Difference in difference with similar units over 2 periods of time I have ESG (Environmental, Social & Governance) scores for 20 companies over a period of 10 years. In the fifth year a policy was introduced and I want to estimate the impact/effect of the policy on ESG scores using difference-in-difference methods.
My questions are:
a) How do I specify the regression model since I am using the same companies (I do not have control and treatment before and after the policy) and my control group is ESG scores before policy and treatment group is ESG scores for the same companies after the policy?
b) Do I have to use propensity score matching (PSM)?
 A: 
a) How do I specify the regression model since I am using the same companies (I do not have control and treatment before and after the policy) and my control group is ESG scores before policy and treatment group is ESG scores for the same companies after the policy?

Difference-in-differences (DiD) methods require your observation of a control group both before and after treatment. Under this approach, you are comparing the before-and-after change in the treatment group with the before-and-after change in the control group. In your setting, however, you only follow one group of companies over time, all of which will be treated in the fifth year. Thus, any DiD application will be invalid.
I have read papers where treatment was "staggered" (i.e., treatment was instituted at different times for different companies) and all units were eventually treated over the course of a panel. The DiD coefficient, in this setting, is estimated using (only) variation in treatment timing. I don't think you can apply this technique, though. As per your post, the policy goes into effect for all companies at the same time; thus, there is no timing to exploit across companies. In sum, I don't see how any DiD method is applicable.

b) Do I have to use propensity score matching (PSM)?

Propensity score matching (PSM) allows you to construct an artificial counterfactual grouping of companies with similar characteristics. It creates more of an "apples to applies" comparison while reducing any bias due to confounding. That being said, PSM requires a pool of non-treated companies to select from. As indicated in your post, you do not observe any non-adopter entities/companies. Your counterfactual is the pre-treatment period for treated entities.
It is hard to say, but your second questions implies there is a group of non-adopter companies. If a subset of "never-receivers" exists, they can be used in a DiD model. I would need more information with respect to how companies were selected into treatment, but there is nothing wrong with using all non-adopter companies in a particular industry as your control group. DiD methods allow for some selection into treatment based upon the time-invariant characteristics of particular companies. I should note that DiD methods can also be combined with various matching algorithms. You can use one or the other, or both.
In sum, DiD evaluations require you to observe a subset of "unexposed" companies/entities before and after treatment. There is also little hope of approximating a random experiment via matching methods when a pool of non-treated companies does not exist.
