Timea -

In general, I do not see any point in testing for heteroscedasticity. What do you do then? If you decide you have homoscedasticity, do you really know the impact of this decision? If you decide there is heteroscedasticity, what do you do about it? How much is there? (White will be addressed below.

That would help, but I have a more consolidated approach.)

Heteroscedasticity is a natural phenomenon. It is a feature, not a bug. See https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity where an idea by Ken Brewer is discussed which explains why you can expect this for finite populations. (See mid-page 111 in Brewer, KRW (2002), Combined survey sampling inference: Weighing Basu's elephants, Arnold: London and Oxford University Press.)

Note also:

https://www.researchgate.net/publication/324706010_Nonessential_Heteroscedasticity

One of the updates to the following project, specifically addresses the uselessness of doing a hypothesis test for heteroscedasticity:

For regressions of form y = y* + e:  

https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression

What needs to be done is to estimate or otherwise arrive at a good value for the "coefficient of heteroscedasticity," and use weighted least squares regression. (If the coefficient of heteroscedasticity is zero, that is homoscedastic.) The following documents give examples of this and then a tool for doing this with your y-data and OLS results to obtain WLS results:

https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity

https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx

For practical purposes, you can see how much results differ when you account for heteroscedasticity, and when you don't. But it can be problematic just to assume there is none.

IF you have enough data, then the predictions do not tend to differ very much, but the difference in the estimated sigma from the estimated residuals will differ by predicted value, which may have a large influence on the estimated variance of the prediction error associated with each prediction.

You could compare your results done this way with what you got with

"White's robust standard error," but I think it best to go with the above procedure. It is more systematic, rather than a patchwork approach. Further, for a smaller data set, the predicted values may sometimes differ substantially. In those cases, the coefficient of heteroscedasticity is less accurate to estimate, but using zero as a default is often substantially in error, and therefore not a good default. The idea of defaults is noted on one of the pages of the spreadsheet above.

Best wishes - Jim

By the way, I don't think it matters what hypothesis test you used when you then decided to use robust standard errors. What may be questionable is how you interpreted the hypothesis test results, and do robust standard errors do all you need to do and well enough? The latter part of the question might at least partly be studied by simulation. But when using the coefficient of heteroscedasticity (i.e., gamma), one is letting the data speak in a way that impacts all results in a smooth and logical manner that I think may be better overall. But comparison of results can be interesting. - It is also often good to be able to estimate the variance of the prediction error associated with each prediction, and to have better predicted values when sample sizes are small enough that this may make a difference. - Also, one can estimate variance of the prediction errors associated with finite population predicted totals. The method involving the coefficient of heteroscedasticity does this very well. - For robust variance estimation of linear functions of beta, see the "sandwich estimator." My reference is pages 126 and 135 of Finite Population Sampling and Inference: A Prediction Approach,  2000, Richard Valliant, Alan H. Dorfman, Richard M. Royall, Wiley Series in Probability and Statistics. There we find that the outside ("bread") of the sandwich involves the x's, and the inside involves estimated residuals. The sandwich estimator can be found in an online search. - Cheers.

thank you!