Background
The typical thing to do is visually inspect the pre-treatment trends for the control and treatment groups. Whether you want it or not, you might be biased when looking at the visual representation. I'd argue that an average university student would claim that neither control 1 nor 2 pass the "parallel trend" assumption. At the same time, many researchers do a more relaxed visual inspection, given that you explain the theory and arguments that support the pre-treatment parallel trends. This, of course, causes some friction, e.g. look at Kearney and Levine (2015) and Jaeger, Joyce and Kaestner (2018), and then again Kearney and Levine (2018).
Mathematical methodology
It is possible to test for the parallel trend assumption more concisely. This answer provides a very nice intro on how to do it.
A formal test which is also suitable for multivalued treatments or several groups is to interact the treatment variable with time dummies. Suppose you have 3 pre-treatment periods and 3 post-treatment periods, you would then regress
$$y_{it} = \lambda_i + \delta_t + \beta_{-2}D_{it} + \beta_{-1}D_{it} + \beta_1 D_{it} + \beta_2 D_{it} + \beta_3 D_{it} + \epsilon_{it}$$
where $y$ is the outcome for individual $i$ at time $t$, $λ$ and $δ$ are individual and time-fixed effects (this is a generalized way of writing down the diff-in-diff model which also allows for multiple treatments or treatments at different times).
The idea is the following. You include the interactions of the time
dummies and the treatment indicator for the first two pre-treatment
periods and you leave out the one interaction for the last
pre-treatment period due to the dummy variable trap. Also now all the
other interactions are expressed relative to the omitted period which
serves as the baseline. If the outcome trends between treatment and
control group are the same, then $β_{−2}$
and $β_{-1}$
should be insignificant, i.e. the difference in differences is not significantly different between the two groups in the pre-treatment period.
What to do?
Always present a graph showing the levels of the two series you are comparing over time, not just their difference. The general rule of thumb is to prefer DiD on a matched sample for this reason – if you can make the levels more similar, readers will be more willing to think the trends will be too.
Kahn-Lane and Lang (2019) on the "failure to reject parallel trends in the pre-treatment data".
Increasingly, researchers point to a statistically insignificant pre-trend test to argue that they therefore accept the null hypothesis of parallel trends. There is no doubt that testing for a common pre-trend plays an important role in validating the parallel trends assumption underlying DiD. However, failing to reject that outcomes in years prior to treatment exhibit parallel trends, should not be confused with establishing the validity of the parallel trends counterfactual. Moreover, clearly, not rejecting the null hypothesis is not equivalent to confirming it.
But also, and in my opinion, more importantly, Kahn-Lane and Lang (2019) note that:
Authors should perform a thorough comparison of the differences between the treatment and control groups including demographic composition, other factors that could have differentially affected each group, and comparison of trends as far back as possible .
Relax parallel trend with Synthetic Control Method
Assuming that the number of pre-intervention periods is large enough, it is likely that unobserved and time varying confounders have similar effects on both treated unit and synthetic counterpart (Kreif et al., 2016). In your case, the synthetic control method might improve the DiD approach by relaxing the parallel trends assumption. However, the longer the pre-treatment data, the better. Moreover, with the DiD there often exists a sort of ambiguity when it comes to the choice of control group as argued by Card (1989) - in case of SCM it is being chosen by a data-driven method.
If you want to read more on Synthetic Control Method, I highly recommend going to the source, i.e. Abadie et al. (2010), Abadie et al. (2011) and Abadie et al. (2015). The application of the SCM for estimating treatment effects in panel settings has become very popular among researchers interested in comparative case studies. Since then, the SCM was applied in various fields and to diverse research topics, establishing an intuitive alternative to creating counterfactuals. According to Athey and Imbens (2017):
Synthetic Control Method is arguably the most important innovation in the policy evaluation
literature in the last 15 years.
Therefore, using SCM as the robustness check for DiD is more than justified.
