Identification of the superoptimal regime with multivalued treatments under longitudinal instrumental variables

Determine whether the longitudinal superoptimal regime can be identified when treatments take more than two values, even when valid instrumental variables are available at every time point.

Background

The paper establishes an identification argument for the longitudinal superoptimal regime when treatments are binary, using sequential identification and Dirac delta functions to recover distributions involving counterfactual natural treatment values. The authors explain that this argument depends crucially on treatment binarity because, with two treatment values, the relevant equations can determine the unknown conditional quantities.

For treatments with more than two values, the authors state that the binary-treatment identification argument no longer applies. The unresolved issue is whether valid instrumental variables at every time point could nevertheless suffice to identify the superoptimal regime under some alternative argument or additional structure.

References

This argument can be extended to $K > 2$ in the same manner. Crucially, this argument relies on the treatments $A_k$ being binary. We do not expect $g{sup}$ to be identified, even when there are valid IVs at every time point, when treatments can take more than two values.

— Optimal sequential decision-making with initiation regimes  (2609.29844 - Laurendeau et al., 24 Sep 2026) in Remark following Section “Identification of the superoptimal regime with binary treatments using Dirac's delta function” (Appendix; immediately after the discussion extending the argument to K > 2)