Interpreting Regression Results: Percentage Change vs. Binomial Approximation

[Algernon81]

Ok, so I am an MA Economics student, just finishing off my thesis, and I am not quite sure whether I am interpreting my findings correctly. It's more of a maths issue than anything, which is why I came here. So, I will explain in a reduced-form way:

logY = a + (beta)logX + e

Ok, my regression is more than that, but I need only a simple version.

logY = log of gdp per capita

logX = log of 'quality of governance', where X is [0,1]

beta = 2.6, when the regression is run


So, I have two countries that I would like to compare, Senegal and Hungary, as they occupy opposite ends of the interquartile range. The basic idea is to show what happens when you increase Senegal's institutions to the level of Hungary's. I am working off the assumption that the log-log form gives me a "one percent increase in X leads to a beta percent increase in Y".

So, do I work out the percentage increase from Senegal's unlogged institutions (0.39) vs Hungary's unlogged institutions (0.72), or the percentage increase between the logged versions?

Let me know if that isn't clear.

 

[Mute]

If $\log Y = A + \beta \log X$, then $Y = D X^\beta$ (where D = (logarithm base)^A ). If X increases by 100c percent, then that means X is changed to X' = X(1+c). So, Y becomes

$$Y' = D (X')^\beta = D X^\beta(1+c)^\beta = Y(1 + c)^\beta.$$

If c is small compared to 1, then $(1 + c)^\beta \approx 1 + \beta c$ (binomial approximation), so an increase of 100c percent in X corresponds to an increase of $100\beta c$ percent in Y, which is your current assumption. If c is not small enough that the binomial approximation is actually a bad approximation to make, then you can find the exact percentage change as follows: let's rewrite things a bit:

$$Y' = Y(1 + c)^\beta + Y - Y = Y + [Y(1 + c)^\beta - Y] = Y(1 + [(1+c)^\beta-1])$$

So, we see that if the change in X is c, the change in Y is $(1+c)^\beta - 1$, or $100[(1+c)^\beta - 1]\%$ .