The Efficiency Gap

TL;DR

This paper investigates the efficiency gap between M- and Z-estimators in semiparametric models for multivariate functionals, revealing that Z-estimators often outperform M-estimators in certain settings.

Contribution

It establishes the existence of an efficiency gap for multivariate functionals like multiple quantiles and (Value at Risk, Expected Shortfall), providing theoretical and numerical evidence.

Findings

Z-estimators can be more efficient than M-estimators for multivariate functionals.

Theoretical proof of the efficiency gap for multiple quantiles and (V@R, ES).

Numerical illustrations confirm the theoretical results.

Abstract

Parameter estimation via M- and Z-estimation is equally powerful in semiparametric models for one-dimensional functionals due to a one-to-one relation between corresponding loss and identification functions via integration and differentiation. For multivariate functionals such as multiple moments, quantiles, or the pair (Value at Risk, Expected Shortfall), this one-to-one relation fails and not every identification function possesses an antiderivative. The most important implication is an efficiency gap: The most efficient Z-estimator often outperforms the most efficient M-estimator. We theoretically establish this phenomenon for multiple quantiles at different levels and for the pair (Value at Risk, Expected Shortfall), and illustrate the gap numerically. Our results further give guidance for pseudo-efficient M-estimation for semiparametric models of the Value at Risk and Expected Shortfall.