Interpretation of coefficient in log-linear model with share predictor

Question

There are several questions on the interpretation of coefficients in log-linear models such as Interpreting regression coefficients of log(y+1) transformed responses
Log linear model interpretation - % Contributions?
Interpretation in log linear regressions with coefficients bigger than 1

but so far I have not found an question that deals with the interpretation of coefficients of explanatory variables presenting shares/percentages. So the model I'm trying to estimate is:

$$\ln (y_s) = \alpha + \beta \frac{\text{number of unemployed}_s}{\text{number of people in the labor force}_s} + \epsilon_s$$

The index $s$ is for states and $\ln(y_s)$ is the natural logarithm of the average income in state $s$. I estimated a beta coefficient of $-3$. The typical interpretation would be that income decreases by $\beta * 100 = -3 * 100$ dollars for a one unit increase in the explanatory variable. But since the explanatory variable here is a share/proportion, would the correct interpretation be one without multiplication of the coefficient by 100? It seems unreasonable that a one unit increase in the share of unemployed to total labor force would decrease average earnings by 300%. A 3% reduction makes more sense but I have yet to understand what would be the correct interpretation.

Thanks a lot in advance and no need to mention the potential endogeneity problems here; I'm really just looking for the interpretation of beta.

Answer 1

The fraction you are interested in is the "labor market tightness" $$\frac{\text{number of unemployed}_s}{\text{number of people in the labor force}_s}$$ You are interested in knowing the effect of an increase in labor market tightness on the GDP, if I have well understood. The labor market tightness is a rate that goes from 0 to 1, not a percentage that goes from 0 to 100. Transform your explanatory variable multiplying it by 100. The best estimate will be probably something like 0.03. The effect on y will be computed as:

$$[exp(0.03)-1]*100$$

After this transformation, the interpretation of the beta will be along these lines: an increase by 1 percentage point in the labor market tightness will decrease the GDP by 3 percentage points. Which seems to make sense since an increase in your x is likely due to an increase in the number of unemployed workers.