# Should inverse probability weighting be used in two-way fixed-effects panel regression?

Let's assume a (balanced) panel data set with two measurement points $$t_0$$ and $$t_1$$, where $$t_0$$ may be considered as the baseline. Some of the ID's are treated at $$t_1$$, i.e. $$D=1$$, the assignment is non-random and uneven, though. The data include many covariates at the baseline, indicated by $$x_1$$ and $$x_2$$, and two outcomes $$y_1$$ and $$y_2$$, where $$y_1$$ is only measured at $$t_0$$, whereas $$y_2$$ is measured at both time points $$t_0$$ and $$t_1$$. Here a schematic representation:

  id time D   y1   y2   x1   x2   ps
1  1    0 0   NA 0.81 0.23 0.61 0.39
2  2    0 0   NA 0.78 0.97 0.37 0.18
3  3    0 0   NA 0.29 0.91 0.36 0.58
4  1    1 0 0.64 0.52   NA   NA 0.39
5  2    1 0 0.71 0.52   NA   NA 0.18
6  3    1 1 0.95 0.87   NA   NA 0.58


For outcome $$y_1$$, to estimate the average treatment effect of the treated $$ATT_1$$ I used a propensity score ($$PS$$) matching approach to account for selection on observables; the $$PS$$ was calculated by logistic regression of treatment $$D$$ on appropriate covariates $$X$$ from the baseline, achieving a good balance. The $$PS$$ was added to the data set for each ID.

For outcome $$y_2$$, since it's measured twice, to estimate $$ATT_2$$ I use ID and time fixed effects panel regression, to account for selection on unobservables and in order to benefit from the panel data structure.

Now, since I already have the $$PS$$'s at hand, I wonder if I could benefit from weighting the fixed effects regression by the inverse of them, i.e. by using inverse probability weighting (IPW). I came across a very similar approach, Kosuke & In Song (2019) are proposing in a method that uses propensity score weighted fixed effects, albeit just for unit fixed effects.

So my question is whether we should use IPW in two-way fixed effects panel regression, to account for selection on both observables and unobservables, or whether this is more of a bad idea and would yield incorrect results?