White Test

Definition

where $N$ is the number of samples, $K$ is the number of explanatory variables

A test used for checking homoskedasticity in regression analysis It tests whether its sample variance is estimable using the model’s explanatory variables. If the square of the regression model’s residual has a trend, it will be estimated by the explanatory variables.

Algorithm

Assume a OLS model $y_i={\beta }_i+{\beta }_2{x}_{2i}+{\beta }_3{x}_{3i}+{ϵ}_i$

1. Estimate the regression model and save the residuals ${\hat{e}}_i$
2. Estimate the auxiliary regression model using the residuals and quadratic terms of the explanatory variables $e_i^2={\alpha }_1+{\alpha }_2{x}_{2i}+{\alpha }_3{x}_{3i}+{\alpha }_4{x}_{2i}^2+{\alpha }_5{x}_{3i}^2+{\alpha }_6{x}_{2i}{x}_{3i}$
3. test the significance using the auxiliary regression’s $R^2$ value By the LRT, $N{R}^2\sim {\chi }^2(K)$, where $N$ is the number of samples, $K$ is the number of the explanatory variables of the auxiliary regression (in this example, $K=5$)