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I used the Breusch-Pagan test for heteroscedasticity, but I have many observations ($\approx 500,\! 000$) and the Breusch-Pagan test uses $nR^2$ as a test statistic where $n$ is the number of observations. But with the Breusch-Pagan test I will always get heteroscedasticity unless my $R^2$ is almost 0. From the residual graphs there seems to be no heteroscedasticity, but according to the Breusch-pagan test there is. So is there any test that is good to use in the case of many observations?

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  • $\begingroup$ You have asked 21 questions, but you haven't upvoted or accepted any of the answers you've received. Have none of the answers been of value for you? If any have helped, please consider upvoting them. If any have resolved a question, consider accepting it by clicking the check mark to its left below the vote total. Upvoting & (possibly) accepting answers is part of the way the SE system is designed to work, & is a nice 'thank you' to those who have helped. Since you are new here, you may want to take our tour, which provides more information about CV. $\endgroup$ Commented Jun 16, 2014 at 16:48
  • $\begingroup$ Answering the query about "good to use" requires knowing why you are performing this test--merely detecting some form of heteroscedasticity usually would be, in itself, pointless. Could you please emend the question to include some of this necessary contextual information? $\endgroup$
    – whuber
    Commented Jun 16, 2014 at 16:56
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    $\begingroup$ Following on from gung's point - the lack of upvotes and accepts suggests to potential answerers of your questions that you don't think any past answers have helped you - not even once in more than 20 questions. This would not encourage anyone to think that answering this question would be likely to help you now. If no past answer has been of any value to you, you should consider whether asking more questions is futile. On the other hand if some previous answers have helped, you should consider changing the impression your lack of upvotes and accepts gives. $\endgroup$
    – Glen_b
    Commented Jun 17, 2014 at 1:21

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Your situation here is analogous to what happens when you test for normality with large samples. That case is discussed in an excellent CV thread here: Is normality testing 'essentially useless'? It may help you to read that, as the basic lessons will carry over to your situation.

Regarding heteroscedasticity, it is generally implausible that the residual variance will be exactly identical everywhere. If it is not, then with enough data you will have very high power (approaching $1$) to detect any variability in the residual variance. Thus, you would essentially always get a significant result.

As @whuber notes, much depends on why you want to conduct such a test. If you just want to test the null hypothesis of homoscedasticity, there is no problem with the fact that you have high power. On the other hand, many people test such assumptions (e.g., normality and homoscedasticity) to assess whether they can trust their $p$-values or if they need to use a robust approach to modeling.

In the latter case, it is worth pointing out that standard linear models (e.g., regression and ANOVA) are quite robust to modest violations of many assumptions. A rule of thumb is that you will be fine if the maximum variance is $\le 4\times$ the minimum variance. If you have a great deal of data and you cannot detect any heteroscedasticity in your residual plots, most likely you will be OK.

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