I had some interesting email from Andrew a while back to do with computing elasticities from log-log regression models, and some related issues.
In his first email, Andrew commented:
"I am interested in the elasticity of H with respect to W, e.g., hours with respect to wages. For simplicity, assume that W is randomly assigned, and that the elasticity is identical for everyone.
Standard practice would be to regress log(H) on a constant and log(W). The coefficient on log(W) then seems to be the elasticity, as it estimates d log(H) / d log(W).
But changes in log( ) are only equal to changes in percent in the limit as the changes go to zero. In practice, one typically uses discrete data. Because the changes in W may be large, the resulting coefficient is just a first order approximation of the elasticity, and is not identical to the true elasticity."
Let's focus on the third paragraph. Keep in mind that log( ), here, refers to "natural" (base 'e') logarithms.
Andrew is quite correct, and this is something that we often overlook when teaching econometrics, or when interpreting someone's regression results. I sometimes refer students to this useful piece by Kenneth Benoit. Here's a key extract from p.4: