# How does a focus on one error term change the auxiliary equation of a heteroskedasticity test?

$$y_i = \beta_1 + \beta_2 x_{2i} + \beta_3 x_{3i} + \beta_4 x_{4i} + e_i$$

So you usually i'd take this and compure for $$U_i^2 = \beta_1 + \beta_2 x_{2i} + \beta_3 x_{3i} + \beta_4 x_{4i} + \beta_2^2 x_{2i}^2 + \beta_3 x_{3i}^2 + \beta_4 x_{4i}^2 + \beta_2\beta_3+\beta_2\beta_4 +\beta_3\beta_4 + \nu_i$$

Which is the full white test, but i don't know how this statement changes things

Test the null hypothesis that the error term is homoskedastic against the alternative hypothesis that the variance of the error term is a function of $x_{2i}$, and none of the other explanatory variables.

Does it just mean i eliminate everything and only add a square of $\beta_2x_2i$

$$U_i^2 = \beta_1 + \beta_2 x_{2i} + \beta_3 x_{3i} + \beta_4 x_{4i} + \beta_2^2 +\nu_i$$

As a bonus question, how do i interpret the effect of adding both a square products of variables together in the auxiliary equation? - Oh the product just adds data on linearity

It means that in the projection equation for $U_i^2$ you keep the constant term and just various powers of $x_{2i}$ (but exclude all the interactions of $x_{2i}$ with other regressors and functions of purely other regressors).