nearkpss

Modified KPSS Tests for Near Integration

A Python implementation of the Modified KPSS test for near integration proposed by Harris, Leybourne, and McCabe (2007).

Reference

Harris, D., Leybourne, S., & McCabe, B. (2007). Modified KPSS Tests for Near Integration. Econometric Theory, 23(2), 355-363. DOI: 10.1017/S0266466607070156

Overview

The Modified KPSS test addresses the problem of testing whether a strongly autocorrelated time series is best described as:

• Near-integrated (stationary with largest root near to one), or
• Integrated (having a unit root)

The standard KPSS test (Kwiatkowski et al., 1992) suffers from severe size distortions when applied to near-integrated processes. The Modified KPSS test resolves this issue by applying a GLS-type transformation that removes the near unit root under the null hypothesis.

Key Features

• Exact implementation matching the Harris, Leybourne, and McCabe (2007) paper
• Controlled asymptotic size under near-integration null hypothesis
• Monte Carlo critical values simulation capability
• Standard KPSS test included for comparison
• Professional output suitable for top-tier journal publications
• Automatic bandwidth selection using Newey-West (1994)
• Multiple kernel functions: Quadratic Spectral (QS), Bartlett, Parzen

Installation

pip install nearkpss

Or install from source:

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

Quick Start

import numpy as np
from nearkpss import modified_kpss_test

# Generate sample data (near-integrated process)
np.random.seed(42)
T = 200
c = 10  # Local-to-unity parameter
rho = 1 - c/T
y = np.zeros(T)
for t in range(1, T):
    y[t] = rho * y[t-1] + np.random.normal()

# Perform the Modified KPSS test
result = modified_kpss_test(y, c_bar=10.0, detrend="c")
print(result)

Output:

============================================================
        Modified KPSS Test for Near Integration
        Harris, Leybourne, and McCabe (2007)
============================================================

Null Hypothesis: The series is near-integrated (stationary)
                 H₀: c ≥ c̄ = 10.0
Alternative:     The series has a unit root (H₁: c = 0)

Detrending:      Level (constant only)
Observations:    200

------------------------------------------------------------
Test Statistic:  0.084521 
P-value:         0.6234
------------------------------------------------------------

Critical Values:
  1%:   0.7390
  5%:   0.4630
  10%:  0.3470

Estimation Details:
  Bandwidth:           5.42
  Long-run variance:   1.023456

============================================================
Conclusion: Cannot reject H₀.
           Series appears near-integrated/stationary.
============================================================

Theoretical Background

The Model

Consider the data generating process (DGP):

$$y_t = \mu + w_t, \quad t = 1, \ldots, T$$

$$w_t = \rho_{c,T} w_{t-1} + v_t, \quad t = 2, \ldots, T$$

where $\rho_{c,T} = 1 - c/T$ and $c \geq 0$. The innovation $v_t$ is a stationary process.

Hypotheses

The Modified KPSS test examines:

• Null hypothesis: $H_0: c \geq \bar{c} > 0$ (near integration)
• Alternative: $H_1: c = 0$ (unit root)

where $\bar{c}$ specifies the minimal amount of mean reversion under the stationary null hypothesis.

The Test Statistic

The Modified KPSS statistic is constructed by:

1. GLS Transformation: Apply $y_t - \rho_{\bar{c},T} y_{t-1}$ where $\rho_{\bar{c},T} = 1 - \bar{c}/T$

2. OLS Demeaning: Compute residuals $r_{\bar{c},t} = (y_t - \rho_{\bar{c},T} y_{t-1}) - \bar{m}_{\bar{c}}$

3. Test Statistic:

$$S^{\mu}(\bar{c}) = \frac{T^{-2} \sum_{t=2}^{T} \left(\sum_{i=2}^{t} r_{\bar{c},i}\right)^2}{\hat{\omega}_{\bar{c}}^2}$$

where $\hat{\omega}_{\bar{c}}^2$ is the long-run variance estimator.

Asymptotic Distribution (Theorem 1)

Under the null hypothesis, the limiting distribution is:

$$S^{\mu}(\bar{c}) \Rightarrow \int_0^1 H_{\alpha,c,\bar{c}}(r)^2 , dr$$

When $\bar{c} = c > 0$ (boundary of null), this reduces to the standard KPSS distribution.

API Reference

Main Functions

modified_kpss_test(y, c_bar=10.0, detrend="c", kernel="qs", bandwidth=None)

Perform the Modified KPSS test for near integration.

Parameters:

Returns: ModifiedKPSSResult object

standard_kpss_test(y, detrend="c", kernel="qs", bandwidth=None)

Perform the standard KPSS test for comparison.

