Interpreting standardized coefficients on a natural log response variable in OLS multiple regression I am working on a housing problem in which I use dichotomous and ratio data to predict
housing production (units constructed in a year-ratio) in a 17 year time period. At this time, I am using OLS and as I get better at stats, I shall attempt this problem using time-series analysis.  That said, I have used R to standardize all of my ratio predicting data and left the dichotomous data raw.  And I have also transformed the response variable to a Natural log to normalize the distribution (i.e. many, many zeros>>yes, I know Poisson or Zero-populated counts in the future).
I have read the post on "interpret coefficients from a quantile regression on standardized data" and also the "convert my unstandardized independent variables to standardized." Based on those, I think that can do the following interpretation based on the following output. The variable region_id is dichotomous, supply is standardized.
Coefficients:
                      Estimate Std. Error t value Pr(>|t|)    
(Intercept)          2.687e+00  2.171e-01  12.379  < 2e-16 ***

region_id            1.805e+00  1.383e-01  13.049  < 2e-16 ***

supply              -2.205e+01  2.204e+00 -10.005  < 2e-16 ***

Region Interpretation:
For every on city that is located in the Houston region, you can expect that annual housing production will increase by 1.8%.  
Supply Interpretation:
For every one-unit increase in the standard deviation of housing supply, you can expect that annual housing production will decrease by -22.05%.
Nota bene.
I am not a stats or math person at all,
but I have been using R for the past three years
and I am quite familiar with OLS, but if you throw
up an equation it will look "appropriately" Greek to me. :)
 A: For binary predictors with large coefficients, the effect of region going from 0 to 1 is $100 \cdot(\exp\{\beta\}-1),$ which means the effect if more like 536% for Houston. When coefficients are small, this transformation will give you roughly the same result as merely looking at them. The intuition is that for a model like $E[y|D]=\exp\{\alpha + \beta D\}\cdot E[\exp{u}],$ where the second term comes from the retransformation back to levels from logs, the effect of D is \begin{equation}\ln
\frac{E[y|D=1]-E[y|D=0]}{E[Y|D=0)} \cdot 100=\frac{\exp\{\alpha + \beta\}-\exp\{\alpha\}}{\exp\{\alpha\}} \cdot 100=(\exp\{\beta\}-1) \cdot 100
\end{equation}
Unfortunately, you can't just plug $\hat \beta$ in for $\beta$. You want to use $$100 \cdot \exp \left( \hat \beta - \frac{1}{2} \hat \sigma_{\beta}^2 \right), $$
where $\hat \sigma_{\beta}$ is the standard error of that coefficient. Similarly, I think you want to multiply the supply coefficient by 100 as well.
These estimates do seem very large, but I really don't know what to expect here.
Take a look at this post by David Giles for formulas and sources. The bias correction he describes (as long as you are willing to assume the errors are normal) will take your effect down a bit, but not substantively so.
