Testing for heteroscedasticity with many observations 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?
 A: 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.  
