"Exponentiate the coefficient, subtract one from this number, and multiply by 100. This gives the percent increase (or decrease) in the response for every one-unit increase in the independent variable. Example: the coefficient is 0.198. (exp(0.198) – 1) * 100 = 21.9. For every one-unit increase in the independent variable, our dependent variable increases by about 22%".
This formula for converting coefficients into percent changes seems to have come out of nowhere. I cannot see why this computes a percent change.
Consider this question, and the top answer just states the following result which appears to compute the same thing in a different way:
log(DV) = Intercept + B1 * IV + Error
"One unit increase in IV is associated with a (B1 * 100) percent increase in DV."
Furthermore, this question has an answer that says
"keep in mind that the interpretation of a "unit change in a logarithm" as a "percent change" is a local approximation."
This just confuses me more. Why do these formulas only produce an approximation?
All of this leads to the question... Why can I interpret a log transformed dependent variable in terms of percent change in linear regression? (And why is it only an approximation?)