Difference-in-difference: common trend I'm new to this concept, and I'm referring to the book Mastering Metrics: The Path from Cause to Effect by Joshua D. Angrist and Jörn-Steffen Pischke.
It states that an important assumption of difference-in-difference is the common trend assumption.
I'm a little confused about its meaning: does it mean that without the treatment both the treatment and control group will have parallel slopes? Or, does it mean that they will both increase/decrease by the same number?
Because clearly these two things don't mean the same, so now I'm confused.
 A: I will focus on your other question.
No. The change in the group averages over time do not have to increase/decrease in the exact same amount for the common trend assumption to hold. I couldn’t imagine a world where the change from period-to-period before some policy/intervention would be exactly similar in both treatment and control groups. To be clear, differences in outcome levels in any period is allowed, but their time variation as exhibited by the trends should be reasonably similar pre-shock. In most papers, demonstrating trend equivalence is often achieved visually with a good plot of the average (group) outcomes over time before some treatment begins. If group trends markedly diverge pre-treatment, or exhibit more volatility in one group and not the other, you could test for group differences with a specification test.
In sum, a perfect clone of the treatment group's time variation is ideal, but is seldom observed in practice. That being said, a visually clear parallelism should be observed in the outcomes before the shock, which we assume will persist in the absence of treatment exposure. A picture is worth a thousand words.
A: You could state the common trend assumption like this: "in a counterfactual world where the treatment has not been implemented, the slope of the outcome in the treatment and control are parallel." The assumption allows you to recover the counterfactual values of the outcome in the treatment group after the treatment is implemented. The treatment effect is then obtained as the difference between the actual values and the counterfactual values of the outcome. 
