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I saw this sentence:

"I use log(income) partly because of skewness in this variable but also because income is better considered on a multiplicative rather than additive scale.

In other words, \$1,000 is worth a lot more to a poor person than a millionaire because \$1,000 is a much greater fraction of the poor person’s wealth"

on page 143 on this link http://www.biostat.jhsph.edu/~iruczins/teaching/jf/ch12.pdf.

But when I check their skewness using library(e1071) in R (as seen below), I found out that the skewness of income is not that high or low. My question is how do I determine if I need to used log transformation in a regression model?

PS the chicago data is in library(faraway)

> skewness(chicago$race)
  [1] 0.5570103
> skewness(chicago$race)
  [1] 0.5570103
> skewness(chicago$fire)
  [1] 1.271188
> skewness(chicago$theft)
  [1] 2.955751
> skewness(chicago$age)
  [1] -0.9210877
> skewness(chicago$income)
  [1] 1.155
> skewness(chicago$involact)
  [1] 0.8079598
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Income is commonly accepted be right skewed, where people making disproportionately large amounts of income pull the mean much higher than the median. Its skewness in that particular data set may not contradict the prevailing norm.

Using log income also lowers the impact of heteroskedasticity. However this is not the best use of it, if heteroskedasticity is a problem you may want to use GLS.

Your primary question: You use log transform for the reason mentioned above, if you believe the increase to be relevant proportionally (+1% income) rather than linearly (+1$ income). This is a question of your theory or functional form. Is a dollar the same for a millionare and for a pauper? Choose linear in this case. If a dollar does nothing for a millionaire but a lot for a pauper, choose ln().

If your dependent variable is also in logs, your coefficient is an elasticity, a very important concept in economics. A coefficient is still called a semielasticity even if the independent variable is in logs and the dependent variable is not!

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  • $\begingroup$ how about the other variables they seem to be skewed to the right also, should I do log transformation to them also? $\endgroup$ – jbest Mar 24 '15 at 5:37
  • $\begingroup$ Only use log if you think the % relationship applies to them. The point of parametric model choice is that you must (always and everywhere) have a theory that justifies the way your data is included. Other things like heteroskedasticity, skew, or significance/size of the coefficients are supposed to be a unintended consequences of a good theory and representative evidence. $\endgroup$ – RegressForward Mar 24 '15 at 14:14
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    $\begingroup$ @RegressForward's advice that you must have a theory is how economists seem to be trained, and it's better advice than the complete opposite. But in practice their definition of theory is as elastic (all puns intended) as anybody else's: often "theory" just means some economist thought this was important before the regression was performed. For everybody else, it can be sensible e.g. to use logs if the relationship is more nearly linear that way, regardless of whether there is theory worthy of the name that predicts precisely that functional form. $\endgroup$ – Nick Cox Mar 24 '15 at 15:27
  • $\begingroup$ @Nick You're right- if the result is better, you will probably use the one that is better for your purposes, and let theory fall by the wayside. But that's letting the power of the dark side corrupt you! $\endgroup$ – RegressForward Mar 24 '15 at 16:04
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    $\begingroup$ If statistics doesn't allow learning from data, we are all doomed here. $\endgroup$ – Nick Cox Mar 24 '15 at 16:21

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