Is difference in differences appropriate here?

[John Biton]

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?

[Clyde Schechter]

The equation you show appears to be a mostly correct specification of the situation you describe, assuming that the intervention was implemented at the same time in each of the counties. The variable you denote T_it is, in fact, independent of t. That is T should be 0 for all observations in control counties and 1 for all observations in treated counties, and should be denoted T_i. Similarly t_it should be independent of i, and denoted t_t (or just t). j Also the fixed effect for county should be independent of t as well: it should be C_i And α_it should be just α.

Moving from the mathematics to the substance, it is important that the control counties have been properly chosen so that their trends in Y were parallel to those of the treatment groups in the pre-intervention era. Also, for difference in differences to identify a causal effect, it is also important that there was no other influence that affected the treatment group but not the controls (or the other way around) at or close to the same time that T took effect.

[Tom Scott]

If you don't have parallel trends, which is rare, you could try matching or switching to an interrupted time series analysis (see Stata's itsa command). Here is a useful read if you can get a copy: https://journals.sagepub.com/doi/abs...62280218814570