The post "How to interpret the difference in log points" shows how to interpret differences in log values still in log form.
As an extension to this, however, I would like to know how to consider an example of regression results where the dependent variable is in log form, and I'm interpreting the impact of the difference between two estimated values within the regression.
The particular example I'm considering is where Acemoglu, Johnson & Robinson 2001 look at the income-institutions relationship in a set of former colonies using settler mortality $M_i$ as an instrumental variable to calculate an index of protection against expropriation $R_i$.
The first stage regression is:
$R_i=\zeta+\beta \ln{M_i} + \delta X_i + v_i$
($X_i$ is a vector of country covariates.)
Then the estimated value for $R_i$ is inserted into the second stage regression to estimate log GDP per capita:
$\ln{y_i} =\mu + \alpha \hat{R_i} + \gamma X_i + \epsilon_i$
How can I interpret different estimates for $R_i$ protection against expropriation, in terms of impact on log GDP per capita? For example Australia has an average protection against expropriation of 9.32, where Argentina's is 6.39. $\alpha$ estimates are all around 1
Does this mean I can say that the rough percentage difference in log GDP per capita should be:
$100( e^{9.32-6.39}−1)$.
Is that how I would estimate based on these numbers? I'd be grateful for advice.