Confusion in and Justification for calculating propensity score for unit-level

[Phuc Nguyen]

Propensity Matching Score has been used widely, however, the way to match firm by firm in PSM in panel data still a mysterious to me.

Background: Using pre-treatment covariates to match
Problem: Any firms will have more than 1 observations, so how to match firm-by-firm?

After doing extensive research, I saw some approachs, these two are prominent:

1. Yang, 2012 on Resource Policy control for year and industry fixed effect beside time-variant factors. However, I did not know how they match firm by firm. But from controlling for year dummy, it is quite sure that they do not use the average of variables during pre-treatment period for matching.
2. Also on journal Resource Policy, Howell, 2016 calculate the average of covariates for each firm during pre-treatment period then do matching.

I am concluding the approach of Howell based on these sentences

Firms in the control group are matched to the treatment group on the basis of the pre-treatment (1998–2001) mean of these variables.

All covariates are measured by the mean before the policy treatment.

I am wondering if I understand the approach of Howell correctly. Apart from that, if I understand correctly, the approach of Howell will not control for unobserved time-specific factors that might affect both the treatment and the outcome (e.g., macroeconomic changes, policy shifts, or other factors that systematically vary over time), leadingh to a suboptimal matching.
Thank you.