From all the data I have worked with, I have noticed that in a linear model with one explanatory variable, taking the ln of that explanatory variable and using the result as the new "independent" variable in the model makes the regression line (with the estimators for the parameters gotten through OLS) a better fit (in terms of a better R^2 value).

Assuming the X's being considered are always positive, my question just has three parts:

1) Is it always the case that the new regression line for "Y on ln(X)" has a more favorable R^2 value as compared to the old regression line for "Y on X"?

2) If no, when is it not?

3) What determines this change in the R^2 value exactly?

All help would be appreciated!

(I know I mentioned ln(X) in this particular question but naturally this doubt would be applicable to most logs with any base, barring some cases)

  • $\begingroup$ Since all logarithms are multiples of one another as you vary the base, there are no cases whatsoever to "bar": you will get identical values of $R^2$ for all logarithms of the independent variable regardless of which base you use. $\endgroup$ – whuber Jul 23 '15 at 17:57
  • $\begingroup$ Taking the log will help when it is appropriate, but not otherwise. To understand the answer to this question, you really just need to understand what it means to take the log of your variable / what the effect is. $\endgroup$ – gung - Reinstate Monica Jul 23 '15 at 17:59
  • $\begingroup$ Please read the linked thread. This may help you as well: In linear regression, when is it appropriate to use the log of an independent variable instead of the actual values? I believe those will provide the information you need to understand this issue. If you still have a question afterwards, come back here & edit your Q to state what you have learned & what you still need to know. Then we will be able to provide the information you need without just duplicating information elsewhere that already didn't help you. $\endgroup$ – gung - Reinstate Monica Jul 23 '15 at 18:02