What does the coefficients of lead indicator mean in parallel trend testing (Diff-in-Diff)?

Question

In a parallel trend testing approach, @Thomas Bilach has an intuitive way to perform by assessing coefficients leads. Intuitively speaking, the specification is

$y_{it} = \alpha_i + \lambda_t + \sum_{\tau = 1}^{q}\theta_{+\tau} d_{i,t+\tau} + \delta D_{it} + u_{it},$

where the model includes unit fixed effects, time fixed effects, a series of lead indicators $d_{it}$, and the contemporaneous policy variable $D_{it}$. The leads should be standardized in a way that $d_{i,t+1}$ is equal to 1 if a treated jurisdiction is 1 year before adoption, 0 otherwise. Similarly, $d_{i,t+2}$ is equal to 1 if a treated jurisdictions is 2 years before adoption, 0 otherwise. The equation generalizes to any number of $q$ leads. The choice of how many leads to include is for you to decide. The estimates of the $\theta_{\tau}$'s should be indistinguishable from 0, which some evaluators investigate using a joint null test. The goal is to assess the "collective significance" of the lead coefficients.

It makes sense to me. However, I have not yet fully understood what does $d_{i,t}$ mean. I am quite confused because from the explanation above, "$d_{i,t+1}$ is equal to 1 if a treated jurisdiction is 1 year before adoption, 0 otherwise" means that at 1 year before adoption, all observations got the value of $d_{i,t+1}$ equalling to 1, so what do we really test here?

Answer 1

The $d$s are not all one since untreated observations will have zeros rather than ones. You are testing the parallel trends assumption in the past when you test that $\theta$s are jointly zero. If it holds in the past, that makes it more credible to hold in the post-treatment period (in the absence of treatment). That is where you need that assumption to be true but cannot test it since treatment has taken place.