# Elasticity with log + 1

I am running time series regressions to estimate the percentage change in quantity to a percentage change in price, with the most basic form being $\ln Q = \beta_0 + \beta_1\ln P + \varepsilon$, where $\ln Q$ and $\ln P$ are the log of quantity and price respectively. My question is if anyone knows about the validity of taking $log(q + 1) = \beta_0 + \beta_1\log(p + 1) + \varepsilon$? The reason for the transformation is obviously to deal with zeros (that is, time periods with zero sales).

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This question may be helpful to you: how-should-i-transform-non-negative-data-including-zeros. – gung Nov 12 '12 at 15:46

You can do this in a GLM framework with a log() link function or with a poisson regression model instead of transforming $q$. Both of these will handle the log of zero issue very well. However, getting the time series aspect of the problem right might be a challenge in this framework. For example, you might have shocks to demand that persist from day to day.
I am not sure why you need to transform $p$ since it is presumably greater than zero at all times.