Effect of adding moments on GMM m-efficiency via projected score

Investigate whether augmenting the set of moment functions with additional linearly independent moments necessarily moves the projection Π(score_ν^⊤ | m) of the true score onto the moment space closer to the true score and thereby improves the m-efficiency bound toward the Cramér–Rao lower bound; establish rigorous conditions under which such efficiency improvement holds.

Background

The paper defines an m-efficiency bound based on projecting the true score onto the space spanned by available moments and shows equality with the Cramér–Rao bound when the score lies in that span. They note that demonstrating equality Π(score_ν^⊤ | m) = score_ν^⊤ is generally challenging.

They explicitly conjecture that adding nonredundant moments should bring the projection closer to the score, hence improving GMM efficiency, but leave formal verification and conditions for this improvement open.