Asymptotically efficient estimation of the conditional expected shortfall

Abstract

A procedure for efficient estimation of the trimmed mean of a random variable conditional on a set of covariates is proposed. For concreteness, the focus is on a financial application where the trimmed mean of interest corresponds to the conditional expected shortfall, which is known to be a coherent risk measure. The proposed class of estimators is based on representing the estimator as an integral of the conditional quantile function. Relative to the simple analog estimator that weights all conditional quantiles equally, asymptotic efficiency gains may be attained by giving different weights to the different conditional quantiles while penalizing excessive departures from uniform weighting. The approach presented here allows for either parametric or nonparametric modeling of the conditional quantiles and the weights, but is essentially nonparametric in spirit. The asymptotic properties of the proposed class of estimators are established. Their finite sample properties are illustrated through a set of Monte Carlo experiments and an empirical application

Conditional Value-at-Risk and Average Value-at-Risk: Estimation and Asymptotics

“…Both estimators show similar behaviors, but RVaR seems a little more stable. Following ideas in McNeil and Frey (2000) and Leorato et al (2012), Operations Research 60(4), pp. 739-756, © 2012 "violation event" is said to occur whenever an observed CDS spread percent change falls below the predicted VaR (we can find a few violation events from Figure 8).…”

“…In both models, unknown parameters were estimated by minimizing the regression quantile loss function. For conditional average value-at-risk, Scaillet (2005) and Cai and Wang (2008) utilized Nadaraya-Watson (NW) type nonparametric double kernel estimation, while Peracchi and Tanase (2008) and Leorato et al (2012) used the semiparametric method.…”

Expectile and quantile regression—David and Goliath?

“…The ES can be written as a function of both, the quantile and the expectile for a level˛. Estimation of the ES has been recently proposed by Leorato et al (2012) by employing the integrated (conditional) fitted quantile regression function (see also Wang and Zhou, 2010). We extend an idea of Taylor (2008) and use the fitted quantiles and expectiles for the estimation of the ES.…”

Bayesian CV@R/super-quantile regression

“…The concept has appeared in the literature before, see for example, Cai and Wang (2008), Chernozhukov and Umantsev (2001), Fermanian and Scaillet (2005), Leorato, Peracchi, and Tanase (2012), Peracchi and Tanase (2008), , Rockafellar, Uryasev and Zabarankin M (2008), Scaillet (2004Scaillet ( , 2005, Wozabal and Wozabal N (2009). A conditional value-at-risk concept known as CAViaR has been proposed by Engle and Manganelli (2004) but its meaning is different compared to , Rockafellar, Uryasev and Zabarankin M (2008), and Rockafellar, Royset and Miranda (2014, RRM).…”