# Elasticities Using GLM

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?

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.
• 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%. – Rob Hyndman Jan 25 '12 at 0:56