @merwanroudane

Citation

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

@article{kapetanios2005,
  title={Unit-root testing against the alternative hypothesis of up to m structural breaks},
  author={Kapetanios, George},
  journal={Journal of Time Series Analysis},
  volume={26},
  number={1},
  pages={123--133},
  year={2005},
  publisher={Wiley Online Library}
}

@software{roudane2024kapetanios,
  author = {Roudane, Merwan},
  title = {kapetanios-test: Python implementation of the Kapetanios unit root test},
  year = {2024},
  url = {https://github.com/merwanroudane/kapetanios}
}

Acknowledgments

• Original methodology by George Kapetanios
• Sequential procedure based on Bai & Perron (1998)
• Gretl package by Andrea E. Sánchez Urbina, Ricardo Ramírez & Daniel Ventosa-Santaulària

Support

For issues, questions, or suggestions:

• Open an issue on GitHub
• Email: merwanroudane920@gmail.com

Changelog

Version 1.0.0 (2024)

• Initial release
• Implementation of Kapetanios (2005) test
• Support for up to 5 structural breaks
• Three model types (A, B, C)
• Multiple lag selection methods
• Comprehensive documentation and examples