<p>You have to add them even though they are in <span class="math-container">$\ln$</span> the model is still linear in the parameters. To see this more clearly you can rewrite your equation as (I use <span class="math-container">$D$</span> for demand, <span class="math-container">$c$</span> for constant, <span class="math-container">$F$</span> for fare and <span class="math-container">$M$</span> and <span class="math-container">$T$</span> for Monday and Tuesday):</p>
<p><span class="math-container">$$\ln(D) = c  + \left(- 0.45  - 0.15(T)  + 0.05(M) \right)\ln(F) + 10(M) + 15(T)$$</span></p>
<p>Hence when both <span class="math-container">$T=0$</span> and <span class="math-container">$M=0$</span> the elasticity for Wednesday is -0.45, when <span class="math-container">$T=1$</span> and <span class="math-container">$M=0$</span> (i.e. its Tuesday) the elasticity is <span class="math-container">$-0.6$</span>, and finally when <span class="math-container">$T=0$</span> and <span class="math-container">$M=1$</span> (i.e. its Monday) the elasticity is <span class="math-container">$-0.4$</span>.</p>
<p>If you want to learn more Wooldridge discusses this in his introduction to econometrics in chapter 7.</p>

CC BY-SA 4.0
2020-07-30T11:13:06.057Z
2020-07-30T11:13:06.057Z
1muflon1
<p>I have the following regression to interpret elasticity of demand:</p>
<p><span class="math-container">$$\ln(demand) = const - 0.6*\ln(fare)$$</span></p>
<p>I understand that a 1% increase in fare results in a 0.6% decrease in demand</p>
<p>I want to add dummy variables for days of week (excluding one of course) and want to add day-of-week <span class="math-container">$* \ln(fare)$</span> interactions so that I can determine the elasticity of demand by day of week (which I already know differs)</p>
<p>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</p>
<p>So now my regression is:</p>
<p><span class="math-container">$$\ln(demand) = const - 0.45*\ln(fare) + 10*(mon) + 15*(tue) + 0.05*\ln(fare)*(mon) - 0.15*\ln(fare)*(tue)$$</span></p>
<p>Here, my elasticity of demand for Wednesday is interpreted as a 1% increase in fare results in a 0.45% decrease in demand</p>
<p>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.</p>
<p>So the question is for Tuesday should The coefficients be added? Or multiplied since it’s logged?</p>

<econometrics><regression>
How to properly interpret logged interaction variables
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# How to properly interpret logged interaction variables

(Asked by: username_0 on 2020-07-30)

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?

**Tags:** econometrics, regression


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## 1 answers

### Answer by username_1 on 2020-07-30. Score: 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.


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