Is it possible to run a generalized difference-in-differences analysis where all units are treated, but at a different intensity?
In general, the generalized difference-in-differences approach requires you to observe a subset of units never exposed to treatment. That being said, it is possible to identify treatment effects even if all entities become exposed to a policy, but the adopters must have different post-treatment phases. Identification is then based upon variation in treatment timing. But as per your comments, the timing of the policy is standardized, and so you cannot exploit the variation present in staggered adoption settings.
Alternatively, it is also possible to subdivide units into different ‘intensity groupings’ (e.g., high versus low), but then I’d suggest forcing the low intensity treatment group to be your reference.
In my situation, a policy was introduced, designed to alter the consumption of antibiotics. Antibiotics are divided into 4 groups based on their potential benefit and resistance profile.
The policy is introduced at the country level but it affects all countries, and at precisely the same time. According to your comments, you don't observe any non-adopter countries. By design, you cannot exploit any variation at this level. All the action is within each country, where antibiotic consumption may vary by batch.
So after time $T$, when the policy is introduced, I can interact the policy variable (=1 from introduction and 0 otherwise), with the group variable (categorical, 1-4).
You can actually use the classical difference-in-difference estimator to simplify your estimation strategy. I addressed a similar concern here, where multiple treatment groups were observed and each group was exposed to treatment at the same time. Simply interact each categorical group with a post-treatment indicator.
In your setting, however, you lack a suitable control group. To achieve something estimable, you'd have to make some ad hoc decision about which category to use as a reference. Suppose you dropped the group where antibiotics are "freely prescribed" to the population. Does the "freely prescribed" group serve as a suitable counterfactual for your other groups? This might be hard to sell.
Am I restricted to a pre/post non-causal design?
Possibly.
However after reading your comments, I am left with more and more questions that I don't think I can adequately address here. I am not quite sure you can proceed with the generalized difference-in-differences estimator, even though it is amenable to a multivalued policy variable. Your comments indicate you include both country and quarter fixed effects. I don't see how this is appropriate unless you acquired quarterly consumption data on countries never espousing the policy. Treatment is not well-defined at this higher level of aggregation.
As for the treatment itself, I would expect the moderately prescribed batch of antibiotics to be consumed more than the "restricted" batch. It seems obvious that consumption is related to how much a product is prescribed to the general population. Is this your goal? And, is consumption your only outcome measure? I should probably leave it up to you to address the mechanism(s) at work.
