# Interpret log-linear with dummy variable

I have the following model: ln(y) = b0 + B1 X1 + B2 ln(X2) + B3 X3

My X1 is a dummy that can take the values 0, 1 and 2.

The coefficient for the dummy 1 is -0.500.

My question is how do I interpret this? Does it mean that Y will be reduced by approximately 50% by changing the dummy variable to 1 instead of 0? And how can I calculate the exact effect on Y and not just an approximation?

All the best,

• There are two papers on the topic which may help: Halvorsen, R. and Palmquist, P., "The Interpretation of Dummy Variables in Semilogarithmic Equations", American Economic Review, Vol. 70, 1980, pp. 474-475. Kennedy, P., "Estimation with Correctly Interpreted Dummy Variables in Semilogarithmic Equations", American Economic Review, Vol. 71, 1981, p. 801. Also check out the formula to interpret beta in this website: econometrics.com/intro/dumlog.htm
– Ana
Jul 27, 2020 at 23:09
• Is it a dummy or a multivalued discrete (or possibly continuous) variable? I'm curious because you noted it takes on the values 0, 1, and 2. I'm sure it was a minor oversight. Jul 28, 2020 at 1:38

## 1 Answer

It means the logarithm of $$Y$$ will be -0.5 higher according to the model, which means that the actual value of $$y$$ will be multiplied by $$\exp(-0.5) \approx 0.6$$, corresponding to a 40% decrease.

• Thanks a lot. So for small changes, say if the coefficient was -0,05 it would be and approximately 5% decrease in Y. Is that correct?
– maS
Apr 30, 2019 at 11:38
• Yes, for small numbers it works that way. But the exact calculation is $\exp(-0.05)\approx 0.95$.
– Gijs
Apr 30, 2019 at 15:39
• Now assume that we are in the same set up, that is: "I have the following model: ln(y) = b0 + B1 X1 + B2 ln(X2) + B3 X3 My X1 is a dummy that can take the values 0, 1 and 2. The coefficient for the dummy 1 is -0.500." Now think that the dummy 2 is 0.30. How to interpret that is it the same formula that we should use for finding the effect of 'switching from 0 to 2'?
– Cenk
Jun 28, 2021 at 17:34