The Efficiency Gap
Abstract
Parameter estimation via M and Zestimation is equally powerful in semiparametric models for onedimensional functionals due to a onetoone 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 onetoone relation fails and not every identification function possesses an antiderivative. The most important implication is an efficiency gap: The most efficient Zestimator often outperforms the most efficient Mestimator. 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 pseudoefficient Mestimation for semiparametric models of the Value at Risk and Expected Shortfall.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.14146
 arXiv:
 arXiv:2010.14146
 Bibcode:
 2020arXiv201014146D
 Keywords:

 Mathematics  Statistics Theory;
 Economics  Econometrics
 EPrint:
 27 pages + 19 pages supplement