If you think this is a duplicate, please have a look at the last paragraph.
In a regression model where both dependent ($Y$) and independent ($X$) variable are in natural logs, what is the exact interpretation of the coefficient of $X$?
Take the following simple model:
$\ln Y = \beta_0 + \beta_1 \ln X + \varepsilon$
I encountered two different rules for the interpretation:
- A $d\,\%$ increase in $X$ is associated with an $d\cdot\beta_1$ percent increase in $Y$. ("Elasticity Interpretation")
- A $d\,\%$ increase in $X$ is associated with an $\left(\exp (\beta_1 \cdot \ln a) -1\right)\cdot 100$ percent increase in $Y$, where $a = (100 + d)/100$.
To rule out that both interpretations are identical, plug in (for example) $d = 50$ and $\beta_1 = 2$:
- Result using rule 1: $Y$ increases by $50\cdot 2 = 100\,\%$.
- Result using rule 2: $Y$ increases by $\left( \exp (2 \cdot \ln 1.5) - 1 \right) \cdot 100 = 125\,\%$.
My guess is that the first interpretation is only approximately correct for small $d$ and $\beta_1$, but this is precisely the question: Which of the two rules is correct? (And why is the other one wrong?)
I am aware of this question, but please note that this is rather a follow-up question than a duplicate. I also know that there is that question which is very similar but unanswered. I considered improving the existing question, but there is a lot of (IMHO) superfluous information in the text that makes it unattractive to answer and I don't see how I could remove these parts without violating the original author's intent.