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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?

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