Using WLS vs making the residuals homoscedastic to fit OLS regression I'm convinced that homoscedasticity (of errors) is not an assumption (at least not explicit) for OLS regression. Also, even though WLS is advocated for heteroscedastic errors, OLS is not particularly bad, at least when the heteroscedasticity is limited as has been discussed by Gung in his answer. However, let's say that I encounter, while fitting an OLS using a feature X and an outcome variable y, heteroscedastic errors and go on to transform the variables, say using log transformation on both - X and y - and thus be able to beget fairly homoscedastic errors. Should I now give the results of OLS credence? Is it even a good practice? Or should I use WLS regression instead?
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EDIT:
Earlier I took X and y for integer counts.
 A: Homoscedasticity is an assumption for the optimality of OLS (i.e., that it is the best linear unbiased estimator of model coefficients) and an assumption for the validity of the usual OLS standard errors.
It is important to note that a model fit using the original features and outc some and a model fit using transformed features and outcome are different models that make different assumptions about the relationship between the features and outcome. The coefficients in one model do not mean the same thing as the coefficients in the other model. The concept of homoscedasticity depends on the specific model used because the errors result from that model. Your first question should be, which model best captures the relationship between the features and the outcome?, or which model allows me to estimate the specific quantity I want to estimate? How to estimate the standard errors comes secondary and is a response to this question.
If your transformed model is the better model (in the sense that it captures the relationship you are after better), then use it, and if the assumption of homoscedasticity of errors is met with this model, then use that assumption. It may still be a good idea to use a robust standard error. As some in the comments have noted, there may be models better than OLS on the transformed variables that suits the data better, like as count models for integer data, such as the (quasi-)Poisson model.
