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In my research, I would like to answer the question whether a regulation change leads to my continous, independent variable (X) affecting my outcome variable to a greater extent for those being treated.

For that, I would like to use a DiD approach, since I have two groups with only one of them being treated in the second period, and then interact this effect with my continous variable. From what I have grasped, this would result in a model specification that looks like a Diff-in-Diff-in-Diff (DDD) model [but is in fact not a DDD like described here ]:

β0 + β1 X + β2 group + β3 period + γ1 X * group + γ2 group * period + γ3 X * period + δ1 X * group * period + ϵ

my questions are:

1) Am I correct in assuming that this model should work for my intentions?

2) What are the identifying assumptions in this case? The common trends assumption like for DD or rather the common difference in trends assumption as described here for the DDD?

3) Is there anything else I should bear in mind?

Thank you very much in advance!

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