# Bootstrap confidence interval for slope of regression line, in the presence of heteroskedasticity

Suppose I have a simple linear regression problem, and I use the bootstrap to obtain a 95% confidence interval for the slope of the regression line. (Maybe I will use this confidence interval to test whether I can reject the null hypothesis that the slope is 0.)

Suppose that the data has heteroskedasticity. I know that when I see heteroskedasticity, I should potentially worry, as this violates the assumptions behind linear regression and a linear model. For example, I know that standard errors for the parameters of the linear model computed using standard methods (linear algebra, least squares, etc.) might be wrong or misleading.

What about when I use the bootstrap, instead of standard methods? Does that avoid those problems? Do I get good 95% confidence intervals for the slope, even in the presence of heteroskedasticity? Or are there caveats and cautions that I need to be wary of?

• Standard bootstrapping does not help much with heteroskedasticity. One type of bootstrapping that is quite effective is the wild bootstrap, but it can take a long time. So, it is more common to use one of the more recent heteroscedasticity-consistent standard errors, HC3-HC5. They sometimes perform just as well as the wild bootstrap. See this paper by Hausman for a good review of the topic dx.doi.org/10.1016/j.econlet.2012.02.007 – Heteroskedastic Jim Nov 14 '18 at 2:23
• Do you have a model for the heteroskedasticity, e.g., the error variance is proportional to the true (conditional) mean of the target variable for each observation? – jbowman Nov 14 '18 at 2:54
• @jbowman, I don't, but that would be a reasonable example situation. An explanation of how/why the bootstrap fails in that situation (or any other example of your choice) would be a reasonable answer. – D.W. Nov 14 '18 at 16:20