BoundedTest: GLS-based Unit Root Tests for Bounded Processes

A Python implementation of unit root tests for bounded time series based on Carrion-i-Silvestre and Gadea (2013).

Overview

boundedtest provides comprehensive tools for testing unit roots in time series that are constrained within bounds. The package implements:

• GLS-detrending with bound-specific non-centrality parameter κ̅(c,c̄)
• OLS-detrending for comparison
• M-type test statistics: MZα, MSB, and MZt
• Parametric and non-parametric long-run variance estimation
• Critical values tailored to different bound configurations

Reference

This package implements the methodology from:

Carrion-i-Silvestre, J.L. and Gadea, M.D. (2013). "GLS-based unit root tests for bounded processes." Economics Letters, 120(2), 184-187.

Installation

pip install boundedtest

Or install from source:

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

Quick Start

import numpy as np
from boundedtest import bounded_unit_root_test

# Generate bounded data
np.random.seed(42)
data = np.cumsum(np.random.randn(200))
data = np.clip(data, -10, 10)  # Apply bounds

# Test for unit root with GLS-BOUNDS method (recommended)
result = bounded_unit_root_test(
    data=data,
    bounds=(-10, 10),
    statistic='mz_alpha',
    detrending='gls_bounds',
    lrv_method='np'
)

print(result)

Main Features

1. Detrending Methods

• 'ols': Standard OLS detrending
• 'gls_ers': GLS detrending with standard κ = -7 (Elliott et al., 1996)
• 'gls_bounds': GLS detrending with bound-specific κ̅(c,c̄) (recommended)

2. Test Statistics

• MZα: Modified Phillips-Perron statistic
• MSB: Modified Sargan-Bhargava statistic
• MZt: Modified t-statistic

3. LRV Estimation

• Non-parametric ('np'): Newey-West with automatic bandwidth selection
• Parametric ('ar'): AR-based with MAIC lag selection (Ng & Perron, 2001)

Examples

Example 1: Basic Usage

import numpy as np
from boundedtest import bounded_unit_root_test

# Generate data
np.random.seed(123)
T = 200
data = np.cumsum(np.random.randn(T))
bounds = (-5, 5)
data = np.clip(data, bounds[0], bounds[1])

# Run test
result = bounded_unit_root_test(
    data=data,
    bounds=bounds,
    detrending='gls_bounds'
)

print(result)
print(f"\nReject H0 at 5%: {result.reject_5pct}")

Example 2: Compare All Test Statistics

results = bounded_unit_root_test(
    data=data,
    bounds=bounds,
    statistic='all',  # Compute all statistics
    detrending='gls_bounds'
)

# Results is a dictionary
for stat_name, result in results.items():
    print(f"\n{stat_name}:")
    print(f"  Statistic: {result.statistic:.4f}")
    print(f"  Critical (5%): {result.critical_values['5%']:.4f}")
    print(f"  Reject H0: {result.reject_5pct}")

Example 3: Compare Detrending Methods

methods = ['ols', 'gls_ers', 'gls_bounds']

for method in methods:
    result = bounded_unit_root_test(
        data=data,
        bounds=bounds,
        statistic='mz_alpha',
        detrending=method
    )
    print(f"\n{method.upper()}:")
    print(f"  MZα = {result.statistic:.4f}")
    print(f"  Reject H0 = {result.reject_5pct}")

Example 4: With Pandas DataFrame

import pandas as pd

# Create DataFrame
df = pd.DataFrame({
    'date': pd.date_range('2020-01-01', periods=200, freq='D'),
    'series': data
})

# Test using the series
result = bounded_unit_root_test(
    data=df['series'],
    bounds=bounds,
    detrending='gls_bounds'
)

Example 5: Unemployment Rate (0-100% bounded)

