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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?

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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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