Here is the situation:
  • There was an intervention T.
  • I am interested in how this intervention changed the minutes spent using a computer in 5 counties: Ca_t, Cb_t, Cc_t, Cd_t, Ce_e
  • I have daily pre/post data on the minutes/spent using a computer on 200 individuals from these 5 counties
  • In addition, I found 'control' groups for each of the 5 treatment counties who showed similar trends to a specific county Ca_c, Cb_c, Cc_c, Cd_c, Ce_c
  • I have daily pre/post data on the minutes/spent using a computer on 200 individuals from these 5 counties, as well
My question is-- do I need to account for the fact that there are different counties other than using fixed effects? In other words, I am expecting the treatment effect to be the SAME across all 5 counties.

Here is what I was thinking. This is estimated at the individual (i) and day (t) level:

Y_it = α_it + βT_it + γt_it + δ (T_it · t_it) + C_it + ε_it

Where
α = constant term
β = treatment group specific effect
γ = time trend common to control and treatment groups
δ = true effect of treatment
C= fixed effect for county

Is this accurate?