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The coefficient on a logged explanatory variable when the dependent variable also is in log form is an elasticity (or the percentage change in the dependent variable if the explanatory variable changes by one percent). Suppose I estimate a regression without logging the dependent variable but I use a log link in a General Linear Model (and family gaussian) while the explanatory variable remains in log form. Is the coefficient on that explanatory variable still an elasticity?

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In the usual case with a log variable, the model is \begin{align} \log(y) &= a + b\log(x) + \varepsilon\newline \text{or}\quad y &= e^a x^b e^\varepsilon, \end{align} where $\varepsilon\sim\text{N}(0,\sigma^2)$ and $b$ is the elasticity.

In the situation you mention, \begin{align} y &= \exp[a + b\log(x)] + \varepsilon \newline \text{or}\quad y &= e^ax^b + \varepsilon, \end{align} where $\varepsilon\sim\text{N}(0,\sigma^2)$. So, ignoring the error, the parameter $b$ is playing the same role in both models and is an elasticity in both cases.

What is different is the assumption on the error distribution. In the first case, the error is assumed to be logNormal and multiplicative while in the second case the error is assumed to be Normal and additive.

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    $\begingroup$ Do you need to multiply b by 100 to get the elasticity? If b=2, is that saying that for a 1% increase in x you get a 2% increase in y? $\endgroup$ – Dimitriy V. Masterov Jan 24 '12 at 17:16
  • $\begingroup$ Let me clarify a bit. Suppose I complicate your model a little bit by adding a binary variable d. Then I fire up R and estimate glm(y ~ factor(d) + x, family=gaussian (link="log")). If b=2, does that mean that for a 1% increase in x you get a 2% increase in y? Or do I need to multiply b by 100 to get this elasticity? Similarly, is the percent change in y from d increasing from 0 to 1 exp(b-1)? I think you don't need the 100, but I just wanted to confirm that. $\endgroup$ – Dimitriy V. Masterov Jan 24 '12 at 17:29
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    $\begingroup$ If b=2, then a 1% increase in x leads to an increase in y by a factor of $(1.01)^2 \approx 1.02$. That is, y increases by approximately 2%. $\endgroup$ – Rob Hyndman Jan 25 '12 at 0:56
  • $\begingroup$ This makes sense to me and agrees with my derivation, but I am trying to reconcile it with David Giles' post on the log linear model: davegiles.blogspot.com/2011/03/dummies-for-dummies.html?m=1. I am not sure why the semielasticity would be 100 times greater. $\endgroup$ – Dimitriy V. Masterov Jan 25 '12 at 4:21

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