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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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.
Unfortunately, the elephant in the room is probably endogeneity. Unless you have a very unusual case, you are not identifying the price elasticity. I would bet dollars to donuts that your price coefficient will be positive. Take a look at these lecture notes to get an idea of what's going wrong.