# Is there a multiple regression model with both percentage and unit changes in $Y$?

In a standard linear model, $Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2$, a unit increase in $X_1$ leads to a $\beta_1$ increase in $Y$ (likewise for $X_2$).

In a log-level model, $\ln(Y) = \beta_0 + \beta_1 X_1 + \beta_2 X_2$, a unit increase in $X_1$ leads to an $\exp(\beta_1) \times 100\%$ increase in $Y$ (likewise for $X_2$).

Question: Is it possible to have a model where a unit increase in $X_1$ leads to a $\beta_1$ increase in $Y$, but a unit increase in $X_2$ leads to a $\exp(\beta_2) \times 100\%$ increase in $Y$?

For example, house price may increase linearly with living area, but being on a high traffic road would probably reduce house prices by a percentage rather than a flat value.

• I am not sure why you say that in the $\ln(Y) = \beta_0 + \beta_1 X_1 + \beta_2 X_2$ model, a unit increase in $X_1$ leads to a $\beta_1\times 100\%$ increase in $Y$? I think, a unit increase in $X_1$ leads to $\exp(\beta_1) \times 100\%$ in $Y$, since $Y=\exp(\beta_0+\beta_1 X_1 + \beta_2 X_2)$. Mar 23, 2017 at 20:48
• I've updated my question. Mar 24, 2017 at 11:26

Your question is this: Is it possible to have a function $Y=f(X_1,X_2)$ such that $f(X_1+1,X_2)=f(X_1,X_2)+\alpha$, and $f(X_1,X_2+1)=\beta\times f(X_1,X_2)$.
I am not sure whether you can find a simple function that satisfies both of the above constraints. But, a way to approach this is to define a function which is linear in $X_1$ but non-linear in $X_2$. For example, $Y=\beta_0+\beta_1 X_1 +\exp(\beta_2X_2)$.