# Which is the best approach? Difference in Difference with Heterogeneous Effects

I am interested in the performance of two competing approaches to estimate a treatment effect, for which one would expect heterogeneous effects.

The point of origin is a treatment conducted from period $t$ onwards (dummy: $d_t$) for the treatment group (dummy: $a_i$). The treatment is expected to only affect that part of the treatment group that has a particular feature (dummy: $f_i$). Agents in both the treatment and the control group can have this feature. The variable of interest (the dependent variable) is $Y$.

Now, there are two ways of estimating the effect of the treatment on the treated:

1. Standard Difference in Differences, estimated seperately first for the subsample that has the feature ($f_i =1$) and then again for that subsample that does not have this feature ($f_i =0$). $$Y = \alpha + \beta_1 a_i + \beta_2 d_t + \beta_3 a_i d_t + \beta_x \mathbf{x}$$ where $\mathbf{x}$ is a vector of controls. One would then obtain two different DiD estimates ($\beta_3$) and the difference between the 2 coeffiecients indicates the different effect of the treatment depending on the feature of the agents.

2. Estimate the effect in one equatin using Difference in Difference in Difference (DDD). $$Y = \alpha + \beta_1 a_i + \beta_2 d_t + \beta_3 a_i d_t + \beta_4 f_i + \beta_5 a_i f_i + \beta_6 d_t f_i + \beta_7 a_i d_t f_i + \beta_x \mathbf{x}$$ where $\beta_7$ tells us to what extent the treatment effect varies with the special feature.

I am now interested in advantages and disadvantages of the two approaches.

Right now, I would prefer the DDD approach. As far as I'm concerned, it has one disadvantage, which stems from the common trend assumption, which is a stronger assumption in the DDD context relative to the normal DiD. But I think this problem could be tackled by including appropriate controls.

• First of all, the subsample of the population with the feature of interest ($f_i=1$) might systematically differ from the rest of the population (significant value for $\beta_4$).
• Second, those agents in the treatment group that posess the feature of interest ($f_i$*$a_i$=1) might systematically differ from other agents, independent from the treatment (significant value for $\beta_5$).
• Third, the treatment might affect all those that have the particular feature, independent from whether they are actually treated or not (significant value for $\beta_6$).