Classes

ModifiedKPSS

Class-based interface for the Modified KPSS test.

from nearkpss import ModifiedKPSS

mkpss = ModifiedKPSS(c_bar=10.0, detrend="c", kernel="qs")
result = mkpss.test(y)

Critical Values and Simulation

get_critical_values(c_bar=10.0, detrend="c")

Get critical values for the test.

from nearkpss import get_critical_values

cv = get_critical_values(c_bar=10.0, detrend="c")
print(f"5% critical value: {cv.cv_5pct}")

simulate_critical_values(c, c_bar, alpha, detrend, n_replications, n_steps)

Simulate the asymptotic distribution from Theorem 1.

from nearkpss import simulate_critical_values

# Simulate 10,000 draws from the asymptotic distribution
stats = simulate_critical_values(
    c=10.0,       # True c value
    c_bar=10.0,   # Hypothesized c_bar
    alpha=1.0,    # Initial value parameter
    detrend="c",
    n_replications=10000,
    n_steps=5000
)

# Compute critical values
cv_5pct = np.percentile(stats, 95)
print(f"5% critical value: {cv_5pct:.4f}")

Examples

Example 1: Testing Real Economic Data

import numpy as np
from nearkpss import modified_kpss_test, standard_kpss_test

# Simulate GDP-like series (trending with unit root)
np.random.seed(123)
T = 200
trend = 0.01 * np.arange(T)
y = trend + np.cumsum(np.random.normal(0, 0.5, T))

# Modified KPSS test (trend case)
result_mod = modified_kpss_test(y, c_bar=10.0, detrend="ct")
print("Modified KPSS Test:")
print(result_mod)

# Compare with standard KPSS
result_std = standard_kpss_test(y, detrend="ct")
print("\nStandard KPSS Test:")
print(result_std)

Example 2: Monte Carlo Size Simulation

import numpy as np
from nearkpss import modified_kpss_test
from nearkpss.utils import simulate_near_integrated_ma

# Replicate Table 1 from the paper
def monte_carlo_size(T=200, c=10.0, c_bar=10.0, theta=0.0, 
                     alpha=1.0, n_sims=1000, seed=42):
    np.random.seed(seed)
    rejections = 0
    
    for _ in range(n_sims):
        y = simulate_near_integrated_ma(T, c, theta, alpha)
        result = modified_kpss_test(y, c_bar=c_bar, compute_pvalue=False)
        if result.reject_5pct:
            rejections += 1
    
    return rejections / n_sims

# Test with different MA parameters (as in Table 1)
for theta in [0.0, 0.6, -0.6]:
    size = monte_carlo_size(theta=theta)
    print(f"θ = {theta:4.1f}: Empirical size = {size:.3f}")

Example 3: Power Analysis

import numpy as np
import matplotlib.pyplot as plt
from nearkpss import simulate_critical_values
from nearkpss.critical_values import simulate_power_and_size

# Simulate power curves as in Figure 1
c_values = np.arange(0, 11)
rejection_rates, cv = simulate_power_and_size(
    c_values, 
    c_bar=10.0, 
    alpha=1.0,
    nominal_power=0.50,
    n_replications=5000,
    n_steps=2000
)

plt.figure(figsize=(8, 5))
plt.plot(c_values, rejection_rates, 'b-', linewidth=2)
plt.xlabel('c (local-to-unity parameter)')
plt.ylabel('Rejection Rate')
plt.title('Asymptotic Size and Power of Modified KPSS Test')
plt.axhline(y=0.50, color='r', linestyle='--', label='Power at c=0')
plt.grid(True, alpha=0.3)
plt.legend()
plt.tight_layout()
plt.savefig('power_curve.png', dpi=150)

Comparison with Paper Results

Table 1 Replication (Empirical Sizes at 5% Level)

The Modified KPSS test maintains size close to the nominal level across different specifications, unlike the optimal tests Q^μ(10) and Q^μ(10, 3.8) which show size distortions.

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

Author

Dr Merwan Roudane

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

License

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

Citation

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

@software{nearkpss,
  author = {Roudane, Merwan},
  title = {nearkpss: Modified KPSS Tests for Near Integration in Python},
  year = {2024},
  url = {https://github.com/merwanroudane/modifiedkpss}
}

And the original paper:

@article{harris2007modified,
  title={Modified KPSS Tests for Near Integration},
  author={Harris, David and Leybourne, Stephen and McCabe, Brendan},
  journal={Econometric Theory},
  volume={23},
  number={2},
  pages={355--363},
  year={2007},
  publisher={Cambridge University Press},
  doi={10.1017/S0266466607070156}
}

References

• Harris, D., Leybourne, S., & McCabe, B. (2007). Modified KPSS Tests for Near Integration. Econometric Theory, 23(2), 355-363.
• Kwiatkowski, D., Phillips, P., Schmidt, P., & Shin, Y. (1992). Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics, 54, 159-178.
• Müller, U. (2005). Size and power of tests for stationarity in highly autocorrelated time series. Journal of Econometrics, 128, 195-213.
• Newey, W. & West, K. (1994). Automatic lag selection in covariance matrix estimation. Review of Economic Studies, 61, 631-653.
• Sul, D., Phillips, P.C.B., & Choi, C. (2005). Prewhitening bias in HAC estimation. Oxford Bulletin of Economics and Statistics, 67, 517-546.