White Test

Overview

Definition

The White Test is a general statistical test used to detect heteroscedasticity in a regression model. Unlike the Breusch-Pagan test, it does not assume any specific form of heteroscedasticity, making it more robust but also less powerful in specific cases.

1. Procedure

Obtain OLS residuals $u_{i}$ .
Run auxiliary regression of squared residuals $u_{i}^{2}$ on:
- All original predictors ( $X_{1} \dots X_{p}$ ).
- The squared terms ( $X_{1}^{2} \dots X_{p}^{2}$ ).
- The cross-products ( $X_{1} X_{2}, \dots$ ).
Compute $L M = n R^{2}$ from this auxiliary regression.

2. Comparison: White vs. Breusch-Pagan

Feature	Breusch-Pagan	White Test
Assumed Form	Linear function of X	General (Quadratic/Interaction)
Sensitivity	Linear Heteroscedasticity	Non-linear Heteroscedasticity and Specification Error
Degrees of Freedom	Lower ( $p$ )	High ( $p + p (p + 1) / 2$ )
Power	Higher (if linear)	Lower (due to high df)

3. Python Implementation Example

from statsmodels.stats.diagnostic import het_white

lm, lm_p, f, f_p = het_white(model.resid, model.model.exog)
print(f"White Test p-value: {lm_p:.4f}")

Breusch-Pagan Test - Alternative heteroscedasticity test.
Multiple Linear Regression - Framework.
Weighted Least Squares (WLS) - Remediation.