I estimated the following specification: $$ Y = a + bX + e , $$
where $Y$ is a dummy variable and has a mean of $.15$. $X$ is log-transformed with a mean of $3.54$ and SD of $.87$; before transformation $X$ had mean $46.87$ and SD $36.82$. The model estimates $b=-.025$. I am trying to interpret correctly coefficient $b$.
Is it correct to say that: the model estimates that a one standard deviation increase in $X$ (36.82) reduces $Y$ by $(36.82*-.025) = -.9205$ percentage points? Then, if correct, how can I make sense of this result given the baseline probability of $.15$? Thank you!
Given a regression model of the form $y = a + b\ln x + \varepsilon$, let $\hat{y} = \hat{a} + \hat{b} \ln x $ be the estimated model. Taking the derivative with respect to $x$, we have
$$ d{\hat{y}} = \hat{b} \frac{dx}{x} . $$
This implies that a 1% increase in $x$ is associated with a $\frac{\hat{b}}{100}$ increase in $\hat{y}$.
So, in the context of this problem, given some starting value of $x$, if you increase that value by $1\%$, then the predicted value of $y$ decreases by $0.00025$.
Generally, with log transformations in regressions, this type of percent increase interpretation works so long as the percent changes are near 0 - the interpretation depends on the approximation $\ln(1+x)\approx x$. For bigger changes in $x$, the percent change interpretation becomes less accurate.