Homoscedasticity Testing

Understanding Homoscedasticity

Homoscedasticity means residuals have constant variance across all predictor values.

Why It Matters

• Biased standard errors: Confidence intervals and p-values are incorrect
• Inefficient estimates: Coefficients are unbiased but not optimal
• Hypothesis testing issues: T-tests and F-tests may be invalid

Available Tests

Test	Best For
Breusch-Pagan	General heteroscedasticity detection
White	Detects complex forms of heteroscedasticity
Goldfeld-Quandt	Heteroscedasticity that increases/decreases monotonically

Example: Single Test

import pandas as pd
import numpy as np
from lrassume import check_homoscedasticity

np.random.seed(42)
X = pd.DataFrame({
    'x1': np.linspace(1, 100, 100),
    'x2': np.random.randn(100)
})
y = pd.Series(2 * X['x1'] + 3 * X['x2'] + np.random.randn(100))

# Run Breusch-Pagan test
test_results, summary = check_homoscedasticity(
    X, y,
    method="breusch_pagan",
    alpha=0.05
)

print(f"Conclusion: {summary['overall_conclusion']}")
print("\nTest Results:")
print(test_results)

Example: All Tests

# Run all three tests
test_results, summary = check_homoscedasticity(X, y, method="all")

print(f"Overall: {summary['overall_conclusion']}")
print(f"Tests agreeing: {summary['tests_in_agreement']}")
print("\nDetailed Results:")
print(test_results)

Expected Output:

Overall: homoscedastic
Tests agreeing: 3

Detailed Results:
            test  statistic  p_value     conclusion  significant
0  breusch_pagan      1.234   0.5391  homoscedastic        False
1          white      2.156   0.3407  homoscedastic        False
2  goldfeld_quandt 0.987   0.4123  homoscedastic        False

Using Pre-fitted Models

from sklearn.linear_model import LinearRegression

# Fit your model
model = LinearRegression().fit(X, y)

# Test with fitted model
test_results, summary = check_homoscedasticity(
    X, y,
    fitted_model=model,
    method="breusch_pagan"
)

Custom Significance Level

# 99% confidence (stricter)
test_results, summary = check_homoscedasticity(
    X, y,
    alpha=0.01
)

# 90% confidence (more lenient)
test_results, summary = check_homoscedasticity(
    X, y,
    alpha=0.10
)

Solutions for Heteroscedasticity

1. Transform the target variable

   # Log transformation
   y_log = np.log(y + 1)
   
   # Square root transformation
   y_sqrt = np.sqrt(y)

2. Weighted Least Squares (WLS)

   from statsmodels.regression.linear_model import WLS
   
   # Weight by inverse variance
   weights = 1 / residuals**2
   model_wls = WLS(y, X, weights=weights).fit()

3. Robust Standard Errors

   import statsmodels.api as sm
   
   model = sm.OLS(y, X).fit()
   robust_results = model.get_robustcov_results(cov_type='HC3')

4. Use a different model
   • Generalized Linear Models (GLM)
   • Quantile regression
   • Tree-based models

Visualizing Heteroscedasticity

import matplotlib.pyplot as plt
from sklearn.linear_model import LinearRegression

model = LinearRegression().fit(X, y)
residuals = y - model.predict(X)
fitted = model.predict(X)

plt.scatter(fitted, residuals)
plt.axhline(y=0, color='r', linestyle='--')
plt.xlabel('Fitted Values')
plt.ylabel('Residuals')
plt.title('Residual Plot')
plt.show()

Good pattern: Random scatter around zero
Bad pattern: Funnel shape (variance increases/decreases)