When regressing DID, we normally run pre-trend analysis. But in some papers recently using "generalized" DID for testing the impact of staggered laws on some dependent variables, the authors did not conduct the pre-trend analysis (here and here). This raises a question to me that why and when we do not need to perform a pre-trend analysis. Any suggestion or document and explanation is much appreciated.
A pre-trend analysis isn't a requirement, though it improves the credibility of a study. Typically, evaluators plot the outcome trends for the treatment group and the control group over time. We should expect the inter-temporal evolution of the trends to be reasonably parallel before the law/policy is enacted.
For example, the authors in the paper you referenced plot the mean change in asset growth in both groups in the 2 years before and the 5 years after the adoption of a leniency law. We should expect the trends in asset growth in the treatment group and the control group to be moving in tandem before the passage of the leniency law. If the two groups were on completely different trajectories, then we may wrongly infer a "treatment effect" when, in fact, divergent paths were already emerging in the pre-treatment epoch.
Technically, a pre-trend analysis was conducted in the work you cite. In section 4.2 they investigate a "dynamic response" to treatment. In particular, they examined whether some countries "anticipated" a change in their leniency laws. Media outlets may report on impending law/policy changes, and firms within those countries may start to change their behavior before the law is actually passed. To investigate anticipatory effects, the authors include dummies from 1 to 4 years before the passage of the law.
Including indicator variables for the periods before the shock serve two purposes in my estimation. First, they model potential anticipatory effects. Second, it is a form of pre-trend testing. With respect to pre-trends, we shouldn't be observing strong non-zero effects in the years before the law is passed, unless we suspect anticipatory behavior. And even if anticipation is suspected, the anticipation phase is usually limited to the periods close to the treatment's start date.