Fourier Cointegration Test

Python implementation of the Fourier approximation test for cointegration proposed by Tsong et al. (2016).

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

📖 Overview

This package implements the cointegration test developed by Tsong et al. (2016), which tests the null hypothesis of cointegration against the alternative of no cointegration, while allowing for structural breaks of unknown form in the deterministic trend using Fourier approximation.

Key Features

✅ Tests for cointegration with structural breaks
✅ No need to estimate break dates or number of breaks
✅ Handles unknown forms of structural changes
✅ Implements both OLS and DOLS estimation
✅ Automatic optimal Fourier frequency selection
✅ Critical values from Tsong et al. (2016) Table 1
✅ F-test for Fourier significance
✅ Compatible with Windows, Linux, and macOS

📚 Reference

Tsong, C.C., Lee, C.F., Tsai, L.J., & Hu, T.C. (2016).
The Fourier approximation and testing for the null of cointegration.
Empirical Economics, 51(3), 1085-1113.
DOI: 10.1007/s00181-015-1028-6

🔧 Installation

From PyPI (coming soon)

pip install fouriercoint

From source

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

Requirements

• Python >= 3.7
• NumPy >= 1.19.0
• SciPy >= 1.5.0

🚀 Quick Start

Basic Usage

import numpy as np
from fouriercoint import fourier_cointegration_test

# Generate sample I(1) data
np.random.seed(42)
T = 200
x = np.cumsum(np.random.randn(T, 1))  # I(1) regressor
y = 2 + 0.5 * x + np.cumsum(np.random.randn(T, 1) * 0.1)  # Cointegrated

# Run the test
results = fourier_cointegration_test(
    y=y,
    x=x,
    m=1,              # 1 = constant + trend + Fourier
    kmax=3,           # Maximum Fourier frequency
    significance_level=0.05,
    use_dols=True     # Use DOLS estimation
)

# Print results
print(f"Test Statistic: {results['test_statistic']:.4f}")
print(f"Critical Value (5%): {results['critical_value']:.4f}")
print(f"Cointegration Evidence: {not results['reject_null']}")
print(f"\n{results['conclusion']}")

Output

======================================================================
FOURIER COINTEGRATION TEST RESULTS
Tsong et al. (2016)
======================================================================
Sample size (T):              200
Number of regressors (p):     1
Model specification (m):      1 (constant + trend + Fourier)
Estimation method:            DOLS
DOLS leads/lags (q):          5
Kernel:                       bartlett
----------------------------------------------------------------------
Optimal Fourier frequency:    k* = 1
Test statistic CI^m_f:        0.0234
Critical value (5%):          0.0480
Long-run variance:            0.0089
----------------------------------------------------------------------
Critical values:
  10%:  0.0420
  5%:   0.0480
  1%:   0.0630
----------------------------------------------------------------------
F-test for Fourier:           F = 8.2341
F critical value (5%):        4.0190
Fourier significant:          True
======================================================================
CONCLUSION: Do not reject the null hypothesis of cointegration at 5.0% level. 
Evidence supports cointegration with structural breaks (Fourier frequency k=1).
======================================================================

📊 Methodology

The Model

The cointegration regression is specified as:

y_t = d_t + x'_t β + η_t

where the deterministic component is:

d_t = δ_0 + δ_1 t + α_k sin(2πkt/T) + β_k cos(2πkt/T)

Test Statistic

The test statistic is:

CI^m_f = T^{-2} ω̂^{-2}_1 Σ_{t=1}^T S^2_t

where:

• S_t = Σ_{i=1}^t ε̂_i is the partial sum of residuals
• ω̂^2_1 is the long-run variance estimate
• m = 0 for constant + Fourier
• m = 1 for constant + trend + Fourier

Hypothesis Testing

• H₀: Cointegration with structural breaks (σ²_u = 0)
• H₁: No cointegration (σ²_u > 0)

Reject H₀ if CI^m_f > critical value → No cointegration
Do not reject H₀ if CI^m_f ≤ critical value → Evidence of cointegration

📖 Detailed Usage

Model Specifications

# Model 0: Level shifts only (constant + Fourier)
results_m0 = fourier_cointegration_test(y, x, m=0)

# Model 1: Level and trend shifts (constant + trend + Fourier)
results_m1 = fourier_cointegration_test(y, x, m=1)

Multivariate Case

# Multiple regressors
x_multi = np.cumsum(np.random.randn(T, 3), axis=0)  # 3 I(1) regressors
y = 2 + x_multi @ np.array([[0.5], [0.3], [-0.2]]) + np.random.randn(T, 1)

results = fourier_cointegration_test(y, x_multi, m=1)

