Efficiency of sequential versus joint procedures under over-identification

Determine whether, in semiparametric models with over-identification (including both row and rank over-identification as defined via the rank of the variance–covariance matrix of the moment functions), the multi-step plug-in (sequential) identification/estimation procedure achieves the same asymptotic efficiency for the parameter of interest as the joint (one-step) procedure, or precisely characterize conditions under which such efficiency equivalence holds.

Background

The paper studies influence functions and the relationship between local robustness and efficiency in joint (one-step) versus sequential (multi-step) procedures. When both parameters are just-identified, the authors show equivalence of joint and sequential influence functions. However, in the presence of over-identification (defined via the rank properties of the variance–covariance matrix of the moments), it is not immediate that sequential estimation matches the efficiency of joint estimation.

This question motivates later developments in Section 4 where the authors provide sufficient conditions for equivalence via orthogonal projections and compatibility assumptions, but the initial formulation explicitly notes the uncertainty in general over-identified settings.