How to properly interpret logged interaction variables

Question

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?

Answer 1

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.