# Interpretation of the coefficient of a dummy variable in a regression with log-transformed outcome

I want to interpret the models (2) Pool variable. It equals 1 if the house has a pool and 0 if not. The relation between the dependent variable and the Pool variable is a log-linear that means ∆y/y = b1*∆X whereby X is a dummy variable D so the change ∆D can only be 1 or -1 if I am not wrong. So if a house has a pool the price will rise by 7.1% and if house has already a pool and it gets destroyed or something like this the price will decrease by 7.1%. Is this interpretation correct?

• Welcome to Cross Validated! Do you take $\log(y)$ and then run a linear regression using those values, or do you run a generalized linear model with a $\log$ link function?
– Dave
Commented Feb 11, 2023 at 16:02
• Ty Dave. I didnt ran this regression it is from the Stock and Watson "Introduction to Econometrics "book. I think you take ln(y) and then run a regression with the listed variables. So the model is just ln(y) = b0 + b1*ln(size) + b2*Pool + ... + the other dummies. Commented Feb 11, 2023 at 16:05
• Your language is potentially confusing, because it strongly suggests the model is causal. It is not. In fact, if a pool were destroyed, that would likely hugely change the value of the house, depending on the circumstances. The term only estimates the average difference in log prices between houses with and without pools, accounting for ("controlling for") all other factors.
– whuber
Commented Feb 11, 2023 at 16:54

The interpretation of the coefficient of pool in the second model is: Compared to houses that have no pool, the geometric mean of the price for houses with pools is $$100(\exp(0.071) - 1)\% = 7.36\,\%$$ higher, all else being held constant. In other words, $$\exp(\beta)$$ is the ratio of the geometric means of pool/no pool (or generally, current level/reference level of the categorical variable). For more information, see this page.