Interpretation of the second order variable (Group*post) in three-way interaction regression

[Pranshu Tripathi]

Dear members,

I am working on a three-way interaction regression equation. Following is the TWFE equation:

Performance_it = b₀ + b₁ (Group_i*Post_t*Size_it) + b₂ (Group_i*Post_t) +b₃ (Group_i*Size_it) + b₄ (Post_t*Size_it) + b₅ Size_it + b₆ (control_it) + e_it

Group is a time invariant binary variable and Post is a time based binary variable which is 0 for pre-policy period and 1 for post-policy period and hence are absorved by firm-fixed effects and time-fixed effects respectively.

I am confused about the interpretation of b₂ and b_5.

Please help in the interpretation.

Thanks

[George Ford]

b₅ is always there. Effect of size (one of them).
b₂ is the intercept shift for treated/post.

I think this is right. Check it.

Control group, Pre-policy (Group=0, Post=0):
= b₀ + b₅·Size

Control group, Post-policy (Group=0, Post=1):
= b₀ + b₄·Size + b₅·Size
= b₀ + (b₄ + b₅)·Size

Treatment group, Pre-policy (Group=1, Post=0):
= b₀ + b₃·Size + b₅·Size
= b₀ + (b₃ + b₅)·Size

Treatment group, Post-policy (Group=1, Post=1):
= b₀ + b₁·Size + b₂ + b₃·Size + b₄·Size + b₅·Size
= b₀ + b₂ + (b₁ + b₃ + b₄ + b₅)·Size

[Pranshu Tripathi]

Dear George
Thanks for the reply.
I understood your comment.

Suppose I want to interpret them individually (not in totality) like
1) Three-way interaction (b₁) shows the differential impact of the policy on the sensitivity of performance to size in treated group compared to the control group.
2) Group*Size (b₃) shows the treated firms' sensitivity of performance to size in pre-policy period
3) Post_t*Size_it (b₄) shows the control firms' sensitivity in post-Policy period compared to the pre-policy period

Now following this:
4) Would Group_i*Post_t (b₂) show the normal DiD result as in case of two-way interaction or basic DiD? (I mean how the interpretation is different when I use three-way interaction)
5) would Size_it (b₅) show the control firms' sensitivity of performance to size in pre-policy period?

In this way it is easier to explain the results considering some interactions are significant whereas some are not.

[George Ford]

b2 is the shift in the intercept, and is the treatment effect only if Size = 0.
b5 is the slope on Size before treatment for the untreated.

[George Ford]

if you standardized Size (or centered),then b2 would be the mean treatment effect.

[Pranshu Tripathi]

Thanks a lot George for clearing the confusin.