Differences-in-Differences Parallel Trends I want to measure whether the impact of a company's headquarter country on my dependent variable (goodwill paid) is stronger during recessions. After some researching, I found out that the differences-in-differences analysis could solve my problem. However, in the internet they always show a diagram (see Figure 1 in Difference-in-Difference Estimation by Columbia Public Health) with the "treatment" and "parallel trends". So two lines that increase or decrease in the same way until the treatment and then one line increase/decreases more than the other. My question now is what is my treatment and what is my control variable in my example? The treatment cannot be recessions because otherwise I just have the treatment group after the treatment and the control group before the recessions. If you think another statistical test may be better, I would be happy to consider that.
Furthermore, I just want to make sure that I created my model correctly: Goodwil Paid=B0+B1recession+B2Country+B3recessionCountry Would that tell me whether the impact of the country is stronger during recessions?
Variable descriptions:
-Goodwil paid (dependent variable): Is about how much is paid for a company in acquisitions.
-Recessions: 1 if the acquisition was during the crisis and 0 otherwise
-Country: The country the acquired company is bought in. It can have a value from 1 to 10 and is based on credit ratings of the countries. Most countries have the rating 1.
Thanks a lot for your help. Let me know if you need further information.

Your situation is slightly more complicated than a standard diff-in-diff (DiD) that has two groups. However, the gist is the same.
A standard DiD would ask something like if US and UK firms have a different impact on goodwill paid in recessions vs normal times. That is achieved by interacting an indicator variable for US vs UK with an indicator for recession vs a normal economy.$^{\dagger}$ You then test this interaction term.
You, however, seem to have companies in multiple nations, meaning that you have multiple groups instead of a control and treatment group. Consequently, you would want to interact the recession/normal dummy with all country indicator variables (with one country being subsumed by the intercept, as is typical for ANOVA derivatives). You then test if the interaction between recession/normal and the country indicator is significant by using a chunk test of nested models. The full model has an intercept, an indicator for recession/normal, indicators for the various countries (with one country omitted and subsumed by the intercept), and the interaction between the recession/normal variable and all of the country indicator variables. Nested within that model is a model with just the intercept, recession/normal indicator, and country indicator (with that same one country omitted and subsumed by the intercept). This then tests if any of the countries behave differently in recession times va normal times, assuming the counties meet the parallel trend assumption of DiD, where the counties move in parallel before the recession hits.
In many regards, this is an ANCOVA with an interaction, just with the traditional continuous covariate replaces with the binary recession/normal indicator variable.
One of the reasons people like DiD is because it helps draw causal inferences. I have major reservations about that here, with comments to the OP making what I believe to be good points, but that chunk test should cover the pure statistics of performing a test.
$^{\dagger}$I have multiple concerns about this approach. First, different countries have different economies, so the US could experience a normal economy while another country experiences a recession. Then the comparison is not on equal footing, since the global economy during those times would be the same. Second, “recession” is not a binary variable.
