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The generalized difference-in-differences (DiD) estimator is DiD with multiple time periods and multiple groups. A very common example is when the impact of laws on firms happens at different times, in part because the laws are passed at different times in different countries. A good example can be seen from Dasgupta, 2019, where he examined the impact of anti-collusion laws on firms' asset growth in the global context.

I am trying to find a way to graph the aggregate trends or do pre-trend analysis (satisfying the parallel trend assumption). However, I cannot find any reasonable way to do so. The way Dasgupta did in his paper is unable to be replicated from my side because I cannot create the control group's relative time to the event (window [-2;+5] around the implementation years). I am wondering if there is any other way to deal with parallel trend testing in generalized DiD?

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    $\begingroup$ If you’re doing a test, then you don’t need to define this window for the control group; the values are consistently 0. The coefficients on the lead indicators should be bounded around zero. $\endgroup$ Jun 17 '21 at 4:08
  • $\begingroup$ @ThomasBilach , thank you for your help, as we mentioned in this post. It seems to be a dead-end from my side in plotting the Figure like what Dasgupta did, so I am looking for an alternative test to verify the parallel assumption of DID setting. The coefficient plotting seesm to be for supporting the stability trends but not satisfy the parallel assumption from my point of view. Please correct me if I fall into any fallacy. $\endgroup$
    – Louise
    Jun 17 '21 at 4:18
  • $\begingroup$ As you mentioned in that post "It's one way to demonstrate the stability of the trends in the pre-period. It doesn't prove it, it's just evidence to support it.:" $\endgroup$
    – Louise
    Jun 17 '21 at 4:21
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    $\begingroup$ The coefficient plot is a popular way to support common trends before some exposure of interest. Simply report the coefficients on some of the lead indicators. $\endgroup$ Jun 17 '21 at 4:36
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    $\begingroup$ Yes. But you don’t have to do it that way. Some even recommend limiting the data to the pre-period and running the test that way. $\endgroup$ Jun 17 '21 at 4:54
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Plotting the aggregate trends across groups is one way to proceed, but when the adoption years vary so widely across $i$ then this approach is a bit messy.

Assessing coefficient leads in one approach. Consistent with a Granger-type causality test, leading values of the policy variable should not predict the current outcome. Here is one specification:

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

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. Note this could fail in practice for many reasons. Firms may change their behavior in response to impending regulations. Law enforcement agencies may design interventions in response to emerging crime patterns. And the list goes on.

In my opinion, it's more common in practice to observe evaluators estimate leads and lags of the policy (treatment) variable. This approach offers a more complete picture of how effects evolve in the pre- and post-adoption periods. In other words, we can assess anticipatory effects and phase-in (phase-out) effects in a single regression equation such as the one specified below:

$$ y_{it} = \alpha_i + \lambda_t + \sum_{\tau = 1}^{q}\theta_{+\tau} d_{i,t+\tau} + \sum_{\tau = 0}^{m}\delta_{-\tau}d_{i,t-\tau} + u_{it}. $$

The sums on the right-hand side of the equation allow for $q$ leads (i.e., $\theta_{+1}, \theta_{+2}, \theta_{+3},...,\theta_{+q}$) and $m$ lags (i.e., $\delta_{-1}, \delta_{-2}, \delta_{-3},..., \delta_{-m}$) of the policy variable. In applied work, evaluators typically plot the estimates of the $\theta_{\tau}$'s and the $\delta_{\tau}$'s over time. You'd hope your estimates on the adoption leads bound zero. If they do, it's evidence in support of common trends in the pre-policy epoch.

I more often see the latter equation estimated in practice. It affords you the opportunity to inspect the adoption leads and assess whether treatment effects vary with time since exposure. The decision regarding how many leads and/or lags to include outside of the immediate adoption period is often arbitrary, though it will depend, to some degree, on the number of pre- and post-policy adoption periods.

In short, I would choose the number of adoption leads (lags) and then estimate a model. Don't modify your results to tell a certain story. If you choose to estimate both equations, then report the estimates of the adoption leads from both equations! Miller and Chillar 2021 synthesize results from both approaches in a fairly digestible way.

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