QFADF - Quantile Fourier ADF Unit Root Test

A Python implementation of the Quantile Fourier ADF unit root test proposed by Li & Zheng (2018), which is robust to both non-Gaussian conditions and smooth structural changes.

Reference

Li, H., & Zheng, C. (2018). Unit root quantile autoregression testing with smooth structural changes. Finance Research Letters, 25, 83-89.

DOI: 10.1016/j.frl.2017.10.008

Features

• Quantile Fourier ADF Test (qr_fourier_adf): Unit root test at a specified quantile with Fourier approximation for structural changes
• Bootstrap Critical Values (qr_fourier_adf_bootstrap): Bootstrap procedure for finite-sample critical values
• QKS and QCM Statistics (qks_qcm_statistics): Quantile Kolmogorov-Smirnov and Cramér-von Mises tests over a range of quantiles
• Optimal Frequency Selection (estimate_optimal_k): Data-driven selection of Fourier frequency
• Pre-computed Critical Values: Based on extensive Monte Carlo simulations
• Publication-ready Output: LaTeX tables and formatted results

Installation

From PyPI (when available)

pip install qfadf

From Source

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

Dependencies

• Python >= 3.8
• NumPy >= 1.20.0
• Pandas >= 1.3.0
• SciPy >= 1.7.0
• Matplotlib >= 3.4.0 (optional, for plotting)

Quick Start

Basic Unit Root Test

import numpy as np
from qfadf import qr_fourier_adf

# Generate a random walk (unit root process)
np.random.seed(42)
y = np.cumsum(np.random.randn(200))

# Perform test at median (tau=0.5)
results = qr_fourier_adf(y, model=1, tau=0.5, k=3, pmax=8)

# Output:
# ======================================================================
#          QUANTILE FOURIER ADF UNIT ROOT TEST
#          Li & Zheng (2018, Finance Research Letters)
# ======================================================================
#   Model:                Constant
#   Fourier frequency k:  3
#   Lags (pmax):          8
#   Sample size:          191
# ----------------------------------------------------------------------
#   Test Results:
# ----------------------------------------------------------------------
#   Quantile (τ):         0.500
#   ρ(τ):                 0.998xxx
#   t_f(τ) statistic:     -1.xxxx
# ----------------------------------------------------------------------

Bootstrap Critical Values

from qfadf import qr_fourier_adf_bootstrap

# Bootstrap for critical values (recommended for inference)
boot_results = qr_fourier_adf_bootstrap(
    y, 
    model=1, 
    tau=0.5, 
    n_boot=1000,
    random_state=42
)

# Access results
print(f"Test statistic: {boot_results['tn']:.4f}")
print(f"Bootstrap p-value: {boot_results['p_value']:.4f}")
print(f"Critical values (1%, 5%, 10%): {boot_results['cv']}")
print(f"Reject H0 at 5%: {boot_results['reject_5pct']}")

Multiple Quantiles Analysis

from qfadf import qr_fourier_adf, qks_qcm_statistics, multiple_quantile_results_table

# Test at multiple quantiles
tau_values = [0.1, 0.25, 0.5, 0.75, 0.9]
results_list = []

for tau in tau_values:
    result = qr_fourier_adf(y, model=1, tau=tau, print_results=False)
    results_list.append(result)

# Create comparison table
comparison_df = multiple_quantile_results_table(results_list)
print(comparison_df)

# QKS and QCM statistics
qks_qcm = qks_qcm_statistics(y, model=1, tau_range=(0.1, 0.9))
print(f"QKS_f: {qks_qcm['QKS_f']:.4f}")
print(f"QCM_f: {qks_qcm['QCM_f']:.4f}")

Optimal Fourier Frequency

from qfadf import estimate_optimal_k

# Estimate optimal k using residual sum of squares criterion
k_opt = estimate_optimal_k(y, model=1, tau=0.5, k_max=5)
print(f"Optimal Fourier frequency: k = {k_opt}")

Model Specifications

Model 1: Constant Only

$$y_t = \phi \tilde{y}_{t-1} + \alpha_0 + \alpha_k \sin(2\pi kt/T) + \beta_k \cos(2\pi kt/T) + \mu_t$$

Model 2: Constant and Trend

$$y_t = \phi \tilde{y}_{t-1} + \alpha_0 + \alpha_k \sin(2\pi kt/T) + \beta_k \cos(2\pi kt/T) + \gamma t + \mu_t$$

Test Statistics

t_f(τ) Statistic

The quantile t-ratio test statistic at quantile τ:

$$t_f(\tau) = \frac{\hat{f}(F^{-1}(\tau))}{\sqrt{\tau(1-\tau)}} (Y'{-1} M_x Y{-1})^{1/2} (\hat{\phi} - 1)$$

QKS_f Statistic

$$QKS_f = \sup_{\tau \in \mathcal{T}} |t_f(\tau)|$$

QCM_f Statistic

$$QCM_f = \int_{\tau \in \mathcal{T}} t_f(\tau)^2 d\tau$$

API Reference

Main Functions

Critical Values

Utilities

Parameters

Output

The test returns a dictionary with:

• tn: Test statistic t_f(τ)
• rho_tau: Estimated AR coefficient ρ(τ)
• tau: Quantile level
• model: Model specification
• k: Fourier frequency
• n: Effective sample size
• cv: Critical values (if bootstrap)
• p_value: Bootstrap p-value (if bootstrap)
• reject_*pct: Decision at various significance levels

Citation

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

@article{li2018unit,
  title={Unit root quantile autoregression testing with smooth structural changes},
  author={Li, Haiqi and Zheng, Chaowen},
  journal={Finance Research Letters},
  volume={25},
  pages={83--89},
  year={2018},
  publisher={Elsevier},
  doi={10.1016/j.frl.2017.10.008}
}

Author

Dr Merwan Roudane

• Email: merwanroudane920@gmail.com
• GitHub: https://github.com/merwanroudane/quantilefourierunitroot

License

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

Related Packages

• Original GAUSS code by Saban Nazlioglu (TSPDLIB)
• statsmodels - Standard ADF tests
• arch - Unit root tests with various specifications

Changelog

Version 1.0.0 (2025)

• Initial release
• Implementation of QR-Fourier-ADF test
• Bootstrap critical values
• QKS and QCM statistics
• Pre-computed critical value tables
• Comprehensive documentation