dyncusum

CUSUM Tests for Structural Change in Dynamic Models

A Python library implementing the CUSUM tests for structural change as described in:

Krämer, W., Ploberger, W., & Alt, R. (1988). "Testing for Structural Change in Dynamic Models." Econometrica, Vol. 56, No. 6, pp. 1355-1369.

Overview

This library provides a complete implementation of the CUSUM (Cumulative Sum) test for detecting structural change in linear regression models with lagged dependent variables. The implementation follows the original paper exactly, including:

1. Dynamic CUSUM Test: The straightforward CUSUM test applied to dynamic models
2. Dufour Test: Modified procedure from Dufour (1982) that first eliminates dynamics
3. Critical Values: Both tabulated values from BDE (1975) and Monte Carlo simulation
4. Power Analysis: Tools to analyze test power under various structural change scenarios
5. Visualization: Publication-quality plots for CUSUM processes
6. Tables: Formatted output for academic publications

Installation

pip install dyncusum

Or install from source:

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

Quick Start

Basic Usage

import numpy as np
from dyncusum import dynamic_cusum_test, plot_cusum, print_test_summary

# Generate sample data
np.random.seed(42)
T = 100
y = np.zeros(T)
u = np.random.normal(0, 1, T)

# AR(1) process: y_t = 0.5 * y_{t-1} + u_t
for t in range(1, T):
    y[t] = 0.5 * y[t-1] + u[t]

# Prepare data for test
y_current = y[1:]
y_lagged = y[:-1]
X = np.ones((T-1, 1))  # Constant term

# Perform the Dynamic CUSUM test
result = dynamic_cusum_test(y_current, X, y_lagged, significance_level=0.05)

# Print summary
print_test_summary(result)

# Plot the CUSUM process
fig = plot_cusum(result, save_path='cusum_plot.png')

Comparing Tests

from dyncusum import dynamic_cusum_test, dufour_test, plot_comparison

# Run both tests
result_dc = dynamic_cusum_test(y_current, X, y_lagged)
result_duf = dufour_test(y_current, X, y_lagged)

# Compare results
fig = plot_comparison(result_dc, result_duf, save_path='comparison.png')

Theoretical Background

The Model

The library considers the dynamic linear regression model:

$$y_t = \gamma y_{t-1} + \beta_1 x_{t1} + \cdots + \beta_K x_{tK} + u_t \quad (t = 1, \ldots, T)$$

where:

• $y_t$ is the dependent variable
• $\gamma$ is the coefficient of the lagged dependent variable ($|\gamma| < 1$)
• $x_{t1}, \ldots, x_{tK}$ are exogenous regressors
• $u_t$ are iid$(0, \sigma^2)$ disturbances

The CUSUM Test

The test statistic is based on recursive residuals:

$$w_r = (y_r - z'_r \hat{\delta}^{(r-1)}) / f_r$$

where $f_r = (1 + z'_r(Z^{(r-1)'}Z^{(r-1)})^{-1}z_r)^{1/2}$

The CUSUM process is:

$$W^{(r)} = \frac{1}{\hat{\sigma}} \sum_{t=K+2}^{r} w_t$$

And the test statistic is:

$$S = \max_{K+1 < r \leq T} \left| \frac{W^{(r)}}{\sqrt{T-K-1}} \right| / \left(1 + 2\frac{r-K-1}{T-K-1}\right)$$

Key Theorems

Theorem 1: Both the Dynamic CUSUM test and the Dufour test retain their asymptotic nominal significance levels in dynamic models.

Theorem 2: If the structural change $g(z)$ is orthogonal to the mean regressor $d$ for almost all $z$, the limiting rejection probability equals the nominal significance level (no power).

Critical Values

Critical values are based on Brownian motion theory (BDE, 1975):

API Reference

Core Functions

dynamic_cusum_test(y, X, y_lagged=None, significance_level=0.05, sigma_method='harvey', n_simulations=10000)

Perform the Dynamic CUSUM test for structural change.

