Elasticity of a 0-1-2 dummy variable in a log-log model We know from the following sources,

*

*"Halvorsen, R. and Palmquist, P., The Interpretation of Dummy
Variables in Semilogarithmic Equations, American Economic Review,
Vol. 70, 1980, pp. 474-475.",

*Dave Giles' blog
(https://davegiles.blogspot.com/2011/03/dummies-for-dummies.html)
and,

*Jeffrey Wooldridge's texts (p. 233, 4th edition), that

in a log-log model with dummies such as the following,
$$ln(Y) = a + \beta ln(X) + cD,$$
where $D$ can take on a value of either 0 or 1, the elasticity of the dummy is as follows:

*

*swithching from 0 to 1: $100 \times[exp(c)-1]$,

*swithching from 1 to 0: $100 \times [exp(-c)-1]$.

My question is whether the formula changes when the dummy takes on values of 0, 1, and 2? I mean, does the formula change depending on the number of categories for the dummy variable?
Take the following example,
$$
ln(wage) = a + \beta ln(tenure) + cD, 
$$
where $D = 0$ for Asians, $D = 1$ for Europeans, and $D = 2$ for Africans. And, the dataset is similar to the following,
Obs    WAGE    TENURE    D
1      1200    22        2
2      1450    25        0
3      984     15        1
4      1050    19        2
5      ....    ..        .
..     ....    ..        .
N      ....    ..        .


When you have a dummy variable with multiple categories, you in fact have (n-1) dummy variables, where n is the number of categories.
Take for example, a variable with three levels as follows:
Obs Category
1   A
2   B
3   A
4   B
5   C

In the equation for your linear model, this would be transformed into a 5X2 Matrix as follows:
B    C
0    0
1    0
0    0
1    0
0    1

So you now have two different "dummy" variables, the dummy variable that expresses whether an observation is part of category B, and a dummy variable that expresses whether an observation is part of category C.
The example you gave of going from 0 to 1 to 2 is NOT a dummy variable, it's just another continuous variable.
