Correcting data for heteroscedasticity in a regression model

Question

I applied OLS on a regression model that looks as follows: $$ y = b_0 + b_1x_1 + b_2x_2 $$ and found that signs of heteroscedasticity. In an econometrics text book, I found that I can divide each variable by the square root of an independent variable. So the new model looks as follows: $$ \frac{y}{\sqrt{x_1}} = \frac{b_0}{\sqrt{x_1}} + \frac{b_1x_1}{\sqrt{x_1}} + \frac{b_2x_2}{\sqrt{x_1}} $$ This is only for $\sqrt{x_1}$ but I will have to do it for $\sqrt{x_1}$ as well. The problem I have is that some of the $x_1$ or $x_2$ values is zero. So what do I do if I want to divide by the $\sqrt{0}$?

Answer 1

One way to think about the assumptions in a statistical model, such as the assumption of homoscedasticity in ordinary least squares regression, is that these properties are desirable. After all, the validity of a modeling method's results are threatened is such assumptions are not met. Researchers would like to check their assumptions (as you have done) and proceed accordingly.

But another way of thinking about such assumptions, including that of homoscedasticity, is that they describe limits of the usefulness of a particular kind of model given how interesting nature actually is. Put another way: heteroscedasticity is interesting, and variance, including dependencies in variance, has substantive importance (not just relevance to the statistical machinery) in countless areas of research. Researchers might check assumptions (as you have done for ordinary least squares), decide that that specific modeling strategy is not quite up to the task of describing those parts of nature that you are interested in, and then move on to models which are up to that task. For example, mixed models will permit you to simply include a description of heteroscedasticity in your regression, and contribute to your understanding of why variance in your outcome has some interesting dependencies.

Answer 2

There are many ways of trying to correct for / account for heteroscedasticity (I list a number of them in my answer here: Alternatives to onle-way ANOVA for heteroscedastic data). The approach recommended in your econometrics book is a kind of data-transformation. Transformation is often the first recommendation in such textbooks because it is perhaps the most straightforward. However, it is often not the best approach. I would recommend using a different strategy. Which would be best is a difficult question and it depends on your situation, your data and your goals. But perhaps the simplest is just to use heteroscedasticity-consistent 'sandwich' standard errors. That might be my first choice in your case.

Answer 3

Or extend the usefulness of regression models in the face of heteroscedasticity, and utilize quantile regression.