How to properly interpret logged interaction variables I have the following regression to interpret elasticity of demand:
$$\ln(demand) = const - 0.6*\ln(fare)$$
I understand that a 1% increase in fare results in a 0.6% decrease in demand
I want to add dummy variables for days of week (excluding one of course) and want to add day-of-week $* \ln(fare)$ interactions so that I can determine the elasticity of demand by day of week (which I already know differs)
For simplicity in writing this out let’s pretend there are 3 days: mon, tue, wed and I will leave Wednesday out as my base
So now my regression is:
$$\ln(demand) = const - 0.45*\ln(fare) + 10*(mon) + 15*(tue) + 0.05*\ln(fare)*(mon) - 0.15*\ln(fare)*(tue)$$
Here, my elasticity of demand for Wednesday is interpreted as a 1% increase in fare results in a 0.45% decrease in demand
But my question is how to interpret elasticity of demand for Tuesday. I know that the interaction and even the dummies are relative to the base.
So the question is for Tuesday should The coefficients be added? Or multiplied since it’s logged?
You have to add them even though they are in $\ln$ the model is still linear in the parameters. To see this more clearly you can rewrite your equation as (I use $D$ for demand, $c$ for constant, $F$ for fare and $M$ and $T$ for Monday and Tuesday):
$$\ln(D) = c  + \left(- 0.45  - 0.15(T)  + 0.05(M) \right)\ln(F) + 10(M) + 15(T)$$
Hence when both $T=0$ and $M=0$ the elasticity for Wednesday is -0.45, when $T=1$ and $M=0$ (i.e. its Tuesday) the elasticity is $-0.6$, and finally when $T=0$ and $M=1$ (i.e. its Monday) the elasticity is $-0.4$.
If you want to learn more Wooldridge discusses this in his introduction to econometrics in chapter 7.