# Different ways to do Panel DID?

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 $$\beta_3$$.

and

$$y_{it} = \gamma(\text{treatment}_{it}) + \eta_{it}$$ 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.