Difference-in-difference-in-differences panel data

Basically, I want to know which interaction term(s) is needed when building a DDD model for panel data? In a DD model I would only include a binary program indicator (like after*treatment) when regressing via FE (and not dummies for after and treatment), however, do I only include the tripple interaction when adding another control group and thereby obtain a DDD model? Or do I also need to include the DD interaction terms?

A:
$Y = a*b*c + yeardummies + controls + error$

or

B:
$Y = a*b + b*c + a*c + a*b*c + yeardummies + controls + error$

Hope it makes sense.

The DDD formula is slightly different than both of those. It is

$Y = \alpha + a + b + c + a*b + a*c + b*c + a*b*c + controls + \epsilon$

where $\alpha$ is the intercept, and $\epsilon$ is the is the error term.

That is, you also have to include the main effects, $a$, $b$, and $c$. Assuming you are examining a single event that occurs at a point in time, one of these three main effects could be modeled by year dummies. See equation 1.3 on page 2 of the Wooldridge Imbens lecture and a related post on CV here.

In one approach, the Stata command would be something like

reg y i.a##i.b##i.c controls, vce(cluster clustvarname)


$c_i$ is unobserved heterogeneity of the observational unit, while $\lambda_t$ is unobserved heterogeneity of time. In the basic two-period model, $\lambda_t$ is a binary variable, but in a multi period environment, you have the ability to more flexibly model time. You would also have a third aspect of unobserved heterogeneity. These (the main effects) can be modeled by adding fixed effects, but for a DDD, you will also need to model the interaction (second order) effects and of course include a term for your triple interaction (the variable of interest).

• Thank you for your time! However, if you read the Wooldridge lecture you will realise that the model you suggest is for cross sectional data and not panel data. The panel data model (section 4) does not include those main effects, and this is what make me question whether I have to include the interaction terms in a DDD version of the panel data model. – Mld May 26 '16 at 13:12
• DD and DDD do not make sense in a cross section. There must be time or something like time in order for there to be an "event." equations 4.1 and 4.5 in the lectures, which cover DD, both have main effects, $c_i$ for the individuals and $\lambda_t$ in equation 4.5 for more than two time periods. – lmo May 26 '16 at 14:31
• All this time I have been reading c_i as the unobserved heterogeneity. Thank you! So, would – Mld May 26 '16 at 19:11
• So, I would need to include the interaction terms ab ac bc and abc in the Stata command for a DDD FE model, but not a and b (given c is yeardummies) right? – Mld May 26 '16 at 19:13
• I updated my answer to clarify some of this. Take a look when you get a chance. – lmo May 26 '16 at 19:25