Is parallel trends assumption necessary in difference-in-differences analysis? Reading the literature on the subject, I haven't encountered clear reasoning why the parallel trends assumption must hold. In fact, there have been recent papers on ways to relax this assumption (see Rambachan and Roth (2019), Bilinski and Hatfield (2019), Freyaldenhoven, et al. (2019)).
To me, it seems like the parallel trends assumption is solving a problem that doesn't exist. The goal in DiD analysis is to estimate the average treatment effect on the treated, either as an absolute change or percentage change. Does it matter if the baseline means and trends for the outcomes differ for the treatment and control groups, if we're only interested in comparing the changes in those trends in the post-treatment time period?
For example, consider the following statistically significant linear trends for made-up monthly medical cost data (with unspecified but unequal intercepts):
$$
\begin{array}{c|lcr}
\text{Period} & \text{Control group} & \text{Treatment group} \\
\hline
\text{Pre} & y_{ctrl} = \beta_0 + 10*t & y_{trmt} = \beta_2 + 20*t \\
\text{Post} & y_{ctrl} = \beta_1 + 15*t & y_{trmt} = \beta_3 + 25*t \\
\end{array}
$$
The pre/post trend in cost increases by 50% in the Control group and increases by 25% in the Treatment group.
The baseline trends are not the same but we're still seeing a significantly lower pre/post trend change for the Treatment group.

It is important to distinguish two parts:

*

*The counterfactual parallel trend assumption (PTA) at the time of treatment T+1

*The testable parallel trend assumption before the treatment t<T if you have more than one pre-treatment period.

What is necessary is to have the counterfactual PTA at T+1 to hold in order to make a causal claim about the Diff-diff, but this is untestable by definition. So usually people test for the testable PTA at t<T. The idea is that if the testable PTA t<T holds, this gives more support to the wishful and unverifiable claim that the counterfactual PTA T+1 holds.
So you are right, the testable PTA t<T is neither necessary nor sufficient for the counterfactual PTA T+1 to hold:

*

*It is not necessary since despite not having PTA at t<T, PTA could still hold at T+1

*It is not sufficient since despite having PTA at t< does not imply that PTA holds at T+1

