Last updated on September 27, 2018
A few months ago, I wrote a post titled “What to Do Instead of ln(x+1),” in which I discussed the inverse hyperbolic sine transformation (IHS, or more specifically arcsinh), which is frequently used in lieu of the logarithmic transformation in order to transform a variable that has a substantial number of zero-valued observations.
The day the post appeared, I received an email from Casey Wichman, from Resources for the Future,* who wrote:
Jonathan Eyer and I used the IHS transformation in our JEEM paper on water scarcity and electricity generation, and I’ve actually been thinking about your final point since then. That is, how to account for elasticities when using the IHS transformation. I would be interested to do a bit more digging to see if anyone has written anything on it. If not (and I can’t remember finding anything when I was looking), I think there would be value in writing a short note … Any interest?
There definitely was interest, and over the last few months, Casey and I wrote an paper titled “Elasticities and the Inverse Hyperbolic Sine Transformation,” in which we characterize the x-elasticity of y (i.e., the dependent variable’s sensitivity to a 1-percent change in the explanatory variable) for situations where (i) a linear y is regressed on the IHS of x, (ii) the IHS of y is regressed on a linear x, (iii) the same, but with a dichotomous explanatory variable, and (iv) the IHS of y is regressed on the IHS of x.
To be clear, we thought this was obvious, we thought that everyone else thought it was obvious, and we thought that everyone thought everyone else thought this is obvious. But upon looking, no one had derived these elasticities to show that this indeed is obvious, and our own judgment and discussion with colleagues and students at workshops, after class, etc. suggested that there’d be value in stating the “obvious.”
Here is the abstract:
Applied econometricians frequently apply the inverse hyperbolic sine (i.e., arcsinh) transformation to a variable because it approximates the natural logarithm of that variable and allows retaining zero-valued observations. We provide derivations of elasticities in common applications of the inverse hyperbolic sine transformation and show empirically that the difference in elasticities driven by ad hoc transformations can be substantial. We conclude by offering practical guidance for applied researchers.
Once again, comments are not only most welcome, they are eagerly solicited.
* After I ran this post by Casey, he wrote “You could probably also file this under ‘how to coauthor with senior colleagues by sending emails out of the blue.’ I would personally file this under ‘how to coauthor with someone whose work you’ve admired since you first met them when they were a job-market candidate.’!