I have some results from a recent paper and I want to make sure my reasoning is correct in its interpretation of the coefficients.
In order to disguise the inflammatory nature of the model, I will relabel variables with synonymous, yet still relevant, labeling.
I have the following model specification:
$y_{idrt} = \alpha_{0} \ + \ \alpha_{i} \ + \ \alpha_{d} \ + \alpha_{r} \ + \ \delta_{1} G_{idrt} \ + \ \delta_{2} May_{i} \ + \ \delta_{3} June_{i} \ + \ \delta_{4} July_{i} \ + \ \delta_{5} August_{i} \ + \ \delta_{6} September_{i} \ + \delta_{7} May_{i} \times G_{idrt} \ + \ \delta_{8} June_{i} \times G_{idrt} \ + \ \delta_{9} July_{i} \times G_{idrt} \ + \ \delta_{10} August_{i} \times G_{idrt} \ + \ \delta_{11} September_{i} \times G_{idrt} \ + \ \epsilon_{idrt}$
Just to give summary, as I probably messed up the correct notation of the model, the model equates to the following:
We have a subject $i$ that goes through four rounds ($r$) of rating decisions where in each decision they are given a randomly assigned piece of output $d$ to rate producing a rating $y$. We do this for 6 months, hence, the indication of $t$ as a higher-order index. In each month $t$, the subject $i$ is unique to that month, hence, $t$ is constant within subject $i$. This implies that the model cannot produce the set of month-specific main effects because each month is collinear with the fixed effects of $\alpha_{i}$. Lastly, we randomly assign a treatment $G$ to each output in each round for each subject in each time-period, that is, the treatment is randomly assigned to the lowest level variable. So broadly we are dealing with a HDFE estimation where we would cluster standard errors on the subject $i$ (multi-way clustering with $t$ is not necessary in this case).
If we run this model in STATA we would really just be running:
$y_{idrt} = \alpha_{0} \ + \ \alpha_{i} \ + \ \alpha_{d} \ + \alpha_{r} \ + \ \delta_{1} G_{idrt} \ + \delta_{2} May_{i} \times G_{idrt} \ + \ \delta_{3} June_{i} \times G_{idrt} \ + \ \delta_{4} July_{i} \times G_{idrt} \ + \ \delta_{5} August_{i} \times G_{idrt} \ + \ \delta_{6} September_{i} \times G_{idrt} \ + \ \epsilon_{idrt}$
So my question is this: how do we interpret the treatment-month interactions? If this is a "reparameterization" of the model due to collinearity in month-specific main effects and subject-specific fixed effects then are all of these "interaction" effects really just simple effects conditioned on a particular month? For example would the following inferences be correct:
$\delta_{1}$ : The treatment effect on ratings of those that received a treatment in the base month of April.
$\delta_{1}$ + $\delta_{2}$ : The treatment effect on ratings of those that received a treatment in May.
$\delta_{1}$ + $\delta_{3}$ : The treatment effect on ratings of those that received a treatment in June.
$\delta_{1}$ + $\delta_{4}$ : The treatment effect on ratings of those that received a treatment in July.
$\delta_{1}$ + $\delta_{5}$ : The treatment effect on ratings of those that received a treatment in August.
$\delta_{1}$ + $\delta_{6}$ : The treatment effect on ratings of those that received a treatment in September.
Any comments would be greatly appreciated.
