Interaction with DID impact

[Frank Sherman]

Hello, everyone; thank you for your attention.

I am working on a TWFE DID model. For individual i in year t :

Y_it = constant + D_it + X_it + YearDummies + e_it,
where D_it is 1 when individual i received treatment in year t , 0 otherwise; X_it is a vector of control variables.



By theory, I suspect that a variable Z_it (not included in X_it) is interacting with the impact of the treatment on Y_it. I want to test/show this point in the model. In a normal regression, one can add an interaction term between Z_it and the variable it interacts with into the model.



But how to do it in a DID model as above? May I add the interaction such as:

Y_it = constant + D_it + D_it*Z_it + Z_it + X_it + YearDummies + e_it,

and, if the coefficient of D_it*Z_it is significantly positive, may I say when Z_it is higher, the treatment effect is increased positively?

[Clyde Schechter]

May I add the interaction such as:

Y_it = constant + D_it + D_it*Z_it + Z_it + X_it + YearDummies + e_it,

and, if the coefficient of D_it*Z_it is significantly positive, may I say when Z_it is higher, the treatment effect is increased positively?

It depends, and, it depends.

It depends on whether Z_it also interacts with some of the X_it variables and whether ignoring that might bias the estimate of the interaction with D_it. Of course, you can always get around this limitation by adding the problematic Z#X interaction terms to the model, assuming your data set is large enough to tolerate that many additional variables in the regression and also large enough to adequately power the test of the Z#X interaction.

Do remember that, as a rule of thumb, if your data set is just large enough to adequately power an interaction for D, you would need a sample 4 times as large to adequately power a test of an interaction Z#D.

[Frank Sherman]

It depends, and, it depends.

It depends on whether Z_it also interacts with some of the X_it variables and whether ignoring that might bias the estimate of the interaction with D_it. Of course, you can always get around this limitation by adding the problematic Z#X interaction terms to the model, assuming your data set is large enough to tolerate that many additional variables in the regression and also large enough to adequately power the test of the Z#X interaction.

Do remember that, as a rule of thumb, if your data set is just large enough to adequately power an interaction for D, you would need a sample 4 times as large to adequately power a test of an interaction Z#D.

Thank you, Clyde.

That means the limitations are the sample size and potential z#x, but the basic principle of adding interaction of D#Z as well as the interpretation of this interaction's coefficient are the same as in a non-DID model, correct?

[George Ford]

Yes, but might include Zit as a variable as Clyde suggests.

Also, make sure you have the cross section fixed effect in there too..

[Frank Sherman]

Yes, but might include Zit as a variable as Clyde suggests.

Also, make sure you have the cross section fixed effect in there too..

Got it. Thanks!

[George Ford]

You may want to zero center Zit as well.

[Frank Sherman]

Thank you, I will try zero centering.