Q:

Difference-in-difference-in-differences estimator

Two towns, $X$ and $Y$. In each town:

Pool cross-sections of male and female hourly wages, one from the year before a wage-discrimination policy took effect and one from the year after.

Consider the following model:

$wage_i=\beta_0+\beta_1after_i+\beta_2female_i+\beta_3X_i +\beta_4after_iX_i +\beta_5after_ifemale_i +\beta_6X_ifemale_i +\beta_7after_ifemale_iX_i+u_i$

where

$after_i=1$ if date after gender-wage discrimination policy; $0$ if date before gender-wage discrimination policy. The gender wage discrimination policy only applies to females in town $X$.
$female_i=1$ if female; $0$ if male.
$X_i=1$ if town $X$; $0$ if town $Y$.

The treatment group is females from town $X$.
Does this model measure the effect of a gender-wage discrimination policy on the average wage of women relative to men in town $X$ compared to the average wage of women relative to men in town $Y$?
In this model, the affect of the policy is captured by $\beta_7$:the 'difference-in-difference-in-differences' estimator.
For $\hat{\beta_7}$ to be interpreted as the causal effect of the policy on wages in town $X$, $E(u_i|female_i, after_i, X_i)=0$ has to be assumed. 
Does this mean that once gender, date of policy, and town have been controlled for, there exist no unobserved factors which change over both town and time that affect wages?

A:

What you propose here is actually difference in difference in differences (DDD) instead of the usual difference in differences (see these lecture notes by Imbens and Wooldridge (2007) on the first two pages). This method can potentially account for the unobserved trends in wages of women across your two towns and the wage changes of both male and female workers in the treatment town. In this sense DDD is a more robust approach to the DD approach you had in mind previously.
Let $\overline{y}$ be average income and index periods $1,2$, females and men $F,M$, and towns $X,Y$, then your DDD coefficient gives you
$$
\begin{align}
\widehat{\beta}_7 &= (\overline{y}_{X,F,2} - \overline{y}_{X,F,1}) \enspace\quad \text{the time change in $\overline{y}$ for women in town X} \newline
&- (\overline{y}_{Y,F,2}-\overline{y}_{Y,F,1}) \qquad \text{the time change in $\overline{y}$ for women in town Y} \newline
&-(\overline{y}_{X,M,2}-\overline{y}_{X,M,1}) \enspace\quad \text{the time change in $\overline{y}$ for men in town X}
\end{align}
$$
so your treatment effect will still be the effect of the policy on female wages relative to male wages in town $X$. The second term subtracts the potential wage trend of females that has nothing to do with the policy. This relates to what I mentioned in my response to your other question that you would need to know the common trends between male and female wages. This was to exclude the possibility that male and female wages are subject to systematically different changes that have nothing to do with the policy. Having untreated females in another town allows to you take out this potential female wage trend that might be different from the male wage trend.
Read the Imbens and Wooldridge lecure very carefully because they do a great job on explaining this (especially see page 2 after equation 1.4).
Even though this method is more robust, it does not mean that there is nothing left which can bias your estimates. If there is another policy at the same time, for instance a law for mothercare that affects female wages, this might still be picked up by your regression. In your work it will be your job to show that nothing else is going on in town $X$ besides your discrimination policy.

