difference-in-differences with multiple time periods - parallel trend assumption

I am performing this resgression: $$y_{it} = \beta_{0} + \beta_{1}\text{Treat}_{i} + \sum_{j \neq k} \lambda_{j} \text{Year}_{t=j} + \sum_{j \neq k} \delta_j \left( \text{Treat}_i \cdot \text{Year}_{t=j} \right) + X_{it}'\gamma + \epsilon_{it}.$$

Yit - is a binary variable time periods t=1,2,...,k,...,T
the treatment happens between k and k+1 (so time k is my last pre-treatment period).

My question is how to present to parallel trend assumption.
I understand that there are 2 methods:
1. If coefficients δ before treatment are essentially zero. (If I get this right the 2 option are that they are equal or close to 0 and statistically significant or they are not equal to 0 and not statistically significant).
2. Run the regression separately for the treatment and control groups. Instead of a series of treat*quarter coefficients, we have just quarter coefficients for each group, and then plot those on the same graph.

Do I understand it correctly? what is the proper way to present it?

would appreciate your help, Thank you!

Both methods are doing something very similar: comparing trends between treatment and control. The first has the advantage of being able to easily test the joint null that all the pre $$\delta$$s are zero, rather than just comparing the coefficients visually, one by one. Moreover, the pooled model will probably yield more precise estimates than two separate models.

The second might be easier to explain to folks that don't understand interactions and makes it easier to plot the coefficients (rather than their difference).

• Thank you! so is it correct I should check if the coefficients δ before treatment are equal or close to 0 and statistically significant or they are not equal to 0 and not statistically significant? I saw in some places that they should only be not statistically significant. now i'm a bit confused
– XYZ
Commented May 11, 2020 at 20:23
• The size and insignificance are related. The null would be that $\delta_1 =\delta_2 =...=\delta_{k}=0$, and if you reject, you don't have parallel trends (sort of). You want them all to be zeros, or least not significantly differently from zero given their size and SEs. It would be hard for them to be zero and significantly different from zero at the same time. Commented May 11, 2020 at 20:37
• I get δ that are in the range of [-0.014,-0.004] none of them is statistically significant. are they different enough from 0 to say that the parallel trend exist?
– XYZ
Commented May 11, 2020 at 20:47
• @ThomasBilach This model here is for $E[Y \vert X]$, which is a proportion when the outcome is binary. The $\delta$ coefficients here correspond to differences in time FEs between treatment and control. I think when you plot individual binary data the plot does get messy, but here you are not plotting the raw data. Commented May 11, 2020 at 23:44
• This means you have similar pre trends between T and C. You actually need trends to be similar in the periods after treatment if there was no treatment, which is not testable since you don’t get to observe that state of the world. However, seeing it in the pre periods usually makes you believe that it would have held in the post in the absence of treatment. Commented Jun 3, 2020 at 18:28