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Within a difference in difference (DD) strategy trends of an outcome of interest are compared for the treatment and the control groups before and after an intervention. The crucial assumption is that the trend observed for the control group must not differ from the one of the treatment in the pre-intervention period (parallel trends).

However, what if within a given population one can convincingly state that only a defined sub-sample of observation qualify to be part of the control group, another part as the treatment, and finally, some observations are dropped?

For example, suppose the following empirical setting: In 2018, the government of country X introduces a minimum salary. This policy is not retroactive and thus is effective only for new labor contracts.
We have longitudinal information, let's say five waves of which one refers to the post-implementation period (2019), and we want to see the effect of the policy on individual income levels. Noteworthy, the policy was implemented at the national level without any temporal lags between administrative units.

Thus the sample can be diveded in three groups:

  1. Observations always being unemployed in the pre-treatment period;
  2. Individuals never being unemployed in the pre-treatment period;
  3. Observations sometimes employed and sometimes unemployed in the pre-treatment period.

Because the policy could change the group belonging of an individual (e.g., the minimum salary could increase involuntary unemployment), the groups for the DD are defined according to the group belonging before the policy implementation described above.

Group 1 and 2 show different levels of income, but the trends in income are parallel. While the trend of group 3 is not parallel to the ones of groups 1 and 2. For obvious reasons group 3 does not show parallel trends since the individuals belonging to it jump from unemployment to employment status across waves.

Can thus one safely exclude group 3 from the DD analysis? Would only the interpretation of the results change? In the example above: the implementation of a minimum salary increases/decreases income for those in prolonged unemployment compared to the never-unemployed. Or does the sample restriction outlined above introduce any sample-selection bias?

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