Interpretation regression coefficients

Question

I have a problem with my co-author, we do not agree on the inerpretation of the regression coeffcients. In our regression (part of a more complicated model, which does not matter here) we defined the dependent variable as log difference, that is, ln(Gt) - ln(Gt-1), where G are local investment. The most relevant independent variable is in level (St, surcharge of a tax). We are struggling with the interpretation of the regression coefficient, beta, on St. In my opinion, an increase in St of one unit leads t a beta percentage points increase in G; while he interprets it in percentage terms.

Could help us?

Answer 1

You have marginally (excluding other variables except the intercept) $$ \log \left(\frac{G_t}{G_{t-1}}\right) = \beta \ \cdot \mathrm{Surcharge} + C $$ meaning

$$ G_t = G_{t-1} \cdot \exp \left(\beta \ \cdot \mathrm{Surcharge}\right) \cdot e^C $$

Here $e^C$ has the interpretation of a `base rate' of growth. A marginal unit increase in $\mathrm{Surcharge}$ yields a multiplication by $e^\beta$. You can interpret this as a $e^\beta$-percent growth (or decrease, if $e^\beta$ < 1) since the last measured value $G_{t-1}$, after the base rate has been applied. For example, an estimated $\hat{\beta} = .5$ would mean the previous value $G_{t-1}$ is first multiplied by the basal rate $e^C$, and then is additionally multiplied by $e^{\hat{\beta}} \simeq 1.648$, a $165\%$ growth compared to the base-rate growth (or a $65\%$ increase with respect to basal growth).