Higher order elicitability and Osband’s principle

By Tobias Fissler, J. Ziegel. 2015

394 citations recorded by the source corpus.

A statistical functional, such as the mean or the median, is called elicitable if there is a scoring function or loss function such that the correct forecast of the functional is the unique minimizer of the expected score. Such scoring functions are called strictly consistent for the functional. The elicitability of a functional opens the possibility to compare competing forecasts and to rank them in terms of their realized scores. In this paper, we explore the notion of elicitability for multi-dimensional functionals and give both necessary and sufficient conditions for strictly consistent scoring functions. We cover the case of functionals with elicitable components, but we also show that one-dimensional functionals that are not elicitable can be a component of a higher order elicitable functional. In the case of the variance this is a known result. However, an important result of this paper is that spectral risk measures with a spectral measure with finite support are jointly elicitable if one adds the `correct' quantiles. A direct consequence of applied interest is that the pair (Value at Risk, Expected Shortfall) is jointly elicitable under mild conditions that are usually fulfilled in risk management applications.

Subfields: Finance, Management Science and Operations Research, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/46369549f9128b93545efa2c6026b96a996d7711

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Citations in the corpus

Elicitability and backtesting: Perspectives for banking regulation (2016, background): “[2015] give a recent account of the pros and cons of popular risk measures with an attempt to determine the best risk measure in practice. On the other hand, Cont et al. [2010] highlight the need to consider the entire “risk measurement procedure”, which includes not just the choice of a risk measur”
Regression diagnostics meets forecast evaluation: conditional calibration, reliability diagrams, and coefficient of determination (2021, methodology): “In the setting of Fissler and Ziegel (2016), if V is an identification function, then so is (x, y) 7→ h(x)V (x, y) whenever h(x) 6= 0 for all x ∈ R.”
Multinomial VaR Backtests: A simple implicit approach to backtesting expected shortfall (2016, neutral): “Fissler & Ziegel (2016) show that VaR and ES are jointly elicitable in the sense that they jointly minimize an appropriate bi-dimensional scoring function; this allows the comparison of different forecasting methods that give estimates of both VaR and ES.”
A joint quantile and expected shortfall regression framework (2017, positive): “This regression approach relies on the class of strictly consistent joint loss functions introduced by Fissler and Ziegel (2016), which permits the joint elicitation of the quantile and the ES.”
Expected Shortfall is jointly elicitable with Value at Risk - Implications for backtesting (2015, methodology): “ity for backtesting risk measure estimates. In particular, we propose the use of Diebold-Mariano tests, and show how they can be implemented for Expected Shortfall (ES), based on the recent result of Fissler and Ziegel (2015) that ES is jointly elicitable with Value at Risk. There continues to be li”
Forecasting Value at Risk and Expected Shortfall Using a Semiparametric Approach Based on the Asymmetric Laplace Distribution (2019, methodology): “Fissler and Ziegel (2016) explained that VaR and ES are jointly elicitable, even though ES is not elicitable individually.”
Extremiles: A New Perspective on Asymmetric Least Squares (2018, background): “Yet, it is known that competing forecastingmethods of expected shortfall may be compared and assessed thanks to Fissler and Ziegel’s (2016) important result that certain spectral risk measures are jointly elicitable with quantiles.”
Forecast combinations for value at risk and expected shortfall (2020, background): “Taylor (2019) uses the score to estimate dynamic joint models for VaR and ES, and this proposal is given theoretical support by the recent work of Patton, Ziegel, and Chen (2019).”
Forecasting risk with Markov-switching GARCH models:A large-scale performance study (2018, background): “However, Fissler and Ziegel (2016) recently showed that, in the case of a constant conditional distribution, the pair (VaR, ES) can be elicited jointly as the values of vt and et that minimize the sample average of the following loss”