cpr-econometrics 0.0.2

Residual-based Cointegration and Non-Cointegration Tests for Cointegrating Polynomial Regressions

CPR: Cointegrating Polynomial Regressions

A Python library for residual-based cointegration and non-cointegration tests for cointegrating polynomial regressions (CPRs).

This package implements the econometric methods developed in:

Wagner, M. (2023). Residual-based cointegration and non-cointegration tests for cointegrating polynomial regressions. Empirical Economics, 65, 1-31. https://doi.org/10.1007/s00181-022-02332-3

Features

• FM-OLS Estimation: Fully Modified OLS estimator for cointegrating polynomial regressions following Wagner & Hong (2016)
• CT Test: KPSS-Shin type test with null hypothesis of cointegration
• PU Test: Phillips-Ouliaris type test with null hypothesis of no cointegration
• Long-run Variance Estimation: Multiple kernel functions (Bartlett, Parzen, Quadratic Spectral, etc.) with automatic bandwidth selection (Andrews 1991, Newey-West 1994, Andrews-Monahan 1992)
• Critical Values: Pre-computed critical values for up to 4 integrated regressors and power 4
• Publication-Ready Output: Formatted output suitable for academic publications

Installation

pip install cpr-econometrics

Or install from source:

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

Quick Start

Basic Example: Environmental Kuznets Curve

import numpy as np
from cpr import fm_cpr, ct_test, pu_test, CPRTestSummary

# Generate sample data (simulating EKC relationship)
np.random.seed(42)
T = 150  # Typical sample size for annual macro data

# GDP per capita (I(1) process)
gdp = np.cumsum(np.random.randn(T) * 0.1) + 10

# Generate AR(1) errors (serially correlated as in real economic data)
errors = np.zeros(T)
for t in range(1, T):
    errors[t] = 0.5 * errors[t-1] + np.random.randn() * 0.3

# Emissions with quadratic relationship (inverted U-shape EKC)
emissions = 2.0 + 0.5 * gdp - 0.02 * gdp**2 + errors

# Create deterministic components (intercept and trend)
deter = np.column_stack([np.ones(T), np.arange(1, T + 1)])

# FM-OLS estimation
result = fm_cpr(
    y=emissions, 
    x=gdp.reshape(-1, 1), 
    orders=2,  # Quadratic specification
    deter=deter,
    kern='ba',  # Bartlett kernel
    band='NW'   # Newey-West bandwidth
)

print("FM-OLS Coefficients for polynomial terms:", result.beta_fm)
print("t-statistics:", result.t_beta)

# CT Test (H0: Cointegration)
ct_result = ct_test(
    u_plus=result.u_plus, 
    omega=result.Omega_udotv, 
    d=1,  # intercept and trend
    m=1,  # 1 integrated regressor
    p=2   # quadratic specification
)
print(ct_result)

# PU Test (H0: No Cointegration)
pu_result = pu_test(
    y=emissions, 
    x=gdp.reshape(-1, 1), 
    d=1, 
    m=1, 
    orders=2
)
print(pu_result)

# Combined Summary
summary = CPRTestSummary(ct_result, pu_result)
print(summary)

Expected Results:

• CT statistic: ~0.03-0.10 (should be < 0.106 for cointegration at 5%)
• PU statistic: ~50-150 (should be > 52.95 for cointegration at 5%)

API Reference

FM-OLS Estimation

from cpr import fm_cpr

result = fm_cpr(
    y,                    # Dependent variable (T,)
    x,                    # Integrated regressors (T, m)
    orders,               # Polynomial orders (int or list)
    w=None,               # Stationary regressors (optional)
    deter=None,           # Deterministic terms (optional)
    kern='ba',            # Kernel: 'tr', 'ba', 'pa', 'bo', 'da', 'qs'
    band='NW',            # Bandwidth: 'And91', 'NW', 'AM92', or numeric
    deme=0                # Demeaning: 0 or 1
)

