Breusch-Pagan Test

Quick Reference

Class: panelbox.validation.heteroskedasticity.breusch_pagan.BreuschPaganTest H₀: \(\text{Var}(\varepsilon_i) = \sigma^2\) (homoskedasticity) H₁: \(\text{Var}(\varepsilon_i) = h(X_i)\) (variance depends on regressors) Statistic: LM = \(nR^2_{\text{aux}}\) ~ \(\chi^2(k)\) Stata equivalent: estat hettest R equivalent: lmtest::bptest()

What It Tests

The Breusch-Pagan (1979) test is a Lagrange Multiplier (LM) test for heteroskedasticity that checks whether the variance of the errors depends on the values of the independent variables. It regresses the squared residuals on the original regressors and tests if the resulting coefficients are jointly zero.

Unlike the White test, the Breusch-Pagan test assumes a specific linear functional form for the variance function, making it a parametric test.

Quick Example

from panelbox import FixedEffects
from panelbox.datasets import load_grunfeld
from panelbox.validation.heteroskedasticity.breusch_pagan import BreuschPaganTest

# Estimate model
data = load_grunfeld()
fe = FixedEffects(data, "invest", ["value", "capital"], "firm", "year")
results = fe.fit()

# Run Breusch-Pagan test
test = BreuschPaganTest(results)
result = test.run(alpha=0.05)

print(f"LM statistic: {result.statistic:.3f}")
print(f"P-value:      {result.pvalue:.4f}")
print(f"Degrees of freedom: {result.df}")
print(result.conclusion)

# Metadata
print(f"R² (auxiliary): {result.metadata['R2_auxiliary']:.4f}")
print(f"N observations: {result.metadata['n_obs']}")
print(f"N regressors:   {result.metadata['n_regressors']}")

Interpretation

p-value	Decision	Interpretation
< 0.01	Strong rejection	Strong evidence that variance depends on regressors
0.01 -- 0.05	Rejection	Variance is not constant; use robust SE
0.05 -- 0.10	Borderline	Weak evidence; consider robust SE as precaution
> 0.10	Fail to reject	No evidence of heteroskedasticity of the assumed form

Important Caveat

Failing to reject does not prove homoskedasticity. The Breusch-Pagan test only detects heteroskedasticity of the specific functional form \(\sigma^2_i = h(X_i\gamma)\). The errors could still be heteroskedastic in a way not captured by this parametric model. For a more general test, use the White test.

Mathematical Details

Auxiliary Regression

Given model residuals \(\hat{e}_{it}\), the test estimates:

\[\hat{e}_{it}^2 = X_{it}\gamma + v_{it}\]

where \(X_{it}\) is the design matrix from the original regression (including the constant).

Hypotheses

\[H_0: \gamma = 0 \quad \text{(homoskedasticity)}\]

\[H_1: \gamma \neq 0 \quad \text{(variance depends on } X \text{)}\]

LM Statistic

\[LM = n \times R^2_{\text{aux}} \sim \chi^2(k)\]

where:

• \(n\) is the total number of observations
• \(R^2_{\text{aux}}\) is the R-squared from the auxiliary regression
• \(k\) is the number of regressors excluding the constant (degrees of freedom)

Intuition

If \(\gamma = 0\) (homoskedasticity), the squared residuals should not be systematically related to \(X\). A high \(R^2_{\text{aux}}\) indicates that \(X\) explains a significant portion of the variation in \(\hat{e}^2\), suggesting the error variance depends on \(X\).

Configuration Options

Parameter	Type	Default	Description
alpha	float	0.05	Significance level

Result Metadata

Key	Type	Description
R2_auxiliary	float	R-squared from auxiliary regression
n_obs	int	Number of observations
n_regressors	int	Number of regressors in design matrix

Diagnostics

Comparing Breusch-Pagan and White Tests

Running both tests helps assess the nature of heteroskedasticity:

from panelbox.validation.heteroskedasticity.breusch_pagan import BreuschPaganTest
from panelbox.validation.heteroskedasticity.white import WhiteTest

bp_result = BreuschPaganTest(results).run()
white_result = WhiteTest(results).run(cross_terms=True)

print(f"Breusch-Pagan: LM={bp_result.statistic:.3f}, p={bp_result.pvalue:.4f}")
print(f"White test:    LM={white_result.statistic:.3f}, p={white_result.pvalue:.4f}")

Reading the Comparison

• Both reject: Heteroskedasticity present, linear form is sufficient to detect it
• BP rejects, White does not: The variance has a simple linear relationship with \(X\); White test has lower power due to extra parameters
• BP does not reject, White rejects: Heteroskedasticity exists but in a nonlinear form (squares or cross-products of \(X\))
• Neither rejects: No evidence of heteroskedasticity (or both lack power)

Common Pitfalls

Common Pitfalls

1. Design matrix required: The test needs the original design matrix \(X\). If the model does not store it, a ValueError is raised.
2. Parametric assumption: The BP test assumes \(\text{Var}(\varepsilon) = h(X\gamma)\). It misses heteroskedasticity that depends on nonlinear functions of \(X\) or omitted variables. Use the White test for a more general alternative.
3. Constant column detection: The test automatically detects and adjusts for a constant column in the design matrix. Degrees of freedom are \(k - 1\) if a constant is present, \(k\) otherwise.
4. Non-negative LM: The LM statistic is clamped to be non-negative (\(\geq 0\)). In rare numerical cases where \(R^2\) is slightly negative, the statistic is set to zero.
5. Large samples: With many observations, even trivially small heteroskedasticity becomes statistically significant. Always pair the test with practical measures of heteroskedasticity severity (e.g., the Modified Wald variance ratio).

See Also

• Heteroskedasticity Tests Overview -- comparison of all tests
• Modified Wald Test -- groupwise heteroskedasticity for FE models
• White Test -- model-free heteroskedasticity test
• Robust Standard Errors -- HC0--HC3 corrections

References

• Breusch, T. S., & Pagan, A. R. (1979). "A simple test for heteroscedasticity and random coefficient variation." Econometrica, 47(5), 1287-1294.
• Greene, W. H. (2018). Econometric Analysis (8^th ed.). Pearson.
• Koenker, R. (1981). "A note on studentizing a test for heteroscedasticity." Journal of Econometrics, 17(1), 107-112.