Fourier GLS Unit Root Tests

A Python implementation of the Flexible Fourier Form and Local Generalised Least Squares De-trended Unit Root Tests.

Reference

Rodrigues, P. M. M and Taylor, A. M. R. (2012)
"The Flexible Fourier Form and Local Generalised Least Squares De-trended Unit Root Tests."
Oxford Bulletin of Economics and Statistics, 74(5), 736-759.

Features

• Fourier GLS Test: Local GLS de-trended unit root test with Fourier approximation for unknown structural breaks
• Fourier DF Test: OLS de-trended Dickey-Fuller test with Fourier terms (Enders & Lee, 2009)
• Fourier LM Test: First-difference de-trended LM test with Fourier terms (Schmidt & Phillips, 1992)
• F-Test for Linearity: Test significance of Fourier terms
• Data-Driven Frequency Selection: Automatic selection using Davies (1987) procedure
• Critical Values: Complete tables from the original paper (Tables 1-3)
• Publication-Ready Output: Formatted results suitable for academic papers

Installation

pip install fourier-gls

Or install from source:

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

Quick Start

import numpy as np
from fourier_gls import fourier_gls

# Generate sample data (unit root with structural break)
np.random.seed(42)
T = 200
t = np.arange(1, T + 1)
y = np.cumsum(np.random.randn(T)) + 5 * np.sin(2 * np.pi * t / T)

# Run Fourier GLS test
result = fourier_gls(y, model=2)  # model=2 includes constant and trend

Output:

================================================================================
                    Fourier GLS Unit Root Test Results
================================================================================
Reference: Rodrigues & Taylor (2012), Oxford Bulletin of Economics and Statistics

Model:               Constant + Trend
Sample Size:         200
Selected Frequency:  k = 1
Selected Lags:       2

--------------------------------------------------------------------------------
Test Statistic:      -3.2456
--------------------------------------------------------------------------------

Critical Values:
    1%:   -4.5930
    5%:   -4.0410
    10%:  -3.7490

--------------------------------------------------------------------------------
Conclusion (5% level): Fail to reject H0: No evidence against unit root at 5% level
================================================================================

API Reference

Main Functions

fourier_gls(y, model=2, pmax=8, fmax=5, ic=3, verbose=True)

Performs the Fourier GLS unit root test with automatic frequency selection.

Parameters:

• y : array-like - Time series data
• model : int - 1 = Constant only, 2 = Constant and linear trend (default)
• pmax : int - Maximum number of lags (default: 8)
• fmax : int - Maximum Fourier frequency, 1-5 (default: 5)
• ic : int - Information criterion: 1=AIC, 2=SIC, 3=t-stat (default: 3)
• verbose : bool - Print results (default: True)

Returns: FourierGLSResult object

fourier_gls_f_test(y, model, k, p=0, verbose=True)

Tests significance of Fourier terms (H0: γ₁ = γ₂ = 0).

gls_detrend(y, z, cbar, return_ssr=False)

Performs GLS detrending using the Elliott, Rothenberg & Stock (1996) procedure.

Result Object

The FourierGLSResult object contains:

Methods:

• summary() : Returns formatted summary string
• to_dict() : Converts results to dictionary

Critical Values

from fourier_gls import get_cbar, get_fourier_gls_critical_values

# Get c-bar parameter (Table 1)
cbar = get_cbar(model=2, k=1)  # Returns -22.00

# Get critical values (Table 2)
cv = get_fourier_gls_critical_values(T=200, model=2)

Model Specifications

Model 1: Constant Only

$$y_t = \delta_0 + \gamma_1 \sin\left(\frac{2\pi k t}{T}\right) + \gamma_2 \cos\left(\frac{2\pi k t}{T}\right) + x_t$$

Model 2: Constant and Trend

$$y_t = \delta_0 + \delta_1 t + \gamma_1 \sin\left(\frac{2\pi k t}{T}\right) + \gamma_2 \cos\left(\frac{2\pi k t}{T}\right) + x_t$$

where $x_t = \rho x_{t-1} + u_t$ and $u_t \sim \text{iid}(0, \sigma^2)$.

Examples

Basic Usage

import numpy as np
from fourier_gls import fourier_gls

# Load your data
y = np.loadtxt('your_data.csv')

# Test with constant + trend model
result = fourier_gls(y, model=2)
print(f"Test statistic: {result.statistic:.4f}")
print(f"Selected frequency: k = {result.frequency}")

Comparing Different Tests

from fourier_gls import fourier_gls, fourier_df, fourier_lm

y = np.random.randn(200).cumsum()

# GLS de-trended test
gls_result = fourier_gls(y, model=2, verbose=False)

# OLS de-trended test  
df_result = fourier_df(y, model=2, verbose=False)

# LM test
lm_result = fourier_lm(y, verbose=False)

print(f"GLS statistic: {gls_result.statistic:.4f}")
print(f"DF statistic:  {df_result.statistic:.4f}")
print(f"LM statistic:  {lm_result.statistic:.4f}")

Fixed Frequency Test

from fourier_gls import fourier_gls_fixed_k

# Use k=1 (single break-like behavior)
result = fourier_gls_fixed_k(y, model=2, k=1)

Testing for Fourier Term Significance

from fourier_gls import fourier_gls_f_test

# Test if Fourier terms are significant
f_result = fourier_gls_f_test(y, model=2, k=1, p=2)

Simulating Critical Values

from fourier_gls import simulate_critical_values

# Generate custom critical values
cv = simulate_critical_values(T=150, model=2, k=1, n_simulations=10000, seed=42)
print(f"Critical values (1%, 5%, 10%): {cv}")

Critical Values Tables

Table 1: Local GLS De-trending Parameters (c̄)

Table 2: Critical Values for t^{ERS}_f Tests

Sample critical values for T = 200, Constant + Trend model:

Compatibility

This implementation is designed to be fully compatible with the original GAUSS code by Saban Nazlioglu and matches the methodology in Rodrigues & Taylor (2012).

Requirements

• Python >= 3.8
• NumPy >= 1.20.0
• SciPy >= 1.7.0

Author

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

License

MIT License - see LICENSE for details.

Citation

If you use this package in academic work, please cite:

@article{rodrigues2012flexible,
  title={The Flexible Fourier Form and Local Generalised Least Squares De-trended Unit Root Tests},
  author={Rodrigues, Paulo MM and Taylor, A M Robert},
  journal={Oxford Bulletin of Economics and Statistics},
  volume={74},
  number={5},
  pages={736--759},
  year={2012}
}

@software{roudane2024fouriergls,
  author = {Roudane, Merwan},
  title = {fourier\_gls: Python Implementation of Fourier GLS Unit Root Tests},
  year = {2024},
  url = {https://github.com/merwanroudane/fouriergls}
}

Related Papers

1. Elliott, G., Rothenberg, T. J., & Stock, J. H. (1996). Efficient tests for an autoregressive unit root. Econometrica, 64, 813-836.

2. Enders, W., & Lee, J. (2012). A Unit Root Test Using a Fourier Series to Approximate Smooth Breaks. Oxford Bulletin of Economics and Statistics, 74(4), 574-599.

3. Becker, R., Enders, W., & Lee, J. (2006). A stationarity test in the presence of an unknown number of smooth breaks. Journal of Time Series Analysis, 27, 381-409.