Main Idea I want to estimate the effect of a treatment that affected a group of individuals that are scattered over a larger geographic area in a short matter of time (a week) via DID. I have reason to believe that the treatment is (almost) exogenous. Also I expect any potential remaining selection into the treatment to be driven by individual characteristics.
Data and Treatment Period I have data for the years 1994, 1995, 1996. Part of my data is based on a survey, which is conducted every year. However, the interviews are done over the span of several months, i.e. not every individual is asked at the same time. More precisely: my treatment starts at 5th of September 1995 and lasts several days. At this point most of the individuals have already been interviewed.
Treatment and Control Group My idea is to lump those individuals that were interviewed before the actual start of the treatment and those that are never going to be treated (because they do not reside in that geographic area) together as a control group.
My treatment group would then be comprised of all individuals who (a) belong to this certain geographic area and (b) were interviewed after the treatment started.
The Model As mentioned above I want to use DID to estimate the average treatment effect on the treated. The base model I have in mind is the following:
Where $y$ denotes the outcome on the individual level, $postperiod$ is a dummy that equals 1 if the individual has been interviewed after the 5th of September 1995, $Treatment$ is a dummy equaling 1 if the unit is treated after the 5th of September 1995 and is residing in mentioned geographic area, $𝜂_t$ denotes the time effect and $𝛾_i$ the individual fixed effects.
Questions My question is whether my approach is internally valid. More specifically, can I put people who I know are going to be treated but were in fact not treated yet at the time of data collection in the control group without accounting for this in any special way?
Also, as I know exactly when the treatment started, how would I define the post-treatment period? 1995 or the remainder of 1995 after the 5th of September? What implications does that have for the time fixed effect? Would I still control for 1994 and 1995 or do I have to split the sample around the 5th of September?
I appreciate any help, even if it answers just details!