If you know what ln(1+income) is, and why it’s a headache, you should read this post

Time for everyone to put on their propeller hats.

I bring you the inverse hyperbolic sine transformation: log(yi+(yi2+1)1/2)

According to a ranting Canadian economist,

Except for very small values of y, the inverse sine is approximately equal to log(2yi) or log(2)+log(yi), and so it can be interpreted in exactly the same way as a standard logarithmic dependent variable.

Here is more.

What’s the fuss about? If you want to look at the effect of an event or policy on income or employment or actions of some sort, you face a problem: the variable has a long upper tail (there are millionaires and workaholics) and so you want to take out the skew (or else have very imprecise estimates).

But transforming income with a natural logarithm doesn’t work, because there are many people with no income or employment or no acts, and ln(0) is undefined. So you either have to drop the zero income folks or (unholy of unholies) give all the zero earners $1. Which rumor has it will get your paper rejected by the AER.

Doubtful? Me too, but rumor also has it this function has the blessing of Cardinal David Card.

h/t Rema Hanna.