I am attempting to conduct a difference-in-differences analysis in R, though not in the traditional sense. My study involves three groups that were treated simultaneously, without a true control group. However, I anticipate that the treatment's impact will be much milder for one group ("Non-tradable Sector") compared to the other two, which I expect to experience more significant effects ("Export Oriented Sector" and "Import Oriented Sector"). This is why I thought of treating the "Non-tradable Sector" as a control group and the latter two as treatment groups.
My main objective is to explore the differences between these groups, particularly focusing on the comparison between the "treated" groups and my "untreated" group. With my setting, I think that I can only explore the differences in average treatment effects between the groups rather than the overall ATE, right? The relationship I'm examining can be described by the following model (referenced from this post)
$$ y_{it} = \alpha + \beta_1 \text{Treat}_i^{\text{exp}} + \beta_2 \text{Treat}_i^{\text{imp}} + \beta \text{Post}_t + \delta_1 (\text{Treat}_i^{\text{exp}} \times \text{Post}_t) + \delta_2 (\text{Treat}_i^{\text{imp}} \times \text{Post}_t) + \gamma_t + \mu_i + \epsilon_{it} $$
where $\text{Treat}_i^{\text{exp}}$ is an indicator for the export oriented group and $\text{Treat}_i^{\text{imp}} $is an indicator for the import oriented group.$\text{Post}_t$ is a dummy indicating the post-treatment period. $\gamma_t$ is the time fixed effect and $\mu_i$ the group fixed effect.
I am measuring how support for independence – a dummy Variable – changes pre- and post-treatment. I expect that the treatment will make the export-oriented group more opposed to independence, while the import-oriented group will likely become more supportive. As for the non-tradable sector, I anticipate their reaction will be similar to the export-oriented group, though less pronounced.
Since I lack panel or longitudinal data and only have cross-sectional data from different time periods and individuals, I created a pseudo-panel based on specific sociodemographic variables and the sectors in which individuals work. This means that individuals can not move from one group to another. However, given that all groups were treated, I am concerned about potential spillover effects.
One challenge is that, although all groups were treated simultaneously, the treatment was disseminated through the media. This means that the treatment's reach could depend on the media individuals consumed and how intensely they engaged with it, leading to varying degrees of exposure. (This could imply that the treatment effect is not homogeneous, though I am unsure if this heterogeneity refers more to the variation in how groups respond to the treatment.) To address this, I plan to additionally conduct a subgroup analysis based on media consumption. I thought of subsetting the data (leaving one of my two treatment groups out of the analysis respectively) and doing this:
$$y_{it} = \alpha + \gamma_1 \text{Treat}_i^{\text{exp}} + \lambda \text{Post}_t + \delta_1 (\text{Treat}_i^{\text{exp}} \times \text{Post}_t) + \delta_2 (\text{Treat}_i^{\text{exp}} \times \text{Post}_t \times \text{Media}) + \gamma_t + \mu_i + \epsilon_{it} $$
$$ y_{it} = \alpha + \gamma_1 \text{Treat}_i^{\text{imp}} + \lambda \text{Post}_t + \delta_1 (\text{Treat}_i^{\text{imp}} \times \text{Post}_t) + \delta_2 (\text{Treat}_i^{\text{imp}} \times \text{Post}_t \times \text{Media}) + \gamma_t + \mu_i + \epsilon_{it} $$
A final consideration about my data: my group sizes are uneven, with 76 individuals in the "Export Oriented Sector," 110 in the "Import Oriented Sector," and 558 in the "Non-tradable Sector". I hope this does not pose too many problems for the analysis.
As I am new to difference-in-differences analysis and navigating a complex setup, I would greatly appreciate any guidance or advice.
