Difference-in-differences estimator Consider the following model:


*

*$wage_i=\beta_0+\beta_1after_i+\beta_2female_i+\beta_3after_ifemale_i+u_i$


where


*

*$after_i=1$ if date after gender-wage discrimination policy; $0$ if date before gender-wage discrimination policy.

*$female_i=1$ if female; $0$ if male.
The 'treatment group' is females.
This model measures the effect of a gender-wage discrimination policy on the average wage of women relative to men.
The affect of the policy is captured by $\beta_3$ which can be estimated as follows:


*

*$\hat{\beta_3}=(E[wage_i|after_i=female_i=1]-E[wage_i|after_i=1, female_i=0])-(E[wage_i|after_i=0, female_i=1]-E[wage_i|after_i=female_i=0])$


which is known 'difference-in-differences' estimator.
Questions:


*

*If $\hat{\beta_3}>0$, then does this means that the policy caused women's earning to increase relative to mens?

*For $\hat{\beta_3}$ to be interpreted as the causal effect of the policy on wages, does $E(u_i|female_i, after_i)=0$ have to be assumed?


Any guidance would be very much appreciated. Thank-you.
 A: The way you have described the problem, females are not the control but the treatment group. Then a positive $\widehat{\beta}_3$ would imply that women's earnings have increased more relative to men's earnings in response to the discrimination policy. Equivalent your expression, this will then be captured by the coefficient:
$$
\begin{align}
\widehat{\beta}_3 &= (E[\text{wage}_i | \text{female}=1, \text{after}=1] - E[\text{wage}_i | \text{female}=1, \text{after}=0]) \newline
&- (E[\text{wage}_i | \text{female}=0, \text{after}=1] - E[\text{wage}_i | \text{female}=0, \text{after}=0])
\end{align}
$$
With respect to your second question: yes, you need this assumption. And it will be very difficult for you to make this case with two time periods.
If the earnings difference between men and women is only due to unobservable factors that remain constant over time, then you will be able to identify the causal effect of your policy. The problem is: you cannot test this. What people usually do is to show that the outcomes of the treatment and the control group moved parallel (or at least very similar) before the treatment occurred. Then you can make the case that the outcomes trends would have continued in the same way in the absence of the treatment.
The difference between this hypothetical continuation in the pre-treatment trend in the outcome for the treatment group after the treatment and the new trend in the outcome for the treatment group after the treatment is your difference in differences estimate, i.e. your treatment effect.
If there are time-varying factors which are unobserved and that affect your gender gap in earnings, difference in differences can't help you. For instance, if employers cut female wages in recessions by more than male wages, this is a problem. In order to make an argument that this gap is only affected due to time-invariant factors besides the female dummy, you would need to show this common pre-treatment trend in earnings between men and women.
