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I am a bachelor student and I try to grasp the intuition of a difference-in-differece-in-differences estimator. The standard difference in difference estimator is usually explaned using a very simple graph (like here).

Is there an equivalent graphical representation of the triple differences estimator. Can you recommend any learning material?

Thank you very much!

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Here's a toy example to get the point across from Jeff Wooldridge's book. The plot is mine. All data is made up.

Suppose the state of Florida implements some change in health care policy aimed at the elderly (people 65 and older) that improves a health outcome, like free flu shots.

One possibility is to use data only on people FL, both before and after the change, with the control group being people under 65 and the treatment group being people 65 and older. The potential problem with this DD analysis is that other factors unrelated to FL new policy might affect the health of the elderly relative to the younger population, for example, changes in Medicare reimbursements for medical care providers.

Another type of DD analysis would be to use another state as the control group and use the elderly from the non-policy states as the control group. Here the problem is that changes in the health of the elderly might be systematically different across states due reasons other than the policy change.

The DDD estimate starts with the time change in averages for the elderly in FL and then nets out the change in means for elderly in the control states and the change in means for the non-elderly in the treatment state of FL.

The dream is that this controls for two kinds of potentially confounding trends: changes in health status of elderly across states (that would have nothing to do with the policy) and changes in health status of all people living in FL (possibly due to other policies that affect everyone’s health, like hurricanes).

This is the graphical representation of the first DD and DDD estimation strategies. $\Delta_t =Y_{after}-Y_{before}$, the change in the outcome over time. Florida is orange, non-FL is blue, elderly are solid and the young are dashed.

enter image description here

The counterfactuals are harder to draw here because there are too many lines. I tried to explain the identifying assumption for this example with slightly different notation in this question. Essentially, as long as the change in unobservables over time for the elderly in the FL compared to the elderly elsewhere is similar is magnitude to the same quantity for the non-elderly, then the DDD identifies the correct effect. So instead for a common trend, it's a common difference in trends assumption, which is somewhat weaker.

And don't forget to cluster your errors!

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