In the following fixed-effects model, $EI$ is a dummy variable indicating an economic integration agreement in place between $i$ and $j$. $A$ is used to index the specific agreement an $i, j$ pair belongs to. $Y_{i,j}$ represents the volume of exports. Assume $A$ = {1,2,3}, meaning there are 3 different agreements in the sample. The goal is to obtain an average partial effect for each agreement in the sample ($\beta_{1, A}$).
$\begin{aligned} Y_{i j, t}= & \exp \left(\eta_{i, t}+\psi_{j, t}+\gamma_{{i j}}+\sum_A \beta_{1, A} E I_{i j, t}\right) \\ & +\varepsilon_{i j, t}\end{aligned}$
Fleshed out, the regression is as follows:
$\begin{aligned} Y_{i j, t}= & \exp \left(\eta_{i, t}+\psi_{j, t}+\gamma_{{i j}}+ \beta_{1, 1} E I_{i j, t}+ \beta_{1, 2} E I_{i j, t}+ \beta_{1, 3} E I_{i j, t}\right) \\ & +\varepsilon_{i j, t}\end{aligned}$
I am unsure how to obtain the specific partial effects. My initial thought was to run the regression for each $A$, but that does not appear to fit the above regression model -- furthermore, these regressions are not informative if $EI$ is time in-varying for a given $i, j$ pair. I was curious if using an interaction between $A$ and $EI$ would be useful, but I believe this is equivalent to running the regression for each individual $A$ and reporting the resulting coefficients.
My other thought is creating a dummy variable for each agreement in $A$. But even then, it is not clear how to proceed with this regression, since this would also require some interaction between the new dummy variable and $EI$.
It is not clear to me how separate $\beta_{1, A}$'s are achieved without some form of interaction of use of $A$ on the explanatory variable $EI$.
