Difference-in-difference to estimate gradual (i.e. slow) policy effect I want to evaluate the impact of a policy on a variable $Y$. In the figure, I plot the mean of $Y$ over time in both the control and treatment groups (the vertical line represents the reform). 
The mean difference between the two groups is slowly increasing over time. When running a standard difference-in-difference the treatment effect turns out insignificant, but I suspect it's because of this "slow" effect. What would be the best specification in this case to show the impact of the reform?
 
 A: I don't know if it is "the best" specification, but you could simply test the presence of an effect of the policy in 2014q3-2016q3, and on 2016q3-2018q3. Alternatively, testing not only an additive impact, but also investigating an impact on time trend.
A: Visually, the treatment group "trend" is showing less pre-treatment volatility. If you're employing the archetypical difference-in-differences (DD) approach, I would investigate the assumption of trend equivalence by interacting your treatment dummy with separate indicators for time (e.g., months). You could do this for the full series, but that could be very computationally demanding if you have a long panel.
Ryan and colleagues (2015) investigated specification choices on the accuracy of estimates in DD settings (see Model 4, p. 1220). You could account for differences in group trends by including pre-period dummies into the basic DD specification and interact those with treatment as well. Coefficients on these interactions should be indistinguishable from zero.
If you're only interested in how the effect of treatment is evolving after the exposure is implemented, you could do the following:
$$
y_{iq} = \gamma T_{i} + \lambda_{1} (T_{i}*\mathbf{I}_{q = 2014_{Q1}}) + \lambda_{2} (T_{i}*\mathbf{I}_{q = 2014_{Q2}}) + \lambda_{3} (T_{i}*\mathbf{I}_{q = 2014_{Q3}}) + \lambda_{4} (T_{i}*\mathbf{I}_{q = 2014_{Q4}}) + \lambda_{5} (T_{i}*\mathbf{I}_{q = 2015_{Q1}}) + \epsilon_{iq}.
$$
where you observe $i$ units (i.e., entities) over $q$ quarters. Here, you would interact separate indicators for each post-treatment period (i.e., separate quarter-year dummies for all periods beyond the vertical line) with your treatment variable, giving you a separate 'treatment effect' for each post-treatment period. Your post-treatment epoch spans many quarters so must multiply $T_{i}$ with a concatenated version of quarter and year. In other words, you are estimating separate $\lambda$'s for each quarter after the reform. I stopped at the first quarter in 2015 but you should include interactions up to the last quarter in 2018, or whenever your panel ends. This way you could investigate how the effect is growing/decaying over time. This may be of substantive interest. I would also recommend plotting the estimates of each $\lambda$.
The DD approach isn't the only method out there for estimating the effects of policy changes, but trying these different approaches could certainly help!
