Since you will be reporting the results of the logged regression you would report those values. Best wishes David Booth

(The absence of) multicollinearity is not an assumption of OLS regression. OLS regression can handle (even strongly) correlated independent (predictor) variables--although very high multicollinearity may cause issues with high standard errors, lack of power, suppression, etc.

An actual assumption that you did not list, but that is very important, is the assumption of independent observations (i.e., no dependencies due to clustered data etc.)

And to add, the independence of the error term and the linearity assumption should be kept in mind...

To answer what I think is the heart of the question: Once you transform your dependent variable, you use that transformed variable for all the analyses and any assumption checks you do. The one exception is that sometimes people will back-transform their results so that they can present results in their original units.

- I should now continue testing multicollinearity and homoskedasticity with this new log variable?

Yes, you should. The (nonlinear) transformation changes both: the functional relationship (a former exponential relationship will now be linear, a former linear relationship will now be logarithmic); if the forme variance was increasing with the expectation, is may now be constant, and if the former variance independent of the expectation, it may now not be.

- should I include the variable before data transformation or the log?

I'd describe the data in a form that is most commonly used or best unterstood. For instance, the (polynomial) relationship of stopping distances of cars depending on their speed ist log/log linear, but one would not report log distances in practice for any interester reader.

I like quantiles as descriptive summaries. You might give the first, second and thirde quartiles to provide a crude impression where the values are and how they are dispersed. For instance, I would summarize the age of people living in Rotterdam 2020 with 37 (22, 54) years. The first value is the second quartile (=median), the two values in parentheses are the first and the third quartile. I would yuse this to describe the data even when age would be use transformed in a model.

Using mean and standard deviation is often a less good way to summarize data. When the distribution of the variable is not symmetric, this gives rise to wrong ideas or interpretations of the variable. The distribution of "log(age)" might be mor symmetric, but who wants to read that Rotterdam people are 3.4 ± 0.2 log years old?

Thank you all for your answers!

Julie Schipper -

For heteroscedasticity, even after a transformation - which I would only do if necessary, and not to address heteroscedasticity - there can still be heteroscedasticity. I recall an example among those that Penn State puts on the internet where a transformation had been done for purposes of addressing heteroscedasticity in a real estate problem, and they still had substantial heteroscedasticity remaining. You can have a kind of artificial, I say "nonessential" heteroscedasticity, which may be the result of a problem with your model, but the natural, I say "essential" heteroscedasticity that you should expect can be impaired, again, by a problem with your model. Data quality issues can also be involved, etc. See https://www.researchgate.net/publication/354854317_WHEN_WOULD_HETEROSCEDASTICITY_IN_REGRESSION_OCCUR.

To get an idea of your level of heteroscedasticity, see https://www.researchgate.net/publication/333642828_Estimating_the_Coefficient_of_Heteroscedasticity

With enough data, you can measure it well using the following:

https://www.researchgate.net/publication/333659087_Tool_for_estimating_coefficient_of_heteroscedasticityxlsx

Defaults are mentioned in one sheer of that Excel file, and it would be reasonable for the coefficient of heteroscedasticity as defined here to be from 0.5 to 1.0. See https://www.researchgate.net/publication/320853387_Essential_Heteroscedasticity, as determined by Ken Brewer.

For more information, see the following:

https://www.researchgate.net/project/OLS-Regression-Should-Not-Be-a-Default-for-WLS-Regression, and the various updates and references there. 

If you are working with data which have already been transformed, then that changes things in ways perhaps both good and problematic. The problematic can also include problems of interpretation, as Jochen noted. I suggest only using transformations if you absolutely must, even if you are going to "back-transform" as Salvatore mentioned. I think that the simpler you can keep things, the better.

Best wishes - Jim