  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.
