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Lets say I have a random variable $y$ that is expected to be:

$$y_i \sim poisson(\lambda_i)$$

$$log(\lambda_i) = \beta_0 + \beta_1x_i$$

But, for every reasons, I am running the following linear regression using OLS:

$$log(y_i) = \beta_0 + \beta_1log(x_i)$$

Given the log-log transformation, I could than say "a 1% change in $x$ causes a $\beta_1$% change in $y$". Switching back to the poisson context, how would I set up a regression model where a similar interpretation (% change in $x$ causes $\beta_1$% change in $y$) is possible?

Since the log is the common link-function for a poisson glm, I do not need to transform the lhs of the equation. So transforming $x_1$ should be sufficient?

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    $\begingroup$ The Poisson GLM and the log-log (elasticity) model are generally two different approaches. They can both work but the choice depends on the problem. An answer to a similar question has been provided here. $\endgroup$ – Digio Jul 30 '17 at 20:16
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The transform on $X$ is not a key difference between the two methods, because, like you have noticed, you can also do it in Poisson regression without a problem. The essential difference is about $Y$: transform is not link (GLM). You see the difference clearly when you write the formulas as a conditional mean.

Linear model transformed with log is:

$$E(\log(Y)|X)=\beta_0+\beta_1X$$

GLM with a log link is (as in Poisson regresion):

$$\log(E(Y|X))=\beta_0+\beta_1X$$

Even if they look the same, they are not the same at all (because $\log$ is not linear).

If you are interested in having no bias on $E(Y|X)$ then GLM is the model to choose. With a transformed linear model, there is a (usually strong) bias. The subtlety happens in the way the noise ($\epsilon$) is transformed in a non linear way. The noise has mean 0, but when transformed by $\log$ it modifies the mean of the estimation.

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    $\begingroup$ +1. I would only like to point out that the log-log model is not unimportant at all. It is a fundamental model in econometrics which gives a different coefficient interpretation than Poisson regression. $\endgroup$ – Digio Jul 31 '17 at 8:40
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Actually, this is a pretty interesting topic in price elasticity modeling. Real-world sales data is often contaminated with zero values in the dependent variable due to no-sales days. Since the log-log regression cannot handle zero values properly without applying the log(c+x) transformation, Poisson regression or any other count data models seem to be a relevant alternative but cannot be interpreted as easily as log-log regressions.

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