# Unemployment rate bounded between 0 and 100
unemployment = np.random.uniform(3, 10, size=100)  # Example data
unemployment += np.cumsum(np.random.randn(100)) * 0.2

result = bounded_unit_root_test(
    data=unemployment,
    bounds=(0, 100),
    detrending='gls_bounds'
)

Understanding Results

The output includes:

• Test Statistic Value: The computed test statistic
• Critical Values: At 1%, 5%, and 10% significance levels
• Rejection Decision: Whether to reject the unit root null hypothesis
• Bounds Information: Original bounds and estimated c-parameters
• Non-centrality Parameter (κ̅): Used in GLS detrending
• LRV Estimate: Estimated long-run variance

Interpretation

• MZα and MZt: Left-tail tests. Reject H₀ if statistic < critical value.
• MSB: Right-tail test. Reject H₀ if statistic > critical value.
• H₀: Series has a unit root (is I(1))
• H₁: Series is stationary (is I(0))

Advanced Usage

Generating Bounded Processes

from boundedtest.regulated_ou import generate_bounded_ar1

# Generate bounded AR(1) process
data = generate_bounded_ar1(
    T=200,
    bounds=(-5, 5),
    rho=1.0,  # Unit root
    sigma=1.0,
    burnin=100,
    seed=42
)

Custom Critical Values via Simulation

from boundedtest.regulated_ou import compute_critical_values

# Compute critical values for specific bounds
cv = compute_critical_values(
    c_lower=-0.5,
    c_upper=0.5,
    kappa=0,
    n_sim=10000,
    alpha=0.05
)

print(cv)

Accessing Individual Components

from boundedtest import (
    ols_detrend,
    gls_detrend,
    compute_mz_alpha,
    estimate_lrv_np,
    get_kappa
)

# Manual detrending
y_tilde, _, _ = ols_detrend(data)

# Compute LRV
from boundedtest.statistics import compute_ar1_residuals
residuals = compute_ar1_residuals(y_tilde)
s2 = estimate_lrv_np(residuals)

# Compute statistic
mz_alpha = compute_mz_alpha(y_tilde, s2)

API Reference

Main Function

bounded_unit_root_test(
    data,           # array-like: Time series data
    bounds,         # tuple: (lower_bound, upper_bound)
    statistic='mz_alpha',  # str: Test statistic to compute
    detrending='gls_bounds',  # str: Detrending method
    lrv_method='np',  # str: LRV estimation method
    deterministics='constant'  # str: Deterministic components
) -> BoundedTestResult

Monte Carlo Simulations

The package includes utilities for Monte Carlo experiments:

from boundedtest.regulated_ou import simulate_bounded_process

# Simulate under H0
data_h0 = simulate_bounded_process(
    T=200,
    b_lower=-5,
    b_upper=5,
    alpha=1.0,  # Unit root
    sigma=1.0
)

# Simulate under H1  
data_h1 = simulate_bounded_process(
    T=200,
    b_lower=-5,
    b_upper=5,
    alpha=0.95,  # Stationary
    sigma=1.0
)

Requirements

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

Citation

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

@article{carrion2013gls,
  title={GLS-based unit root tests for bounded processes},
  author={Carrion-i-Silvestre, Josep Llu{\'\i}s and Gadea, Mar{\'\i}a Dolores},
  journal={Economics Letters},
  volume={120},
  number={2},
  pages={184--187},
  year={2013},
  publisher={Elsevier}
}

License

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

Author

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

Contributing

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

Acknowledgments

This implementation is based on the methodology developed by Josep Lluís Carrion-i-Silvestre and María Dolores Gadea. The original MATLAB code and methodological guidance were instrumental in developing this Python package.

See Also

• Original paper: Economics Letters (2013)
• Elliott, Rothenberg & Stock (1996): "Efficient Tests for an Autoregressive Unit Root"
• Ng & Perron (2001): "Lag Length Selection and the Construction of Unit Root Tests"
• Cavaliere (2005): "Limited Time Series with a Unit Root"