I am using a basic diff-in-diff strategy and have a couple of questions regarding the common trends assumption.
I use panel level data at the individual level and graph the average of the outcome between the two groups across the years. Does this make sense? My diff-in-diff regression will be an model of within player estimates, whilst the graph will show between player estimates. So can I really visually inspect for the common trends assumption? Further, my model includes time dummies and covariates, so it is harder to say that even though the pre-treatment trends don't look similar, it might be that after controlling for time dummies and covariates, the pre-treatment trends should be similar. I can't really 'graph' this arguement though.
I'm trying to test the assumption. I am using the approach given in this stackexchange post "Difference in Difference method: how to test for assumption of common trend between treatment and control group?" however that model does not make much sense to me as it has time dummies plus a treatment dummy interacting with a time dummy which would seem to indicate colinearity problems. In terms of other tests, I have seen online that I should delete all my post treatment data, and just keep my pre-treatment data. Then, using just my pre-treatment data I should put in 'placebo years'. That is, run exactly the same model (with time dummies + covariates) but with a different treatment year, and see whether the DiD coefficient is 0 and significant. Is this the correct understanding?
My model at the moment is: $y_{it}= u_i(\text{individual fixed effect}) +\beta_1* \text{treatment group} + \beta_2*\text{treatment period} + \text{time dummies} + \text{covariates} + e_{it}$
Thank you