Goodman, 2019 is a very famous paper decomposing the coefficients of staggered DiD. Here I do not talk about the paper itself, I talk about the note accompanying with this paper.
Pre-trend testing with multiple periods and groups have been discussed as in the note of Pischke,2005, page 7. Pischke's way of pre-trend test is summarized at running the regression $$y_{it} = \lambda_i + \delta_t + \beta_{-2}D_{it} + \beta_{-1}D_{it} + \beta_1 D_{it} + \beta_2 D_{it} + \beta_3 D_{it} + \epsilon_{it}$$
where $y$ is the outcome for individual $i$ at time $t$, $\lambda$ and $\delta$ are individual and time fixed effects
And then see how close $\beta_{-i}$ (i=1,2) to 0 and how $\beta_{j}$(j=1,2,3) varies. It is also used by a lot of papers and discussion in Cross Validated by Andy, Thomas Bilach.
The thing is Goodman, 2019, page 8 proposed a way of testing pre-trend, but the way he describe things quite ambiguous to me
The first two sentences seem to align with what Pischke, 2005 did above. However, I do not understand what does the last three sentences mean?
