Common trend assumption

Question

I am using a basic diff-in-diff strategy and have a couple of questions regarding the common trends assumption.

1. 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.

2. 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

Answer 1

For your first point, plotting the average of the outcome for the treatment and control group over time would be the right thing to do in order to see the unconditional evolution of the outcomes in both groups over time. Your statement that you are essentially using between player variation is not correct though. When using difference-in-differences (DiD) you are essentially comparing the group averages. In a simple setting with one pre- and one post-treatment period you can compute the DiD coefficient as $$ \delta_{did} = \left[ E(y_{it}|g=1,t=1)-E(y_{it}|g=1,t=0) \right] - \left[ E(y_{it}|g=0,t=1)-E(y_{it}|g=0,t=0) \right] $$ where $g=1$ is the treatment group, and $t=1$ is the post-treatment period (see here for further explanation). So given that the computation of the treatment effect ultimately happens at the group level, plotting the group averages over time is a good indication for whether the parallel trends assumption holds.

For your second question, doing the procedure in the answer you linked is actually very similar to including placebo treatments. Suppose you have time periods $t = 1, 2,...,k,...,T$ periods where the treatment happens between $k$ and $k+1$ (so time $k$ is your last pre-treatment period). In your setting, you could run the following regression: $$ y_{it} = \text{individual Fe}_i + \text{time FE}_t + \sum_{j\neq k} \delta_j \left( \text{treatmentgroup}_i \cdot I(t=j) \right) + X'\gamma + \epsilon_t $$

i.e. you are interacting the treatment group indicator with time dummies for all periods except for period $k$ (because you need to leave one interaction out as otherwise there will be perfect multicollinearity) which are the $I(\cdot)$ terms in the regression equation.

Then all the $\delta_j$ with $j<k$ are placebo tests for whether the treatment had an effect on the outcome between the two groups. This should not happen because if the treatment has an effect before it even occurs, then this casts doubts on the parallel trends assumption. Plotting these coefficients is basically the "conditional" outcome distribution plot as compared to plotting the unconditional outcome evolution over time as discussed for point 1.
The nice thing is that the $\delta_j$ coefficients with $j>k$ then show you how the treatment effect evolves over time, i.e. how long it takes to fade away or whether it is persistent. An example of how such a coefficient plot would look like is shown below. There is a nice command available in Stata (and I think also in R) called coefplot which does this for you.

Here the omitted time period is 1940 which is the last pre-treatment period). None of the coefficients before are statistically significant and afterwards you see a permanent effect of the treatment (in this particular case).