Linear regression, heteroscedasticity, White's test interpretation?

Question

I am trying to test whether my regression has an issue of heteroscedasticity. After running a regression, I can clearly see that the residual plot has a pattern. After taking a log of the dependent variable the pattern is much, much reduced. The White's test on the original formula returns a p-value of 0.0004 before the transformation (the model with strong pattern in residuals), and a p-value of 0.08 after the log transformation.

I can see that the second model has less heteroscedasticity on the plot, but how do I interpret the results of White's test? Does the first value mean that we can reject that there is heteroscedasticity at (100-0.0004)% significance, while in the second model, we can reject it at, say, 95% confidence?

Answer 1

The original White paper where the test statistic was proposed is an enlightening read. This excerpt I think is of interest here:

...the null hypothesis maintains not only that the errors are homoskedastic, but also that they are independent of the regressors, and that the model is correctly specified... Failure of any of these conditions cal lead to a statistically significant test statistic.

Assuming that the model is correctly specified your results indicate that for non-transformed case there is a clear presence of heteroskedasticity, and in the log case there is no heteroskedasticity at 5% significance level, but there is at 10%. This means that in the log case further tests should be made, since the test "barely" accepts the null hypothesis of no heteroskedasticity. For me personally this would be an indication that maybe model specification is not correct and other heteroskedasticity tests should be made. Incidentally White gives an overview of alternative tests in its article: Godfrey, Goldfeld-Quandt, etc.

Answer 2

This does not answer the question of how to use the test. However, you should know that most economists generally never run those tests -- especially, applied microeconomists. Instead, you just use the Huber-White adjusted standard errors which corrects for various misspecifications in the distribution of your error terms.

That's not a sharp "statistics" answer, but it's how most practitioners in economics handle it. Godfrey Goldfeld-Quant or White's tests are barely ever used or discussed.