Customize Test Options

results = fourier_cointegration_test(
    y=y,
    x=x,
    m=1,
    kmax=5,                    # Test frequencies 1 through 5
    q=3,                       # DOLS leads/lags (default: T^{1/3})
    kernel='qs',               # 'bartlett' or 'qs' (Quadratic Spectral)
    bandwidth=10,              # Manual bandwidth (default: automatic)
    significance_level=0.01,   # 1%, 5%, or 10%
    use_dols=True,             # DOLS vs OLS
    verbose=True               # Print detailed output
)

Access Detailed Results

# Test statistics for all frequencies
for k in range(1, 4):
    stat = results['all_frequencies'][k]['statistic']
    print(f"k={k}: Statistic = {stat:.4f}")

# Cointegrating vector
beta_hat = results['optimal_results']['beta']
print(f"Estimated β: {beta_hat.flatten()}")

# Residuals and diagnostics
residuals = results['optimal_results']['residuals']
lrv = results['optimal_results']['lrv']

F-test for Fourier Significance

The package automatically performs an F-test to check if the Fourier component is necessary:

f_test = results['F_test']
print(f"F-statistic: {f_test['F_statistic']:.4f}")
print(f"Fourier needed: {f_test['reject_null']}")

🧪 Testing Framework

Running Tests

# Example: Test with known cointegrated data
import numpy as np
from fouriercoint import fourier_cointegration_test

np.random.seed(123)
T = 150

# Generate cointegrated data with structural break
t = np.arange(T)
break_point = T // 2
trend_shift = np.where(t < break_point, 0, 3)  # Level shift at midpoint

x = np.cumsum(np.random.randn(T, 1))
y = 2 + 0.5 * x + trend_shift.reshape(-1, 1) + np.random.randn(T, 1) * 0.5

# Test should find cointegration with Fourier term
results = fourier_cointegration_test(y, x, m=0, kmax=3)
assert not results['reject_null'], "Should not reject cointegration"
assert results['fourier_significant'], "Fourier should be significant"

💡 Practical Application: Fiscal Sustainability

Following Tsong et al. (2016) empirical application:

import pandas as pd
from fouriercoint import fourier_cointegration_test

# Load fiscal data (revenue and expenditure)
data = pd.read_excel('fiscal_data.xlsx')
revenue = data['revenue'].values.reshape(-1, 1)
expenditure = data['expenditure'].values.reshape(-1, 1)

# Test for cointegration (fiscal sustainability)
results = fourier_cointegration_test(
    y=revenue,
    x=expenditure,
    m=1,  # Allow for both level and trend breaks
    kmax=3,
    significance_level=0.05
)

# Interpretation
if not results['reject_null']:
    print("Evidence of fiscal sustainability (cointegration found)")
    print(f"Structural breaks approximated with Fourier k={results['optimal_k']}")
else:
    print("No evidence of fiscal sustainability")

📊 Comparison with Alternative Tests

🔬 Technical Details

Critical Values

Critical values are provided for:

• Model specifications: m ∈ {0, 1}
• Fourier frequencies: k ∈ {1, 2, 3}
• Regressor dimensions: p ∈ {1, 2, 3, 4}
• Significance levels: {1%, 5%, 10%}

The package automatically interpolates for other values.

Long-run Variance Estimation

Supports multiple kernel methods:

• Bartlett (Newey-West): Recommended for most cases
• Quadratic Spectral: Better for highly persistent processes

Automatic bandwidth selection using Andrews (1991) method.

DOLS Estimation

Dynamic OLS (Saikkonen 1991) is used to handle endogenous regressors:

y_t = d_t + x'_t β + Σ_{i=-q}^{q} Δx'_{t-i} φ_i + ε*_t

📝 Citation

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

@article{tsong2016fourier,
  title={The Fourier approximation and testing for the null of cointegration},
  author={Tsong, Ching-Chuan and Lee, Cheng-Feng and Tsai, Li-Ju and Hu, Te-Chung},
  journal={Empirical Economics},
  volume={51},
  number={3},
  pages={1085--1113},
  year={2016},
  publisher={Springer}
}

@software{roudane2024fouriercoint,
  author = {Roudane, Merwan},
  title = {fouriercoint: Fourier Cointegration Test in Python},
  year = {2024},
  url = {https://github.com/merwanroudane/fouriercoint}
}

🤝 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

📄 License

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

📧 Contact

Dr. Merwan Roudane
Email: merwanroudane920@gmail.com
GitHub: @merwanroudane

🙏 Acknowledgments

• Original methodology: Tsong, Lee, Tsai, and Hu (2016)
• Inspired by R implementation and econometric theory
• Built for the Python econometrics community

⚠️ Disclaimer

This package is provided for research and educational purposes. Users should verify results for critical applications.

Made with ❤️ for the econometrics community