# 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?