# Elasticity of a 0-1-2 dummy variable in a log-log model

We know from the following sources,

1. "Halvorsen, R. and Palmquist, P., The Interpretation of Dummy Variables in Semilogarithmic Equations, American Economic Review, Vol. 70, 1980, pp. 474-475.",
2. Dave Giles' blog (https://davegiles.blogspot.com/2011/03/dummies-for-dummies.html) and,
3. 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.

• I see, so, should we apply the same formula to transform the coefficients for finding the elasticity ?
– Cenk
Jun 29, 2021 at 3:55
• "Elasticity" makes me think you're an economist, but I don't see that specific example referenced in your question. Jun 29, 2021 at 3:59
• Alright, ln(WAGE) = a + ln(TENURE) + cD, and D = 0 for Asians, D=1 for Europeans, D=2 for Africans. In this situation do you still need to transform c by the above-mentioned formula to reach the elasticity value? Please also consider the information in references (I mean Wooldridge's book, Dave's blog, and the paper of Halvorsen and Palmquist.
– Cenk
Jun 29, 2021 at 4:12
• So in this case you equation is actually; ln(WAGE)=a + ln(TENURE) +c1D +c2D. Where c1 would be the coefficient for Europeans and c2 would be the coefficient for Africans. Any stats software should give you this form of output. So yes, still use the equations. Just make sure you plug in the correct coefficient. Jun 29, 2021 at 4:20