Do these additional OLS regressions showcase that High-ESG funds have higher absolute levels of flows in the pre-period or does it tell something about the pre-trends?
In my review of this section, the author is showcasing mean differences across sustainability fund groups. Note in Table 3 the authors estimate separate equations within the different timing epochs. The dummy variables simply indicate whether a fund had a high, above average, below average, or low Morningstar sustainability rating as of December 2019. In other words, the estimates reflect average differences across groups within a specified time window; each group is compared to a baseline, which is average (i.e., 3 globes) funds. We can confirm this by reviewing their estimating equation. Equation 2 on page 15 doesn't include any interactions with a post-crisis indicator since they already subdivided the sample by time (e.g., pre- versus post-COVID).
Technically, they're averaging across the weeks in the different sub-periods. In the pre-shock era, for example, all we can say is high ESG funds received relatively higher weekly net flows. In my opinion, it's more indicative of higher absolute levels in the weeks starting at the beginning of the year and ending in the week prior to the onset of the stock market crash on February 20. But note how they're averaging over the weeks pre-shock, so we're not observing the week-over-week flow trends. The real work is shown in Figure 1, where we observe relatively parallel weekly average flow trends across sustainability rating groups. Parallel paths should be observed across groups and time periods.
How can you formally test the parallel trend assumption in a generalized difference-in-difference setting like this, where you have weekly-panel-data and the crisis affects all groups however with different intensities?
Technically, this is a 'classical' difference-in-differences setting. The treatment epochs (e.g., "crash" and "stimulus" periods) seem very well-defined. Note how the post-COVID era affects all funds at the same time. Once the post-treatment indicators are defined in software, we simply interact those with the high/low ESG dummies. Now say the shock affects all funds; in other words, all units in the sample are treated. In this case, we need a reference group, say the average fund. The coefficients on the interactions of a post-COVID indicator with the high/low ESG dummies estimate how much more flows high or low sustainability funds receive after the onset of the COVID-19 shock relative to before, as compared to the average fund.
It's also worth highlighting that the "intensity groups" you are referring to are categorical in nature. To assess parallel trends in this setting, I recommend plotting raw and/or normalized net flows by intensity group. In other words, plot average weekly retail net flows of high (i.e., 5 globes), average (i.e., 3 globes), and low (i.e., 1 globe) sustainability funds over time—separately. Figure 1 offers a nice illustration of this.
Third question: Do the higher flows in the High-ESG class during the pre-period (as indicated in question 1) cause problems when you try to formally test this (with for example an event-study)?
In general, no.
While a level difference is allowed, diverging time variation is not. As indicated in my other post, if average weekly flows for high ESG funds were on a different growth trajectory before the economic shock, then this biases the estimated treatment effect. Put more simply, it's permissible for average net flows among high ESG funds to be higher than low ESG funds in any one week, but their movement over time (i.e., week-over-week) should be reasonably similar as the pandemic nears.
