3 related questions about DDD (TD, triple-diff) estimators In Jeff Wooldridge's Econometric Analysis (2nd edition), he derives the expression for the difference-in-difference-in-differences (DDD) estimator on page 151 for the two period case where state B implements a health care policy change aimed at the elderly.
First, I am puzzled by why the equation (6.56) does not have a fourth term of
$$(\bar y_{A,N,2} - \bar y_{A,N,1}),$$
which would correspond to the change for the mean health outcomes for the non-elderly (group N) in the states that do not change their policy (group A).
He cites Gruber (1994) as using this method, but my reading of table 3 in that paper is that it is a difference of two DDs, so you need the fourth term to have that (otherwise you get $\delta_3 + \delta_0$ instead of just $\delta_3$). 
I have already checked the errata for the second printing, and this did not come up, so I must be missing something here. It also appears in his 2007 NBER lecture notes in the same form.

My second question is that in the case with more than two time periods, JW suggests a regression that includes:


*

*a full set of dummies for type of state (A or B)

*a full set of dummies for age category (E or N)

*dummies for all time periods

*pairwise interactions between the previous three

*a policy dummy that takes the value of 1 for groups and time periods subject to the policy, which is the DDD parameter of interest


JW writes "full sets of dummies" and "all time periods", but I am not sure how that can be done without falling into the dummy variable trap. It might seem natural to drop the type A state one and the non-elderly one (group N), but say I have 10 time periods, and treatment happens in period 5. How does one pick which time dummy to drop to avoid the dummy variable trap? This choice seems to alter the DDD parameter and its interpretation, but I not sure if one is best. Here's another question where there's a natural choice because there's a single pre period that serves as the baseline.

Finally, what exactly is the identifying assumption with DDD, analogous to common trends with plain DD? Are there any ways to test/bolster it with multiple periods?  
In Myoung-jae Lee's Micro-Econometrics for Policy, Program, and Treatment Effects, the condition (translated into JW's example) is listed as
$$
\delta_3 + E[u_{1,2} - u_{0,1}\vert E=1,B=1]-E[u_{0,2} - u_{0,1}\vert E=1,A=1]-{E[u_{0,2} - u_{0,1}\vert N=1,B=1]-E[u_{0,2} - u_{0,1}\vert N=-=1,A=1]},
$$ 
where the first subscript indexes the potential outcome (1 treated, 0 if not) and the second is time (post is 2, pre is 1). I interpret this as saying that as long as the change in unobservables over time for the elderly in the treated state compared to the elderly elsewhere is similar is magnitude to the same quantity for the non-elderly, then the DDD identifies the correct effect. This seems weaker than common trends, which would be sufficient, but not necessary for DDD. Is this correct? 
 A: *

*You're right. In this 2011 slideshow, equation (4), slide 6, shows the missing term. As you mention it, the interpretation to keep in mind is that a triple difference is a difference between two differences in differences. 

*You're right to mention that the estimation requires you to omit a reference category for your time, state and age group fixed effects (to avoid collinearity). However, the choice of the reference category should not matter for the estimator of the triple-difference term. Note that you can write directly your dummies as interactions between time periods, states, and age groups, see the equation in Pischke's lecture notes (bottom of p.16). 

*One way to frame the identification assumption is the following. In standard DiD, you would like your two groups to have evolved in a similar way if treatment had not existed. In triple difference, you would like the gap between your treated stated and your states to evolve similarly over time for older and younger individuals, in the absence of the treatment. You could also frame it switching states and age groups. The way you would empirically test this would first be to eyeball the trends before treatment happened (if you have data before it happens). In a DiD case, you would just plot the treated and control average for each year before treatment. In the triple difference case, you could do the same with four lines or, more conveniently, you could plot gaps between treated and control states, for each age group and year, and check whether they are parallel. 

