ADD: Tests depend on objectives, three tests for three different objectives. Introduce DZ test as only general test. Discuss IMPOSSIBILITY of completely general test. There is relevant material in my textbook.
The Eicker-White estimator of the OLS covariance matrix is completely general – it produces a first order consistent estimate for all types of heteroskedasticity (satisfying minimal regularity conditions). This has led to a widespread misconception that the associated test is equally general. That is, the Eicker-White test[1] will detect all types of heteroskedasticity. In the fact the opposite is true. If we assume that there are no misspecifications other than heteroskedasticity, then the White test is a very specialized test suitable for one purpose only: to validate an OLS regression analysis in presence of potential heteroskedasticity. The object of this section is to clarify the precise nature of the White test both heuristically and analytically. We first provide a mathematical characterization of the Eicker-White test.
[1] For example, a recent leading textbook claims that “The White test is extremely general; to carry it out, we need not make any assumptions about the nature of the heteroskedasticity.”This test was also introduced by Eicker (1965), so we deviate from standard usage in econometrics and re-label it the Eicker-White (EW) test for heteroskedasticity.