Effect Size log linear model when predictor is in percent

Question

When performing a linear regression with a log-transformed dependent variable, one has to exponentiate the estimated coefficient of a predictor, subtract it by one and multiply it by 100 in order to get the estimated percentage point change of the dependent variable for a one unit increase in the predictor.

In contrast, log-log models are directy interpretable as elasiticities. Therefore, the estimated coefficient of a predictor can be interpreted as a percent point change in the dependent variable for a one percentage in the predictor.

However, which case applies if the predictor is scaled in percent (possible values from 0 to 100 in %)? I assume the first case applies since, in order to get rid of the log transformation of the dependent variable, the complete regression equation has to be exponentiated. However, I'm not sure and I can't find any literature that discusses this specific case.

Answer 1

In the following model

\begin{equation} \log(y_i) = \beta_0 + \beta_1 x_i + \dots + u_i , \end{equation}

where $x$ is a proportion (with values from 0 to 1 - i.e. not 0 to 100% as in your question), $\beta_1$ may be interpreted as follows: "Given one percentage point change in $x$ (ceteris paribus), $y$ increases by $\beta_1$ percent." (Non-negativity assumptions apply to $y$ observations)

See Wooldridge, Introductory Econometrics, Chapter 6, 5th or 6th ed.