I have a Pseudo-Panel of economic sectors, some of which were subjected to a tax relief. I'm trying to model the effect of this reform through a Diff-in-Diff with Matching. However, I'm uncertain about the Common Trend assumption in this case. Even if the selection of a sector to the tax relief is based on observables, and I rightfully estimate its propensity score, different economic sectors will naturally have different time trends due to its market idiosyncrasies. Is it right to say that, if time trends are in fact different, DiD with Matching is not the answer here?
Suppose you have identified a treatment and control group on the basis of propensity score. Here it doesnot matter if different economic sectors have different time trends (about which you are correct that they surely will), what matters is whether those time trends are parallel or not before the tax relief is introduced in the treated economic sectors. You may refer to the discussion on the following link to test how to do this: Difference in Difference method: how to test for assumption of common trend between treatment and control group?