# log-log transformation

I am making a linear regression model for house prices. My data set includes price per square foot of a property and floorspace so I have multiplied them to get the total price of each property. My independent variable is: Total Price = price $\cdot$ floorspace. My regression equation is: $$log(total price) = c + log(floorspace) + \text{other explanatory variables} + \dotso + e$$ And: $$log(price \cdot floorspace) = c + log(floorspace) + \dotso +e$$ And: $$log(price) + log(floorspace) = c + log(floorspace) + \dotso +e$$ Is this a problem? I am essentially using log(floorspace) on both sides of the equation? Should I use a different transformation for floorspace such as floorspace$^2$?

• Apart from the problem you mention which is central, logarithms have a rationale here in modelling what is likely to be more nearly a multiplicative relationship (and quite possibly as promoting more equal scatter etc.). Squaring anything couldn't possibly help with those. Dec 16, 2015 at 14:11

## 1 Answer

Generally no it does not make any sense to include the same variable on both of sides of the equation. However, this is not really what you are doing.

Certainly it is hard to argue that floor space should not enter into a regression model for the total price of a house. It does however appears as if you are trying to explain the price of the house, by the price pr. square foot - this is a bad idea.

I think you might consider why the data came in price pr. floor space form. This allows you to focus on the attributes of the house (like hedonic regression), such that you (under certain conditions) can calculate the marginal effect of (say) the distance to the freeway on price pr. square foot.