Breusch-Godfrey Test

Overview

Definition

The Breusch-Godfrey Test is a Lagrange Multiplier test for autocorrelation in the errors in a regression model. Unlike the Durbin-Watson Test, it is valid in the presence of lagged dependent variables and can test for higher-order serial correlation (up to lag $p$ ).

1. Procedure

Estimate the OLS regression: $Y_{t} = X_{t} β + u_{t}$
Obtain residuals: ${\hat{u}}_{t}$
Run auxiliary regression: Regress ${\hat{u}}_{t}$ on original $X_{t}$ and lagged residuals ${\hat{u}}_{t - 1}, \dots, {\hat{u}}_{t - p}$ .
Calculate $L M = (N - p) R^{2}$ from the auxiliary regression.

2. Comparison: BG vs. DW

Feature	Durbin-Watson	Breusch-Godfrey
Order defined	First-order (AR(1)) only	Any order $p$
Lagged Dependent Variables	Invalid (Biased)	Valid
Distribution	Exact Bounds tables	Asymptotic Chi-Square

Recommendation: Breusch-Godfrey is generally preferred for its flexibility.

3. Python Implementation Example

import statsmodels.stats.diagnostic as dg

# Perform Test (up to lag 2)
# Results: [LM Stat, LM p-value, F-Stat, F p-value]
bg_test = dg.acorr_breusch_godfrey(model, nlags=2)

print(f"LM Statistic: {bg_test[0]:.3f}")
print(f"LM p-value: {bg_test[1]:.4f}")

Durbin-Watson Test - Simpler alternative.
Multiple Linear Regression - Framework.