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# Time-Varying Difference-in-Difference

One research question of interest is the effect of the ACA’s Medicaid expansion on health outcomes.  One way to look at these outcomes would be to do a difference in difference analysis.  Suppose states implemented the Medicaid expansion in 2012.  We could then compare the change in health outcomes for people in states where the Medicaid expansion occurred to people in states where the Medicaid expansion did not occur.

This simple approach works well if few individuals switch states after the expansion.  Consider the case where states with high economic growth are more likely to expand Medicaid and motivated individuals are more likely to move to those states to get jobs.  In this case, the effect of the Medicaid expansion will likely be overestimated because the people who moved to the state with the Medicaid expansion likely would have had better health outcomes even if they had stated in their current state (i.e, one that did not expand Medicaid  eligibility).

How does one address this issue?  A paper by Lee and Kim (2014) present Difference-in-Difference for stayers with time varying qualification approach to address this issue.  They divide the world into 4 types of people:

• Out-stayers: Remain in a non-Medicaid expansion in both years (M2011= 0, M2012= 0)
• Out-movers: Move from a state that had a Medicaid expansion to one that did not (M2011= 1, M2012= 0)
• In-movers: Move from a state that did not expand Medicaid to one that did (M2011= 0, M2012= 1)
• In-stayers: Remained in a state with a Medicaid expansion in both periods (M2011= 1, M2012= 1)

To avoid bias that occurs from endogenous moves, the authors propose that the difference-in-difference should drop from the sample people who switched states.  If the health outcome of interest is represented by the variable Y, the estimator is:

E[Y2012– Y2011| M2011= 1, M2012= 1] – E[Y2012– Y2011| M2011= 0, M2012= 0].

This approach is most robust when most people are stayers.  In the extreme case where 100% of individuals are movers, this modified DiD is not even possible.  For instance, if one examines the effect of Medicare with a DiD, all newly qualitified individuals will be movers since age is perfectly correlated with time.

Their approach can even address the issue of Ashenfelter’s dip where the treatment group improves solely due to regression to the mean.  The authors write:

The so-called ‘Ashenfelter (1978) dip’ for job trainings is that the treatment group experience a dip (i.e., a low Y2  in earnings) just before getting treated. Because the ‘dip’ is transitory by definition, the treatment group is bound to have a higher Y3  even without the treatment—an untreated moving effect of a sort. Considering the effect on in-stayers takes care of the Ashenfelter dip problem…

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