What is the difference between these two specifications, when performing an analysis on the impact of a policy in a panel data environment?

$$y_{it} = \beta_1 (\text{treatgroup}_{i}) + \beta_2 (\text{treatment}_t) + \beta_3 (\text{treatgroup}_{i} \cdot \text{treatment}_t) + \epsilon_{it}$$

where $\text{treatgroup}_{i}$ is a dummy for whether individual $i$ is in the treatment group, $\text{treatment}_t$ is a dummy for the post-treatment period, and the interaction between the two captures the treatment effect in $\beta_3$.


$$y_{it} = \gamma(\text{treatment}_{it}) + \eta_{it}$$ where $\text{treatment}_{it}$ is a dummy variable which equals one if individual $i$ is in the treatment group AND time $t$ is at or after the treatment date.

Are $\beta_3$ and $\gamma$ estimating the same parameter? The parameter being the average effect of treatment over individuals and over treatment-time $$ \frac{1}{I}\sum_i^I\frac{1}{T}\sum_t^{T}\eta_{it} $$ where $\eta$ is the true time- and individual-variant effect of the treatment, $I$ is the number of population individuals ans $T$ is the number of periods of interest.

Note that I used freely pieces of @Andy 's answer in this post Measuring length of intervention effect to formulate my answer more quickly and also to build on the same topic.


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