# Results include:
# - result.beta_fm: FM-OLS coefficients for polynomial terms
# - result.t_beta: t-statistics
# - result.u_plus: FM-OLS residuals
# - result.Omega_udotv: Conditional long-run variance

CT Test (Cointegration Test)

from cpr import ct_test

ct_result = ct_test(
    u_plus,               # FM-OLS residuals
    omega,                # Long-run variance estimate
    d,                    # Deterministics: -1, 0, or 1
    m,                    # Number of integrated regressors (1-4)
    p                     # Highest power (1-4)
)

# Interpretation:
# - Reject H0 (cointegration) if statistic > critical value
# - Low statistic → evidence FOR cointegration

PU Test (Non-Cointegration Test)

from cpr import pu_test

pu_result = pu_test(
    y,                    # Dependent variable
    x,                    # Integrated regressors
    d,                    # Deterministics: -1, 0, or 1
    m,                    # Number of integrated regressors
    orders,               # Polynomial orders
    kern='ba',            # Kernel function
    band='NW'             # Bandwidth
)

# Interpretation:
# - Reject H0 (no cointegration) if statistic > critical value
# - High statistic → evidence FOR cointegration

Deterministic Specifications

d	Description	Model
-1	No deterministics	y = β₁x + β₂x² + ... + u
0	Intercept only	y = c + β₁x + β₂x² + ... + u
1	Intercept and trend	y = c + δt + β₁x + β₂x² + ... + u

Kernel Functions

Code	Kernel	Use Case
'tr'	Truncated	Not recommended
'ba'	Bartlett	Standard choice (default)
'pa'	Parzen	Alternative to Bartlett
'bo'	Bohman	Smooth kernel
'da'	Daniell	Spectral analysis
'qs'	Quadratic Spectral	Optimal for some purposes

Bandwidth Selection

Code	Method	Reference
'NW'	Newey-West (1994)	Default, data-driven
'And91'	Andrews (1991)	AR(1)-based
'AM92'	Andrews-Monahan (1992)	VAR prewhitening
Numeric	Fixed bandwidth	User-specified

Decision Rule for EKC Analysis

Following Wagner (2023):

CT Test	PU Test	Interpretation
Fail to reject	Reject	Evidence FOR cointegration
Reject	Fail to reject	Evidence AGAINST cointegration
Reject	Reject	Conflicting evidence
Fail to reject	Fail to reject	Conflicting evidence

Critical Values

Critical values are embedded in the package for:

• m = 1, 2, 3, 4 integrated regressors
• p = 1, 2, 3, 4 polynomial powers
• d = -1, 0, 1 deterministic specifications
• Significance levels: 1%, 2.5%, 5%, 10%

from cpr import get_ct_critical_value, get_pu_critical_value

# Get specific critical value
cv_ct = get_ct_critical_value(d=1, m=1, p=2, alpha=0.05)
cv_pu = get_pu_critical_value(d=1, m=1, p=2, alpha=0.05)

Applications

This package is suitable for:

• Environmental Kuznets Curve (EKC) analysis
• Material Kuznets Curve (MKC) studies
• Intensity-of-use analysis
• Exchange rate target-zone models
• Any nonlinear cointegrating relationship involving powers of I(1) variables

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{wagner2023residual,
  title={Residual-based cointegration and non-cointegration tests for cointegrating polynomial regressions},
  author={Wagner, Martin},
  journal={Empirical Economics},
  volume={65},
  pages={1--31},
  year={2023},
  publisher={Springer},
  doi={10.1007/s00181-022-02332-3}
}

License

MIT License - see LICENSE for details.

Author

Dr Merwan Roudane
Email: [e-mail]
GitHub: https://github.com/merwanroudane/cpr

Acknowledgments

This implementation is based on the MATLAB code accompanying Wagner (2023) and follows the econometric methodology developed in Wagner & Hong (2016).