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to interpret the White-Test, it is recommended to use the LM-statistic = N*R-squared of the auxiliary regression which follows a Chi-squared distribution with df = number of restrictions.

But I wondered why. Wouldn't it be just fine to use the F-Test? The null hypothesis of the F-Test would be that all parameters of the auxiliary regression are 0 (excluding the constant). Of course, we have interactions terms and explanatory variables^2 , but the restrictions of the null hypothesis are still linear: ß_1 = ß_2 = ... = 0. So I don't see a problem using the F-statistic. And the advantage of the F-statistic would be that we don't rely on the asymptotic properties of LM-tests.

Why isn't this done normally? Why is only the LM-statistic used for the White test?

Joe94
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    Hi: According to the link below, you can use an F-test. But the F-test statistic is not the you'd get from a test of $\beta_i = 0 ~\forall i $. This probably has something to do with the fact that White's test has squares and products of the $X_i$ in the regression where $\epsilon^2_{i}$ is the response so one can't use the regular F-test because there are other terms in there. The standard F-test assumes one just has single coefficients in the model. https://www.dummies.com/article/business-careers-money/business/economics/test-for-heteroskedasticity-with-the-white-test-154899 – mlofton Jan 16 '22 at 00:22

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You can use an asymptotic F-test just fine. In fact, for an asymptotic F-test we have:

$F=\frac{LM}{n-LM}\frac{n-k}{q}\overset{d}{\rightarrow}F\left ( q, n-k \right )$

so it won't be a much different concept anyway.

And the advantage of the F-statistic would be that we don't rely on the asymptotic properties of LM-tests.

It seems that your confusion stems from wanting to use an exact statistic as opposed to an approximate statistic. If you want to go down that route - be my guest, however, you often fail to to know the exact distribution of the statistic when the assumptions involved are not met. Moreover, «exact» does not mean much on its own and certainly does not automatically equal «better».

Always Right Never Left
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