Parameters:

• y: Dependent variable vector
• X: Exogenous regressors matrix
• y_lagged: Lagged dependent variable (optional)
• significance_level: Significance level (default: 0.05)
• sigma_method: 'harvey' or 'ols' (default: 'harvey')
• n_simulations: Monte Carlo simulations for p-value

Returns: CUSUMResult object

dufour_test(y, X, y_lagged, significance_level=0.05, sigma_method='harvey', n_simulations=10000)

Perform the Dufour (1982) modified CUSUM test.

Power Analysis

monte_carlo_power(gamma, psi, z_star, b, T=120, n_simulations=1000, ...)

Compute power via Monte Carlo simulation.

replicate_table_1(T=120, n_simulations=1000, ...)

Replicate Table I from the original paper.

Visualization

plot_cusum(result, title=None, figsize=(10, 6), ...)

Plot the CUSUM process with critical boundaries.

create_summary_figure(result, figsize=(14, 10), ...)

Create a comprehensive summary figure with multiple panels.

Examples

Detecting Structural Change

import numpy as np
from dyncusum import dynamic_cusum_test, generate_dynamic_data

# Generate data with a structural break at t = 60
T = 100
gamma = 0.5
beta = np.array([2, 10])

# Break: coefficients change by 20% after observation 60
structural_break = {
    'z_star': 0.6,  # Break at 60% of sample
    'delta_delta': np.array([0, 0.4, 2.0])  # Change in [γ, β_1, β_2]
}

y, X, y_lagged = generate_dynamic_data(T, gamma, beta, 
                                        structural_break=structural_break, 
                                        seed=42)

# Test for structural change
result = dynamic_cusum_test(y, X, y_lagged)
print(f"Test Statistic: {result.statistic:.4f}")
print(f"P-value: {result.p_value:.4f}")
print(f"Reject H0: {result.reject_null}")

Power Analysis

from dyncusum import analyze_power_vs_angle, plot_power_curve

# Analyze how power varies with the angle ψ
results = analyze_power_vs_angle(gamma=0.0, z_star=0.5, b=12, T=120)

# Plot power curve
fig = plot_power_curve(
    np.array(results['angles']), 
    np.array(results['powers']),
    save_path='power_curve.png'
)

Replicating Paper Results

The library can replicate the Monte Carlo results from Table I:

from dyncusum import replicate_table_1, format_power_table

# Note: This takes several hours with default settings
results = replicate_table_1(
    T=120,
    n_simulations=100,  # Use 1000 for publication
    gamma_values=[-0.5, 0, 0.5],
    psi_values=[0, 30, 60, 90],
    z_star_values=[0.3, 0.5, 0.7],
    b_values=[3, 6, 12]
)

# Format as table
print(format_power_table(results))

Citation

If you use this library in your research, please cite both the original paper and this implementation:

@article{kramer1988testing,
  title={Testing for Structural Change in Dynamic Models},
  author={Kr{\"a}mer, Walter and Ploberger, Werner and Alt, Raimund},
  journal={Econometrica},
  volume={56},
  number={6},
  pages={1355--1369},
  year={1988}
}

@software{roudane2024dyncusum,
  author = {Roudane, Merwan},
  title = {dyncusum: CUSUM Tests for Structural Change in Dynamic Models},
  year = {2024},
  url = {https://github.com/merwanroudane/cusum}
}

References

• Brown, R. L., Durbin, J., & Evans, J. M. (1975). Techniques for Testing the Constancy of Regression Relationships over Time. Journal of the Royal Statistical Society, Series B, 37, 149-163.
• Dufour, J. M. (1982). Recursive Stability Analysis of Linear Regression Relationships. Journal of Econometrics, 19, 31-76.
• Harvey, A. (1975). Comment on the Paper by Brown, Durbin and Evans. Journal of the Royal Statistical Society, B, 37, 179-180.

License

MIT License - see LICENSE file for details.

Author

Dr. Merwan Roudane

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

Contributing

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