<p>Let’s denote <span class="math-container">$T:=\text{treatment}, F:=\text{female}$</span>.</p>
<p>For females we have</p>
<p><span class="math-container">$\mathbb{E}[y|F=1] = \beta_0 + (\beta_1 + \beta_3) T + \beta_2$</span></p>
<p>On the other hand, for males we have</p>
<p><span class="math-container">$\mathbb{E}[y|F=0] = \beta_0 + \beta_1 T$</span></p>
<p>Reconditioning by <span class="math-container">$T=0$</span>,</p>
<p><span class="math-container">$\mathbb{E}[y|F=1,T=0] = \beta_0 + \beta_2$</span></p>
<p>and</p>
<p><span class="math-container">$\mathbb{E}[y|F=0,T=0] = \beta_0$</span></p>
<p>Note these two expressions only differ by <span class="math-container">$\beta_2$</span>.</p>
<p>Therefore, <span class="math-container">$\beta_2$</span> is the average difference between untreated females and untreated males, ceteris paribus.</p>

CC BY-SA 4.0
2023-08-29T10:35:50.260Z
2023-08-29T10:35:50.260Z
Nicolas Torres
<p>I have a regression:</p>
<p>y=b_0+b_1x(treatment)+b_2x(female)+b_3x(treatmentxfemale)+e</p>
<p>the effect of the treatment for the female is b_1+b_3. So that, what is the interpretation for coefficient b_2?</p>

<regression>
coefficient interpretation
userF
# coefficient interpretation

(Asked by: username_0 on 2023-08-29)

I have a regression:

y=b\_0+b\_1x(treatment)+b\_2x(female)+b\_3x(treatmentxfemale)+e

the effect of the treatment for the female is b\_1+b\_3. So that, what is the interpretation for coefficient b\_2?

**Tags:** regression

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## 1 answers

### Answer by username_1 on 2023-08-29. Score: 2

Let’s denote $T:=\text{treatment}, F:=\text{female}$.

For females we have

$\mathbb{E}[y|F=1] = \beta\_0 + (\beta\_1 + \beta\_3) T + \beta\_2$

On the other hand, for males we have

$\mathbb{E}[y|F=0] = \beta\_0 + \beta\_1 T$

Reconditioning by $T=0$,

$\mathbb{E}[y|F=1,T=0] = \beta\_0 + \beta\_2$

and

$\mathbb{E}[y|F=0,T=0] = \beta\_0$

Note these two expressions only differ by $\beta\_2$.

Therefore, $\beta\_2$ is the average difference between untreated females and untreated males, ceteris paribus.


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